Wong Paper2

8/13/2019 Wong Paper2

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Chiral kinetic theory and anomalies in arbitrary dimensions

VATSAL DWIVEDI

University of Illinois,

Department of Physics

1110 W. Green St.

Urbana, IL 61801 USA

E-mail: [email protected]

MICHAEL STONE

University of Illinois,

Department of Physics

1110 W. Green St.

Urbana, IL 61801 USA

E-mail: [email protected]

Abstract

We derive the collisionless classical Boltzmann equation for a gas of massless Weyl fermions

moving in a background gauge field in (2N+ 1) + 1 spacetime dimensions. We show how classical

versions of the gauge and Abelian chiral anomalies arise from the Chern character of the non-

Abelian Berry connection that parallel transports the spin degree of freedom in momentum space.

PACS numbers: 11.10.Kk, 11.15.-q, 12.38.Aw, 12.38.Mh, 71.10.Ca

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I. INTRODUCTION

The axial anomalya non-conservation of currents associated with chiral fermionsis

usually regarded as a purely quantum effect so it is rather surprising that it is possible to

extract the abelian anomaly from a purely classical Hamiltonian phase-space calculation

[1]. The authors of [1] did so by considering the dynamics of a finite-density gas of Weyl

fermions and arguing that the incompressibility of phase space allows the anomalous influx

of particles from the negative-energy Dirac sea into the positive-energy Fermi sea to be

reliably counted by keeping track of the density flow only near the Fermi surface. Near

the Fermi surface, and therefore well away from the dangerously quantum Dirac point,

a classical Boltzmann equation becomes sufficiently accurate for this purpose. The only

quantum input required is knowledge of how to normalize the phase space measure and asimple computation of the momentum-space gauge field that accounts for the gyroscopic

effect of the Weyl particles spin. The gauge field is a Berry-phase effect that subtly alters

the classical canonical structure so that x and p are no longer conjugate variables, and

d3pd3xis no longer the element of phase space volume [2, 3].

In a previous paper [4] we extended the analysis of [1] and considered the motion of Weyl

particles in a background non-Abelian gauge field. In this way we obtained the non-Abelian

gauge anomaly in 3+1 dimensions. Our calculation, like that in [1], relied on a number of

simplifications peculiar to three spatial dimensions in particular that the Berry phase

is indeed a phase and not a more general unitary matrix. In the present paper we derive

both the Abelian and the gauge anomaly in any even number (2N+ 1) + 1 of space-time

dimensions. Apart from the intrinisic interest of higher dimension anomalies, it turns out

that the structure of the calculation becomes more transparent when we discard the special

features of the lower dimensional dynamics.

In section II we review the action that describes the motion of a Weyl particle in a

background 3 +1 Abelian gauge field. We then explain how this action can be extended so as

obtain the motion of a Weyl particle in any even-dimensional spacetime in a background non-

Abelian field. In section III we review how the differential form formulation of Hamiltonian

dynamics is modified when the underlying symplectic form becomes time dependent. We

use this language to extend Liouvilles theorem on phase-space volume conservation to the

time dependent case, and identify a vulnerablity in the proof that allows for anomalous

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conservation Laws. In IV we assume that the time dependent Liouville theorem is not

compromised and show that it leads to the expected conservation of the Abelian number

current and the covariant conservation of the gauge current. Section V then shows how the

non-zero Chern-character of the internal momentum space spin-transport gauge field leads

to a violation of Liouvilles theorem and hence to the singlet anomaly and the covaraint

form of the Gauge anomaly.

II. A CLASSICAL ACTION FOR WEYL FERMIONS

It was shown in [1] that the classical phase-space action for a 3+1 dimensional Weyl

fermion moving in a background electromagnetic field is

S[x, p] = dt (p x |p| e(x) +eA x a p) . (1)HereeA x is the standard coupling of the Maxwell vector potential Ato the velocity xofthe chargeeparticle. The combinatione(x, t)+ |p|, where(=A0) is the scalar potentialand|p| is the kinetic energy of the massless fermion, is the classical Hamiltonian H(x, p).The a pterm accounts for the gyroscopic effect of the spin angular momentum, which fora right handed massless particle is forced to point in the direction of the momentum.

A. Berry connection in momentum space

The momentum-space gauge field a(p) appearing in (1) is the adiabatic Berry connection

[6] which has components

ak = ip, +| pk

|p, +. (2)

These components are obtained from the E = +|p| eigenvector|p, + of the quantumHamiltonian

Hp= p, (3)

and the resulting Berry curvature b= a possesses a monopole singularity

b= 23(p) (4)

at p= 0. When p0 and the 2|p| energy gap is large we can ignore the negative energyeigenstate|p, and safely make the adiabatic approximation that allows us to forget most

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quantum effects. We retain only the Abelian Berry phase, now interpreted as a classical

gauge potential. The adiabatic approximation fails near p= 0, however, and (4) should

be understood only as shorthand indicating the presence of a non-zero Berry flux through

surfaces surrounding (but distant from) the origin.

We desire to generalize (1) from 3 to 2N+1 space dimensions. The principal complication

is that in dimensions greater than three the Berry phaseis replaced by a unitary evolution

matrixand a(p) by a non-Abelian gauge field. Extracting the effects of the gauge field by

taking a suitable limit of the quantum system is complicated, and is ultimately equivalent to

the conventional Feynman diagram calculations of anomalies that we wish to avoid. Instead

we build on the results of [1] by letting symmetry and gauge invariance dictate the necssary

modification to the Abelian action functional (1).

In 2N+ 1 space dimensions the quantum Weyl Hamiltonian (3) becomes

Hp=2N+1i=1

ipi, (5)

where the i are a set of 2N-by-2N Dirac gamma matrices (or more accurately Dirac alpha

matrices) obeying

{i, j}= 2ij. (6)

The gamma matrices will also obey

i1i2 i2N+1 =iNI2Ni1i2...i2N+1, (7)

where sign depends the Weyl particles helicityi.e. which of the two inequivalent irre-ducible representations of the Clifford algebra (6) is selected.

The eigenvalues ofHp remain|p|, but each energy level is now 2N1-fold degenerate.The positive energy eigenspaces V+(p) form the fibres of a non-trivial Spin(2N) bundle over

momentum space minus its origin. If

|p, , +

, = 1, . . . , 22N1 form a basis forV+(p), the

natural non-Abelian Berry connection connection on the bundle has components [7].

a,k = ip, , +| pk

|p, , +. (8)

Its curvature tensor is

Fij =ajpi

aipj

i[ai, aj]. (9)

We anticipate that a(p) is to be replaced by the matrix-valued a(p).

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It is often convenient to expand the matrix-valued connection and curvature as

ai =n


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if quantized would give us back the charge representation. How to do this is explained in

detail in [4, 25]. Briefly stated, a representation ofG with highest weight determines a

Lie algebra element that lies in the Cartan sub-algebra. The quantum representation

Hilbert space is replaced by a classical phase space

O that is the adjoint orbit of. This

orbit can be identified with the coset G/H, where the isotropy group H is the subgroup

of G whose adjoint action gg1 leaves fixed. The matrix-valued gauge fieldcomponentsA andFare replaced by functionsO R as

AA def= t r (QA),FF def= t r (QF), (15)

where

Q=gg1 =Qaa, (16)

and the trace is taken in some fixed faithful representation (most conveniently the defining

representation ofG). We can use this trace to define a metric

ab= tr {ab} (17)

that we will use to raise or lower indices on Lie-algebra tensors such as the structure con-

stants fabc. In particularQa =abQ

c is the classical analogue of the generator a, and the

commutation relations (14) are replaced by the classical Poisson-bracket relations

{Qa, Qb}= ifcabQc. (18)

The obvious way to treat non-Abelian Berry phase is to repeat this scheme and intro-

duce (t) Spin(2N) to acommodate the spin dynamics. We therefore have an elements Cartan(Spin(2N)) that is determined by the particles spin s, an orbitOs and thesubstitution

ai ai def= t r (Sai),FijFij def= t r (SFij), (19)

where

Sdef= s

1 = S(m)X(m). (20)

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C. Wong action

We now assemble these ingredients to write down a classical action that should describe

motion of a general-dimension Weyl fermion in the background non-Abelian gauge field:

S[x, p, g , ] =

dt

pixi |p| +itrg1 ddt

i(A0+ xiAi) g itrs1 ddt

ixiai (21)

This action is clearly invariant under the gauge transformation

g(t) h1(x(t), t)g(t),A h1Ah+ih1

xh. (22)

It is also invariant under a p-dependent change of basis

|p, , +

|p, , +

U(p) that

takes

(t) U1(p)(t),aj U1ajU+ iU1

pjU. (23)

The equations of motion that arise from varyinggg(1 + g1g) and(1 + 1)are

[, g

1

(t i(A0+ xi

Ai)g] = 0,[s,

1(t i piai)] = 0. (24)

These equations do not uniquely determine g and . Indeed any solution g(t) and (t)

can be mutiplied on the right by an arbitrary time-dependent element of the corresponding

isotropy subgroup and still satisfy the equation. As a result g and must be thought of as

living in the cosetsO andOs. The time evolution of the Lie-algebra-valued quantitiesQand Sis insensitive to the ambiguity, and from (24) we find that

Q =i[Q, A0+ xiAi],S =i[S, aipi]. (25)

In components these equations read

Qc = fabcQa(Ab0+ x

iAbi),

S(c) = f(a)(b)(c)S(a)a

(b)i p

i. (26)

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The remaining equations of motion

pi = tr {Q(Fi0+Fijxj)},xi = pi tr {SFij} pj. (27)

are more straightforward, although in principal we still have to solve them for xi and pi as

we did in [4].

III. A GENERALIZED LIOUVILLE THEOREM

Both the gauge and abelian chiral anomalies can be understood physically as a spectral

flow of states from the infinitely deep negative-energy Dirac sea, through the diabolical

[5] Dirac point, and into the finite-depth positive-energy Fermi sea. If we monitor thephase-space density only in the positive energy region, the influx of states will appear in

the Liouville phase-space volume conservation law as a source term located at p = 0. We

therefore begin by recalling how Louvilles theorem is modified when the symplectic structure

is allowed to depend on time.

A. Extended phase space

Consider a general even-dimensional phase space M with co-ordinates = (1, . . . , 2n)

and equipped with an action functional

S[] =

dt

2ni=1

i(, t)i H(, t)

. (28)

Demanding that S= 0 under a variation i i +i results in the equations of motionji

ij

j =

H

i +

it

. (29)

We will use the notationij =

ji

ij

. (30)

For the equations of motion to be solvable for i without constraints, the matrix ij must

be invertible at every point in M. We assume that this condition is satisfied. Now, from

(29) and the antisymmetry ofij, we see that the i automatically satisfy the condition

i

H

i +

it

= 0. (31)

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As with the usual Hamiltonian formalism, many results are most compactly obtained with

the tools of vector fields and differential forms. To use these in the time dependent setting it

is convenient to extend the even-dimensional phase space Mto the odd-dimensional space

M = M

R, where R is the time co-ordinate. We may then combine the i and the

hamiltonianHinto a one-form

H=2ni=1

idi Hdt, (32)

on TM. Let H=dHbe its exterior derivative

H=1

2ijd

idj

it

+ H

i

di dt. (33)

Define a vector field

v= t

+ j j

. (34)

and take the interior product ofv with the two-formHto get

ivH=

ijj + H

i +

it

di i

H

i +

it

dt. (35)

From (29) and (31) we see that the equation ivH = 0 is equivalent to the equations of

motion.

Any odd-dimensional two-form such as H must possess at least one null vector. Here,

the matrixij being invertible ensures that the null space ofH ispreciselyone dimensional.

Collectively the null spaces compose thecontact bundleoverM, and the v(, t) determined

by the equations of motion is the unique vector in the fibre over (, t) that has unity as the

coefficient of/t.

B. The Liouville measure

We can define a volume (2n+ 1)-form on the extended phases space by

def

= 1

n!nHdt,

= 1

n!ndt. (36)

Here

=1

2ijd

idj, (37)

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and the second line of (36) follows from the first because the explicit factor ofdt excludes

all didtterms in nH.

We can use the first line of (36) to compute Lie derivative of with respect to v. We

find

Lv = (ivd+div),= div, (d = 0 because is a top form)

= 1

n!d((iv

nH)dt+

nH), (iv is an antiderivation, and

nH is even)

= 1

n!d(nH), (iv

nH nn1H ivH= 0 by the equations of motion)

= 0. (d2H= 0 provided H is nowhere-singular) (38)

We can alternatively compute the Lie derivative of from the second line of (36). We

observe that1

n!ndt=

d1 d2ndt, (39)

where

det() is the Pfaffian Pf() of the skew-symmetric matrix ij. From the

derivation property of the Lie derivative we now find

LV =

Lv

d1 d2ndt+

Lvd1 d2ndt

= t + i

i d1 d2ndt+ ii d1 d2ndt=

t +

i

i

d1 d2ndt. (40)

Comparing the two computations shows that our generalized Hamilton equations lead to

t +

i

i = 0. (41)

This last equation is the time-dependent version of Liouvilles theorem. In our application,

the manipulations in (38) will fail at the last step because a generalization of the Berry-phase

monopole leads nH to be singular. Consequently (41) will be violated by a source term at

the Dirac point. The remaining sections of this paper will be devoted to finding the strength

of this source term in terms of the external fields acting on our particles.

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IV. THE BOLTZMANN EQUATION

Now we apply the formalism of section III to the action functional (21). The extended

phase space we need is M = R2N+2 O Os R with R2N+2 being the particles (x, p)co-ordinates,O,Os the internal gauge and spin spaces, and R being time. We will, asmuch as possible, avoid using explicit co-ordinatesi on the internal spaces, and instead use

intrinsic geometric quantities.

We begin by writing (21) as an integral along the phase-space trajectory. We need to

define the differential forms

A= Aidxi +A0dt, a= aidp

i, (42)

as well asQR =dgg

1, SR =d1, (43)

which are pullbacks to the trajectory in (,g,x,p) space of the right-invariant Maurer-Cartan

forms onGand on Spin(2N) repectively. We then have

S[x, p, g , ] =

pidxi |p|dt+ tr {Q(idgg1 +A)} tr {S(id1 +a)}

=

pidxi |p|dt+ tr {Q(iQR + A)} tr {S(iSR+ a)}

. (44)

Now (44) is of the general form (28) with

H=pidxi |p|dt+ tr {Q(iQR+ A)} tr {S(iSR+ a)}. (45)

UsingF =dA iA2, F= da ia2 anddR = (R)2 we find that

H=dpidxi d|p|dt+F F itr {Q(QR iA)2} +itr {S(SR ia)2}, (46)

and the volume form (36) becomes

= 1

M!MHdt, (47)

whereM= (2N+ 1) +m+ms with m= dim(O)/2,ms= dim(Os)/2.There are potentially many terms in the high power ofH appearing in (47). A consid-

erable simplification arises, however, because all terms involving explicit As and as must

cancel. This cancellation can be seen by observing that at any chosen point in x, p space

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we can make gauge transformations so that both A and a (but not their derivatives) van-

ish. The transformations (22) and (23) that return us to the original gauge will leave this

(a= A = 0) volume form unchanged either because traces are invariant under adjoint

actions, or because the inhomogeneous g1dg and1d terms will seek to introduce extra

dxs, dts, or dpis that are forbidden by antisymmetry.

Thus we find that

1

M!MH =

1

(2N+ 1)!(dpidxi +F F)2N+1dds (48)

where

d = 1

m![itr {(QL )2}]m, ds=

1

ms![itr {s(SL )2)}]ms, (49)

are, up to factors, the Kirillov-Kostant measures on the co-adjoint orbits

O and

Os re-

spectively. We have taken advantage of the absence of the gauge fields to introduce the

left-invariant Maurer Cartan forms QL =g1dg andSL =

1d.

We also find that

v=

t+ xi

xi+ pi

xi+ (g1g)aLgaugea + (

1)(a)Lspin(a) (50)

is the appropriate form of the vector field from section III. Here Lgaugea is the left-invariant

vector field dual to the left-invariant Maurer-Cartan form. We have

QL (Lgaugea ) =a, (51)

so QL (v) = (g1g)aa =g

1g is the analogue ofdxi(v) = xi. SimilarlyLspin(a) is dual to the

Spin(2N) Maurer-Cartan form:

SL (Lspin(a) ) =X(a) (52)

In order to write down the Liouville theorem we will need to know how to compute the

Lie derivative of the adjoint orbit measures. After a little effort and the use of (26) we find

Lvd= fabcQaAbixi

Qcd+. . . , (53)

where the dots indicate terms involving dxi ordt that have no effect in ddx1 dx2N+1dt.

There is a similar expression for the Lie derivative ofs.

We now define

by

=

ddsdx1 dx2N+1dp1 dp2N+1dt (54)

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Iffor the duration of this section onlywe ignore the singularity at p= 0 and follow the

proceedure in section III we end up with withLv = 0 being equivalent to

t

+ xi

xi + pi

pi = 0, (55)

where

t = t

+ fabcAa0Q

b

Qc,

xi = xi

+fabcQaAbi

Qc,

pi = pi

+f(a)(b)(c)a

(a)k S

(b)

S(c), (56)

are covariant derivatives that ensure invariance under gauge transformations.

We introduce a phase-space density f(x, p, Q,S, t) and let it be advected with the flow

t+ xi

xi+ pi

pi+ Qa

Qa+ S(a)

S(a)

f= 0. (57)

This advection condition is the collisionless Boltzmann equation for our system.

We can use (26) and (56) to group these terms ast+ xixi+ pipi f= 0, (58)and so find that

Lv(f) = 0, (59)

which expresses the conservation of probability.

A. Conservation laws

The continuity equation (59) for the phase space density is the origin of various conser-

vation laws. From (59) we can derive the conservation

J0

t +

Ji

xi = 0 (60)

of the particle-number current

J0(t, x) =

f(x, p, Q,S, t)

dds

d2N+1p

(2)2N+1,

Ji(t, x) =

xif(x, p, Q,S, t)

dds

d2N+1p

(2)2N+1, (61)

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and the covariant conservation

J0at

fabcAb0J0c + Jiaxi

fabcAbiJic = 0 (62)

of the gauge current

J0a(t, x) =

Qaf(x, p, Q,S, t)

dds

d2N+1p

(2)2N+1,

Jia(t, x) =

Qax

if(x, p, Q,S, t)

ddsd2N+1p

(2)2N+1. (63)

In each of these definitions the integral is over all of momentum space and over both adjoint

orbitsO,Os. While writing down the expressions for the currents we have taken theoportunity to normalize the measure factors. We have put a 1/2with eachdp and rescaled

the adjoint orbit measure so that

d = 1

(2)mm![i tr {(QL )2}]m, ds=

1

(2)msms![i tr {s(SL )2)}]ms, (64)

This rescaling is one place where we need knowledge of quantum mechanics: the normalized

Liouville measure counts (approximately) one quantum state per unit volume of the classical

phase space. Consequently the exclusion principle says that the maximum allowed value of

f(x,p,Q, S, t) is unity.

To see that (59) leads to the conservation laws (60) and (62) recall that if we have a q-

dimensional manifoldMand integrate a p-form over a smooth p-dimensional regionN(),

each point of which is being advected by a flow v = dx/d, then Liebniz integral formula

readsd

d

N()

=

N()

Lv. (65)

An immediate corollary is that when we integrate a q-form over the entirety ofM we

have

M Lv = 0. (66)Now (60) is equivalent to the vanishing of

R2N+1R

(x, t)

J0

t +

Ji

xi

d2N+1x dt (67)

for any test function (x, t). ButLv(f) = 0 and (66) gives us

0 =

M

Lv(f)

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=

M

t + xi

xi

f

=R2N+1R

(x, t)

J0

t +

Ji

xi

d2N+1xdt. (68)

A similar calculation gives (62).

V. THE LIOUVILLE ANOMALY

We see that in

1

(2N+ 1)!(dpidxi +F F)2N+1dt= dx1 dx2N+1dp1 dp2N+1dt (69)

the measure factor

is the Pfaffian of the (4N+ 2)-by-(4N+ 2) matrix

ij =F I

I Fij

(70)

The factor dt excludesF01 from being an entry in the block submatrixFappearing here,so it has the same number of rows and columns asF. The Schur determinant formula nowshows that

det

F II F

= det(I FF) (71)

and so the Liouville measure is given by the double Pfaffian

= Pf(I FF) N

k=0

I2k

Pf( FI2k)Pf(FI2k).Here eachI2k is a cardinality-2k subset of the indices on the matrices. The term with k = 0

is understood to be unity.

When we expand out MH/M! there is no longer an explicit dt and soF01 is allowed. Inparticular one of the many terms that appear inMh /M! is

1N!(2)N

F2N 1

(N+ 1)!F

2N+1 dds (72)

where the 2s come from the factors of 1/2 that we put with each dpi.

It is this term that causes dMH/M!= 0 in the last line of (38) and hence gives rise to theanomaly. We show in the appendix that the integral

1

N!(2)N

S

tr (FN) (73)

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of the Berry-curvature Chern character over a closed 2N-surface S in momentum space is

1 (depending on the helicity) when Sencloses p= 0, and is zero otherwise. As with theAbelian monopole, we can conveniently abbreviate this statement as

d 1N!(2)Ntr FN =2N+1(p)dp1 dp2N+1, (74)exhibiting a higher-dimensional version of (4). The second factor in (72) is closed

d

1

(N+ 1)!tr

F

2

N+1= 0 (75)

by well known properties of characteristic classes.

In (74) of course we have the matrix-valued curvature F and a trace over its quantum

indices. Similarly in (75). In (72), however, we have the function-valuedFandF and areto integrate over the adjoint orbitsOs andO. However we also show in the appendix thatprovided we take integral not over the nave orbit associated with the greatest weight of

the spin representation but instead over the orbit associated with the Weyl shifted weight,

then the quantum and classical traces coincide. Similarly the classical traceO is alwayspropertional to the quantum trace, and so (75) remains true when Fis replaced byF.

One additional ingredient is required. It is not the integral ofLv that is needed inthe conservation law, but the integral ofLv(f). If we are to get the standard expression

for the anomaly we must have fbe identically unity over and within a surface sufficiently

far from p = 0 that the adiabatic approximation and the resultant classical machinary be

applicable. Thus we need a Fermi sea that is deep enough that finite temperature effects do

not depopulate the sea too close to the Dirac point.

Assuming that f is indeed unity in the region where it is needed, we immediately find

that contribution of the delta function (74) toLv modifies the conservations laws to read

J0

t

+ Ji

xi =

O1

(N+ 1)! F

2N+1

d (76)

and

J0at

fabcAb0J0c + Jiaxi

fabcAbiJic =O

Qa1

(N+ 1)!

F2

N+1d. (77)

The phase space integrals are classical approximations to the symmetrized tracesO

1

(N+ 1)!

F

2

N+1d 1

(N+ 1)!str

F

2

N+1 (78)

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and O

Qa1

(N+ 1)!

F

2

N+1d 1

(N+ 1)!str

a

F

2

N+1 (79)

taken in the representation with greatest weight vector . With this substition (76) is the

familar expression for the singlet anomaly in 2N+ 2 spacetime dimensions, and (77) is thecovariant form of the non-abelian anomaly.

It is well known [25], and exhibited explicity in [4] in the case G = SU(3), that the

accuracy of the classical approximation to the symmetrized trace is greatly increased if the

integral is actually taken over the orbit corresponding to the Weyl-shifted weight + ,where the Weyl vector is half the sum of the positive roots.

VI. DISCUSSION

VII. ACKNOWLEDGEMENTS

This project was supported by the National Science Foundation under grant DMR 09-

03291.

Appendix A: Spin Chern character

Here we compute the Chern character for the bundle of positive-energy eigenstates of

Hp=2N+1i=1

ipi (A1)

over the 2N-sphere.

H has eigenvalues|p| and the projectors on to the positive and negative energyeigenspacesV+(p),V(p) are

P =

|p, , +p, , +|=1

2(I+ pi

i),

P =

|p, , p, , |= 12

(I pii), (A2)

respectively. Here p is the unit vector p/|p|.It is easy to see that

PdPdP = i4

(I+ pii)1

4i[i, j]dp

idpj

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= 1

i

,

|p, , +Fp, , +| (A3)

has matrix elements only in V+(p), and coincides there with the matrix elements of the

Berry curvature F= da

ia2 divided byi.

If we take

2N+1=

I 0

0 I

, (A4)

and we consider the fibre over p= (0, . . . , 0, 1) then at that point

dPdP =2N

i,j=1

i

4

Xij,+ 0

0 Xij,

dpidpj. (A5)

where the Xij, are the matrices representing the generator Xij in the two inequivalent spin

representations of Spin(2N). The projector Pensures that the trace over the 2N indices in

PdPdPcoincides with the trace over the 2N1 indices in the + representation

The Chern character is given by

chN(V+) def

= 1

N!

1

2

NS2N

tr +(FN)

= 1

N!

i

2

NS2N

tr {(PdPdP)N} (A6)

To evaluate this set

X=2N+1=1

pii (A7)

so that X=P P and

(2 4N)tr {(PdPdP)N} = tr {X(dX)2N}=iN2Ni1,...,i2N+1 pi1dpi2 dpi2N+1

=

iN2N(2N!)d[Area on S2N] (A8)

Thus

1

2N(2N)!

S2N

tr {X(dX)2N} =iN|S2N|

=iN2N+1/2/

2N+ 1

2

. (A9)

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Thus, on using x(x) = (x+ 1) etc. we find

chN(V+) = 1

N!

1

2i

NS2N

tr {(PdPdP)N} (A10)

=

1

2(N!) 18iN

S2n tr {X(dX)2N}=1. (A11)

Appendix B: Classical and quantum traces for Spin(2N)

We see that the computation of the Chern character in appendix A required us to compute

the trace

C+ =i1i2i2N1i2Ntr +

{Xi1,i2Xi3i4

Xi2N1i2N

} (B1)

where

Xij = 1

4i[i, j] (B2)

is the matrix representing the Spin(2N) generatorXij in the positive-helicity representation.

Using

i1i2 i2N+1 =iNI2Ni1i2...i2N+1 (B3)

found that

C = i1i2...i2N1i2Ntr {Xi1i2Xi3i4 X2N1,2N}

= (i)N2N(2N)!tr {P1 . . . 2N}=2N(2N)!1

2tr {I2N}

=(2N)!/2. (B4)

What we need for the classical calculation is to compute the corresponding phase space

integral

i1i2i2N1i2NSi1i2Si3i4 Si2N1i2Nd+ (B5)We begin by summarizing some basic facts about the Lie algebra of SO(2N). This algebra

coincides with that of Spin(2N), which is the double cover of SO(2N). The Hermitian matrix

generators in the defining vector representation of SO(2N) areXij =i(eij eji). Here eijis the matrix with unity at site i, j, so that

eijekj =jkeil. (B6)

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Thus the commutation relations are

[Xij , Xmn] =i(miXjn mjXin Xmijn +Xmjin). (B7)

These relations can be understood as saying that Xmn transforms a skew 2-tensor.

With traces taken in the defining vector representation, we have

tr {XijXkl}= 2(ikjl iljk). (B8)

If we take as our basis set the Xij with i < j we therefore have

gij,kl= tr {XijXkl}= 2ikjl . (B9)

The Cartan algebra is generated by

hn= X2n1,2n, n= 1, . . . , N , (B10)

and the roots areei ej for a suitable orthogonal basis ei. In the same basis the Nfundamental weights are

1 = (1, 0, . . . , 0, 0, 0) 2Ndefining vector rep.

2 = (1, 1, . . . , 0, 0, 0),

...

N2 = (1, 1, . . . 1, 0, 0),

+ = (1, 1, . . . , 1, 1, 1)/2, +,

= (1, 1, . . . 1, 1, 1)/2, .

The Weyl vector is defined to be half the sum of the positive roots, or equivalently the

sum of the findamental weights. Therefore

= (N 1, N 2, . . . , 1, 0). (B11)

Consider the representation with highest weight s and highest weight vector|s. TheCartan algebra element s is defined so that

tr {sX}=s|X|s (B12)

for anyXin the Lie algebra. Thus in the representation s= + we have

s=1

4(X12+X34+ +X2N1,2N) (B13)

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We take the orbit coordinates to be xij ,i < j given byi


22/24

For the spin representationswe haveXij =Pij/2i, whereP = (15)/2. Here

5 = (i)N1 . . . 2N (B23)

obeys 25 = I2NThe exact trace we need is

C = tr {Xi1i2Xi3i4 X2N1,2N}i1i2...i2N

= (i)N2N(2N)!tr {P1 . . . 2N}=2N(2N)!1

2tr {I2N}

=(2N)!/2, (B24)

We see that the quantum corrections dominate the leading terms for the basic spin reps,

and the proportionality factor between the quantum trace and classical trace grows as N!.

However the Weyl shift s s + leads to the exact result.

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