SYMMETRY CLASSES OF DISORDERED FERMIONS by P ...sentation theory, symmetric spaces 1. Introduction In a famous and inﬂuential paper published in 1962 (“the threefold way: algebraic

SYMMETRY CLASSES OF DISORDERED FERMIONS

by

P. Heinzner, A. Huckleberry & M.R. Zirnbauer

Abstract. — Building upon Dyson’s fundamental 1962 article known in random-matrixtheory asthe threefold way, we classify disordered fermion systems with quadratic Hamil-tonians by their unitary and antiunitary symmetries. Important physical examples are af-forded by noninteracting quasiparticles in disordered metals and superconductors, and byrelativistic fermions in random gauge field backgrounds.

The primary data of the classification are a Nambu space of fermionic field operatorscarrying a representation of some symmetry group. Our approach is to eliminate all of theunitary symmetries from the picture by transferring to an irreducible block of equivarianthomomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector spaceV, a space of endomorphisms inEnd(V�V�), a bilinear form onV�V� which is either symmetric or alternating, and oneor two antiunitary symmetries that may mixV with V�. Every such set of block data isshown to determine an irreducible classical compact symmetric space. Conversely, everyirreducible classical compact symmetric space occurs in this way.

This proves the correspondence between symmetry classes and symmetric spaces con-jectured some time ago.

Keywords: disordered electron systems, random Dirac fermions, quantum chaos; repre-sentation theory, symmetric spaces

1. Introduction

In a famous and influential paper published in 1962 (“the threefold way: algebraicstructure of symmetry groups and ensembles in quantum mechanics” [D]), FreemanJ. Dyson classified matrix ensembles by a scheme that became fundamental to severalareas of theoretical physics, including the statistical theory of complex many-bodysystems, mesoscopic physics, disordered electron systems, and the field of quantumchaos. Being set in the context of standard quantum mechanics, Dyson’s classificationasserted that “the most general matrix ensemble, defined with a symmetry group thatmay be completely arbitrary, reduces to a direct product of independent irreducible

2 P. HEINZNER, A. HUCKLEBERRY & M.R. ZIRNBAUER

ensembles each of which belongs one of three known types.” These three ensembles,or rather their underlying matrix spaces, are nowadays known as the Wigner-Dysonsymmetry classes of orthogonal, unitary, and symplectic symmetry.

Over the last ten years, various matrix spaces beyond Dyson’s threefold way havecome to the fore in random-matrix physics and mathematics. On the physics side, suchspaces arise in problems of disordered or chaotic fermions; among these are the Eu-clidean Dirac operator in a stochastic gauge field background [VZ ], and quasiparticleexcitations in disordered superconductors or metals in proximity to a superconductor[A]. In the mathematical research area of number theory, the study of statistical corre-lations in the values of the Riemann zeta function, and more generally of families ofL-functions, has prompted some of the same extensions [K ].

A brief account of why new structures emerge on the physics side is as follows.When Dirac first wrote down his famous equation in 1928, he assumed that he waswriting an equation for thewavefunctionof the electron. Later, because of the insta-bility caused by negative-energy solutions, the Dirac equation was reinterpreted (viasecond quantization) as an equation for thefermionic field operatorsof a quantumfield theory. A similar change of viewpoint is carried out in reverse in the Hartree-Fock-Bogoliubov mean-field description of quasiparticle excitations in superconduc-tors. There, one starts from the equations of motion for linear superpositions of theelectron creation and annihilation operators, and reinterprets them as a unitary quan-tum dynamics for what might be called the quasiparticle ‘wavefunction’.

In both cases – the Dirac equation and the quasiparticle dynamics of a superconduc-tor – there enters a structure not present in the standard quantum mechanics underlyingDyson’s classification: the fermionic field operators are subject to a set of conditionsknown as thecanonical anticommutation relations, and these are preserved by thequantum dynamics. Therefore, whenever second quantization is undone (assuming itcanbe undone) to return from field operators to wavefunctions, the wavefunction dy-namics is required to preserve some extra structure. This puts a linear constraint onthe allowed Hamiltonians. A good viewpoint to adopt is to attribute the extra invariantstructure to the Hilbert space, thereby turning it into a Nambu space.

It was conjectured some time ago [A] that extending Dyson’s classification to theNambu space setting, the relevant objects one is led to consider are large familiesof symmetric spacesof compact type. Past understanding of the systematic natureof the extended classification scheme relied on the mapping of disordered fermionproblems to field theories with supersymmetric target spaces [Z] in combination withrenormalization group ideas and the classification theory of Lie superalgebras.

An extensive review of the mathematics and physics of symmetric spaces, coveringthe wide range from the basic definitions to various random-matrix applications, hasrecently been given in [C]. That work, however, offers no answers to the questionas towhy symmetric spaces are relevant for symmetry classification, and under whatassumptions the classification by symmetric spaces is complete.

SYMMETRY CLASSES OF DISORDERED FERMIONS 3

In the present paper, we get to the bottom of the subject and, using a minimal set oftools from linear algebra, give a rigorous answer to the classification problem for disor-dered fermions. The rest of this introduction gives an overview over the mathematicalmodel to be studied and a statement of the main result obtained.

We begin with a finite- or infinite-dimensional Hilbert spaceV carrying a unitaryrepresentation of some compact Lie groupG0 – this is the group of unitary symmetriesof the disordered fermion system. We emphasize thatG0 need not be connected; infact, it might be just a finite group.

Let W = V �V �, called the Nambu space of fermionic field operators, be equippedwith the inducedG0-representation. This means thatV is equipped with the givenrepresentation, andg( f ) := f Æ g�1 for f 2 V �, g 2 G0. Let C : W ! W be theC -antilinear involution determined by the Hermitian scalar producth ; iV on V . Inphysics this operator is called particle-hole conjugation. Another canonical structureonW is the symmetric complex bilinear formb : W �W ! C defined by

b(v1+ f1;v2+ f2) := f1(v2)+ f2(v1) :

It encodes the canonical anticommutation relations for fermions, and is related to theunitary structureh ; i of W by b(w1;w2) = hCw1;w2i for all w1;w2 2W .

It is assumed thatG0 is contained in a groupG – the total symmetry group of thefermion system – which is acting onW by transformations that are either unitary orantiunitary. An elementg2G either stabilizesV or exchangesV andV �. In the lattercase we say thatg2 G mixes, and in the former case we say that it is nonmixing.

The groupG is generated byG0 and distinguished elementsgT which act as anti-unitary operatorsT : W ! W . These are referred to as distinguished ‘time-reversal’symmetries, orT-symmetries for short. The squares of thegT lie in the center of theabstract groupG; we therefore require that the antiunitary operatorsT representingthem satisfyT2 = �Id. The subgroupG0 is defined as the set of all elements ofGwhich are represented as unitary, nonmixing operators onW .

If T andT1 are distinguished time-reversal operators, thenP :=TT1 is a unitary sym-metry.P may be mixing or nonmixing. In the latter case,P is in G0. Therefore, moduloG0, there exist at most two differentT-symmetries. If there are exactly two such sym-metries, we adopt the convention thatT is mixing andT1 is nonmixing. Furthermore,it is assumed thatT andT1 either commute or anticommute, i.e.,T1T =�TT1.

As explained throughout this article, all of these situations are well motivated byphysical considerations and examples. We note that time-reversal symmetry (and allotherT-symmetries) of the disordered fermion system may also be broken; in this caseT andT1 are eliminated from the mathematical model andG0 = G.

GivenW and the representation ofG on it, the object of interest is the real vectorspaceH of C -linear operators in End(W ) that preserve the canonical structuresb andh ; i of W and commute with theG-action. Physically speaking,H is the space of‘good’ Hamiltonians: the field operator dynamics generated byH 2 H preserves both


the canonical anticommutation relations and the probability in Nambu space, and iscompatible with the prescribed symmetry groupG.

When unitary symmetries are present, the spaceH decomposes byblocksassoci-ated with isomorphism classes ofG0-subrepresentations occurring inW . To formal-ize this, recall that two unitary representationsρi : G0 ! U(Vi), i = 1;2, are equiv-alent if and only if there exists a unitaryC -linear isomorphismϕ : V1 ! V2 so thatρ2(g)(ϕ(v)) = ϕ(ρ1(g)(v)) for all v2 V1 and for allg2 G0. Let G0 denote the spaceof equivalence classes of irreducible unitary representations ofG0. An elementλ 2 G0is called an isomorphism class for short. By standard facts (recall that every represen-tation of a compact group is completely reducible) the unitaryG0-representation onVdecomposes as an orthogonal sum over isomorphism classes:

V =�λVλ :

The subspacesVλ are called theG0-isotypic components ofV . Some of them may bezero. (Some of the isomorphism classes ofG0 may just not be realized inV .)

For simplicity suppose now that there is only one distinguished time-reversal sym-metryT, and for any fixedλ 2 G0 with Vλ 6= 0, consider the vector spaceT(Vλ). IfT is nonmixing, i.e.,T : V ! V , thenT(Vλ) � V must coincide with the isotypiccomponent for the same or some other isomorphism class. (Since conjugation bygT isan automorphism ofG0, the decomposition intoG0-isotypic components is preservedby T.) If T is mixing, i.e.,T : V ! V �, thenT(Vλ) = V �

λ0, still with someλ0 2 G0.Now define the blockBλ to be the smallestG-invariant space containingVλ�V �

λ .Note that if we are in the situation of nonmixing andT(Vλ) 6= Vλ, then

Bλ =�Vλ�T(Vλ)

��Vλ�T(Vλ)

��:

On the other hand, if we are in the situation of mixing andT(Vλ) 6= V �λ , then

Bλ =�Vλ�T(V �

λ )��V �

λ �T(Vλ)�

:

The blockBλ is halved ifT(Vλ) = Vλ resp.T(Vλ) = V �λ .

Note that if there are two distinguishedT-symmetries, the above discussion is onlyslightly more complicated. In any case we now have the basicG-invariant blocksBλ.

Because different blocks are built from representations of different isomorphismclasses, the good Hamiltonians do not mix blocks. Thus everyH 2 H is a direct sumover blocks, and the structure analysis ofH can be carried out for each blockBλseparately. IfVλ is infinite-dimensional, then to have good mathematical control wetruncate to a finite-dimensional spaceVλ �Vλ and form the associated blockBλ �W .The truncation is done in such a way thatBλ is aG-representation space and is Nambu.

The goal now is to compute the space of Hermitian operators onBλ which commutewith theG-action and respect the canonical symmetricC -bilinear formb induced fromthat onV �V � (such a space of operators realizes what is called asymmetry class).


For this, certain spaces ofG0-equivariant homomorphisms play an essential role,i.e., linear mapsS: V1 !V2 betweenG0-representation spaces which satisfy

ρ2(g)ÆS= SÆρ1(g)

for all g2 G0, whereρi : G0 ! U(Vi), i = 1;2, are the respective representations. If itis clear which representations are at hand, we often simply writegÆS= SÆg or S=gSg�1. Thus we regard the space HomG0(V1;V2) of equivariant homomorphisms as thespace ofG0-fixed vectors in the space Hom(V1;V2) of all linear maps. IfV1 =V2 =V,then these spaces are denoted by EndG0(V) and End(V) respectively.

Roughly speaking, there are two steps for computing the relevant spaces of Her-mitian operators. First, the blockBλ is replaced by an analogous blockHλ of G0-equivariant homomorphisms from a fixed representation spaceRλ of isomorphismclassλ and/or its dualR�λ to Bλ. The spaceHλ carries a canonical form (called ei-thers or a) which is induced fromb. As the notation indicates, although the originalbilinear form onBλ is symmetric, this induced form is either symmetric or alternating.

Change of parity occurs in the most interesting case when there is a nontrivialG0-isomorphismψ : Rλ ! R�λ. In that case there exists a bilinear formFψ : Rλ�Rλ ! C

defined byFψ(r; t) = ψ(r)(t), which is either symmetric or alternating. In a certainsense the formb is a product ofFψ and a canonical form onHλ. Thus, ifFψ is alter-nating, then the canonical form onHλ must also be alternating.

After transferring to the spaceHλ, in addition to the canonical bilinear forms or awe have a unitary structure and conjugation by one or two distinguished time-reversalsymmetries. Such a symmetryT may be mixing or not, and bothT2 = Id andT2 =�Idare possible. The second main step of our work is to understand these various cases,each of which is directly related to a classical symmetric space of compact type. Suchare given by a classical Lie algebrag which is eithersun, usp2n, or son(R).

In the notation of symmetric spaces we have the following situation. Letg be theLie algebra ofantihermitianendomorphisms ofHλ which are isometries (in the senseof Lie algebra elements) of the induced complex bilinear formb= s or b= a. This isof compact type, because it is the intersection of the Lie algebra of the unitary groupof Hλ and the complex Lie algebra of the group of isometries ofb. Conjugation by theantiunitary mappingT defines an involutionθ : g! g.

The good Hamiltonians (restricted to the reduced blockHλ) are theHermitianoper-atorsh2 ig such that at the level of group action the one-parameter groups e�ith satisfyTe�ith = e+ithT, i.e., ih2 g must anticommute withT. Equivalently, ifg= k�p is thedecomposition ofg into θ-eigenspaces, the space of operators which is to be computedis the(�1)-eigenspacep. The space of good Hamiltonians restricted toHλ then is ip.Since the appropriate action of the Lie groupK (with Lie algebrak) on this space is justconjugation, one identifies ip with the tangent spaceg=k of an associated symmetricspaceG=K of compact type.

It should be underlined that there is more than one symmetric space associated to aCartan decompositiong = k� p. We are most interested in the one consisting of the


physical time-evolution operators e�ith; if G (not to be confused with the symmetrygroupG) is the semisimple and simply connected Lie group with Lie algebrag, this isgiven as the image of the compact symmetric spaceG=K under the Cartan embeddingintoG defined bygK 7! gθ(g)�1, whereθ : G! G is the induced group involution.

The following mathematical result is a conseqence of the detailed classification workin Sects. 3, 4 and 5.

Theorem 1.1. — The symmetric spaces which occur under these assumptions are ir-reducible classical symmetric spacesg=k of compact type. Conversely, every irre-ducible classical symmetric space of compact type occurs in this way.

We emphasize that here the notionsymmetric spaceis applied flexibly in the sensethat depending on the circumstances it could mean either the infinitesimal modelg=kor the Cartan-embedded compact symmetric spaceG=K.

Let us mention that direct sumsg=k�g=k may occur in examples where the originalsituation is irreducible, e.g., when the initial blockV�V� is invariant under two dis-tinguished time-reversal symmetries. But the main object of interest would seem to bethe irreducible classical symmetric space of compact type.

Theorem 1.1 settles the question of symmetry classes in disordered fermion systems;in fact every physics example is handled by one of the situations above.

The paper is organized as follows. In Sect. 2, starting from physical considerationswe motivate and develop the model that serves as the basis for subsequent mathemati-cal work. Sect. 3 proves a number of results which are used to eliminate the group ofunitary symmetriesG0. The main work of classification is given in Sect. 4 and Sect.5. In Sect. 4 we handle the case where there is at most one distinguished time-reversaloperator present, and in Sect. 5 the case where there are two. There are numerous sit-uations that must be considered, and in each case we precisely describe the symmetricspace which occurs.

Various examples taken from the physics literature are listed in Sect. 6, illustratingthe general classification theory.

2. Disordered fermions with symmetries

‘Fermions’ is the physics name for the elementary particles which all matter is madeof. The goal of present article is to establish a symmetry classification of Hamiltonianswhich arequadratic in the fermion creation and annihilation operators. To motivatethis restriction note that at the fundamental level, any Hamiltonian for fermions isof Dirac type; thus it is always quadratic in the fermion operators, albeit with time-dependent coefficients that are themselves operators. At the nonrelativistic or effectivelevel, quadratic Hamiltonians arise in the Hartree-Fock mean-field approximation formetals and the Hartree-Fock-Bogoliubov approximation for superconductors. By the


Landau-Fermi liquid principle, such mean-field or noninteracting Hamiltonians givean adequate description of physical reality at very low temperatures.

In the present section, starting from a physical framework, we develop the appropri-ate model that will serve as the basis for the mathematical work done later on. Please beadvised thatdisorder, though advertised in the title of the section and the title of paper,will play no explicit role here. Nevertheless, disorder (and/or chaos) are the indispens-able agents thatmust be presentin order to remove specific and nongeneric featuresfrom the physical system and make a classification by basic symmetries meaningful.In other words, what we carry out in this paper is the first step of a two-step program.This first step is to identify in the total space of Hamiltonians some linear subspacesthat are relevant (in Dyson’s sense) from a symmetry perspective. The second step isto put probability measures on these spaces and work out the disorder averages andstatistical correlation functions of interest. It is this latter step that ultimately justifiesthe first one and thus determines the name of the game.

2.1. The Nambu space model for fermions. —The starting point for our considera-tions is the formalism of second quantization. Its relevant aspects will now be reviewedso as to introduce the key physical notions as well as the proper mathematical language.

Let i = 1;2; : : : label an orthonormal set of quantum states for a single fermion.Second quantizing the many-fermion system means to associate with eachi a pairof operatorsc†

i andci , which are called fermion creation and annihilation operators,respectively. These operators are subject to thecanonical anticommutation relations

c†i c†

j +c†j c

†i = 0 ;

cicj +cjci = 0 ;

c†i cj +cjc

†i = δi j ;

for all i; j. They act in a Fock space, i.e., in a vector space with a distinguished vec-tor, called the ‘vacuum’, which is annihilated by all of the operatorsci (i = 1;2; : : :).Applying n creation operators to the vacuum one gets a state vector forn fermions. Afield operatorψ is a linear combination of creation and annihilation operators,

ψ = ∑i

�vi c

†i + fi ci

�;

with complex coefficientsvi and fi .To put this in mathematical terms, letV be the complex Hilbert space of single-

fermion states. (We do not worry here about complications due to the dimension ofV being infinite. Later rigorous work will be carried out in the finite-dimensionalsetting.) Fock space then is the exterior algebra^V , with the vacuum being theone-dimensional subspace of constants. Creating a single fermion amounts to exte-rior multiplication by a vectorv2 V and is denoted byε(v). To annihilate a fermion,one contracts with an elementf of the dual spaceV �; this operation on V is written


ι( f ). In that framework the canonical anticommutation relations read

ε(v)ε(v)+ ε(v)ε(v) = 0 ;

ι( f )ι( f )+ ι( f )ι( f ) = 0 ;

ι( f )ε(v)+ ε(v)ι( f ) = f (v) :

They can be viewed as the defining relations of an associative algebra generated by thevector spaceW := V �V � which is isomorphic to the space of field operatorsψ.

This algebra, called the Clifford algebraC (W ), comes with a natural grading by thedegree:

C (W ) = C �C 1(W )�C 2(W )� : : : ;

whereC 1(W ) �= W . In the sequel, we shall focus on the componentsC 1(W ) andC 2(W ). Since we only consider Hamiltonians that are quadratic in the creation andannihilation operators, we will be able to reduce the second-quantized formulation tostandard single-particle quantum mechanics, albeit on the doubled spaceW carryingsome extra structure.W is sometimes referred to asNambu spacein physics.

On W there exists a canonical symmetric complex bilinear formb defined by

b(v+ f ; v+ f ) = f (v)+ f (v) = ∑i( fi vi + fi vi) :

The significance of this bilinear form in the present context lies in the fact that it en-codes onW the canonical anticommutation relations (CAR) obeyed by the Cliffordalgebra generators inC 1(W ). Indeed, we can view a field operatorψ = ∑i(vi c

†i + fi ci)

either as a vectorψ = v+ f 2V �V �, or equivalently as an operatorψ = ε(v)+ ι( f )2C 1(W ). Adopting the operator perspective, we get from CAR that

ψψ+ ψψ = f (v)+ f (v) = ∑i

�fi vi + fi vi

�:

Switching to the vector perspective we have the same answer fromb(ψ; ψ). Thus

ψψ+ ψψ = b(ψ; ψ) :

Definition 2.1. — In the Nambu space model for fermions one identifies the spaceC 1(W ) of field operatorsψ = ε(v)+ ι( f ) with the complex vector spaceW = V �V �

equipped with its canonical unitary structureh ; i and canonical symmetric complexbilinear form b.

Remark. — Having already expounded the physical origin of the symmetric bilinearform, let us now specify the canonical unitary structure ofW . The complex vectorspaceV , being isomorphic to the Hilbert space of single-particle states, comes with aHermitian scalar product (or unitary structure)h ; iV . Givenh ; iV define aC -antilinearbijectionC : V ! V � by

Cv= hv; �iV ;


and extend this to an antilinear transformationC : W !W by the requirementC2= Id.ThusCjV � = (CjV )�1. The operatorC is calledparticle-hole conjugationin physics.UsingC, transfer the unitary structure fromV to V � in the natural way:

h f ; f iV � := hC f;Cf iV = hCf ;C fiV :

The canonical unitary structure ofW is then given by

hv+ f ; v+ f i= hv; viV + h f ; f iV � = ∑i

�vi vi + fi fi

�:

Thush ; i is the orthogonal sum of the Hermitian scalar products onV andV �.

Proposition 2.2. — The canonical unitary structure and symmetric complex bilinearform ofW are related by

hψ; ψi= b(Cψ; ψ) :

Proof. — Given an orthonormal basisc†1;c1;c

†2;c2; : : : this is immediate from

C∑i(vi c†i + fi ci) = ∑i(vi ci + fi c

†i )

and the expressions forh ; i andb in components.

Returning to the physics way of telling the story, consider the most general Hamilto-nianH which is quadratic in the single-fermion creation and annihilation operators:

H = 12 ∑i j Ai j

�c†

i cj �cjc†i

�+ 1

2 ∑i j

�Bi j c†

i c†j + Bi j cjci

�:

The HamiltoniansH act on the field operatorsψ by the commutator,ψ 7! [H;ψ], andthe evolution with time is determined by the Heisenberg equation of motion,

i~dψdt

= [H;ψ] ;

where~ is Planck’s constant. By the canonical anticommutation relations, this dynami-cal equation is equivalent to a system of linear differential equations for the coefficientsvi and fi :

i~vi = ∑ j

�Ai j vj +Bi j f j

�;

�i~ fi = ∑ j

�Bi j vj + Ai j f j

�:

If these are assembled into a column vectorv, the evolution equation takes the form

v = Xv ; X =�i~

�A B�B �A

�:

The matrix elements ofX obey the relationsBi j =�Bji (from cicj =�cjci) andAi j =Aji (from the physical requirement of self-adjointness ofH).

To recast all this in concise mathematical terms, recall the grading of the CliffordalgebraC (W ) = C �C 1(W )�C 2(W )� : : :. From the Clifford algebra perspective,the HamiltonianH is viewed as an operator in the degree-two componentC 2(W ).


For our purposes it is therefore important to accumulate some information aboutC 2(W ). It is well known [B] thatC 2(W ) is a Lie algebra with the commutator playingthe role of the Lie bracket; in factC 2(W) is canonically isomorphic to the complexorthogonal Lie algebraso(W ;b) associated with the vector spaceW = V �V � andits canonical symmetric complex bilinear formb. Here the Lie algebraso(W ;b) isdefined to be the subspace of elementsE 2 End(W ) satisfying the condition

b(Eψ; ψ)+b(ψ;Eψ) = 0 :

If E is any endomorphism in End(V �V �), decompose it into blocks as

E =

�A B

C D

�;

whereA 2 End(V ), B 2 Hom(V �;V ), C 2 Hom(V ;V �) andD 2 End(V �). Let theadjoint (or transpose) ofA 2 End(V ) be denoted byAt 2 End(V �).

Proposition 2.3. — An endomorphism E=

�A B

C D

�2 End(V �V �) lies in the com-

plex orthogonal Lie algebraso(V �V �;b) if and only ifB;C are skew andD=�At.

Proof. — Consider first the caseB= C= 0, and letψ = v+ f andψ = v+ f . Then

b(Eψ; ψ) = b(Av�At f ; v+ f ) = f (Av)�At f (v)

= At f (v)� f (Av) =�b(v+ f ;Av�At f ) =�b(ψ;Eψ) :

A similar calculation for the caseA= 0 gives

b(Eψ; ψ) = b(B f +C f ; v+ f ) = f (B f )+Cv(v)

= � f (B f )�Cv(v) =�b(v+ f ;B f +Cv) =�b(ψ;Eψ) :

Since these two cases complement each other, the statement follows.

By fixing orthonormal basesc†1;c

†2; : : : of V andc1;c2; : : : of V � as before, we assign

matrices with matrix elementsAi j , Bi j , Ci j to the linear operatorsA, B, C. A straight-forward computation using the canonical anticommutation relations then yields:

Proposition 2.4. — TheC -linear mapping fromso(W ;b) to C 2(W ) given by�A B

C �At

�7! 1

2 ∑i j Ai j (c†i cj �cjc

†i )+

12 ∑i j (Bi j c†

i c†j +Ci j cicj)

is an isomorphism of Lie algebras.

In addition to acting on itself by the commutator, the Lie algebraC 2(W ) acts (stillby the commutator) on all of the componentsC k(W ) of degreek� 1 of the CliffordalgebraC (W ). In particular,C 2(W ) acts on the degree-one componentC 1(W ). Bythe isomorphismsC 2(W ) �= so(W ;b) andC 1(W ) �= W , this action coincides withthe fundamental representation ofso(W ;b) on its defining vector spaceW . In otherwords, taking the commutator of the HamiltonianH 2 C 2(W ) with a field operator


ψ 2 C 1(W ) yields the same answer as viewingH as an element ofso(W ;b), then

applyingH =

�A B

C �At

�to the vectorψ = v+ f 2 V �V � by

H � (v+ f ) = (Av+B f )+(Cv�At f ) ;

and finally reinterpreting the result as a field operator inC 1(W ).The closure relation[C 2(W );C 1(W )] � C 1(W ) and the isomorphismsC 1(W ) �=

W andC 2(W )�= so(W ;b) make it possible to reduce the dynamics of field operatorsto a dynamics on the Nambu spaceW . As we have seen, after reduction the generatorsX 2 End(V �V �) of time evolutions of the physical system are of the special form

X =�i~

�A B

B� �At

�;

whereB 2 Hom(V �;V ) is skew, andA= A� 2 End(V ) is self-adjoint w.r.t.h ; iV .

Proposition 2.5. — The one-parameter groups of time evolutions t7! etX in the Nambuspace model preserve both the canonical unitary structureh ; i and the canonical sym-metric complex bilinear form b ofW = V �V �.

Proof. — By Prop. 2.3 the generatorX is an element of the complex Lie algebraso(W ;b). Hence the exponentialUt = etX lies in the complex orthogonal Lie groupSO(W ;b), which is defined to be the set of solutionsg in End(W ) of the conditions

b(gψ;gψ) = b(ψ; ψ) ; and Det(g) = 1 :

SinceA= A�, andB� 2Hom(V ;V �) is the adjoint ofB 2Hom(V �;V ), the gener-

atorX is antihermitian with respect to the unitary structure ofW . The exponentiatedgeneratorUt therefore lies in the unitary group U(W ), which is to say that

hUtψ;Utψi= hψ; ψi

for all realt. ThusUt preserves bothb andh ; i.

Remark. — In physical language, the invariance ofb under time evolutions meansthat the canonical anticommutation relations for fermionic field operators do not changewith time. Invariance ofh ; i means that probability in Nambu space is conserved. (Ifthe quadratic HamiltonianH arises as the mean-field approximation to some many-fermion problem, the latter conservation law holds as long as quasiparticles do notinteract and thereby are protected from decay into multi-particle states.)

We now distill the essence of the information conveyed in this section. The quantumtheory of many-fermion systems is set up in a Hilbert space called the fermionic Fockspace in physics (or the spinor representation in mathematics). The field operators ofthe physical system span a vector spaceW = V �V �, which generates a CliffordalgebraC (W ) whose defining relations are the canonical anticommutation relations.


Since[C 2(W );C 1(W )]� C 1(W ), the discussion of the field operator dynamics forthe important case of quadratic HamiltoniansH 2 C 2(W ) can be reduced to a discus-sion on the Nambu spaceW �= C 1(W ). Via this reduction, the vector spaceW inheritstwo natural structures: the canonical symmetric complex bilinear formb encoding theanticommutation relations, and a canonical unitary structureh ; i determined by theHermitian scalar product ofV . Both of these structures are invariant, i.e., are pre-served by physical time evolutions. Under the reduction toW , the commutator actionof C 2(W ) onC 1(W ) becomes the fundamental representation ofso(W ;b) onW .

2.2. Symmetry groups. —Following Dyson, the classification of disordered fermionsystems will be carried out in a setting that prescribes two pieces of data:

� One is given a Nambu spaceW = V �V � equipped with its canonical unitarystructureh ; i and canonical symmetricC -bilinear formb.

� On W there acts a groupG of unitary and antiunitary operators (the joint sym-metry group of a multi-parameter family of fermionic quantum systems).

Given this setup, one is interested in the linear space of HamiltoniansH with the prop-erty that they commute with theG-action onW , while preserving the invariant struc-turesb andh ; i of W under time evolution by e�itH=~. Such a space of Hamiltonians isof course reducible in general, i.e., the Hamiltonian matrices decompose into blocks.The goal of classification is to enumerate all thesymmetry classes, i.e., all the types ofirreducible block which occur in this way.

In the present subsection, we provide some information on what is meant by unitaryand antiunitary symmetries in the present context. We begin by recalling the basicnotion of a symmetry group in quantum Hamiltonian systems.

In classical mechanics the symmetry groupG0 of a Hamiltonian system is under-stood to be the group of symplectomorphisms that commute with the phase flow of thesystem. Examples are the rotation group for systems in a central field, and the groupof Euclidean motions for systems with Euclidean invariance.

In passing from classical to quantum mechanics, one replaces the classical phasespace by a complex Hilbert spaceV , and assigns to the symmetry groupG0 a (projec-tive) representation by unitaryC -linear operators onV . While the consequences due toone-parameter continuous subgroups ofG0 are particularly clear from Noether’s theo-rem, the components ofG0 notconnected with the identity also play an important role.A prominent example is provided by the operator for space reflection. Its eigenspacesare the subspaces of states with positive and negative parity, and they reduce the matrixof any reflection-invariant Hamiltonian to two blocks.

Not all symmetries of a quantum mechanical system are of the canonical, unitarykind: the prime counterexample is the operationgT of inverting the time direction –called time reversal for short. In classical mechanics this operation reverses the sign ofthe symplectic structure of phase space; in quantum mechanics its algebraic propertiesreflect the fact that the timet enters in the Dirac, Pauli, or Schrödinger equation as


i~d=dt: there, time reversalgT is represented by anantiunitaryoperatorT, which is tosay thatT is complex antilinear:

T(zv) = zTv (z2 C ; v2 V ) ;

and preserves the Hermitian scalar product up to complex conjugation:

hv; viV = hTv;TviV :

Another example of such an operation is charge conjugation in relativistic theories.Further examples are provided by chiral symmetry transformations (see Sect. 2.3).

By the symmetry groupG of a quantum mechanical system with HamiltonianH,one then means the group of all unitary and antiunitary transformationsg of V thatleave the Hamiltonian invariant:gHg�1 = H. It should be noted that finding the totalsymmetry group of a quantization of some Hamiltonian system is not always straight-forward. The reason is that there may exist nonobvious quantum symmetries such asHecke symmetries, which are of number-theoretic origin and have no classical limit.For our purposes, however, this complication will not be an issue. We take the groupG and its action on the Hilbert space to befundamental and given, and then ask whatis the linear space of Hamiltonians that commute with theG-action.

For technical reasons, we assume the groupG0 to be compact; this is an assumptionthat covers most (if not all) of the cases of interest in physics. The noncompact groupof space translations can be incorporated, if necessary, by wrapping the system arounda torus, whereby translations are turned into compact torus rotations.

What we have sketched – a symmetry groupG acting on a Hilbert spaceV – is theframework underlying Dyson’s classification. As was explained in Sect. 2.1, we wishto enlarge it so as to capture all examples that arise in disordered fermion physics.

For this, recall that in the Nambu space model for fermions, the Hilbert space is notV but the space of field operatorsW = V �V �. The givenG-representation onVtherefore needs to be extended to a representation onW . This is done by the conditionthat the pairing betweenV andV � (and thus the pairing between fermion creation andannihilation operators) be preserved. In other words, ifU : V ! V andA : V ! Vare unitary resp. antiunitary operators, their induced representations onV � (which westill denote by the same symbols) are defined by requiring that

(U f )(Uv) = f (v) = (A f)(Av)

for all v2 V and f 2 V �. In particular theG0-representation onV � is thedualone,

U( f ) = f ÆU�1 :

Equivalently, theG-representation onW is defined so as to be compatible with particle-hole conjugationC : W !W in the sense that operations commute:

CU =UC ; and CA= AC :


Indeed, if f = Cv then f (v) = hv; vi and from the invariance of the pairing betweenV andV � one infers the relationshv; vi= (U f )(Uv) = hU�1C�1UCv; vi andhv; vi =(A f)(Av) = hA�1C�1ACv; vi.

While the framework so obtained is flexible enough to capture the situations thatarise in the nonrelativistic quasiparticle physics of disordered metals, semiconductorsand superconductors, it is still slightly too narrow to accommodate some much studiedexamples that have emerged from elementary particle physics. Let us explain this.

2.3. The Euclidean Dirac operator. —An important development in random-matrixphysics over the last ten years was the formulation [VZ ] and study of the so-calledchiral ensembles, which model Dirac fermions in a random gauge field background,and lie beyond Dyson’s 3-way classification. From the viewpoint of applications, theserandom-matrix models have the merit of capturing some universal features of the Diracspectrum of quantum chromodynamics (QCD) in the low-energy limit. In the presentsubsection we will demonstrate that, but for one minor difference, they fit naturallyinto our fermionic Nambu space model with symmetries.

Let M be a four-dimensional Euclidean space-time (more generally,M could bea Riemannian 4-manifold with spin structure), and consider overM a unitary spinorbundleStwisted by a moduleR for the action of some compact gauge groupK. Denoteby V the Hilbert space ofL2-sections of the twisted bundleSR.

Now letDA be a self-adjoint Dirac operator forV in a given gauge field background(or gauge connection)A. AlthoughDA is not a Hamiltonian in the strict sense of theword, it has all the right mathematical attributes in the sense of Sect. 2.1; in particularit determines a Hermitian form, called the action functional, on differentiable sectionsψ 2 V . In physics notation this functional is written

ψ 7!Z

Mψ(x) � (DAψ)(x)d4x ; DA = iγµ(∂µ�Aµ) ;

whereγµ = γ(eµ) are the gamma matrices [i.e., the Clifford actionγ : T�M ! End(S)evaluated on the dualeµ of an orthonormal coordinate frameeµ of TM], the operators∂µ are the partial derivatives corresponding to theeµ, andAµ(x) 2 Lie(K) are the com-ponents of the gauge field. If the physical situation calls for a mass, then one adds acomplex number im (times the unit operator onV ) to the expression forDA.

The Dirac operators of prime interest to low-energy QCD have zero (or small) mass.To express the massless nature ofDA one introduces an object called thechirality op-eratorΓ in mathematics [B], or γ5 = γ0γ1γ2γ3 in physics.Γ = γ5 is a section of End(S)which is self-adjoint and involutory (Γ2 = Id) and anticommutes with the Clifford ac-tion (Γγµ+ γµΓ = 0). By the last property one has

ΓDA+DAΓ = 0

in the massless limit. This relation is called chiral symmetry in physics. Note, however,that chiral ‘symmetry’is not a symmetryin the sense of the present paper. (Symmetries


alwayscommutewith the Hamiltonian, never do they anticommute with it!) Nonethe-less, we shall now recognize chiral symmetry as being equivalent to a true symmetry,by importing the Dirac operator into the Nambu space model as follows.

As before, take Nambu space to be the sumW = V �V � equipped with its canoni-cal unitary structureh ; i and symmetric complex bilinear formb. The antilinear bijec-tionC : V ! V � andC : V �!V is still defined byhw1;w2i= b(Cw1;w2).

Now extend the Dirac operatorDA 2 iu(V ) to an operatorDA that acts diagonallyon W = V �V �, by requiringDA to satisfy the commutation lawC iDA = iDAC, orequivalentlyCDA =�DAC. Thus,

DA 2 End(V )�End(V �) ,! End(W ) ;

andDA on End(V �) is given by�DtA. The diagonally extended operatorDA lies in

the intersection ofso(W ;b) with iu(W ) – as is required in order for the statement ofProp. 2.5 to carry over to the one-parameter groupt 7! eitDA. The property thatDAdoes not mixV andV � can be attributed to the existence of a U1 symmetry group thathasV andV � as inequivalent representation spaces.

To implement the chiral symmetry of the massless limit, extend the chirality operatorΓ to a diagonally acting endomorphism in End(V )�End(V �) by CΓC�1 = Γ. Theextended operators still satisfy the chiral symmetry relationΓDA +DAΓ = 0. Thendefine an antiunitary operatorT by T := CΓ. Note that this isnot the operation ofinverting the time but will still be called the ‘time reversal’ for short.

BecauseDA anticommutes with bothC andΓ, one has

TDAT�1 = DA :

ThusT is a true symmetry of the (extended) Dirac operator in the massless limit.Note thatCT = TC from CΓ = ΓC. As was announced above, the situation is the

same as before but for one difference: while the time reversal in Sect. 2.2 was anoperatorT : V ! V andT : V � ! V �, the present one is an operatorT : V ! V �

andT : V �! V . We refer to the latter type asmixing, and the former asnonmixing.To summarize, physical systems modelled by the Euclidean (or positive signature)

Dirac operator are naturally incorporated into the framework of Sects. 2.1 and 2.2.The Hilbert spaceV here is the space ofL2-sections of a twisted spinor bundle overEuclidean space-time, and the role of the Hamiltonian is taken by the quadratic ac-tion functional of the Dirac fermion theory. When transcribed into the Nambu spaceW = V �V �, the chiral ‘symmetry’ of the massless theory can be expressed as a trueantiunitary symmetryT, with the only new feature being thatT mixesV andV �.

The most general situation occurring in physics may exhibit, besideT, one or severalother antiunitary symmetries. In the example at hand this happens if the representationspaceRcarries a complex bilinear form which is invariant under gauge transformations(see Sects. 6.2.2 and 6.2.3 for the details). The Dirac operatorDA then has one extraantiunitary symmetry, sayT1, which is nonmixing. Forming the composition ofT1 with


T we get amixing unitary symmetry P= TT1 : V $ V �. This fact leads us to adoptthe final framework described in the next subsection.

2.4. The mathematical model. —The following model is now well motivated.We are given a Nambu space(W ;b;h ; i) carrying the action of a compact group

G. The groupG0 is defined to be the subgroup ofG which acts by canonical unitarytransformations, i.e., unitary transformations that preserve the decompositionW =V �V �. The full symmetry groupG is generated byG0 and at most two distinguishedantiunitary time-reversal operators. If there is just one, we denote it byT, and ifthere are two, byT andT1. In the latter case we adopt the convention thatT mixes,i.e.,T : V ! V �, while T1 is nonmixing. The distinguished time-reversal symmetriesalways satisfyT2 =�Id andT2

1 =�Id. In the case that there are two, it is assumed thatthey commute or anticommute, i.e.,T1T = �TT1. Consequently the unitary operatorP = TT1 (which mixes) also satisfiesP2 = �Id. In the situation withP present welet G1 denote theZ2-extension ofG0 defined byP and refer to it as the full group ofunitary symmetries.

We emphasize that the original action ofG0 on V has been extended toW via itscanonically induced action onV �. In other words, iff 2V � theng( f )(v)= f (g�1(v)).This is equivalent to requiring that a unitary operatorU 2G0 commutes with particle-hole conjugationC : W !W . In fact we require that all operators ofG commute withC. Whereas the unitary operators preserve the Hermitian scalar producth ; i, for anantiunitary operatorA we have thathAw1;Aw2i= hw1;w2i for all w1;w2 2 W .

If U is an operator coming fromG0 andT is a distinguished time-reversal symmetry,thenTUT�1 is unitary and nonmixing, i.e., it is inG0. Thus, for the correspondingoperatorgT in G, we assume thatgT normalizesG0 andg2

T is in the center ofG0.According to Prop. 2.5 the time evolutions of the physical system leave the struc-

ture of Nambu space invariant. The infinitesimal version of this statement is that theHamiltoniansH lie in the intersection of the complex orthogonal Lie algebraso(W ;b)with iu(W ), the Hermitian operators onW .

Let us summarize our situation in the language and notation introduced above.

Definition 2.6. — The data in the Nambu space model for fermions with symmetries is(W ;b;h ; i;G), where the compact group G is called the symmetry group of the system.G is represented onW = V �V � by unitary and antiunitary operators that preservethe structure ofW ; i.e., for every unitary U and antiunitary A one has

hψ; ψi= hUψ;Uψi= hAψ;Aψi ; b(ψ; ψ) = b(Uψ;Uψ) = b(Aψ;Aψ)

for all ψ; ψ 2 W . The space of ‘good’ Hamiltonians is theR-vector spaceH ofoperators H inso(W ;b)\ iu(W ) that commute with the G-action:

UHU�1 = H = AHA�1 :


At the group level of time evolutions this means that

Ue�itH=~ = e�itH=~U ; Ae�itH=~ = e+itH=~A ;

for all unitary U, antiunitary A, H2 H , and t2 R.

We remind the reader that the subgroup of unitary operators which preserves thedecompositionW = V �V � is denoted byG0, and the full group of unitaries byG1.

Several further remarks are in order. First, for a unitaryU 2G1 (resp. antiunitaryA),the compatibility ofb with theG-action is a consequence of Prop. 2.2 and the commu-tation lawCU =UC andCA= AC. Second, it is possible that the fermion system doesnot have any antiunitary symmetries andG= G0. When some antiunitary symmetriesare present,G is generated byG0 and one or at most two distinguished time-reversalsymmetries as explained above. Third, motivated by the prime physics example of timereversal, we have assumed that the (one or two) distinguished time-reversal symmetriesT satisfyT2 =�Id. The reason for this can be explained as follows.

The operatorT has been chosen to represent some kind ofinversionsymmetry.Since this means that conjugation byT2 represents the unit operator,T2 must be aunitary multiple of the identity on any subspace ofW which is irreducible under timeevolutions of the fermion system. Thus for all practical purposes we may assume thatT is a projective involution, i.e.,T2 = z� Id with za complex number of unit modulus.

Proposition 2.7. — If a projective involution T: W ! W of a unitary vector spaceW is antiunitary, then either T2 =+IdW or T2 =�IdW .

Proof. — A projective involutionT has squareT2 = z� Id with z2 C n f0g. SinceT is antiunitary,T2 is unitary, and hencejzj = 1. But an antiunitary operator isC -antilinear, and therefore the associative lawT2 �T = T �T2 forcesz to be real, leavingonly the possibilitiesT2 =�Id.

Since this work is meant to simultaneously handle symmetry at both the Lie algebraand Lie group level, a final word should be said about the notion that a bilinear formF is respected by a transformationB. At the group level whenB is invertible and isregarded as being in GL(W), whereW is the underlying vector space ofF : W�W!C , this means thatB is an isometry in the sense thatF(Bw1;Bw2) = F(w1;w2) for allw1;w2 2W. On the other hand, at the Lie algebra level whereB2 End(W), this meansthat for allw1;w2 2W one has

ddt

��t=0

F(etBw1;etBw2) = F(Bw1;w2)+F(w1;Bw2) = 0 :


3. Reduction to the case ofG0 = fIdg

Recall that our main goal, e.g., on the Lie algebra level, is to describe the space ofG0-invariant endomorphisms which on a block in Nambu space are compatible withthe unitary structure, time reversal and the symmetricC -bilinear form.

Here we prove results which allow us to transfer this space to a certain space ofG0-equivariant homomorphisms. The unitary structure, time reversal and the bilinearform are essentially canonically transferred, and as before, compatibility with thesestructures is required. However, in the new settingG0 acts trivially. This is of coursean essential simplification.

3.1. Spaces of equivariant homomorphisms. —Throughout this sectionλ 2 G0denotes a fixed isomorphism class (i.e., an equivalence class of irreducible repre-sentations ofG0), andλ� denotes its dual. Ablock is determined by a choice of finite-dimensionalG0-invariant subspaceV =Vλ (in the given Hilbert spaceV ) such that allof its irreducible subrepresentations have isomorphism classλ.

The full groupG of (unitary and antiunitary) symmetries is generated byG0 and atmost two distinguished time-reversal symmetries. In this section it is assumed through-out that these time-reversal operators stabilize the truncated subspaceW = V�V� ofNambu space. The case where one or both time-reversal symmetries do not stabilizeW, i.e., where a larger block is generated, is handled in later sections.

If h ; iV is the initial unitary structure onV, one definesC : V !V� by C(v)(w) =hv;wiV . Taking CjV� to be the inverse of this map, one obtains the associatedC -antilinear isomorphismC : W !W. All symmetries inG are assumed to commutewith C. We remind the reader thatG0 acts onV� by g( f ) = f Æg�1.

Let R be a fixed irreducibleG0-representation space which is inλ. Denote bydits dimension. Of courseR� is a representative ofλ�. We fix an antilinear bijectionι : R! R� which is defined by aG0-invariant unitary structureh ; iR onR.

In the sequel we will often make use of the following consequence of Schur’sLemma.

Proposition 3.1. — If two irreducible G0-representation spaces R1 and R2 are equi-variantly isomorphic byψ : R1 ! R2, then HomG0(R1;R2) = C �ψ, i.e., the linearspace of G0-equivariant homomorphisms from R1 to R2 has complex dimension oneand every operator in it is some multiple ofψ.

The following related statement was essential to Dyson’s classification and will playa similarly important role in the present article.

Lemma 3.2. — If an irreducible G0-representation space R is equivariantly isomor-phic to its dual R� by an isomorphismψ : R! R�, then ψ is either symmetric oralternating, i.e., eitherψ(r)(t) = ψ(t)(r) or ψ(r)(t) =�ψ(t)(r) for all r; t 2 R.


Proof. — It is convenient to think ofψ as defining an invariant bilinear formB(r; t) =ψ(r)(t) onR. We then decomposeB into its symmetric and alternating parts,B=S+A,where

S(r; t) = 12

�B(r; t)+B(t; r)

�and A(r; t) = 1

2

�B(r; t)�B(t; r)

�:

Both areG0-invariant, and consequently their degeneracy subspaces are invariant.Since the representation spaceR is irreducible, it follows that each is either nonde-generate or vanishes identically. But both being nondegenerate would violate the factthat up to a constant multiple there is only one equivariant isomorphism in End(R).ThereforeB is either symmetric or alternating as claimed.

Now let H := HomG0(R;V) be the space ofG0-equivariant linear mappings fromR toV. Its dual space isH� = HomG0(R

�;V�). The key space for our first considerations is(HR)� (H�R�). Note thatG0 acts on it by

g(h r + f t) = hg(r)+ f g(t) :

We can applyh 2 H to r 2 R to form h(r) 2 V. Sinceh is G0-equivariant we haveg�h(r) = h(g(r)). The same goes for the corresponding objects on the dual side. Thusin our finite-dimensional setting the following is immediate.

Proposition 3.3. — If H = HomG0(R;V) and H� = HomG0(R�;V�) the map

ε : (HR)� (H�R�) ! V�V� =W ;

h r + f t 7! h(r)+ f (t) ;

is a G0-equivariant isomorphism.

Transferring the unitary structure fromW to (HR)� (H�R�) induces a unitarystructure onH�H�. For this, note for example that forh1 r1 andh2 r2 in HRwe have

hh1 r1;h2 r2iHR := hh1(r1);h2(r2)iV :

Observe that forh1 and h2 fixed, the right-hand side of this equality defines aG0-invariant unitary structure onR which is unique up to a multiplicative constant. Thuswe defineh ; iH by

hh1 r1;h2 r2iHR = hh1;h2iH � hr1; r2iR :

Given the fixed choice ofh ; iR this definition is canonical.We will in fact transfer all of our considerations forV�V� to the spaceH�H�, the

latter being equipped with the unitary structure defined as above. One of the key pointsfor this is to understand how to express aG0-invariant endomorphism

S2 EndG0(V�V�) �=ε EndG0(HR�H�R�)

as an element of End(H�H�). Also, we must understand the role of time reversal.


In this regard the two casesλ 6= λ� andλ = λ� pose slightly different problems. Be-fore going into these in the next sections, we note several facts which are independentof the case.

First, letV1 andV2 be vector spaces whereG0 acts trivially, and letR1 andR2 bearbitraryG0-representation spaces.

Proposition 3.4. —

HomG0(V1R1;V2R2) = Hom(V1;V2)HomG0(R1;R2) :

Proof. — Note that Hom(V1R1;V2R2) = Hom(V1;V2)Hom(R1;R2), and let(ϕ1; : : : ;ϕm) be a basis of Hom(V1;V2). Then for every elementS of Hom(V1;V2)Hom(R1;R2) there are unique elementsψ1; : : : ;ψm so thatS= ∑ϕi ψi . If S is G0-equivariant, then

S= gÆSÆg�1 = ∑ϕi (gÆψi Æg�1) ;

and the desired result follows from the uniqueness statement.

Our second general remark concerns the way in which a distinguished time-reversalsymmetryT is transferred to an antilinear endomorphism ofH R�H�R�. Letus consider for example the case of mixing where it is sufficient to understandT :H R! H�R�. For that purpose we view End(H R) as End(H)End(R), let(ϕ1; : : : ;ϕm) be a basis of End(H) and write

Γ =CT = ∑ϕi ψi

for ψ1; : : : ;ψm 2 End(R). Now T is equivariant in the sense thatT Æ g = a(g) ÆT,wherea is the automorphism ofG0 determined by conjugation withgT . Thus, sincethe C -antilinear operatorC intertwinesG0-actions, theC -linear mappingΓ = CT isinvariant with respect to the twisted conjugationΓ 7! a(g)Γg�1. Consequently, everyψi is invariant with respect to this conjugation.

This means that theψi : R! R are equivariant with respect to the originalG0-representation on the domain space and the newG0-action,v 7! a(g)(v), on the imagespace. But by Prop. 3.1, up to a constant multiple there is only one such element ofEnd(R), i.e., we may assume that

Γ = ϕψ ;

whereψ is unique up to a multiplicative constant.Note further thatC is also of this factorized form. Indeed, we have

hh r; �iHR= hh; �iH hr; �iR ;

and ifγ : H !H� is defined byh 7! hh; �iH, thenC= γ ι. Furthermore, sinceΓ andCare pure tensors, so isT = CΓ = TH TR , with the factors being antilinear mappingsTH = γÆϕ : H ! H� andTR = ιÆψ : R! R�.


Of course we have only considered a piece ofT, and that only in the case of mixing.However, exactly the same arguments apply to the other piece and also in the case ofnonmixing. Thus we have the following observation.

Proposition 3.5. — The induced map

T : (HR)� (H�R�)! (HR)� (H�R�) ;

is the sum T= A1B1+A2B2 of pure tensors.

In the case of mixing this means thatA1B1 is an antilinear mapping fromH Rto H�R� and vice versa forA2B2.

If T doesn’t mix, thenA1B1 : HR! HR andA2B2 : H�R�! H�R�.In this case we impose the natural condition that theAi andBi be antiunitary. For laterpurposes we note that this condition determines the factors only up to multiplicationby a complex number of unit modulus. Using the formulaC = γ ι and the fact thatC commutes withT, one immediately computesA2B2 from A1B1 (or vice versa).The involutory propertyT2 =�Id also adds strong restrictions. Of course there may betwo distinguished time reversals,T andT1, and we require that they commute withCandT1T =�TT1. These properties are automatically transferred at this level, becausethe transfer process from(HR)� (H�R�) to V�V� is an isomorphism.

Finally, we prove an identity which is essential for transferring the complex bilinearform. For this we begin with

h r + f t 2 (HR)� (H�R�) ;

applyε to obtainh(r)+ f (t), and then apply the linear functionf (t)2V� to the vectorh(r) 2V. The resultf (t)(h(r)) is to be compared to the productf (h) t(r). Recall thatthe dimension of the vector spaceR is denoted byd.

Proposition 3.6. —

f (t)(h(r)) = d�1 f (h) t(r) :

Before beginning the proof, which uses bases for the various spaces, we set the no-tation and prove a preliminary lemma. Letmdenote the multiplicity of the componentV and fix an identification

V�V� = R� : : :�R�R�� : : :�R�

with m summands ofR andR�. Let (e1; : : : ;ed) be a basis ofR and(ϑ1; : : : ;ϑd) be itsdual basis. These define bases(ek

1; : : : ;ekd) and(ϑk

1; : : : ;ϑkd) of the correspondingk-th

summands above. LetI kR andI k

R� be the respective identity mappings.

Lemma 3.7. —

I `R�(I kR) = δk` d :


Proof. — Expressing the operators in the bases, i.e.,

I kR= ∑i ϑ

ki ek

i and I `R� = ∑ j e`j ϑ`

j ;

one hasI `R�(I

kR) = ∑i; j ϑk

i (e`j)ϑ`

j(eki ) = ∑i; j δk`

i j = δk` d ;

which is the statement of the lemma.

Proof of Prop. 3.6. — We expandh2 H = HomG0(R;V) ash= ∑hkI kR , and f 2H� =

HomG0(R�;V�) as f = ∑ fÌ `R� . If r = ∑ riei andt = ∑ t jϑ j , then

h(r) = ∑i;khkri eki and f (t) = ∑ j ;` f`t j ϑ`

j :

Thusf (t)(h(r)) = ∑i jk`

δkì j f`hkt j ri =

�∑k fkhk

�t(r) :

Prop. 3.6 now follows from the above lemma which implies thatf (h) = d ∑ fkhk.

3.2. The case whereλ 6= λ�. — Recall that our goal is to canonically transfer the dataon V �V� to H �H�, thus removingG0 from the picture. In the case whereλ 6= λ�this is a particularly simple task.

First, we apply Prop. 3.4 to transfer elements of EndG0(V�V�). In the case at handHomG0(R;R�) and HomG0(R

�;R) are both zero, and both EndG0(R) and EndG0(R�) are

isomorphic toC . Thus it follows from Prop. 3.4 that

EndG0(V�V�) �= EndG0(HR�H�R�)�= End(H)�End(H�) ,! End(H�H�) :

We always normalize operators in EndG0(H R) to the formϕ IdR and normalizeoperators in EndG0(H

�R�) in a similar way. Thus we identify EndG0(V�V�) withEnd(H)�End(H�) as a subspace of End(H�H�) and have the following result.

Proposition 3.8. — The condition that an operator inEndG0(V �V�) respects theunitary structure on V�V� is equivalent to the canonically transferred operator inEnd(H�H�) respecting the canonically transferred unitary structure on H�H�.

Now let us turn to the condition of compatibility with a transferred time-reversaloperatorT : H R�H�R� ! H R�H� R�. There are a number of cases,depending on whether or notT mixes and which of the conditionsT2 =�Id or T2 = Idare satisfied. The arguments are essentially the same in every case. Let us first gothrough the details in one of them, the mixing case whereT2 =�Id. To be consistentwith the slightly more complicated discussion in the case whereλ = λ�, let us writethis in matrix notation.

ForA2 End(H) andD 2 End(H�), we regard

M =

�A IdR 0

0 D IdR�

�


as the associated transformation in EndG0(H R�H�R�). To construct the trans-ferred time-reversal operator recall the statement of Prop. 3.5. In the setting underconsiderationT squares to minus the identity; it is therefore expressed as

T =

�0 �α�1β�1

αβ 0

�;

whereα : H ! H� andβ : R! R� are complex antilinear. Note that sinceα β =zα z�1β, the mappingsα andβ are determined only up to a common multiplicativeconstantz2 C nf0g. Conjugation ofM in EndG0(HR�H�R�) by T yields

TMT�1 =

�α�1Dα IdR 0

0 αAα�1 IdR�

�:

Clearly, compatibility ofM with T here means thatD = αAα�1.Formulating this in a less detailed way gives the appropriate statement: conjugation

of M in EndG0(H R�H�R�) by T yields the same compatibility condition asconjugating �

A 00 D

�by

�0 �α�1

α 0

�:

Here the sign in front ofα�1 is arbitrary. For definiteness we choose it in such a waythat the transferred time-reversal operator has the same involutory propertyT2 = �Idor T2 = Id as the original operator; in the case under consideration this means that wechoose the minus sign.

Proposition 3.9. — There is a transferred time-reversal operator T: H�H�! H�H� which satisfies either T2 = �Id or T2 = Id. It mixes if and only if the originaloperator mixes, and a canonically transferred mapping inEnd(H �H�) commuteswith it if and only if the original mapping inEndG0(V�V�) commutes with the originaltime-reversal operator.

Proof. — It only remains to handle the case of nonmixing, e.g., whenT2 = �Id. Aswe have seen,T : HR! HR is a pure tensor:

TjHR = αβ ;

which givesT2jHR = α2β2 = �IdH IdR in the case at hand. Since the inducedmapβ : R! R is antiunitary by convention, we haveβ2 = z� IdR with jzj= 1. Asso-ciativity (β2 �β = β �β2) then impliesz=�1. Unlike the case of mixing,β now playsa role through its parity. Ifβ2 = +IdR, the transferred time-reversal operatorα on Hstill satisfiesα2 =�IdH . On the other hand, ifβ2 =�IdR we haveα2 =+IdH instead.Thus the involutory propertyT2 = �Id is passed on to the transferred time-reversaloperator, but depending on the involutory character ofβ the parity may change.

We remind the reader that two distinguished time-reversal symmetries may be present.The above shows that both can be transferred with appropriate involutory properties.


Further, it must be shown that they can be transferred (along withC) so thatTC=CT,T1C =CT1, andT1T =�TT1 still hold. Even if there is just one such operator, it mustbe shown that the transferred operator can be chosen to satisfyTC= CT. Since thediscussion for this is the same as in the case whereλ = λ�, we postpone it to Sect. 3.4.

Finally, we turn to the problem of transferring the complex bilinear form onV�V�

to H�H�. If b denotes the pullback byε of the canonical symmetric bilinear form onV�V�, then by Prop. 3.6

b(h1 r1+ f1 t1;h2 r2+ f2 t2) = d�1( f2(h1)t2(r1)+ f1(h2)t1(r2)) :

Now in this case, i.e., whereλ 6= λ�, theG0-invariant endomorphisms are acting on

HR�H�R� by

�A IdR 0

0 D IdR�

�, where

A�D 2 End(H)�End(H�) ,! End(H�H�) :

Inserting the operatorA�D into the above expression forb we have the following factinvolving the canonical symmetric bilinear forms onH�H�,

s(h1+ f1;h2+ f2) = f1(h2)+ f2(h1) :

Proposition 3.10. — A map inEndG0(V�V�) respects the canonical symmetric bi-linear form if and only if the transferred map inEnd(H)�End(H�) ,! End(H�H�)respects the canonical symmetric bilinear form s on H�H�.

In summary, we have shown that ifλ 6= λ�, then all relevant structures onV �V�

transfer to data of essentially the same type onH �H� (the only exception beingthat the parity of the transferred time-reversal operator may be reversed). In this caseEndG0(V�V�) is canonically isomorphic to End(H)�End(H�) ,! End(H�H�). Anoperator in EndG0(V �V�) respects the original structures if and only if the corre-sponding operator in End(H�H�) respects the transferred structures onH�H�. Thelatter are the transferred unitary structure, induced time reversal and the symmetricbilinear forms.

3.3. The case whereλ = λ�. — Throughout this section it is assumed thatλ = λ�,andψ : R! R� is aG0-equivariant isomorphism. Thus we have the identification

HR�H�R �= HR�H�R�;

h r + f t 7! h r + f ψ(t) :

Applying Prop. 3.4 to each component of an operator in EndG0(H R�H�R) itfollows that

EndG0(HR�H�R�)�= End(H�H�) :


We therefore identify End(H�H�) with EndG0(HR�H�R�) = EndG0(V�V�)by the mapping

M =

�A B

C D

�7!

�A IdR Bψ�1

Cψ D IdR�

�:

Recall the induced unitary structure which is defined, e.g., onHR by

hh1 r1;h2 r2iHR := hh1(r1);h2(r2)iV = hh1;h2iH hr1; r2iR :

It is easy to verify that this defines a unitary structure onH �H� with the desiredproperty: a map in EndG0(V�V�) preserves the given unitary structure onV�V� ifand only if the transferred mapM preserves the induced unitary structure onH�H�.

Now let us consider time reversal. For example, take the case of nonmixing whereT1 : HR! HR. Using Prop. 3.5 we have

T1 =

�αβ 0

0 α β

�;

and conjugating the transformation

�A IdR Bψ�1

Cψ D IdR�

�at the level of operators on

HR�H�R� yields�αAα�1 IdR αBα�1βψ�1β�1

αCα�1 βψβ�1 αDα�1 IdR�

�:

Now, as has been mentioned in Sect. 3.1, the equivariant antiunitary mapsβ andβ areonly unique up to multiplicative constants of unit modulus. They will be chosen inthe next subsection so that the distinguished time-reversal operator(s) and the unitarystructureC commute. These choices having been made, we chooseψ so thatβψβ�1 =ψ. In this way, in the case whereT1 is nonmixing as above, conjugation of the matrixM by T1 is given by �

A B

C D

�7!

�αAα�1 αBα�1

αCα�1 αDα�1

�: (1)

Thus the transferred time-reversal operator is simply given byT1 = α� α onH�H�.If T2 = εT Id (with εT =�1), the compatibility condition for the mixing operator

T =

�0 α�1β�1

εT αβ 0

�

is βψ�1β = εβψ (with εβ =�1). If this holds, conjugation ofM by T is given by�A B

C D

�7!

�α�1Dα εα α�1Cα�1

εα αBα αAα�1

�: (2)


with εα = εβεT . In this case the appropriate transferred operator is given by

T =

�0 α�1

εα α 0

�:

Given the (essentially unique) choices of the tensor-product representations ofT, T1andC which are defined byT1T =�TT1 and by the conditions thatT andT1 commutewith C, we show in Sect. 3.4 that there is a unique choice ofψ so that both of thesecompatibility conditions hold.

If we are in the nonmixing caseβ : R! R, and it so happens thatβ is G0-invariant,then the two alternatives for the involutory property ofT can be distinguished by thetype of the unitary representationR as follows. Definingι : R! R� by r 7! hr; �iR asbefore, consider the unitary mappingψ : R! R� given as the compositionψ = ι Æβ.Sinceβ is G0-invariant,ψ is G0-equivariant, and the statement of Lemma 3.2 applies.Using the antiunitarity ofβ one has

ψ(r)(t) = hβr; tiR= hβ2r;βtiR = ψ(t)(β2r) ;

and therefore the following statement is immediate.

Lemma 3.11. — The parity of an antiunitary and G0-invariant mappingβ : R! R isdetermined by the parity of the irreducible G0-representation space R; i.e.,β satisfiesβ2 = IdR resp.β2 =�IdR if R carries an invariantC -bilinear form which is symmetricresp. alternating.

If β2 = IdR, the transferred time reversal satisfiesT2 =�Id or T2 = Id if the originaltime reversal has these properties. On the other hand, ifβ2 =�IdR, then the propertiesare reversed; e.g., ifT2 = �Id on the original space, then transferred time reversalsatisfiesT2 = Id. We again remind the reader that we must check that the transferredtime-reversal operator(s) andC can be compatibly chosen. It turns out that there is infact just enough freedom in the choice of the constants to achieve this (see Sect. 3.4).

Example. — An example of particular importance in physics is the transfer of the(true) time reversalT in the case where all spin rotations are symmetries. On funda-mental grounds,T is a (nonmixing) operator which commutes with the spin-rotationgroup SU2 and satisfiesT2 =(�1)nId on quantum mechanical states with spinS= n=2.

Let V = H C n+1 be the tensor product of a vector spaceH with the spinn=2 re-presentation space of SU2. For simplicity assume that there are no further symmetries.

Our Nambu space is already in the formV�V� = (HR)� (H�R�). Thus thereduced space isH�H�. Let the time-reversal operator onV = H C

n+1 be writtenT = αβ. The SU2-representation space with spinn=2,C n+1, is known to have parity+1 (symmetric invariant form) forn even, and�1 (alternating invariant form) fornodd. By Lemma 3.11 this impliesβ2 = (�1)nId. The situation on the dual spaceV� isthe same. Thus in this case the transferred time-reversal operatorα : H�H�!H�H�

always satisfiesα2 =+IdH�H� , independent of the spin. �


Now let us turn to the problem of transferring the complex bilinear form. For thisLemma 3.2 is an essential fact. Earlier we identifiedH R�H�R� with V �V�

by the mapε : h r + f t 7! h(r)+ f (t). Using this along with Prop. 3.6 we nowtransfer the canonical symmetric bilinear form onV �V� to H �H�. For this lets(resp.a) denote the canonical symmetric (resp. alternating) form onH�H�.

Proposition 3.12. — Depending onψ being symmetric or alternating, a transferredmap inEnd(H �H�) respects the canonical symmetric form s or alternating form aif and only if the original endomorphism inEndG0(V �V�) respects the canonicalsymmetric complex bilinear form on V�V�.

Proof. — We give the proof for the case whereψ is alternating. The proof in thesymmetric case is completely analogous.

Let M =

�A B

C D

�2 End(H�H�) act as aG0-invariant operator

�A IdR Bψ�1

Cψ D IdR�

�

on H R�H�R� and letb be the symmetric complex bilinear form on this spacewhich is induced from the canonical symmetric form onV�V�. We assume thatM 2GL(H�H�) and give the proof in terms of the isometry propertyb(Mv;Mw)= b(v;w).Let us do this in a series of cases. First, forh1 r1 andh2 r2 in HR,

b(M(h1 r1);M(h2 r2))

= b(Ah1 r1+Ch1ψ(r1);Ah2 r2+Ch2ψ(r2))

= Ch2(Ah1)ψ(r2)(r1)=d+Ch1(Ah2)ψ(r1)(r2)=d

= a(Ah1+Ch1;Ah2+Ch2)ψ(r2)(r1)=d :

WhenM is the identity this becomes

b(h1 r1;h2 r2) = a(h1;h2)ψ(r2)(r1)=d :

Thereforeb(h1 r1;h2 r2) = b(M(h1 r1);M(h2 r2)) if and only if a(h1;h2) =a(M(h1);M(h2)). For f1 t1; f2 t2 2 H�R� the discussion is analogous.

Forh r 2 HR and f t 2 H�R� we have a similar calculation:

b(M(h r);M( f t))

= b(Ah r +Chψ(r);B f ψ�1(t)+D f t)

= D f (Ah) t(r)=d+Ch(B f )ψ(r)(ψ�1(t))=d

= a(M(h);M( f )) t(r)=d :

Of course the analogous identity holds forb(M( f t);M(h r)).

Remark. — To avoid making sign errors and misidentifications in later computations,we find it helpful to transfer the particle-hole conjugation operatorC along with the


complex bilinear form. This is done by insisting that the statement of Lemma 2.2remains true after the transfer. Thus the relationb(Cw1;w2) = hw1;w2i continues tohold in all cases. By an almost identical variant of the computation that led to Lemma3.11, the transferred operatorC has parityC2 =+Id orC2 =�Id depending on whetherthe transferred bilinear form is symmetric or alternating. �

3.4. Precise choice of time-reversal transfer. —Recalling the situation of this sec-tion, we have assumed that the distinguished time-reversal operator(s) stabilize theinitial block V �V�, and we have transferred all structures to the space(H R)�(H�R�) which is isomorphic toV�V�.

The time-reversal operator(s)T and the operatorC are given by(2� 2)-matricesof pure tensors on this space. The space of endomorphismsB that commute with theG0-action is identified with End(H�H�) or End(H)�End(H�) depending on whetheror notλ = λ�. The good HamiltoniansB anticommute withC, and commute with thetime-reversal operator(s)T. If the matrices of pure tensors representing the antiunitaryoperatorsC andT have entriesγδ, this means thatB anticommutes (resp. commutes)with the matrices defined by the operatorsγ. Although the pure tensor decompositionis not unique, this statement is independent of that decomposition.

It has been shown above that the transferred operatorsT andC onH�H�, i.e., thosedefined by the operatorsγ, can be chosen with the desired involutory properties. It willnow be shown that there is just enough freedom to insure thatTC= CT, T1C = CT1,andT1T = �TT1 still hold after transferral. After these conditions have been met,we show as promised thatψ : R! R� can be chosen in a unique way so that thecompatibility conditions of Sect. 3.3 hold, i.e., so that it makes sense to define thetransferred operators by the first factors of the tensor-product representations.

We carry this out in the case whereλ = λ� and two distinguished time-reversaloperators are present. All other cases are either subcases of this or are much simpler.

The operatorC always mixes. We will always choose it to be of the formC= γ ι :HR! H�R� andC = γ�1 ι�1 : H�R�! HR. Of course this is in the casewhereb is symmetric. Ifb is alternating, then we haveC2 = �Id, and we make thenecessary sign change.

Here we restrict to the case whereT2 = T21 = Id. The various other involutory

properties make no difference in the argument. Just as in the case ofC we chooseT = αβ : HR! H�R� andT = α�1β�1 : H�R�! HR. Similarly, wechooseT1 = α1β1 : HR! HR andT1 = α2β2 : H�R�! H�R�.

On (H R)� (H�R�), the operatorsT and T1 commute withC, and we haveT1T = �TT1. We now choose the tensor representations so that the same relationshold for the induced operators on the first factors.

If α, α1, α2, andγ are any choices for the first factors of the tensor-product repre-sentations ofT, T1 andC, then there exist constantsc1, c2 andc3 so thatα2α = c1αα1(from TT1 =�T1T), γα1 = c2α2γ (from CT1 = T1C), andγα�1γ = c3α (CT = TC).


Let α = ξα, γ = ηγ, and αi = ziα (for i = 1;2), whereξ, η and zi are complexnumbers yet to be determined. Just as theci , these constants are of modulus one.

The scaled operators satisfyα2α = χ1c1αα1, γα1 = χ2c2α2γ, andγα�1γ = χ3c3α,whereχ1 = ξ�2z1z2, χ2 = η2(z1z2)

�1, andχ3 = ξ�2η2. Observe that the charactersχisatisfy the relationχ1χ2 = χ3, and that, e.g.,χ2 andχ3 are independent.

The constantsci satisfy an analogous relation. For this first useγα1γ�1 = c2α2andγα�1γ = c3α to obtainγα1α�1γ = (c2=c3)α2α. Then compose both sides of thisequation with the inverse ofα1 on the right and use the relationα2αα�1

1 = c1α toobtainγα1α�1γα�1

1 = (c1c2=c3)α. Now γα�11 = (c2α2)

�1γ. Thus

γα1α�1γα�11 = γα1α�1α�1

2 c�12 γ = c�1

2 γc1α�1γ = (c1c2)�1c3α ;

and hencec1c2=c3 = c3=c1c2, i.e.,c21c2

2 = c23.

Sinceχ2 andχ3 are independent, we can choose the scaling numbers so thatc2 =c3 = 1, thereby arranging thatCT = TC andCT1 = T1C still hold after transferral.To preserve these relations we must now keepχ2 andχ3 fixed at unity, which fromχ1χ2 = χ3 implies thatχ1 = 1. Sincec2

3 = c21c2

2, we then conclude thatc1 takes one ofthe two values�1, and further scaling does not change this constant.

In summary we have the following result.

Proposition 3.13. — The transferred operators T1, T and C can be chosen so thatT1C = CT1, TC= CT, and T1T = �TT1. Assuming that the time-reversal operatorshave been transferred to commute with C in this way, the relation T1T = �TT1 isautomatic and further scaling does not change the sign. Furthermore, theC -linearisomorphismψ : R! R� can be chosen to meet the compatibility conditions whichdetermine the conjugation rules (1) and (2).

Proof. — It remains to prove thatψ can be chosen as stated. For the nonmixing op-eratorT1 the compatibility condition isβ2ψβ�1

1 = ψ. Given some choice ofψ (whichwe will modify) there is a constantc2 C so thatβ2ψβ�1

1 = cψ. This constantc is uni-modular sinceβ1 andβ2 are antiunitary. To satisfy the compatibility condition, replaceψ by ξψ, whereξξ�1c= 1. Note that this choice ofξ only determines its argument.

Turning to the compatibility conditionβψ�1β = εβψ for the mixing operatorT, westart fromcψ = βψ�1β for some otherc2 C , and use theC -antilinearity ofβ to deduceψ�1 = cβ�1ψβ�1. Multiplying expressions givesc= c2 R. Then, rescalingψ to ξψ,the compatibility condition is achieved by settingεβ := c=jcj and solvingjξj2 = jcj.Since this rescaling (withξ 2 R) does not affect the compatibility condition for thenonmixing operator, we have determined the desired isomorphismψ.

Finally, sinceC is a pure tensor, it follows from our representation of the transferredbilinear formb thatcb(Ch1;h2) = hh1;h2i for some constantc. Thus we replaceb bycb and obtain the following final transferred setup onH�H�:

� The canonical bilinear formb which is either symmetric or alternating.


� A unitary structureh ; i which is compatible withb in the sense thatb(Ch1;h2) =hh1;h2i. The operatorC : H $H� satisfies eitherC2 = Id orC2 =�Id, dependingonb being symmetric or alternating.

� Either zero, one, or two time-reversal operators. They are antiunitary and com-mute withC. In the case of two,T is mixing andT1 is nonmixing. In the caseof one, both mixing and nonmixing are allowed. The same involutory propertieshold as before transfer, but signs might change, i.e., ifT2 = Id holds before trans-fer, then it is possible thatT2 =�Id afterwards. Furthermore,T1T =�TT1, andconsequently the unitary productP := TT1 satisfiesP2 =�Id.

In the following sections all of the symmetric spaces which occur in our basic modelwill be described, using the transferred setup. This means that we describe the sub-space of Hermitian operators in End(H)�End(H�) or End(H�H�) which are com-patible withband theT-symmetries. We first handle the case of one or no time-reversaloperator (Sect. 4), and then carry out the classification when bothT andT1 are present(Sect. 5). The final classification result, Theorem 1.1, then follows.

4. Classification: at most one distinguished time reversal

This section is devoted to giving a precise statement of Theorem 1.1 and its proof inthe case where at most one distinguished time-reversal symmetry is present. Combin-ing this with the results of Sect. 3, we obtain a precise description of the blocks thatoccur in the model motivated and described in Sects. 1 and 2.

4.1. Statement of the main result. —Throughout this sectionV denotes a finite-dimensional unitary vector space. The associated spaceW =V�V� is equipped withthe canonically induced unitary structureh ; i andC -antilinear mapC : V !V�, v 7!hv; �i. The results of the previous section allow us to completely eliminateG0 from thediscussion so that it is only necessary to consider the following data:

� The relevant space E of endomorphisms.This is either the full space End(W) orEnd(V)�End(V�) embedded as usual in End(W).

� The canonical complex bilinear form b:W�W! C . This is either the symmetricform s which is given by

s(v1+ f1;v2+ f2) = f1(v2)+ f2(v1) ;

or the alternating forma which is given by

a(v1+ f1;v2+ f2) = f1(v2)� f2(v1) :

Equivalently,C : V ! V� is extended to aC -antilinear mappingC : W !W byC2 =+Id resp.C2 =�Id, andb(Cw1;w2) = hw1;w2i holds in all cases.

� The antiunitary mapping T: W!W, which satisfies eitherT2 =�Id or T2 = Id.We say thatT is nonmixing ifTjV : V !V andTjV� : V�!V�. If TjV : V !V�,


then we refer toT as mixing. In all casesT commutes withC. We also includethe case whereT is not present.

Fixing one of these properties each, we refer to(V;E;b;T) asblock data; e.g.,E =End(W), b= s, T2 =�Id andT being nonmixing would be such a choice.

Our main result describes the symmetric spaces associated to given block data. Letus state this at the Lie algebra level, where for convenience of formulation we onlyconsider the case of trace-free operators. In order to state this result, it is necessary tointroduce some notation.

Given block data(V;E;b;T), let g be the subspace ofE of antihermitian operatorsA which are compatible withb in the sense that

b(Aw1;w2)+b(w1;Aw2) = 0

for all w1;w2 2W. It will be shown thatg is a Lie subalgebra ofE which is invariantunder conjugationA 7! TAT�1 with T. This defines a Lie algebra automorphism

θ : g! g ; A 7! TAT�1 ;

which is usually called aCartan involution. If k := Fix(θ) = fA2 g : θ(A) = Ag andpis the spacefA2 g : θ(A) =�Ag of antifixed points, then

g= k�p

is called the associatedCartan decomposition.The spaceH = H (V;E;b;T) of Hermitian operators which are compatible with the

block data is ip, which is identified with the infinitesimal versionp= g=k.In order to give a smooth statement of our classification result, we recall that the

Lie algebrassun, usp2n, andso2n are commonly referred to as being of typeA, C, andD, respectively. By an irreducibleACD-symmetric space of compact type one meansan (irreducible) compact symmetric space of any of these Lie algebras. With a slightexaggeration we use the same terminology in Theorem 4.1 below. The exaggerationis that the caseso2n=(sop� soq) with p andq odd must be excluded in order for thattheorem to be true. For the overall statement of Theorem 1.1 there is no danger ofmisinterpretation, as the case wherep andq are odd does occur in the situation wheretwo distinguished time-reversal symmetries are present (see Sect. 5).

Theorem 4.1. — Given block data(V;E;b;T), the spaceH = H (V;E;b;T)�= g=k isthe infinitesimal version of an irreducible ACD-symmetric space of compact type. Con-versely, the infinitesimal version of any irreducible ACD-symmetric space of compacttype can be constructed in this way.

There are several remarks which should be made concerning this statement. First,as we have already noted, in order to give a smooth formulation, we have reduced totrace-free operators. As will be seen in the proof, there are several cases where withoutthis assumptiong would have a one-dimensional center.


Secondly, recall that one of the important cases of a compact symmetric space is thatof a compact Lie groupK with the geodesic inversion symmetry at the identity beingdefined byk 7! k�1. Usually one equipsK with the action ofG=K�K defined by left-and right-multiplication, and views the symmetric space asG=K, where the isotropygroup K is diagonally embedded inG. The infinitesimal version is then(k� k)=k,and the automorphismθ : g! g is defined by(X1;X2) 7! (X2;X1). In this setting onespeaks of symmetric spaces of type II.

In our case the classical compact Lie algebras do indeed arise from appropriateblock data, but in the situation whereT does not leave the original spaceW invariant.In that setting,T mapsW = W1 = V1�V�

1 to W2 = V2�V�2 , which has differentG0-

representations from those inW1. Thus the relevant block isW1�W2. Using the resultsof the previous section, in this case we also removeG0 from the picture.

Nevertheless, we are left with a situation where the block isW1�W2 andT : W1 !W2. Thus we wish to allow situations of this type, i.e., whereV�V� is notT-invariant,to be allowed block data. These cases are treated separately in Sect. 4.4.

The case where the symmetric space is just the compact group associated tog alsoarises whenT is not present, i.e., when there is no condition which creates isotropy.

Finally, as has already been indicated in Sect. 1, the appropriate homogeneous spaceversion of Theorem 4.1 is given by replacing the infinitesimal symmetric spaceg=k bythe Cartan-embedded symmetric spaceM �= G=K. HereG is the simply connectedgroup associated tog, a mappingθ : G!G is defined as the Lie group automorphismwhose derivative at the identity is the Cartan involution of the Lie algebra, andM isthe orbit ofe2G of the twistedG-action given byx 7! gxθ(g)�1.

4.2. The associated symmetric space. —In this and the next subsection we work inthe context of simple block data(V;E;b;T) whereW =V�V� is T-invariant.

In the present subsection we prove the first half of Theorem 4.1, namely thatH �=g=k is an infinitesimal version of a classical symmetric space of compact type. Thisessentially amounts to showing that all the involutions which are involved commute.

Let σ : E ! E be theC -antilinear Lie-algebra involution that fixes the Lie algebraof the unitary group inE. If the adjoint operationA 7! A� is defined by

hAw1;w2i= hw1;A�w2i ;

thenσ(A) = �A�. The transformationsS2 E which are isometries of the canonicalbilinear form satisfy

b(Sw1;Sw2) = b(w1;w2)

for all w1;w2 2W. Thus the appropriate Lie algebra involution is theC -linear auto-morphism

τ : E ! E ; A 7! �At ;

whereA 7! At is the adjoint operation defined by

b(Aw1;w2) = b(w1;Atw2) :


Finally, letθ : E ! E be theC -antilinear map defined byA 7! TAT�1.

Proposition 4.2. — The operations A7! A� and A7! At are related by A� =CAtC�1.

Proof. — Fromb(Cw1;w2) = hw1;w2i and the definition ofA 7! A� we have

b(Aw1;w2) = hC�1Aw1;w2i= hC�1w1;(C�1AC)�w2i= b(w1;C

�1A�Cw2) ;

i.e.,At =C�1A�C, independent of the caseb= s or b= a.

Proposition 4.3. — The involutionsσ, τ andθ commute.

Proof. — UsinghC�1A�Cw1;w2i= hw1;C�1ACw2i along withAt =C�1A�C we have

(At)� =C�1AC= (A�)t ;

and consequentlyστ = τσ.SinceT is antiunitary, one immediately shows from the definition ofA� that

hw1;TA�T�1w2i= hTAT�1w1;w2i :

In other words,

θ(σ(A)) =�TA�T�1 =�(TAT�1)� = σ(θ(A)) :

Finally, sinceθ(A) = TAT�1 andT commutes withC, it follows thatθτ = τθ.

Let s := Fix(τ). Sinceθ and σ commute withτ, it follows that they restrict toC -antilinear involutions of the complex Lie algebras. We denote these restrictions by thesame letters. For future reference let us summarize the relevant formulas.

Proposition 4.4. — For A2 s it follows that

σ(A) =CAC�1 and θ(A) = TAT�1:

The parity of C is C2 =+Id for b= s symmetric, and C2 =�Id for b= a alternating.

The spaceg of antihermitian operators inE that respectb is therefore the Lie algebraof σ-fixed points ins. Sinceσ defines the unitary Lie algebra inE, it follows thatg isa compact real form ofs. Let us explicitly describes andg.

If E = End(W) andb= s is symmetric, thens is the complex orthogonal Lie algebraso(W;s) �= so2n(C ). If E = End(W) andb = a is alternating, thens is the complexsymplectic Lie algebrasp(W;a) �= sp2n(C ). If E = End(V)�End(V�), then in bothcases forb it follows that its isometry groupS is SLC (V) acting diagonally by itsdefining representation onV and its dual representation onV�. In this case we haves = sl(V) �= sln(C ). Note that this is a situation where we have used the trace-freecondition to eliminate the one-dimensional center.

For the discussion ofg it is important to note that sinceσ(A) = CAC�1, it followsthatg just consists of the elements ofs which commute withC.

In the symmetric caseb= s, whereC defines a real structure onW, it is appropriate toconsider the set of real pointsWR = fv+Cv : v2Vg. Thinking in terms of isometries,


we regardG = exp(g) as being the group ofR-linear isometries of the restriction ofb= s to WR which are extended complex linearly toW. Note that in this casebjWR =2Reh ; i, and that everyR-linear transformation ofWR which preserves Reh ; i extendsC -linearly to a unitary transformation ofW. Thus, ifE = End(W) andb= s, theng isnaturally identified withso(WR;sjWR)

�= so2n(R).In the alternating caseb= a, if E = End(W), then as in the previous case, sinceσ

definesu(W)� E, it follows that its setg of fixed points ins is a compact real form ofs. Sinces is the complex symplectic Lie algebrasp(W;a)�= sp2n(C ), it follows thatgis isomorphic to the Lie algebrausp2n of the unitary symplectic group.

It is perhaps worth mentioning thatC for b= a defines a quaternionic structure onthe complex vector spaceW. Thus the conditionA = CAC�1 defines the subalgebragln(H ) in End(W). The further conditionA=�A� shows thatg can be identified withthe algebra of quaternionic isometries, another way of seeing thatg�= usp2n.

Finally, in the case whereE = End(V)�End(V�) we have already noted thats =sl(V) which is acting diagonally. It is then immediate that in both the symmetric andalternating casesg= su(V)�= sun. Of courseg acts diagonally as well.

Let us summarize these results.

Proposition 4.5. — In the case where E= End(W) the following hold:

� If b = s is symmetric, theng�= so2n(R).� If b = a is alternating, theng�= usp2n.

If E = End(V)�End(V�), theng is isomorphic tosun and acts diagonally.

Sinceθ commutes withσ, it stabilizesg. Hence,θjg is a Cartan involution whichdefines a Cartan decomposition

g= k�p

of g into its (�1)-eigenspaces. The fixed subspacek= fA2 g : θ(A) = Ag is a subal-gebra andg=k is the infinitesimal version of a symmetric space of compact type.

Recall that, given block data(V;E;b;T), the associated space

H = H (V;E;b;T)�= ip

of structure-preserving Hamiltonians has been identified withp = g=k. Thus we haveproved the first part of Theorem 4.1. The second part is proved in the next section bygoing through the possibilities in Prop. 4.5 along with the various possibilities forT.

It should be noted that ifT =�C, theng= k, i.e., the symmetric space is just a point.Such a degenerate situation, where the set of Hamiltonians is empty, never occurs in awell-posed physics setting.

4.3. Concrete description: symmetric spaces of type I. —Here we describe thepossibilities for each set of block data(V;E;b;T) under the assumption thatW =V�V� is T-invariant. The results are stated in terms of theACD-symmetric spaces, withn := dimCV. The methods of proof of showing which symmetric spaces arise also show


how to explicitly construct them. In the present subsection, all of these are compactirreducible classical symmetric spaces of type I in the notation of [H].

4.3.1. The case E= End(V)�End(V�). — Under the assumptionE = End(V)�End(V�) it follows thatg is just the unitary Lie algebrasu(V) �= sun which is actingdiagonally onW = V�V�. This is independent ofb being symmetric or alternating.Thus we need only consider the various possibilities forT. If T is not present, thesymmetric space isg= sun.

1. T2 =�Id, nonmixing:sun=uspn.SinceT is nonmixing and satifiesT2 = �Id, it follows thathTv1;v2i= a(v1;v2)is aC -linear symplectic structure onV which is compatible withhv1;v2i. Thusthe dimensionn of V must be even here. The facts thatg is acting diagonallyassu(V) and that the elements ofk are precisely those which commute withT,imply thatk= uspn as announced.

2. T2 = Id, nonmixing:sun=son.SinceT andg are acting diagonally, as in the previous case it is enough to onlydiscuss the matter onV. In this caseT defines a real structure onV with VR =fv+Tv : v2Vg, and the unitary isometries which commute withT are just thosetransformations which stabilizeVR and preserve the restriction ofh ; i. Sincehx;yiVR = Rehx;yiV for x;y2VR, it follows thatk= so(VR)�= son(R).

3. T2 =�Id, mixing: sun=s(up�uq).Here it is convenient to introduce the unitary operatorP = CT, which satisfiesP2 = Id or P2 =�Id, depending on the parity ofT. Denote the eigenvalues ofPby u and�u. SinceP does not mix, the condition that a diagonally acting unitaryoperator commutes withT (or equivalently, withP) is just that it preserves theP-eigenspace decompositionV = Vu�V�u. Since the two eigenspacesVu andV�u areh ; i-orthogonal, we havek = s(u(Vu)�u(V�u)), and the desired resultfollows with p= dimVu andq= dimV�u.In the caseP2 = �Id, if there existed a subspaceVR of real points that was sta-bilized byP, thenP would be a complex structure ofVR and the dimensions ofVu andV�u would have to be equal. In general, however, no such spaceVR existsand the dimensionsp andq are arbitrary.

4.3.2. The case E= End(W), b= s. — In this case we have the advantage that wemay restrict the entire discussion to the set of real points

WR = Fix(C) = fv+Cv : v2Vg :

Thusk is translated to being the Lie algebra of the group of isometries of 2Reh ; i onV. Here the Lie algebrag is so(WR). Thus in the case whereT is not present, thesymmetric space isso2n(R).


1. T2 =�Id, nonmixing or mixing:so2n(R)=un.Independent of whether or not it mixes,TjWR : WR ! WR is a complex struc-ture onWR. A transformation in SO(WR) commutes withT if and only if it isholomorphic. Since Reh ; i is T-invariant, this condition defines the unitary sub-algebrak�= un in g�= so2n(R).

2. T2 = Id, nonmixing:so2n(R)=(son(R)� son(R)).SinceTjWR : WR ! WR, we have the decompositionWR = W+

R�W�

Rinto the

(�1)-eigenspaces ofT. We still identify g with the Lie algebra of the groupof isometries ofWR equipped with the restricted form Reh ; i. The subalgebrak,which is fixed byθ : X 7!TXT�1, is the stabilizerso(W+

R)�so(W�

R) of the above

decomposition. Now let us compute the dimensions of the eigenspaces. In thecase at handT defines a real structure on bothV andV�. SinceC commutes withT, it follows that Fix(T) =V

R�V�

RisC-invariant. ThusW+

R= fv+Cv : v2VRg.

A similar argument shows thatW�R

= fv+Cv : v2 iVRg.

3. T2 = Id, mixing: so2n(R)=(so2p(R)� so2q(R)).The exact same argument as above shows thatk = so(W+

R)� so(W�

R). It only

remains to show that the eigenspaces are even-dimensional. For this we considerthe unitary operatorP = CT which leaves bothV andV� invariant. Its(+1)-eigenspaceW+1 is just the complexification ofW+

R. The intersections ofW+1

with V andV� are interchanged byC, and therefore dimC W+1 =: 2p is even. Ofcourse the same argument holds forW�1.

4.3.3. The case E= End(W), b = a. — Since in this caseg is the Lie algebra ofantihermitian endomorphisms which respect the alternating forma on W, it followsthatg�= usp2n. Thus ifT is not present the associated symmetric space isusp2n.

If T is present, we letP := CT. The unitary operatorP always commutes withT, and froma(w1;w2) = hC�1w1;w2i one infers thata(Pw1;Pw2) = a(w1;w2) in allcases, independent ofT being mixing or not.

The classification spelled out below follows from the fact that commutation withTis equivalent to preservation of theP-eigenspace decomposition ofW.

1. T2 =�Id, nonmixing:usp2n=(uspn�uspn).In this caseP2 = Id, andT2 = �Id forcesn to be even. LetW be decomposedinto P-eigenspaces asW =W+1�W�1. If w1 2W+1 andw2 2W�1, then

a(w1;w2) = a(Pw1;Pw2) =�a(w1;w2) = 0 ;

and we see thatW+1 andW�1 area-orthogonal. The mixing operatorP is trace-less. Therefore the dimensions ofW+1 andW�1 are equal, and both of them aresymplectic subspaces ofW. The fact that the decompositionW = W+1�W�1 isalsoh ; i-orthogonal therefore implies thatk= usp(W+1)�usp(W�1).


2. T2 =�Id, mixing: usp2n=(usp2p�usp2q).Here, using the same argument as in the previous case, one shows that theP-eigenspace decompositionW = W+1�W�1 still is a direct sum ofa-orthogonal,complex symplectic subspaces. Since these are alsoh ; i-orthogonal, it followsthatk = usp(W+1)�usp(W�1). Note that in the present case the nonmixing op-eratorP stabilizes the decompositionW =V�V�. Thus, sinceP commutes withC, it follows thatW

+1 =V+1�V�

+1 andW�1 =V�1�V��1.

3. T2 = Id, mixing or nonmixing:usp2n=un.In this caseP2=�Id. Herea(Pw1;Pw2)= a(w1;w2) implies that theP-eigenspacedecompositionW=W+i�W�i is Lagrangian. (This means in particular dimW+i =dimW�i .) Thus its stabilizer insp(W) is the diagonally actinggl(W+i). Since thedecomposition ish ; i-orthogonal, it follows thatk= u(W+i)�= un.

4.4. Concrete description: symmetric spaces of type II. —Recall the original sit-uation where the symmetry groupG0 is still in the picture. As described in Sect. 1 weselect from the given Hilbert space a basic finite-dimensionalG0-invariant subspaceVwhich is composed of irreducible subrepresentations all of which are equivalent to afixed irreducible representationR.

Although the initial block of interest isW = V �V�, it is possible that it is notT-invariant and that it must be expanded. Let us formalize this situation by denotingthe initial block byW1 = V1�V�

1 . We then letP = CT and regard this as a unitaryisomorphism

P : W1 !W2;

whereW2 =V2�V�2 is another initial block.

For i 2 f1;2g, let Ri be theG0-representation onVi which induces the representationonWi . The mapP is equivariant, but only with respect to the automorphisma of G0which is defined bygT-conjugation:PÆg= a(g)ÆP.

Now two situations arise. IfR1�= R2 or R�1

�= R2, then we may build a new blockW = V �V� which is T-invariant so that the results of the previous section can beapplied: ifR1

�= R2, then we letV :=V1�V2 and ifR1�= R�2, thenV :=V1�V�

2 .We assume now that neitherR1

�= R2 norR1�= R�2, and consider the expanded block

W =W1�W2. Recall thatW1 andW2 are in the Nambu spaceW which decomposes asa direct sum of nonisomorphic representation spaces that are orthogonal with respect toboth the unitary structure and the canonical symmetric form. Thus the decompositionW =W1�W2 is orthogonal with respect to both of these structures.

Under the assumption at hand it is immediate that

EndG0(W) = EndG0(W1)�EndG0(W2) :

Thus we are in a position to apply the results of Sect. 3.To do so in the case whereR1

�= R�1, we let ψ1 : R1 ! R�1 denote an equivariantisomorphism, and organize the notation so thatP : V1 !V2. Of courseR1 andR2 are


abstract representations, but we now choose realizations of them inV1 andV2 so thatψ2 := Pψ1P�1 : R2 !R�2 makes sense. Since

Pψ1P�1(g(v2)) = P(ψ1(a�1(g)P�1(v2)))

= P(a�1(g)ψ1(P�1(v2))) = g(Pψ1P�1(v2)) ;

it follows thatψ2 : R2 ! R�2 is aG0-equivariant isomorphism.Assume for simplicity thatψ1 is even, i.e., thatψ1(v1)(v1) = ψ1(v1)(v1). Then

ψ2(v2)(v2) = Pψ1P�1(v2)(v2) = ψ1(P�1(v2))(P

�1(v2))

= ψ1(P�1(v2))(P

�1(v2)) = Pψ1P�1(v2)(v2) = ψ2(v2)(v2) :

The computation in the case whereψ1 is odd is the same except for a sign change.Thusψ1 andψ2 have the same parity.

Now let Ei (for i = 1;2) be the relevant space of endomorphisms that was producedby our analysis ofWi in Sect. 3. Recall that this is either the space End(Hi)�End(H�

i )or End(Hi �H�

i ). Let gi be the Lie algebra of the group of unitary transformationswhich preservebi . The key points now are that the unitary structure onE := E1�E2 isthe direct sum structure, the complex bilinear form onE is b= b1�b2, and the parityof b1 is the same as that ofb2. Thusg1

�= g2.For the statement of our main result in this case, let us recall that the infinitesimal

versions of symmetric spaces of type II are of the formg� g=g, where the isotropyalgebra is embedded diagonally.

Proposition 4.6. — If R1 is neither isomorphic to R2 nor to R�2, then the infinitesimalsymmetric space associated to the T-invariant block data is a type-II ACD-symmetricspace of compact type. Specifically, the classical Lie algebrassun, so2n(R), andusp2narise in this way.

Proof. — Identify g1 andg2 by the isomorphismP. Call the resulting Lie algebrag.The transformations that commute withT are those in the diagonal ing�g. Thus theassociated infinitesimal version of the symmetric space is of type II. The fact that theonly Lie algebras which occur are those in the statement has been proved in 4.2.

This completes the proof of Theorem 4.1. In closing we underline that under theassumptions of Prop. 4.6 the odd-dimensional orthogonal Lie algebra does not appearas a type-II space; only the even-dimensional one does.

5. Classification: two distinguished time-reversal symmetries

Here we describe in detail the situation where both of the distinguished time-reversaloperatorsT andT1 are present. As would be expected, there are quite a few cases. Thework will be carried out in a way which is analogous to our treatment of the case whereonly one time-reversal operator was present. In the first part (Sect. 5.1) we operateunder the assumption that the initial truncated spaceV�V� is invariant under both of


the distinguished operators. In the second part (Sect. 5.2) we handle the general casewhere bigger blocks must be considered.

5.1. The case whereV�V� is G-invariant. — Throughout,T is mixing,T1 is non-mixing andP := TT1. Our strategy in Sects. 5.1.3 and 5.1.4 will be to first compute theoperators which areb-isometries, are unitary and commute withP. This determinesthe Lie algebrag and its action onV�V�. Thenk is determined as the subalgebra ofoperators which commute withT or T1, whichever is most convenient for the proof.The space of Hamiltonians is identified withg=k as before.

In the case ofE = End(V)�End(V�), whereg acts diagonally, the answer forg=kdoes not depend on the involutory properties ofC, T, andT1 individually, but only onthose of the nonmixing operatorsCP= CTT1 andT1. The pertinent Sects. 5.1.1 and5.1.2 are organized accordingly.

5.1.1. The case E= End(V)�End(V�), (CP)2= Id. — Recall that in the case ofE =End(V)�End(V�) it follows that theb-isometry group is SLC (V) acting diagonally.Thus the Lie algebrag consists of those elements of the unitary algebrasu(V) whichcommute with the mixing unitary symmetryP. Equivalently,g is the subalgebra ofsu(V) defined by commutation with the antiunitary operatorCP.

In the present caseCP defines a real structure onV, and we have theg-invariantdecompositionV = VR� iVR. Since the unitary structureh ; i is compatible with thisreal structure, it follows thatg = so(VR). Our argumentation is based aroundT1. Ifit anticommutes withP, then we replaceP by iP so that it commutes. Of course thishas the effect of changing to the case(CP)2 =�Id which is, however, handled below.Hence, in both cases we may assume thatP andT1 commute.

1. T21 = Id: son=(sop� soq).

The space ofCP-real pointsVR is T1-invariant and splits into a sumV+R�V�

Rof

T1- eigenspaces. The Lie algebrak is the stabilizer of this decomposition, whichis h ; i-orthogonal. Thusk= so(V+

R)� so(V�

R).

Observe that in this casen can be any even or odd number and thatp andq arearbitrary with the condition thatn= p+q.

2. T21 =�Id: so2n=un.

In this caseT1 is a complex structure onVR which is compatible with the unitarystructure. Thusk= u(VR;T1) and the desired result follows with 2n= dimC V.

5.1.2. The case E= End(V)�End(V�), (CP)2 = �Id. — The first remarks made atthe beginning of Sect. 5.1.1 still apply:g is the subalgebra of the diagonally actingsu(V) which commutes with the antiunitary operatorCP. But nowCP defines aC -bilinear symplectic structure onW = V �V� by a(w1;w2) := hCPw1;w2i. ActuallyCP is already defined onV and transported toV� by C. Thusg= usp(V).

1. T21 =�Id: usp2n=(usp2p�usp2q).

In this caseΓ := CT : V ! V is a unitary operator which satisfiesΓ2 = Id, and


which defines the eigenspace decompositionV =V+�V�. This decompositionis botha- andh ; i-orthogonal, and consequentlyk= usp(V+)�usp(V�).Note that there is no condition onp andq other thanp+q= n.

2. T21 = Id: usp2n=un.

Let VR be theT1-real points ofV. Thenk is the stabilizer ofVR in g = usp(V).Here the symplectic structurea on V restricts to a real symplectic structureaRonVR. Since the unitary structureh ; i is compatible with this structure,k is themaximal compact subalgebraun of the associated real symplectic algebra.

5.1.3. The case E= End(V�V�), b= s. — Recall that in this caseC2 = Id, and theb-isometry group ofW =V�V� is SO(W). Before going into the various cases, let usremark on the relevance of whether or not time-reversal operators commute withP.

If P2 = u2Id, where eitheru=�1 oru=�i, we consider theP-eigenspace decom-positionW =Wu�W�u. Note dimWu = dimW�u from TrP= 0. The Lie algebrag�so(W) of operators which preserveb= sand commute withP is soR(Wu)�soR(W�u).

An antiunitary operator which commutes withP preserves the decompositionW =Wu�W�u if u=�1, and exchanges the summands ifu=�i. Similarly, if it anticom-mutes withP, then it exchanges the summands inW = W+1�W�1 and preserves thedecompositionW = W+i �W�i . For this reason, as will be clear from the first casebelow, the sign ofTT1 =�T1T has no bearing on our classification.

1. T2 = T21 = Id: (son=(sop� soq))� (son=(sop� soq)).

Suppose first thatP2 = Id, giving theP-eigenspace decompositionW = W+1�W�1. Each of the time-reversal operators commutes withP. To determinekwe consider the unitary operatorΓ = CT1 which is a mixingb-isometry satis-fying ΓP = PΓ andΓ2 = Id. ThusW+1 further decomposes into a direct sumW+1 = W+1

+1 �W�1+1 of Γ-eigenspaces, which are orthogonal with respect to both

b andh ; i. The same discussion holds forW�1. The stabilizer of this refined de-composition isk=

�soR(W

+1+1 )� soR(W

�1+1 )

��soR(W

+1�1 )� soR(W

�1�1 )

�. From

TrP=TrΓ= 0 one infers dimW+1+1 = dimW�1

�1 = pand dimW�1+1 = dimW+1

�1 = q.Now consider the case whereP2 =�Id but the time-reversal operatorsanticom-mute with each other and hence withP. In this situation theP-eigenspace de-compositionW =W+i �W�i is still T-invariant. Therefore we are in exactly thesame situation as above, and of course obtain the same result.This happens in all cases below. Thus, for the remainder of this section we as-sume that the time-reversal operators commute withP.

2. T2 = T21 =�Id: (so2n=(son� son))� (so2n=(son� son)).

The situation is exactly the same as that above, except thatΓ =CT1 now satisfiesΓ2 =�Id. SinceΓ preserves the sets ofC-real points ofW+1 andW�1, Γ definesa complex structure of these real vector spaces. Therefore we have the additionalcondition dimW+i

+1 = dimW�i+1 on the dimensions of theΓ-eigenspaces.


3. T2 =�T21 : (son� son)=son.

The argument to be given is true independent of whetherT2 = Id or T2 =�Id.As usual we consider theP-eigenspace decompositionW =W+i�W�i . SinceP isan isometry of bothb andh ; i, the decomposition isb- andh ; i-orthogonal. Thusg = soR(W+i)� soR(W�i). Now T is antilinear and commutes withP. Thusit permutes theP-eigenspaces, i.e.,T : W+i ! W�i . Sincek consists of thoseoperators ing that commute withT, andT is compatible with both the unitarystructure and the bilinear formb, it follows that(A;B) 2 g is in k if and only ifB = TAT�1. In other words, after applying the obvious automorphism,k is thediagonal ing�= son� son.

5.1.4. The case E= End(V�V�), b= a. — Recall that in this caseC2 =�Id, and theb-isometry group ofW = V�V� is Sp(W). For the same reasons as indicated abovewe may assume that the time-reversal operators commute withP.

1. T2 = T21 = Id: (usp2n=un)� (usp2n=un).

Observe that theP-eigenspace decompositionW = W+1�W�1 is a- and h ; i-orthogonal and that thereforeg = usp(W+1)� usp(W�1). Let the dimension bedenoted by dimC (W+1) = dimC (W�1) = 2n.Now T defines real structures onW+1 andW�1, and these are compatible witha.Hence in both cases the restrictionaR to the setWR

�1 of fixed points ofT is a realsymplectic structure. The algebrak consists of the pairs(A;B) of operators ingwhich stabilizeWR

+1�WR�1. This means thatA, e.g., is in the maximal compact

subalgebra of the real symplectic Lie algebra determined byaR onWR+1, i.e., in a

unitary Lie algebra isomorphic toun. A similar statement holds forB.

2. T2 = T21 =�Id: (usp2n=(usp2p�usp2q)� (usp2n=(usp2p�usp2q)).

The argument made above still shows thatg= usp(W+1)�usp(W�1).Now, to determinek we consider the operatorΓ := CT1 which stabilizes thisdecomposition and satisfiesΓ2 = Id. Thus the further condition to be satisfied inorder for an operator to be ink is that theΓ-eigenspace decomposition of eachsummand must be stabilized, i.e.,k = �ε;δ=�1usp(W

δε ). The dimensions must

match pairwise because TrP= TrΓ = 0.

3. T2 =�T21 : sun=son.

The answer forg=k is the same for the two casesT2 = Id or T2 =�Id.In either case it follows froma(w1;w2) = a(Pw1;Pw2) that the summands of theP-decompositionW=W+i�W�i area-Lagrangian. Thus ana-isometry stabilizesthe decomposition if and only if it is aC -linear transformation acting diagonally,and consequentlyg= su(W+i) (which is acting diagonally as well).Without loss of generality we may assume thatT2 = Id (or else we replaceT byT1). ThenT is a real structure which permutes theP-eigenspaces. Thus the diago-nal action(w+;w�) 7! (Bw+;Bw�) commutes withT if and only if TBT�1 = B.SinceT is compatible with the initial unitary structure, if follows thatB is in the


associated real orthogonal group. For example, if unitary coordinates are chosenso thatT is given by(z;w) 7! (w; z), thenTBT�1 = B simply means thatB= B.

5.2. Building bigger blocks. — BeforeG0-reduction we must determine the basicblock associated to theG0-representation spaceV. This has been adequately discussedin all cases with the exception of the one where there are two time-reversal operators.Here we handle that case by reducing it to the situation where there is only one.

Write the initial block asV1�V�1 and build a diagram consisting of the four spaces

Vi �V�i , i = 1; : : : ;4, with the mapsT, T1, andP emanating from each of them. To be

concrete,T : V1�V�1 ! V2�V�

2 definesV2, T1 : V1�V�1 ! V3�V�

3 definesV3, andT1 : V2�V�

2 !V4�V�4 definesV4. The relationP= TT1 defines the remaining maps.

At this point there is no need to discuss mixing.We also underline that, by the nature of the basic model, any two spacesVi�V�

i andVj �V�

j are either disjoint in the big Nambu space or are equal.Let us now complete the proof of our classification result, Theorem 1.1, by running

through the various cases which occur in the present setting where the initial blockmust be extended. We only sketch this, because given how the extended block casewas handled in the setting of one distinguished time-reversal symmetry (Sect. 4.4) andthe detailed classification results above, the proof requires no new ideas or methods.

1) V1�V�1 is T -invariant and is not T1-invariant. — Here it is only necessary to

considerP : W1 =V1�V�1 !V3�V�

3 =W3. If g is the Lie algebra of unitary operatorswhich commute with theG0-action and respect theb-structure onV1�V�

1 , then thefurther condition of compatibility withP means that the algebra in the present case isg acting diagonally viaP onW1�W3. Thus we have reduced to the case of only onetime-reversal operator onW1, which has been classified above.

Note that this argument has nothing to do with whether or notT is mixing. Hence,in this and all of the following cases there is no need to differentiate betweenT andT1.

2) V1�V�1 is neither T- nor T1-invariant. — Consider the diagram introduced above

where all the spacesWi = Vi �V�i occur. If any of theWi is invariant by eitherT

or T1, then we change our perspective, replaceW1 by that space and apply the aboveargument. Thus we may assume that noWi is stabilized by eitherT or T1. It is stillpossible, however, thatW1 =W4, and in that case it follows thatW2 =W3.

2.1) W1 = W4. — Here bothW1 andW4 areP-invariant. We leave it to the reader tocheck thatP can be transferred to the level of End(H)�End(H�) or End(H�H�) justas we transferred the time-reversal operators. Thus, e.g., it is enough to know the Liealgebra of operatorsg on W1 which are compatible with the unitary structure, areb-isometries and are compatible withP. This has been computed in Sect. 5.1. Of coursewe did this in the case whereV�V� is T- andT1-invariant, but the compatibility withP had nothing to do with time reversal.

In the present case bothT andT1 exchangeW1 andW2. Thus our symmetric spaceis (g�g)=g.


2.2) The spaces Wi are pairwise disjoint. — Here we will go through a number of sub-cases, depending on whether or not there exist (equivariant) isomorphisms betweenvarious spaces. Such an isomorphism is of course assumed to be unitary and to com-mute withC; in particular it is ab-isometry.

2.2.1) W1�= W4. — If ϕ is the isomorphism which does this, thenTϕT�1 =: ψ is an

isomorphism ofW2 andW3. Using these isomorphisms, we buildW := W1�W4 andW :=W2�W3 which are of our initial type; they are stabilized byP and exchanged byT. Thus, as in2.1), if g is the Lie algebra of operators onW which are compatible withthe unitary structure, areb-isometries and are compatible withP, then our symmetricspace is(g�g)=g.

2.2.2) W1�=W2. — For the reasons given above,W3

�=W4 and we buildW andW as inthat case. In the present situationP exchangesW andW. We must then consider twosubcases during our procedure for identifyingg.

The simplest case is whereW andW are not isomorphic. In that setting the Lie alge-brag of unitary operators onW which commute with theG0-action and are compatiblewith b acts diagonally onW�W. This is exactly our algebra of interest.

Thus in this case we can forgetW, and regardg as acting onW. HereT stabilizesW and thus the associated symmetric space isg=k, wherek consists of the operators ing which commute withT. This situation has been classified above; in particular, onlyclassical irreducible symmetric spaces of compact type occur.

Our final case occurs under the assumptionW1�= W2 in the situation whereW and

W are isomorphic. Here we view an operator which commutes with theG0-action as amatrix �

A B

C D

�:

Compatibility with P can then be interpreted asB andD being determined fromAandC by P-conjugation. In this notationA : W !W andC : W ! W. But we mayalso regardC as an operator onW which is transferred to a map fromW to W bythe isomorphism at hand. Therefore the Lie algebra of interest can be identified withthe set of pairs(A;C) of operators onW which are compatible with the unitary andb-structures and commute with theG0-action onW. Hence the associated symmetricspace is the direct sumg=k�g=k wherek is determined by compatibility withT : W!W, i.e., a direct sum of two copies of an arbitrary example that occurs with only oneT-symmetry.

6. Physical realizations

We now illustrate Theorem 1.1 by the two large sets of examples that were alreadyreferred to in Sect. 2: (i) fermionic quasiparticle excitations in disordered normal-and superconducting systems, and (ii) Dirac fermions in a stochastic gauge field back-ground. In each case we fix a specific Nambu spaceW , and show how a variety of


symmetric spaces (each corresponding to a symmetry class) is realized byvarying thegroup of unitary and antiunitary symmetries, G.

The invariable nature ofW is a principle imposed by physics: electrons, e.g., haveelectric chargee=�1 and spinS= 1=2 and these properties cannot ever be changed.What can be changed, however, by varying the experimental conditions, are the sym-metries of the Hamiltonian governing the specific situation at hand. For example,turning on an external magnetic field breaks time-reversal symmetry, adding spin-orbitscatterers to the system breaks spin-rotation symmetry, lowering the temperature en-hances the pairing forces that may lead to a spontaneous breakdown of the global U1charge symmetry, and so on.

6.1. Quasiparticles in metals and superconductors. —The setting here is the onealready described in Sect. 2.1: given the complex Hilbert spaceV of single-electronstates, we form the Nambu spaceW = V �V � of electron field operators. OnWwe then have the canonical symmetric bilinear formb, the particle-hole conjugationoperatorC : W !W , and the canonical unitary structureh ; i.

The complex Hilbert spacesV andV � are to be viewed as representation spacesof a U1 group, which is the global U1 gauge degree of freedom of electrodynamics.Indeed, creating or annihilating one electron amounts to adding one unit of negative orpositive electric charge to the fermion system. In representation-theoretic terms, thismeans thatV carries the fundamental representation of the U1 gauge group whileV �

carries the antifundamental one. Thusz2 U1 here acts onV by multiplication withz,and onV � by multiplication withz.

Extra structure arises from the fact that electrons carry spin 1/2, which implies thatV is a tensor product of spinor space,C 2, with the Hilbert spaceX for the orbitalmotion in real space. The spin-rotation group Spin3 = SU2 acts trivially onX and bythe spinor representation on the factorC

2. (In a framework more comprehensive than isof relevance to the disordered systems setting developed here, the spinor representationwould enter as a projective representation of the rotation group SO3, and SO3 wouldact on the factorX by rotations in the three-dimensional Euclidean space.) On physicalgrounds, spin rotations must preserve the canonical anticommutation relations as wellas the unitary structure ofV . Therefore, by Prop. 2.2 spin rotations commute with theparticle-hole conjugation operatorC.

Another symmetry operation of importance for present purposes is time reversal. Asalways in quantum mechanics, time reversal is implemented as an antiunitary operatorT on the single-electron Hilbert spaceV . Its algebraic properties are influenced bythe spin 1/2 nature of the electron: fundamental physics considerations dictateT2 =�Id. A closely related condition is that time reversal commutes with spin rotations.Textends to an operation onW byCT = TC.

In physics one uses the wordquasiparticlefor the excitations that are created byacting with a fermionic field operator on a many-fermion ground state.


6.1.1. Class D. — In the general context of quasiparticle excitations in metals andsuperconductors, this is the fundamental class whereno symmetries are present.

A concrete realization takes place in superconductors where the order parametertransforms under spin rotations as a spin triplet,S= 1 (i.e., the adjoint representationof SU2), and transforms under SO2-rotations of two-dimensional space as ap-wave(the fundamental representation of SO2). A recent candidate for a quasi-2d (or layered)spin-tripletp-wave superconductor is the compound Sr2RuO4 [M, E]. (A nonchargedanalog is theA-phase of superfluid3He [VW ].) Time-reversal symmetry in such asystem may be broken spontaneously, or else can be broken by an external magneticfield creating vortices in the superconductor. Further realizations proposed in the recentliterature include double-layer fractional quantum Hall systems at half filling [R] (moreprecisely, a mean-field description for the composite fermions of such systems), and anetwork model for the random-bond Ising model [SF].

The time-evolution operatorsU = e�itH=~ in this class are constrained only by therequirement that they preserve both the unitary structure and the symmetric bilinearform of W . If WR is the set of real pointsfv+Cv : v 2 V g, we know from Prop.4.5 that the space of time evolutions is a real orthogonal group SO(WR). In Cartan’snotation this is called a symmetric space of theD family. The HamiltoniansH are suchthat iH 2 so(WR); this means that the Hamiltonian matrices are imaginary skew in asuitably chosen basis (called Majorana fermions in physics).

Note that sinceWR is a real form of(X C 2)� (X C 2)�, the dimension ofWR

must be a multiple of four (for spinless particles it would only be a multiple of two).

6.1.2. Class DIII. — Let now time reversal be a symmetry of the quasiparticle sys-tem. This means that magnetic fields and scattering by magnetic impurities are absent.On the other hand, spin-rotation invariance is again required to be broken.

Known realizations of this situation exist in gapless superconductors, say with spin-singlet pairing, but with a sufficient concentration of spin-orbit impurities to causestrong spin-orbit scattering [SF]. In order for quasiparticle excitations to exist at lowenergy, the spatial symmetry of the order parameter should bed-wave (more precisely,a time-reversal invariant combination of the angular momentuml = +2 andl = �2representations of SO2). A noncharged realization occurs in theB-phase of3He [VW ],where the order parameter is spin-triplet without breaking time-reversal symmetry.Another candidate are heavy-fermion superconductors [S], where spin-orbit scatteringoften happens to be strong owing to the presence of elements with large atomic weightssuch as uranium and cerium.

Time-reversal invariance constrains the set of good HamiltoniansH by H =THT�1.SinceT2 = �Id for spin 1/2 particles, we are dealing with the case treated in 4.3.2.1.The space of time evolutions therefore is SO(WR)=U(V ), which is a symmetric spaceof theDIII family. The standard form of the Hamiltonians in this class is

H =

�0 ZZ� 0

�; (3)


whereZ 2 Hom(V �;V ) is skew. (Note again that the dimension ofWR is a multipleof four, and would be a multiple of two for particles with spin zero).

6.1.3. Class C. — Next let the spin of the quasiparticles be conserved, and let time-reversal symmetry be broken instead. Thus magnetic fields (or some equivalentT-breaking agent) are now present, while the effect of spin-orbit scattering is absent. Thesymmetry group of the physical system then isG= G0 = Spin3 = SU2.

This situation is realized in spin-singlet superconductors in the vortex phase [S4].Prominent examples are the cuprate (or high-Tc) superconductors [T], which are lay-ered and exhibitd-wave symmetry in their copper-oxide planes. It has been speculatedthat some of these superconductors break time-reversal symmetry spontaneously, bythe generation of an order-parameter component idxy or is [S3]. Other realizations ofthis class include network models of the spin quantum Hall effect [G].

Following the general strategy of Sect. 3, we eliminateG0 = SU2 from the pictureby transferring fromV �V � to the reduced spaceX�X�. In the process the bilinearform b undergoes a change of parity. To see this letR= C 2 (a.k.a. spinor space) be thefundamental representation space of SU2. R is isomorphic toR� by ψ : r 7! hiσ2r; �iRwhereσ2 is the second Pauli matrix. This isomorphismψ : R! R� is alternating.Therefore, by Prop. 3.12 the symmetric bilinear form ofV �V � gets transferred tothe alternating forma of X�X�.

From Prop. 4.5 we then infer that the space of time evolutions is USp(X�X�) — asymmetric space of theC family. The standard form of the Hamiltonians here is

H =

�A BB� �At

�;

with self-adjointA2 End(X) and complex symmetricB2 Hom(X�;X).

6.1.4. Class CI. — The next class is obtained by taking spin rotations as well as thetime reversalT to be symmetries of the quasiparticle system. Thus the symmetry groupis G= G0[TG0 with G0 = Spin3 = SU2.

Like in the previous symmetry class, physical realizations are provided by the low-energy quasiparticles of unconventional spin-singlet superconductors [T]. The differ-ence is that the superconductor must now be in the Meissner phase where magneticfield are expelled by screening currents. In the case of superconductors with severallow-energy points in the first Brillouin zone, scattering off hard impurities is needed tobreak additional conservation laws that would otherwise emerge (see Sect. 6.1.5).

To identify the relevant symmetric space, we again transfer fromV �V � to thereduced spaceX�X�. As before, the bilinear formb changes parity from symmetricto alternating under this reduction. In addition now, time reversal has to be transferred.As was explained in the example following Lemma 3.11, the time-reversal operatorchanges its involutory character fromT2 =�IdV�V � to T2 =+IdX�X�.

In the language of Sect. 4 the block data areV = X, E = End(V �V�), b = a, Tnonmixing, andT2 = Id. This case was treated in 4.3.3.2. From there, we know that


the space of time evolutions is USp(X�X�)=U(X) – a symmetric space in theCIfamily. The standard form of the Hamiltonians in this class is the same as that given in(3) but now withZ 2 Hom(X�;X) complex symmetric.

6.1.5. Class AIII. — This class is commonly associated with random-matrix modelsfor the low-energy Dirac spectrum of quantum chromodynamics with massless quarks(see Sect. 6.2.1). Here we review an alternative realization, which has recently beenidentified [A3] in d-wave superconductors with soft impurity scattering.

To construct this realization one starts from classCI, i.e. from quasiparticles in a su-perconductor with time-reversal invariance and conserved spin, and enlarges the sym-metry group by imposing another U1 symmetry, generated by a Hermitian operatorQwith Q2 = Id. The physical reason for the extra conservation law is approximate mo-mentum conservation in a disordered quasiparticle system with a dispersion law thathas Dirac-type low-energy points at four distinct places in the Brillouin zone.

Thus beyond the spin-rotation group SU2 there now exists a one-parameter groupof unitary symmetries eiθQ. The operators eiθQ are defined onV , and are diagonallyextended toW = V �V �. They are characterized by the property that they commutewith particle-hole conjugationC, time reversalT, and the spin rotationsg2 SU2.

The reduction to standard block data is done in two steps. In the first step, weeliminate the spin-rotation group SU2. From the previous section, the transferred dataare known to beE = End(X�X�), b= a, T nonmixing, andT2 = Id.

The second step is to reduce by the U1 group generated byQ. For this consider theC -linear operatorJ := iQ with J2 =�Id, and let theJ-eigenspace decomposition ofXbe writtenX = X+i �X�i . There is a corresponding decompositionX� = X�

+i �X��i .

SinceJ commutes withT, a complex structure is defined by it on the set ofT-realpoints ofX. Therefore dimX+i = dimX�i . Another consequence ofJT = TJ is thattheC -antilinear operatorT exchangesX+i with X�i . ThusT is mixing with respect tothe decompositionsX = X+i �X�i andX� = X�

+i �X��i . TheC -antilinear operator

C mapsX�i to X��i .

The fully reduced block data now areV := X+i �X�+i , E = End(V)�End(V�),

b= a, T mixing, andT2 = Id. The finite-dimensional version of this case was treated in4.3.1.3. Our answer for the space of time-evolution operators was SUp+q=S(Up�Uq),which is a symmetric space in theAIII family.

Unlike the general case handled in 4.3.1.3, it here follows from the fundamentalphysics definition of particle-hole conjugationC and time reversalT that the operatorCT stabilizes a real subspaceVR. We also have(CT)2 =�Id. Therefore, the operatorCT defines a complex structure ofVR, and hence the integersp andq, which are thedimensions of theCT-eigenspaces inV, must be equal.

6.1.6. Class A. — At this point a new symmetry requirement is brought into play:conservation of the electric charge. Thus the global U1 gauge transformations of elec-trodynamics are now decreed to be symmetries of the quasiparticle system. This means


that the system no longer is a superconductor, where U1 gauge symmetry is sponta-neously broken, but is a metal or normal-conducting system. If all further symmetriesare broken (time reversal by a magnetic field or magnetic impurities, spin rotations byspin-orbit scattering, etc.), the symmetry group isG= G0 = U1.

All states (actually, field operators) inV have the same electric charge. Thus theirreducible U1 representations which they carry all have the same isomorphism class,sayλ. States inV � carry the opposite charge and belong to the dual classλ�. Sinceλ 6= λ�, we are in the situation of Sect. 4.3.1, whereE = End(V )�End(V �). With Tbeing absent, the space of time evolutions is U(V ) acting diagonally onV �V �.

In random-matrix theory, and in the finite-dimensional case where U(V )�= UN, onerefers to these matrix spaces as the circular Wigner-Dyson class of unitary symmetry.The Hamiltonians in this class are represented by complex Hermitian matrices.

If we make the restriction to traceless Hamiltonians, the space of time evolutionsbecomes SUN, which is a type-II irreducible symmetric space of theA family.

6.1.7. Class AII. — Beyond charge conservation or U1 gauge symmetry, time rever-salT is now required to be a symmetry of the quasiparticle system. Physical realiza-tions of this case occur in metallic systems with spin-orbit scattering. The pioneeringexperimental work (of the weak localization phenomenon in this class) was done ondisordered magnesium films with gold impurities.

The block data now isE = End(V )�End(V �), b = s, T nonmixing,T2 = �Id.This case was considered in 4.3.1.1. The main point there was that time reversalTdefines aC -linear symplectic structurea on V by a(v1;v2) = hTv1;v2i. Conjugationby T therefore fixes a unitary symplectic group USp(V ) inside of U(V ), and the spaceof good time evolutions isG=K = U(V )=USp(V ). In the finite-dimensional settingwhereG=K�=U2N=USp2N, this is called the circular Wigner-Dyson class of symplecticsymmetry in random-matrix theory. The Hamiltonians in this class are represented byHermitian matrices whose matrix entries are real quaternions. The irreducible partSU2N=USp2N, obtained by restricting to traceless Hamiltonians, is a type-I symmetricspace in theAII family.

6.1.8. Class AI. — The next class is the Wigner-Dyson class of orthogonal symmetry.In the present quasiparticle setting it is obtained by imposing spin-rotation symmetry,U1 gauge (or charge) symmetry and time-reversal symmetry all at once.

Important physical realizations are by disordered metals in zero magnetic field.Families of quantum chaotic billiards also belong to this class.

The group of unitary symmetries here isG0 = U1�SU2. We eliminate the spin-rotation group SU2 from the picture by transferring fromV = X C 2 to the reducedspaceX. Again, the involutory character ofT is reversed in the process: the transferredtime reversal satisfiesT2 = +Id. The parity of the bilinear form also changes, fromsymmetric to alternating; however, this turns out to be irrelevant here, as there is stillthe U1 charge symmetry and we are in the situationλ 6= λ�.


The block data now isE = End(X)�End(X�), b= a, T nonmixing,T2 = Id. Ac-cording to 4.3.1.2 these yield (the Cartan embedding of) U(X)=O(X) as the space ofgood time evolutions. The irreducible part SU(X)=SO(X), or SUN=SON in the finite-dimensional setting, is a symmetric space in theAI family. The Hamiltonian matricesin this class can be arranged to be real symmetric.

6.2. The Euclidean Dirac operator for chiral fermions. — We now explore thephysical examples afforded by Dirac fermions in a random gauge field background.These examples include the Dirac operator of quantum chromodynamics, i.e., the the-ory of strong SU3 gauge interactions between elementary particles called quarks.

The mathematical setting for this has already been described in Sect. 2.3. Recallthat one is given a twisted spinor bundleSR over Euclidean space-time, and thatVis taken to be the Hilbert space ofL2-sections of that bundle. One is interested in theDirac operatorDA in a gauge field backgroundA and in the limit of zero mass:

DA = iγµ(∂µ�Aµ) :

We extend the self-adjoint operatorDA diagonally fromV to the fermionic NambuspaceW = V �V � by the conditionDA =�CDAC�1. The chiral ‘symmetry’ΓDA+DAΓ = 0, whereΓ = γ5 is the chirality operator, then becomes a true symmetryDA =TDAT�1 with an antiunitary operatorT =CΓ = ΓC, which mixesV andV �.

6.2.1. Class AIII. — Let now the complex vector spaceR= C N be the fundamentalrepresentation space for the gauge group SUN with N � 3. (N is called the number ofcolors in this context.) Quantum chromodynamics is the special caseN = 3.

The fact that the extended Dirac operatorDA acts diagonally onW = V �V � is at-tributed to a symmetry groupG0 = U1 which hasV andV � as inequivalent representa-tion spaces. For a generic gauge-field configuration there exist no further symmetries;thus the total symmetry group isG= G0[TG0.

The block data here isV = V , E = End(V)�End(V�), b= s, T mixing, T2 = Id,which is the case considered in 4.3.1.3. Ifn= dimV, we have

p�= sun=s(up�uq) :

The difference of integersp� q is to be identified with the difference between thenumber of right and left zero modes ofD2

A. (‘Right’ and ‘left’ in this context pertainto the (+1)- and (�1)-eigenspaces of the chiralityΓ = γ5.) The latter number is atopological invariant called the index of the Dirac operator.

6.2.2. Class BDI. — We retain the framework from before, but now consider thegauge group SU2, where the number of colorsN = 2. In this case the massless DiracoperatorDA has an additional antiunitary symmetry [V1], which emerges as follows.

Recall that the unitary SU2-representation spaceR= C 2 is isomorphic to the dualrepresentation spaceR� by a C -linear mappingψ : R! R�. Combining the inverseof this with ι : R! R� defined byι(r) = hr; �iR, we obtain aC -antilinear mapping


β := ψ�1Æ ι : R!R. The mapβ thus defined commutes with the SU2-action onR. ByLemma 3.11 it satisfiesβ2 =�IdR sinceψ is alternating.

Now, on the (untwisted) spinor bundleSover Euclidean space-timeM there existsa C -antilinear operatorα, calledcharge conjugationin physics, which anticommuteswith the Clifford actionγ : T�M ! End(S); thusαiγ = iγα. Sinceγ5 = γ0γ1γ2γ3, thisimplies thatα commutes withγ5 = Γ and stabilizes theΓ-eigenspace decompositionS= S+�S� into half-spinor componentsS�. The charge conjugation operator hassquareα2 =�IdS.

For the case of three or more colors, the existence ofα is of no consequence from asymmetry perspective, as the fundamental and antifundamental representations of SUNare inequivalent forN � 3. ForN = 2, however, we also haveβ, andα combines withit to give an antiunitary symmetryT1 = αβ. Indeed,

T1DAT�11 = (αβ)DA(αβ) = α(iγµ)α�1β(∂µ�Aµ)β�1 :

Since gauge transformationsg(x)2SU2 commute withβ, so do the componentsAµ(x)2su2 of the gauge field. ThusβAµβ�1 = Aµ , and sinceα(iγ)α�1 = iγ, we have

T1DAT�11 = DA :

Note that the antiunitary symmetryT1 : V ! V is nonmixing, andT21 = Id. As usual,

the extension to an operatorT1 : W !W is made by requiringCT1 = T1C.Thus we now have two antiunitary symmetries,T and T1. BecauseT is mixing

and T1 nonmixing, the unitary operatorP = TT1 = T1T mixes V with V �. SinceT2 = T2

1 = Id, and(CP)2 = Id, this is the case treated in 5.1.1.1, where we found

p�= so(VR)=(so(V +

R)� so(V �

R)) :

After truncation to finite dimension this issop+q=(sop� soq). The differencep� qstill has a topological interpretation as the index of the Dirac operator.

Although our considerations explicitly referred to the case of the gauge group beingSU2, the only specific feature we used was the existence of an alternating isomorphismψ : R! R�. The same result therefore holds for any gauge group representationRwhere such an isomorphism exists. In particular it holds for the fundamental represen-tation of the whole series of symplectic groups USp2N (which includes SU2 �= USp2).

6.2.3. Class CII. — Now takeR to be the adjoint representation of any compact Lie(gauge) groupK with semisimple Lie algebra. This case is called ‘adjoint fermions’ inphysics. A detailed symmetry analysis of it was presented in [HV ].

The Cartan-Killing form on Lie(K),

B(X;Y) = Trad(X)ad(Y) ;

is nondegenerate, invariant, complex bilinear, and symmetric.B therefore defines anisomorphismψ : R! R� by ψ(X) = B(X; �). SinceB is symmetric, so isψ.

The change in parity ofψ reverses the parity of the antiunitary operatorβ = ψ�1Æ ι,which now satisfiesβ2 =+IdR. By α2 =�Id this translates toT2

1 = (αβ)2 =�Id.


Thus we now have two antiunitary symmetriesT andT1 with T2 = Id = �T21 , and

(CP)2 = (CTT1)2 =�Id. This case was handled in 5.1.2.1 where we found

p�= usp(V )=(usp(V +)�usp(V �)) :

In a finite-dimensional setting this would beusp2p+2q=(usp2p�usp2q).In summary, the physical situation is ruled by a mathematical trichotomy: the iso-

morphismψ : R! R� is either symmetric, or alternating, or does not exist. The cor-responding symmetry class of the massless Dirac operator isCII, BDI, or AIII, respec-tively. As was first observed by Verbaarschot [V], this is the same trichotomy thatruled Dyson’s threefold way.

Acknowledgment. This work was carried out under the auspices of the DeutscheForschungsgemeinschaft, SFB/TR12. Major portions of the article were preparedwhile M.R.Z. was visiting the Institute for Advanced Study (Princeton, USA) and theNewton Institute for Mathematical Sciences (Cambridge, UK). The support of theseinstitutions is gratefully acknowledged.

References

[A] Altland, A. and Zirnbauer, M.R.: Nonstandard symmetry classes in mesoscopic normal-/superconducting hybrid systems, Phys. Rev. B55, 1142-1161 (1997)

[A3] Altland, A., Simons, B.D. and Zirnbauer, M.R.: Theories of low-energy quasiparticlestates in disorderedd-wave superconductors, Phys. Rep.359, 283-354 (2002)

[B] Berline, N., Getzler, E. and Vergne, M.:Heat kernels and Dirac operators(Springer-Verlag, Berlin, 1992)

[C] Caselle, M. and Magnea, U.: Random-matrix theory and symmetric spaces, Phys. Rep.394, 41-156 (2004)

[D] Dyson, F.J.: The threefold way: algebraic structure of symmetry groups and ensembles inquantum mechanics, J. Math. Phys. 3, 1199-1215 (1962)

[E] Eremin, I., Manske, D., Ovchinnikov, S.G. and Annett, J.F.: Unconventional superconduc-tivity and magnetism in Sr2RuO4 and related materials, Ann. Physik13, 149-174 (2004)

[G] Gruzberg, I.A., Ludwig, A.W.W. and Read, N.: Exact exponents for the spin quantumHall transition, Phys. Rev. Lett.82, 4524-4527 (1999)

[H] Helgason, S.:Differential geometry, Lie groups and symmetric spaces(Academic Press,New York, 1978)

[HV] Halasz, M.A. and Verbaarschot, J.J.M.: Effective Lagrangians and chiral random-matrixtheory, Phys. Rev. D51, 2563-2573 (1995)

[K] Katz, N.M. and Sarnak, P.:Random matrices, Frobenius eigenvalues, and monodromy(American Mathematical Society, Providence, R.I., 1999)

[M] Mackenzie, A.P. and Maeno, Y.: The superconductivity of Sr2RuO4 and the physics ofspin-triplet pairing, Rev. Mod. Phys.75, 657-712 (2003)

[R] Read, N. and Green, D.: Paired states of fermions in two dimensions with breaking ofparity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B61, 10267-10297 (2000)


[S] Stewart, G.S.: Heavy-fermions systems, Rev. Mod. Phys.56, 755-787 (1984)[S3] Senthil, T., Marston, J.B. and Fisher, M.P.A.: Spin quantum Hall effect in unconventional

superconductors, Phys. Rev. B60, 4245-4254 (1999)[S4] Senthil, T., Fisher, M.P.A., Balents, L. and Nayak, C.: Quasiparticle transport and local-

ization in high-Tc superconductors, Phys. Rev. Lett.81, 4704-4707 (1998)[SF] Senthil, T. and Fisher, M.P.A.: Quasiparticle localization in superconductors with spin-

orbit scattering, Phys. Rev. B61, 9690-9698 (2000)[T] Tsuei, C.C. and Kirtley, J.R.: Pairing symmetry in the cuprate superconductors, Rev. Mod.

Phys.72, 969-1016 (2000)[V] Verbaarschot, J.J.M.: The spectrum of the QCD Dirac operator and chiral random-matrix

theory: the threefold way, Phys. Rev. Lett.72, 2531-2533 (1994)[V1] Verbaarschot, J.J.M.: The spectrum of the Dirac operator near zero virtuality forNc= 2,

Nucl. Phys. B426, 559-574 (1994)[VW] Vollhardt, D. and Wolfle, P.: The superfluid phases of Helium 3(Taylor & Francis,

London, 1990)[VZ] Verbaarschot, J.J.M. and Zahed, I.: Spectral density of the QCD Dirac operator near zero

virtuality, Phys. Rev. Lett.70, 3852-3855 (1993)[Z] Zirnbauer, M.R.: Riemannian symmetric superspaces and their origin in random-matrix

theory, J. Math. Phys.37, 4986-5018 (1996)

June 10, 2004

P. HEINZNER, A. HUCKLEBERRY, Fakultat fur Mathematik, Ruhr-Universität Bochum, Germany

M.R. ZIRNBAUER, Institut fur Theoretische Physik, Universität zu Koln, GermanyE-mail : [email protected], [email protected],

[email protected]

Documents

SYMMETRY CLASSES OF DISORDERED FERMIONS by P ...sentation theory, symmetric spaces 1. Introduction In a famous and inﬂuential paper published in 1962 (“the threefold way: algebraic