Quantum magnetism and criticalitystaff.ulsu.ru/moliver/ac/ref/sach08.pdf · Quantum magnetism and criticality Magnetic insulators have proved to be fertile ground for studying new

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Quantum magnetism and criticality

Magnetic insulators have proved to be fertile ground for studying new types of quantum many-body state,

and I survey recent experimental and theoretical examples. The insights and methods also transfer to

novel superconducting and metallic states. Of particular interest are critical quantum states, sometimes

found at quantum phase transitions, which have gapless excitations with no particle- or wave-like

interpretation, and control a significant portion of the finite-temperature phase diagram. Remarkably, their

theory is connected to holographic descriptions of Hawking radiation from black holes.

SUBIR SACHDEVDepartment of Physics, Harvard University, Cambridge,Massachusetts 02138, USA

e-mail: subir [email protected]

I. INTRODUCTION

Quantum mechanics introduced several revolutionary ideas intophysics, but perhaps the most surprising to our classical intuitionwas that of linear superposition. This is the idea that any linearcombination of two allowed quantum states of a system is also anallowed state. For single particles, linear superposition enables usto explain experiments such as double-slit interference, in whichelectrons can choose one of two possible slits in an obstacle, butinstead choose a superposition of the two paths, leading to aninterference pattern in their probability to traverse the obstacle.However, the concept of linear superposition had unexpectedconsequences when extended to many particles. In a well-knownpaper, Einstein, Podolsky and Rosen1 (EPR) pointed out that evenwith just two electrons, linear superposition implied non-local andnon-intuitive features of quantum theory. They imagined a statein which the spin of one electron was up and the second down,which we write as |↑↓〉. Now let us take a linear superpositionof this state with one with the opposite orientations to bothelectron spins, leading to the state (|↑↓〉 − |↓↑〉)/

√2. Such a

‘valence-bond’ state is similar to that found in many situationsin chemistry and solid-state physics, for example, the state of thetwo electrons in a hydrogen molecule. EPR imagined a ‘thoughtexperiment’ in which the two electrons are moved very far apartfrom each other, while the quantum states of their spins remainundisturbed. Then we have the remarkable prediction that if wemeasure the spin of one of the electrons, and find it to be |↑〉,for example, the perfect anticorrelation of the spins in the valencebond implies that the spin of the other far-removed electron isimmediately fixed to be |↓〉. EPR found this sort of ‘action-at-a-distance’ effect unpalatable. Today, there are many examples ofthe EPR experiments in the laboratory, and the peculiar non-localfeatures of quantum mechanics have been strikingly confirmed.In recent years, the term ‘entanglement’ has frequently been usedto describe the non-local quantum correlations inherent in thevalence-bond state of two spins.

The purpose of this article is to describe the consequencesof the linear superposition principle of quantum mechanics

when extended to a very large number of electrons (and otherparticles). We are interested in how many electrons entangle witheach other in the low-energy states of hamiltonians similar tothose found in certain transition-metal compounds, and related‘correlated electron’ systems. Whereas there is a precise definitionof entanglement for two electrons that captures the essence of theEPR effect, much remains to be understood in how to define ananalogous measure of ‘quantum non-locality’ for large numbersof electrons. We shall sidestep this issue by restating the question:how can we tell when two quantum states have genuinely distinctsuperpositions of the electron configurations? Furthermore, can weclassify and list the distinct classes of states likely to be found incorrelated electron materials?

A useful strategy is to follow the low-energy states of ahamiltonian that varies as a function of a parameter g . We examinewhether the ground state breaks a symmetry of the hamiltonian.We also examine the quantum numbers of the excitations neededto describe the low-energy Hilbert space. If one or more of such‘discrete’ characteristics is distinct between large and small g , thenwe assert that these limiting states realize distinct quantum phases.The distinction between the phases implies, in particular, that theycannot be smoothly connected as g is varied. Consequently, theremust be at least one special value of g , which we denote gc, at whichthere is a non-analytic change in the characteristics of the groundstate, for example, the ground-state energy. Often some of the mostinteresting quantum states, with non-trivial and non-local typesof entanglement, are found precisely at the quantum critical pointg = gc, where the system is delicately balanced between the stateson either side of gc. This happens when there is a correlation lengththat becomes large near gc, as is the case close to a ‘second-order’phase transition (near strong ‘first-order’ transitions the groundstate simply jumps discontinuously between simple limiting stateson either side of gc). Thus, a description of the hamiltonian as afunction of g exposes two quantum phases and their connectionacross a quantum phase transition, and we present several examplesof such phenomena below.

Experimentally, it is almost never possible to examine theground state of a large quantum system in isolation. There aresmall random perturbations from the environment that eventuallybring the system into equilibrium at a temperature T > 0. So it iscrucial to examine the fingerprints of the ground-state quantumphases and quantum critical points on the non-zero temperaturephase diagram. We will turn to this issue in Section V, where wewill argue that such fingerprints can often be significant and easily

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© 2008 Nature Publishing Group


Figure 1 Neel states. a,b, Ground states of the S= 1/2 antiferromagnet H0 with allJ i j = J on the square (a) and triangular (b) lattices. The spin polarization is (a)collinear and (b) coplanar.

detectable. Indeed, the quantum critical point can serve as the bestpoint of departure for understanding the entire phase diagram inthe g , T plane.

We will begin Section II by describing the rich variety ofquantum phases that appear in two-dimensional insulators, whoseprimary excitations are S = 1/2 electronic spins on the sites of alattice. Interest in such two-dimensional spin systems was initiallystimulated by the discovery of high-temperature superconductivityin the cuprate compounds. However, since then, such quantumspin systems have found experimental applications in a wide varietyof systems, as we will describe below. The critical quantum phasesand points of such materials will be discussed in Section III. SectionIV will extend our discussion to include charge fluctuations,and will describe some recent proposals for novel metallic andsuperconducting states, and associated quantum critical points.

Although we will not discuss these issues here, related ideasalso apply in other spatial dimensions. Similarly, quantum statesappear in three dimensions, although the quantum critical pointsoften have a distinct character. A fairly complete understanding ofone-dimensional correlated states has been achieved, building on‘bosonization’ methods.

II. PHASES OF INSULATING QUANTUM MAGNETS

High-temperature superconductivity appears when insulators suchas La2CuO4 are doped with mobile holes or electrons. La2CuO4 isan antiferromagnet, in which the magnetic degrees of freedom areS = 1/2 unpaired electrons, one on each Cu atom. The Cu atoms

reside on the vertices of square lattices, which are layered atop eachother in the three-dimensional crystal. The couplings between thelayers are negligible (although this small coupling is important inobtaining a non-zero magnetic ordering temperature), and so weneed only consider the S = 1/2 spins Si, residing on the sites ofa square lattice, i. The spins are coupled to each other througha superexchange interaction, and so we will consider here thehamiltonian

H0 =

∑〈ij〉

JijSi ·Sj .

Here, 〈ij〉 represents nearest-neighbour pairs and Jij > 0 is theexchange interaction. The antiferromagnetism is a consequence ofthe positive value of Jij , which prefers an antiparallel alignment ofthe spins. We will initially consider, in Section IIA the case of i onthe sites of a square lattice, and all Jij = J equal, but will generalizein the subsequent sections to other lattices and further interactionsbetween the spins.

Our strategy here will be to begin by guessing possible phasesof H0, and its perturbations, in some simple limiting cases. We willthen describe the low-energy excitations of these states by quantumfield theories. By extending the quantum field theories to otherparameter regimes, we will see relationships between the phases,discover new phases and critical points, and generally obtain aglobal perspective of the phase diagram.

A. NEEL ORDERED STATES

With all Jij = J , there is no parameter, g , that can tune the groundstate of H0; J sets the overall energy scale, but does not modifythe wavefunction of any of the eigenstates. So there is only a singlequantum phase to consider.

For the square lattice, there is convincing numerical andexperimental evidence2 that the ground state of H0 has Neel order,as shown in Fig. 1a. This state spontaneously breaks the spin-rotation symmetry of H0, and is characterized by the expectationvalue

〈Sj〉 = (−1)j8. (1)

Here, (−1)j represents the opposite orientations of the spins on thetwo sublattices as shown in Fig. 1a. The vector 8 represents theorientation and magnitude of the Neel order. We have |8| < 1/2,implying that the ground-state wavefunction is not simply theclassical state shown in Fig. 1a, but has quantum fluctuations aboutit. These fluctuations will entangle the spins with each other, butqualitatively the essential character of the state is captured by thepattern in Fig. 1a. Note that this pattern implies a long-range,and classical, correlation between the spins, but not a significantamount of entanglement because there are no EPR effects betweenany pair of well-separated spins.

Having described the ground state, we now consider theexcitations. This is most conveniently done by writing the formof the Feynman path integral for the trajectories of all the spinsin imaginary time, τ. After taking the long-wavelength limit toa continuous two-dimensional space, this path integral definesa quantum field theory in 2 + 1 dimensional space-time withcoordinates (r,τ). The quantum field theory is for a field 8(r,τ),which is the value of the Neel order when averaged over thesquare-lattice spins, Si, located on sites within some averagingneighbourhood of r. A quick derivation of the effective action forthis quantum field theory is provided by writing all terms in powersand gradients of 8 that are invariant under all the symmetries of thehamiltonian. In this manner, we obtain the action3,4

S8 =

∫d2rdτ

((∂τ8)2

+ v2(∇x8)2+ s82

+u(82)2). (2)

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Here, v is a spin-wave velocity, and s,u are parameters whose valuesare adjusted to obtain Neel order in the ground state. In mean-fieldtheory, this happens for s < 0, where we have |〈8〉| = (−s)/(2u)by minimization of the action S8. A standard computation ofthe fluctuations about this saddle point shows that the low-energyexcitations are spin waves with two possible polarizations and anenergy ε that vanishes at small wavevectors k, ε= vk. These spinwaves correspond to local oscillations of 8 about an orientationchosen by spontaneous breaking of the spin-rotation symmetryin the Neel state, but which maintain low energy by fixing themagnitude |8|. The spin waves also interact weakly with each other,and the form of these interactions can also be described by S8.All effects of these interactions are completely captured by a singleenergy scale, ρs, which is the ‘spin stiffness’, measuring the energyrequired to slowly twist the orientation of the Neel order across alarge spatial region. At finite temperatures, the thermal fluctuationsof the interacting spin waves can have strong consequences. Wewill not describe these here (because they are purely consequencesof classical thermal fluctuations), apart from noting4 that all thesethermal effects can be expressed universally as functions of thedimensionless ratio kBT/ρs.

For future analysis, it is useful to have an alternative descriptionof the low-energy states above the Neel ordered state. For theNeel state, this alternative description is, in a sense, a purelymathematical exercise: it does not alter any of the low-energyphysical properties, and yields an identical low-temperature theoryfor all observables when expressed in terms of kBT/ρs. The key stepis to express the vector field 8 in terms of an S = 1/2 complexspinor field zα, where α=↑↓ by

8 = z∗

ασ αβzβ (3)

where σ are the 2×2 Pauli matrices. Note that this mapping from8 to zα is redundant. We can make a space-time-dependent changein the phase of zα by the field θ(x,τ)

zα → eiθzα (4)

and leave 8 unchanged. All physical properties must thereforealso be invariant under equation (4), and so the quantum fieldtheory for zα has a U(1) gauge invariance, much like that foundin quantum electrodynamics. The effective action for zα thereforerequires the introduction of an ‘emergent’ U(1) gauge field Aµ

(where µ= x,τ is a three-component space-time index). The fieldAµ is unrelated to the electromagnetic field, but is an internalfield that conveniently describes the couplings between the spinexcitations of the antiferromagnet. As we have noted above, in theNeel state, expressing the spin-wave fluctuations in terms of zαand Aµ is a matter of choice, and the above theory for the vectorfield 8 can serve us equally well. The distinction between the twoapproaches appears when we move out of the Neel state acrossquantum critical points into other phases (as we will see later):in some of these phases, the emergent Aµ gauge field is no longeroptional, but an essential characterization of the ‘quantum order’ ofthe phase. As we did for S8, we can write the quantum field theoryfor zα and Aµ by the constraints of symmetry and gauge invariance,which now yields

Sz =

∫d2rdτ

[|(∂µ− iAµ)zα|

2+ s|zα|

2+u(|zα|

2)2

+1

2e20

(εµνl∂νAl)2

]. (5)

For brevity, we have now used a ‘relativistically’ invariant notation,and scaled away the spin-wave velocity v; the values of the couplings

s,u are different from, but related to, those in S8. The Maxwellaction for Aµ is generated from short-distance zα fluctuations,and it makes Aµ a dynamical field; its coupling e0 is unrelatedto the electron charge. The action Sz is a valid description ofthe Neel state for s < 0 (the critical upper value of s will havefluctuation corrections away from 0), where the gauge theory entersa Higgs phase with 〈zα〉 6= 0. This description of the Neel stateas a Higgs phase has an analogy with the Weinberg–Salam theoryof weak interactions—in the latter case, it is hypothesized thatthe condensation of a Higgs boson gives a mass to the W and Zgauge bosons, whereas here the condensation of zα quenches theAµ gauge boson.1. Triangular lattice. There have been numerous recent studies5 ofthe spin excitations of the insulator Cs2CuCl4. Just as in La2CuO4,the dominant spin excitations are S = 1/2 spins on the Cu ions,but now they reside on the vertices of a triangular lattice, asshown in Fig. 1b. Such an antiferromagnet is well described bythe hamiltonian H0, with a nearest-neighbour exchange J and ion the sites of the triangular lattice. From numerical studies ofsuch spin systems6, and also from observations5 in Cs2CuCl4, theground state of H0 also has broken spin-rotation symmetry, but thepattern of spin polarization is now quite different. We now replaceequation (1) by

〈Sj〉 = N1 cos(K · rj)+N2 sin(K · rj), (6)

where ri is the position of site i, and K = (4π/3a)(1,√

3) for theordering pattern in Fig. 1b on a triangular lattice of spacing a. Themost important difference from equation (1) is that we now requiretwo orthogonal vectors N1,2 (N1 ·N2 = 0) to specify the degeneratemanifold of ground states. As for the square lattice, we can writean effective action for N1,2 constrained only by the symmetries ofthe hamiltonian. Minimization of such an action shows that theordered state has N2

1 =N22 fixed to a value determined by parameters

in the hamiltonian, but are otherwise arbitrary. Moving on tothe analogue of the spinor representation in equation (3), we nowintroduce another spinor wα, which parameterizes N1,2 by7

N1 + iN2 = εαγwγσ αβwβ, (7)

where εαβ is the antisymmetric tensor. It can be checked that wα

transforms as an S = 1/2 spinor under spin rotations, and thatunder translations by a lattice vector y wα → e−iK ·y/2wα. Apartfrom these global symmetries, we also have the analogue of thegauge invariance in equation (4). From the relationship of wα tothe physical observables in equation (7), we now find a Z2 gaugetransformation

wα → ηwα, (8)

where η(r, τ) = ±1. This Z2 gauge invariance will play animportant role in the discussion in Section IID. The low-energytheory of the antiferromagnetically ordered state described byequation (6) can now be obtained from the effective action for N1,2

or wα. We will not write it out explicitly here, deferring it also toSection IID.

B. COUPLED-DIMER ANTIFERROMAGNET

This spin model is shown in Fig. 2. We begin with the square-latticeantiferromagnet in Fig. 1a, and weaken the bonds indicated by thedashed lines to the value J/g . For g = 1, this model reduces to thesquare-lattice model examined in Section IIA. For g > 1, the modelcan be understood as a set of spin dimers, with the intra-dimerexchange interaction J , and a weaker coupling between the dimersof J/g . A number of Cu compounds, such as TlCuCl3 (refs 8,9)




Figure 2 The coupled-dimer antiferromagnet. This is described by the hamiltonianH0, with J i j = J on the solid red lines and J i j = J/g on the dashed red lines. a, Thelarge-g ground state, with each ellipse representing a singlet valence bond(|↑↓〉−|↓↑〉)/

√2. b, The S= 1 spin triplon excitation. The pair of parallel spins

on the broken valence bond hops between dimers using the J/g couplings.

and BaCuSi2O6 (ref. 10), realize such coupled spin-dimer models,although not in precisely the spatial pattern shown in Fig. 2.

A simple argument shows that the large-g ground state isqualitatively distinct from the g = 1 Neel state. For g = ∞, thehamiltonian separates into decoupled dimers, which have theobvious exact ground state∏

〈ij〉∈A

1√

2

(|↑〉i |↓〉j −|↓〉i |↑〉j

), (9)

where A represents the set of links with solid lines in Fig. 2.This is the product of S = 0 spin singlets on the dimers, and sopreserves full rotational invariance, unlike the Neel state. There isa gap towards S = 1 excitations that are created by breaking thespin-singlet valence bonds, as shown in Fig. 2b. This gap ensuresthat perturbation theory in 1/g is well defined, and so the samestructure is preserved for a finite range of 1/g . The most significantchange at non-zero 1/g is that the triplet excitations becomesmobile, realizing an S = 1 quasiparticle, sometimes called thetriplon. Triplons have been clearly observed in a number of neutronscattering experiments on spin-dimer11 and related12 states.

It is interesting to understand this state and its triplonexcitations from the perspective of quantum field theory. By thesame procedure as in Section IIA, we can argue that the theory inequation (2) also applies in the present phase. However, we nowneed s > 0 to preserve spin-rotation invariance. The excitations ofS8 about the 8 = 0 saddle point consist of the three polarizations

Figure 3 Valence-bond solid states. a, Columnar VBS state of H0 +H1 with J i j = Jon all bonds. This state is the same as in Fig. 2a, but the square-lattice symmetryhas been broken spontaneously. Rotations by multiples of π/2 about a lattice siteyield the four degenerate states. b, Four-fold degenerate plaquette VBS state withthe rounded squares representing an S= 0 combination of four spins.

of 8 oscillations: we can identify these oscillations with anS = 1 quasiparticle, which clearly has all of the characteristics ofthe triplon.

So we now have the important observation that S8 describesthe g = 1 Neel state for s < sc (the critical value sc = 0 in mean-fieldtheory), and the large-g coupled-dimer antiferromagnet for s> sc.This suggests that there is a quantum phase transition betweenthese two states of H0 at a critical value g = gc that is also describedby S8. This is indeed the case, as we will discuss in Section IIIA.

C. VALENCE-BOND SOLIDS

The state in equation (9), and its triplon excitations, are alsoacceptable caricatures of a valence-bond solid (VBS) phase.However, the crucial difference is that the VBS appears as theground state of a hamiltonian with full square-lattice symmetry,and the space-group symmetry is spontaneously broken, as shownin Fig. 3a. Thus, H0 has all nearest-neighbour exchanges equal, andfurther exchange couplings, with full square-lattice symmetries,are required to destroy the Neel order and restore spin-rotationinvariance. Such VBS states, with spin-rotation symmetry, butlattice symmetry spontaneously broken, have been observedrecently in the organic compound13 (C2H5)(CH3)3P[Pd(dmit)2]2

and in14 ZnxCu4−x(OD)6Cl2. In the underdoped cuprates,bond-centred modulations in the local density of statesobserved in scanning tunnelling microscopy experiments15

have a direct interpretation in terms of predicted dopedVBS states16–18.




We frame our discussion in the context of a specificperturbation, H1, studied recently by Sandvik19 and others20,21:

H1 = −Q∑〈ijkl〉

(Si ·Sj −

1

4

)(Sk ·Sl −

1

4

),

where 〈ijkl〉 refers to sites on the plaquettes of the square lattice.The ground state of the hamiltonian H0 + H1 will depend onthe ratio g = Q/J . For g = 0, we have the Neel ordered state ofSection IIA. Recent numerical studies19–21 have shown convincinglythat VBS order is present for large g (VBS order had also been foundearlier in a related ‘easy-plane’ model22). We characterize the VBSstate by introducing a complex field, Ψ , whose expectation valuemeasures the breaking of space-group symmetry:

Ψ = (−1)jx Sj ·Sj+x + i(−1)jy Sj ·Sj+y . (10)

This definition satisfies the requirements of spin-rotationinvariance, and measures the lattice symmetry broken by thecolumnar VBS state because under a rotation about an evensublattice site by an angle φ= 0,π/2,π,3π/2, we have

Ψ → eiφΨ . (11)

Thus, Ψ is a convenient measure of the Z4 rotation symmetryof the square lattice, and 〈Ψ 〉 = 0 in any state (such as theNeel state) in which this Z4 symmetry is preserved. From thesedefinitions, we see that 〈Ψ 〉 6= 0 in the VBS states; states witharg(〈Ψ 〉) = 0,π/2,π,3π/2 correspond to the four degenerateVBS states with ‘columnar’ dimers in Fig. 3a, whereas states witharg(〈Ψ 〉) = π/4,3π/4,5π/4,7π/4 correspond to the plaquetteVBS states in Fig. 3b.

VBS order with 〈Ψ 〉 6= 0 was predicted in early theoreticalwork23 on the destruction of Neel order on the square lattice.We now present the arguments, relying on the theory Sz inequation (5) as a representation of the excitations of the Neel state.We mentioned earlier that Sz describes the Neel state for s < sc

(with sc = 0 in mean-field theory). Let us assume that Sz remainsa valid description across a quantum phase transition at s = sc,and ask about the nature of the quantum state24 for s > sc. Aconventional fluctuation analysis about the zα = 0 saddle pointreveals two types of excitation: (1) the S =1/2 zα quanta that realizebosonic quasiparticles known as spinons (Fig. 4), and (2) the Aµ

‘photon’ that represents a spinless excitation we can identify withthe resonance between different valence-bond configurations25.More precisely, the Aµ ‘magnetic flux’ determines the relativephases of different possible pairings of the electrons into valencebonds. It would also seem that the s > sc phase of Sz breaks nosymmetries of the underlying spin hamiltonian H0 +H1. However,there are well-known arguments26 that this phase of Sz breaks a‘hidden’ or ‘topological’ symmetry. Let us define the topologicalcurrent by

Jµ = εµνl∂νAl, (12)

where εµνl is the totally antisymmetric tensor in three space-timedimensions, so that Jt is the ‘magnetic’ flux. The conservation law,∂µJµ = 0, is a trivial consequence of this definition, and representsthe conservation of Aµ flux in the s > sc phase of Sc. However,by defining

Jµ = ∂µζ, (13)

where ζ is a free massless scalar field at low energies, we canalso identify Jµ as the Noether current associated with the shiftsymmetry ζ→ ζ+constant. This shift symmetry is spontaneouslybroken in the s > sc phase of Sz , and ζ is the associatedGoldstone boson. (Indeed, this Goldstone boson is, of course,

Figure 4 Caricature of a spin-liquid state. The valence bonds are entangledbetween different pairings of the spins, only one of which is shown. Also shownare two unpaired S= 1/2 spinons, which can move independently in thespin-liquid background.

just the Aµ photon that has only a single allowed polarization in2 + 1 dimensions.) Nevertheless, these observations seem purelyformal because they only involve a restatement of the obviousdivergencelessness of the flux in equation (12), and the shiftsymmetry is unobservable.

The key point made in ref. 23 was that the above shift symmetryis, in fact, physically observable, and is an enlargement of the Z4

rotation symmetry of the square lattice. They focused attention onthe monopole operator, V , which changes the total Aµ flux by 2π(by the Dirac argument, because total flux is conserved, any suchtunnelling event must be accompanied by a thin flux tube carryingflux 2π—this string is unobservable to the zα quanta that carry unitAµ charge). From equation (13), the flux density is measured by∂tζ, and so its canonically conjugate variable under the Maxwellaction in equation (5) is ζ/e2

0; consequently, the operator that shiftsthe flux by 2π is

V = exp

(i2πζ

e20

).

In terms of the underlying spin model, this monopole is atunnelling event between semiclassically stable spin textures thatdiffer in their ‘skyrmion’ number. By a careful examination of thephase factors27 associated with such tunnelling events, it is possibleto show that V transforms just like Ψ in equation (11) under alattice Z4 rotation (and also under other square-lattice space-groupoperations). However, this is also the action of V under the shiftsymmetry, and so we may identify the Z4 rotation with a particularshift operation. We may also expect that there are further allowedterms that we may add to Sz that reduce the continuous shiftsymmetry to only a discrete Z4 symmetry, and this is indeed the case(equation (17)). With this reduction of a continuous to a discretesymmetry, we expect that the Goldstone boson (the photon) willacquire a small gap, as will become clearer in Section IIIB. Thesearguments imply that we may as well identify the two operators23,28:

V ∼Ψ (14)

with a proportionality constant that depends on microscopicdetails. The breaking of shift symmetry in the s > sc phase of Sz

now implies that we also have 〈Ψ 〉 6=0, and so reach the remarkableconclusion that VBS order is present in this phase. Thus, it seems




that Sz describes a transition from a Neel to a VBS state withincreasing s, and this transition will be discussed further in SectionIIIB. We will also present there a comparison of the numericalobservation of VBS order in H0 +H1 with present theory.

D. Z2 SPIN LIQUIDS

The discussion in Section IIC on the entanglement between valencebonds shown in Fig. 4 raises a tantalizing possibility. Is it possiblethat this resonance is such that we obtain a ground state of anantiferromagnet that preserves all symmetries of the hamiltonian?In other words, we restore the spin-rotation invariance of the Neelstate but do not break any symmetry of the lattice, obtaining a statecalled a resonating valence-bond (RVB) liquid. Such a state wouldbe a Mott insulator in the strict sense, because it has an odd numberof electrons per unit cell, and so cannot be smoothly connectedto any band insulator. For phases such as those described by Sz ,with gapped spinons and a U(1) photon, the answer is no: themonopole operator V , identified with the VBS order Ψ , has long-range correlations in a state with a well-defined U(1) photon.

The remainder of the next two sections will discuss a variety ofapproaches in which this monopole instability can be suppressed toobtain RVB states.

The route towards obtaining a RVB state in two dimensions waspresented in refs 29–31. The idea is to use the Higgs mechanism toreduce the unbroken gauge invariance from U(1) to a discrete gaugegroup, Z2, and so reduce the strength of the gauge-flux fluctuations.To break U(1) down to Z2, we need a Higgs scalar, Λ, that carriesU(1) charge 2, so that Λ → e2iθΛ under the transformation inequation (4). Then a phase with 〈Λ〉 6= 0 would break the U(1)symmetry, in the same manner that the superconducting orderparameter breaks electromagnetic gauge invariance. However, agauge transformation with θ = π, while acting non-trivially onthe zα, would leave Λ invariant: thus, there is a residual Z2

gauge invariance that constrains the structure of the theory in theHiggs phase.

What is the physical interpretation of the field Λ, and howdoes its presence characterize the resulting quantum state, that is,what are the features of this Z2 RVB liquid that distinguish it fromother quantum states? This is most easily determined by writingthe effective action for Λ, constrained only by symmetry and gaugeinvariance, including its couplings to zα. In this manner, we expandthe theory Sz to Sz +SΛ, with

SΛ =

∫d2rdτ

[|(∂µ−2iAµ)Λa|

2+ s|Λa|

2+ u|Λa|

4

− iΛaεαβz∗

α∂a z∗

β+c.c.

]. (15)

We have introduced multiple fields Λa, with a spatial index a,which is necessary to account for the space-group symmetry ofthe underlying lattice—this is a peripheral complication that wewill gloss over here. The crucial term is the last one coupling Λa

and zα: it indicates that Λ is a molecular state of a pair of spinonsin a spin-singlet state; this pair state has a ‘p-wave’ structure, asindicated by the spatial gradient ∂a. It is now useful to examinethe mean-field phase diagram of Sz +SΛ, as a function of thetwo ‘masses’ s and s. We have two possible condensates, and hencefour possible phases18. (1) s < 0, s > 0: this state has 〈zα〉 6= 0 and〈Λ〉 = 0. We may ignore the gapped Λ modes, and this is just theNeel state of Section IIA. (2) s > 0, s > 0: this state has 〈zα〉 = 0and 〈Λ〉= 0. Again, we may ignore the gapped Λ modes, and this isthe VBS state of Section IIC. (3) s < 0, s < 0: this state has 〈zα〉 6= 0and 〈Λ〉 6= 0. Because of the zα condensate, this state breaks spin-rotation invariance, and we determine the spin configuration byfinding the lowest-energy zα mode in the background of a non-zero

〈Λ〉 in equation (15), which is

zα =(wαei〈Λ〉·r

+ εαβw∗

βe−i〈Λ〉·r

)/√

2, (16)

with wα being a constant spinor. Inserting equation (16) intoequation (3), we find that 8 is space dependent so that 〈Si〉 obeysequation (6) with N1,2 given by equation (7) and the wavevectorK = (π,π) + 2〈Λ〉. Thus, this state is the same as the coplanarspin-ordered state of Section IIA1! The Z2 gauge transformation inequation (8) is the same as the Z2 θ= π transformation we notedearlier in this subsection. (4) s> 0, s < 0: this state has 〈zα〉= 0 and〈Λ〉 6=0. This is the Z2 spin-liquid (or Higgs) state we are after. Spin-rotation invariance is preserved, and there is no VBS order becausemonopoles are suppressed by the Λ condensate, 〈Ψ 〉 ∼ 〈V 〉 = 0.

We are now in a position to completely characterize this Z2

spin liquid. One class of excitations of this state are the zα spinonsthat now carry a unit Z2 ‘electric’ charge. A second class arevortices in which the phase Λ winds by 2π around a spatial point:these are analogous to Abrikosov vortices in a Bardeen–Cooper–Schrieffer superconductor, and so carry Aµ flux π. Furthermore,because the Aµ flux can be changed by 2π by monopoles, onlythe flux modulo 2π is significant. Consequently there is only asingle type of vortex, which is often called a vison. Two visons canannihilate each other, and so we can also assign a Z2 ‘magnetic’charge to a vison. Finally, from the analogue of the Abrikosovsolution of a vortex, we can deduce that a Z2 electric chargepicks up a factor of (−1) when transported around a vison.This Z2 × Z2 structure is the fundamental characterization of thequantum numbers characterizing this spin liquid, and its electricand magnetic excitations. An effective lattice gauge theory (an ‘odd’Z2 gauge theory) of these excitations, and of the transition betweenthis phase and the VBS state was described in ref. 32 (a detailedderivation was published later33). We also note that Z2 × Z2 is thesimplest of a large class of topological structures described by Baisand collaborators34,35 in their work on Higgs phases of discretegauge theories.

Since the work of refs 29–31, the same Z2 × Z2 structurehas been discovered in a variety of other models. The mostwell known is the toric code of Kitaev36, found in an exactlysolvable Z2 gauge theory, which has found applications as atopologically protected quantum memory37. An exactly solvablemodel of S = 1/2 spins in a Z2 spin-liquid phase was alsoconstructed by Wen38. Senthil and Fisher39 have described Z2 spin-liquid states in easy-plane magnets and superconductors. Moessnerand Sondhi40 have studied quantum dimer models that also exhibitZ2 spin-liquid phases. Finally, Freedman et al.41 have presented aclassification of topological quantum phases using Chern–Simonsgauge theory in which the Z2 ×Z2 state is the simplest case.

III. QUANTUM CRITICAL POINTS AND PHASES

In Section II, we saw several examples of insulating quantummagnets that were tuned between two distinct quantum phasesby a coupling g . We will now turn our attention to the transitionbetween these phases, and show that the critical points realize non-trivial quantum states, as was promised in Section I. In Section IIIBwe will also describe interesting cases where such critical quantumstates are found not just at isolated points in the phase diagram, butextend into a finite critical quantum phase.

A. LANDAU–GINZBURG–WILSON CRITICALITY

We begin by considering the coupled-dimer antiferromagnet ofSection IIB. With increasing g , this model has a phase transitionfrom the Neel state in Fig. 1a to the dimer state in Fig. 2a. Wealso noted in Section IIB that this transition is described by thetheoryS8 in equation (2) by tuning the coupling s. From numerical




studies2, we know that the transition in the dimer antiferromagnetoccurs at g = 1.9107 . . ., and the measured critical exponentsare in excellent agreement with those expected from S8. Thequantum state at g = gc is described by a conformal field theory(CFT), a scale-invariant structure of correlations that describes arenormalization group fixed point (known as the Wilson–Fisherfixed point) of S8. The ground state breaks no symmetries, andthere is a spectrum of gapless excitations whose structure is notcompletely understood: the spectrum is partly characterized by thescaling dimensions of various operators in the CFT, which can becomputed either numerically or using a number of perturbativefield-theoretic methods.

We place this critical quantum state in the category ofLandau–Ginzburg–Wilson (LGW) criticality, because the aboveunderstanding followed entirely from identification of the brokensymmetry in the Neel phase as characterized by the field 8,and subsequent derivation of the effective action for 8 usingonly symmetry to constrain the couplings between the long-wavelength modes. This procedure was postulated by LGW forclassical thermal phase transitions, and it works successfully for thepresent quantum transition.

Striking experimental evidence for this picture has been foundrecently42 in the coupled-dimer antiferromagnet TlCuCl3. One ofthe primary predictions of S8 is that the low-lying excitationsevolve from two spin-wave modes in the Neel state, to the threegapped triplon particles in the dimer state. The extra modeemerges as a longitudinal spin excitation near the quantumcritical point, and this mode has been observed in neutronscattering experiments.

B. CRITICAL U(1) SPIN LIQUIDS

Consider the phase transition in the model H0 + H1, as a functionof the ratio g = Q/J , in a model that preserves full square-latticesymmetry. The crucial difference from the model of Section IIIAis that there is only one S = 1/2 spin per unit cell, and so thereis no trivial candidate for an S = 0 ground state (like that inequation (9)) in the non-Neel phase. Indeed, this argument issufficient to exclude S8 from being the appropriate quantum fieldtheory for the transition, because S8 has a trivial gapped groundstate for large s.

Instead, as we discussed in Sections IIA and IIC, the groundstate of H0 + H1 evolved with increasing Q/J from the Neel statein Fig. 1a to the VBS states in Fig. 3. We also argued that the low-energy theory for the vicinity of this transition was given by thetheory Sz in equation (5) of S = 1/2 spinons coupled to a U(1)gauge field Aµ. The Neel order parameter 8 was related to thespinons by equation (3), whereas the monopole operator for theAµ gauge field was identified with the VBS order parameter Ψin equation (14). Notice that this low-energy effective theory isnot expressed directly in terms of either the Neel order 8 or theVBS order Ψ . This proposal therefore runs counter to the LGWprocedure, which would lead directly to a theory of coupled 8, Ψfields, such as those presented in ref. 43. Instead, the low-energytheory involves an emergent U(1) gauge field Aµ.

Now we ask about the fate of this U(1) gauge field acrossthe Neel–VBS transition. We noted in Section IIA that Aµ wasquenched by the Higgs phenomenon in the Neel state, whereas inSection IIC we pointed out that Aµ was gapped in the VBS stateby the reduction of the continuous shift symmetry of the field ζin equation (13) to a Z4 lattice-rotation symmetry. However, is itpossible that Aµ becomes gapless and critical in the transition fromthe Neel to the VBS state?

An affirmative answer to the last question would yield a mostinteresting U(1) spin-liquid state28,44. For Aµ to remain gapless, it isrequired that the continuous shift symmetry in ζ be preserved. The

Im (Ψ )

Re (Ψ )

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APS

Figure 5 Histogram of the VBS order parameter. The VBS order is defined inequation (10), and its histogram was measured in the numerical study19 by Sandvikof H0 +H1. The circular symmetry is emergent and implies a Goldstone boson thatis the emergent photon Aµ . Reprinted with permission from ref. 19.

connection with the Z4 lattice-rotation symmetry implies that theresulting U(1) spin-liquid state actually has a continuous spatial-rotation symmetry. In other words, the wavefunction describing theresonating valence bonds has an emergent circular symmetry, eventhough the spins on the valence bonds reside on the sites of a squarelattice. Recent numerical studies by Sandvik19 and others21 on themodel H0 + H1 have observed this remarkable phenomenon. Theycomputed the histogram in their simulation for the VBS order Ψnot too far from the transition, and the results are shown in Fig. 5.The most striking feature is the almost perfect circular symmetryof the histogram; similar results have also been obtained in ref. 21,and earlier for an easy-plane model22. This circular symmetry alsomeans that the system has not made a choice between the VBSorders in Fig. 3a and b, for which the distribution in Fig. 5 wouldbe oriented along specific compass directions (N, E, S, W and NE,SE, SW, NW respectively). These numerical results strongly supportour claim that the theory Sz is the proper low-energy theory for thetransition from the Neel to the VBS state, rather than the analogueof the theory in ref. 43.

Let us restate the emergence of circular symmetry in moreformal terms. The theory Sz has a global SU(2) spin-rotationsymmetry, and also a global U(1) symmetry associated with therotation in phase of the monopole V due to the shift symmetry inmassless scalar ζ (noted below equation (13)). The underlying spinmodel H0 +H1 also has a global SU(2) spin-rotation symmetry, butonly a Z4 lattice-rotation symmetry. The simplest term we can addto Sz that breaks its U(1) symmetry down to Z4 is

SV = l

∫d2rdτ

[V 4

+c.c.], (17)

and a non-zero l gives a mass to the scalar ζ. The circular symmetryof Fig. 5 implies that l scales under renormalization to smallvalues4,28 over the length scales studied by the simulation.

A separate question is whether the Neel–VBS transition issecond-order, with a critical point where l scales all the wayto zero. Numerical results are inconclusive at present19–21,45,46,




Figure 6 Bosons with repulsive interactions on a square lattice at filling f= 1.a,b, Snapshots of the wavefunction of the superfluid (a) and insulating (b) states.

with indications of both a second-order and a weakly first-ordertransition. In any case, it is already clear from the present results(Fig. 5) that there is a substantial length scale, much larger thanthe spin correlation length, over which monopoles described byequation (17) are suppressed, and the spin-liquid theory of zαspinons and the Aµ gauge field applies. Indeed, the length scale atwhich ζ acquires a mass, and the zα spinons are confined, is largerthan all system sizes studied so far.

A separate class of critical U(1) spin liquids appear in modelsin which the spinons are represented as fermions. These states ofantiferromagnets were proposed by Rantner and Wen47, buildingon early work by Affleck and Marston48, and a complete theorywas presented by Hermele et al.49,50. They are closely analogousto the U(1) spin liquid discussed above but with two importantdifferences: (1) the bosonic zα spinons are replaced by masslessDirac fermions ψα, and (2) they are postulated to exist notat isolated critical points between phases but as generic criticalquantum phases. The latter difference requires that the CFT of thesespin liquids has no relevant perturbations that are permitted bythe symmetries of the underlying antiferromagnet. This has so faronly been established in a 1/N expansion in models with SU(N)spin symmetry. These spin liquids have also been proposed51 ascandidate ground states for the kagome-lattice antiferromagnetobserved recently in ZnxCu4−x(OH)6Cl2.

IV. SUPERFLUIDS AND METALS

Our discussion of quantum phases and critical points has so farbeen limited to examples drawn from quantum magnetism, inwhich the primary degrees of freedom are stationary unpairedelectrons that carry spin S = 1/2. We have seen that a plethora of

interesting quantum states are possible, many of which have foundrealizations in experimental systems. In this section, we will widenthe discussion to include systems that also allow for motion of massand/or charge. Thus, in addition to insulators, we will find statesthat are metals or superfluids/superconductors. We will find thatmany phases and transitions can be understood by mappings andanalogies from the magnetic systems.

The simplest examples of a quantum phase transition involvingmotion of matter are found in systems of trapped ultracold atoms,which have been reviewed earlier52. Here, the degrees of freedomare neutral atoms, and so the motion involves mass, but not charge,transport. Bosonic atoms are placed in a optical lattice potentialcreated by standing waves of laser light, and their ground state isobserved as a function of the depth of the periodic potential. For aweak potential, the ground state is a superfluid, whereas for a strongpotential, the ground state may be an insulator.

A crucial parameter determining the nature of the insulator,and of the superfluid–insulator transition, is the filling fraction, f ,of the optical lattice. This is the ratio of the number of bosons tothe number of unit cells of the optical lattice.

We begin by considering the simplest case, f = 1, which isshown in Fig. 6. For a weak potential, the bosons move freelythrough the lattice, and form a Bose–Einstein condensate, leadingto superfluidity. In the site representation shown in Fig. 6, thesuperfluid state is a quantum superposition of many-particleconfigurations, only one of which is shown. The configurationsinvolve large fluctuations in the number of particles on eachsite, and it is these fluctuations that are responsible for easymotion of mass currents, and hence superfluidity. For a strongpotential, the bosons remain trapped in the lattice wells, oneeach per unit cell, forming the simple ‘Mott insulator’ shown inFig. 6. We can describe quantum phase transition between thesestates by using the LGW order-parameter approach, just as wedid for the coupled-dimer antiferromagnet in Section IIIA. Theorder parameter is now the condensate wavefunction, which isrepresented by a complex field Φ. Indeed, this field has two realcomponents but it is otherwise similar to the three-component field8 considered in Section IIA. The LGW theory for the superfluid–insulator transition53 is then given by the action in equation (2),with the replacement 8 →Φ.

Now let us move to f = 1/2, which is shown in Fig. 7. Thesuperfluid state is similar to that at f = 1, with large numberfluctuations on each site. However, the nature of the insulatoris now very different. Indeed, in a single-band ‘boson Hubbardmodel’ with purely onsite repulsive interactions between theparticles, the ground state remains a superfluid even at infinitelylarge repulsion (‘hard-core’ bosons): this is clear from Fig. 7,where it is evident that number fluctuations are possible at siteseven when double occupancy of the sites is prohibited. However,once offsite interactions are included, interesting insulating statesbecome possible. The nature of these phases can be understood bya very useful mapping between hard-core bosons and the S = 1/2antiferromagnets studied in Sections IIA and IIC; the Hilbert spacesof the two models are seen to be identical by identifying an emptysite with a down spin, for example, and a site with a boson withan up spin. Furthermore, in this mapping, an easy-plane Neel stateof Section IIA maps onto the superfluid state of the bosons. It isthen natural to ask for the fate of the VBS states in Fig. 3 underthe mapping to the hard-core bosons. A simple analysis yields theinsulating states shown in Fig. 7. The same reasoning also enablesus to analyse the superfluid–insulator transition for f =1/2 bosons,by connecting it to the Neel–VBS transition in Section IIIB. Inthis manner, we conclude that the low-energy quantum states nearthe superfluid–insulator transition are described by the U(1) gaugefield theory Sz in equation (5); the only change is that a further




Figure 7 Bosons with repulsive interactions on a square lattice at filling f= 1/2. a,b, Snapshots of the wavefunction of the superfluid (a) and two possible insulating (b)states. The single bosons on the links in the first insulator resonate between the two sites on the ends of the link, and this insulator is similar to the VBS state in Fig. 3a.Similarly, the two bosons on the plaquette in the second insulator resonate around the plaquette, and this insulator is similar to the VBS state in Fig. 3b.

‘easy-plane’ term is permitted that lowers the global symmetry fromSU(2) to U(1). The zα quanta of this theory are now emergent‘fractionalized’ bosons that carry boson number 1/2 (in unitsin which the bosons shown in Fig. 7 carry boson number 1); amicroscopic picture of these fractionalized bosons can be obtainedby applying the above mapping to the spinon states in Fig. 4.

The crucial result we have established above is that LGWcriticality does not apply at f = 1/2. By extensions of the aboveargument, it can be shown that LGW criticality applies only atinteger values of f . Superfluid–insulator transitions at fractional fare far more complicated.

The above gauge theory for bosons at f =1/2 has been extendedto a wide variety of cases, including a large number of other fillingfractions54, and models of paired electrons on the square lattice55,56

also at general filling fractions. The resulting field theories are morecomplicated than Sz , but all involve elementary degrees of freedomthat carry fractional charges coupled to emergent U(1) gauge fields.

In the hole-doped cuprate superconductors, there are a varietyof experimental indications57 of a stable insulator in somecompounds at a hole density of x = 1/8. As we will discuss furtherin Section V, it is useful to consider a superfluid–insulator transitionat x = 1/8, and to build a generalized phase diagram for thecuprates from it. The low-energy theory for such a transition issimilar56 to that for bosons (representing paired holes) at f = 1/16.We show the spatial structure of some of the large numberof allowed insulating states that are possible near the onset ofsuperfluidity in Fig. 8; these are analogues of the two distinctpossible states for f = 1/2 in Fig. 7. It is our proposal17,18,54 that thebond-centred modulations in the local density of states observedby Kohsaka et al.15 (Fig. 10) can be described by theories similar tothose used to obtain Fig. 8.

Having described the superfluid–insulator transition, we brieflyturn our attention to the analogues of the ‘exotic’ spin-liquidstates of magnets discussed earlier in Section IID and III. Thetopological order associated with such states is not restrictedto insulators with no broken symmetry, but can be extendednot only into insulators with conventional broken symmetries,but also into superfluids39. Indeed, an exotic metallic state isalso possible58, called the fractionalized Fermi liquid (FFL). Thesimplest realization of an FFL appears in the two-band model(a Kondo lattice), in which one band forms a Z2 spin liquid, whereasthe other band is partially filled and so forms a Fermi surface ofordinary electron-like quasiparticles. One unusual property of anFFL is that the Fermi surface encloses a volume that does not countall of the electrons: a density of exactly one electron per unit cell

is missing, having been absorbed into the Z2 spin liquid. Thereis, thus, a close connection between the quantum entanglementof a spin liquid and deviations from the Luttinger counting ofthe Fermi surface volume. The FFL also has interesting quantumcritical points towards conventional metallic phases. These havebeen reviewed in ref. 59 and have potential applications to quantumphase transitions in the rare-earth ‘heavy fermion’ compounds.

Another class of metallic and superconducting states, the‘algebraic charge liquids’60, are obtained by adding mobile carriersto antiferromagnets described bySz . In this case, the charge carriersare spinless fermions that carry charges of the Aµ gauge field. Thesefermions can form Fermi surfaces or pair into a superconductor.These exotic metallic and superconducting states have a numberof interesting properties that have been argued to describe theunderdoped cuprates60.

V. FINITE-TEMPERATURE QUANTUM CRITICALITY

We now turn to an important question posed in Section I: what isthe influence of a zero-temperature quantum critical point on theT > 0 properties?

Let us address this question in the context of the superfluid–insulator transition of bosons on a square lattice at f = 1 discussedin Section IV. The phase diagram of this model is shown in Fig. 9as a function of T and the tuning parameter g (the strength ofthe periodic potential). Note the region labelled ‘quantum critical’:here, the physics is described by taking the critical hamiltonianat g = gc and placing it at a temperature T . The deviationof the coupling g from gc is a subdominant perturbation—onexamining the quantum fluctuations with decreasing energy scales,the excitations of the quantum critical state are first damped out bythermal effects, before the system is able to decide which side of thetransition it resides on, that is, before it is sensitive to whether g issmaller or larger than gc. Thus, we reach the seemingly paradoxicalconclusion that the window of influence of the quantum criticalpoint actually widens as the temperature is increased. Of course,there is also an upper bound in temperature, the value of whichdepends on all details of the particular experimental system, abovewhich the quantum criticality is irrelevant because thermal effectsare so strong that neighbouring quantum degrees of freedom barelydetect each other’s presence.

The ‘transport’ properties of systems in the quantum criticalregion are of particular interest. For the superfluid–insulatortransition, this means the study of the electrical conductivity. Moregenerally, we can consider the response functions associated with




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APS

y

x

Figure 8 Structures of possible insulating states on the square lattice for pairedelectrons at a hole density of x= 1/8 (f= 1/16). Reprinted with permission fromref. 54. The shading is a measure of the charge density; the density is modified onbonds or plaquettes for configurations such as those of the insulating states in Fig. 7.

any globally conserved quantity, such as the total spin of thequantum magnetic systems of Section II. In traditional condensed-matter physics, transport is described by identifying the low-lyingexcitations of the quantum ground state, and writing ‘transportequations’ for the conserved charges carried by them. Often,these excitations have a particle-like nature, such as the ‘triplon’particles of Section IIB, in which case we would describe theircollisions by a Boltzmann equation. In other cases, the low-lyingexcitations are waves, such as the spin waves of the Neel statesof Section IIA, and their transport is described by a nonlinearwave equation (such as the Gross–Pitaevski equation). However, formany interesting quantum critical points, including the superfluid–insulator transition in two dimensions, neither description ispossible, because the excitations do not have a particle-like or wave-like character.

Despite the absence of an intuitive description of the quantumcritical dynamics, we can expect that the transport propertiesshould have a universal character determined by the quantum field

g

T

gc00

InsulatorSuperfluid

Quantumcritical

TKT

Figure 9 Phase diagram of the superfluid–insulator transition in two dimensions.Similar crossover diagrams apply to most of the quantum phase transitionsdiscussed in this review article. The quantum critical point is at g= gc, T= 0. Thedashed lines are crossovers, whereas the solid line is a phase transition at theKosterlitz–Thouless temperature TKT > 0.

theory of the quantum critical point. In addition to describingsingle excitations, this field theory also determines the S-matrix ofthese excitations by the renormalization group fixed-point value ofthe couplings, and these should be sufficient to determine transportproperties61. So the transport coefficients, like the conductivity,should have universal values, analogous to the universality ofcritical exponents in classical critical phenomena.

Let us now focus on critical points described by CFTs, andon the transport properties of CFTs at T > 0. CFTs in spatialdimension d = 1 are especially well understood and familiar, andso we consider, for example, the superfluid–insulator transition inone-dimensional systems. A phase diagram similar to that in Fig. 9applies: the quantum critical region of interest here is retained, butthe critical temperature for the onset of superfluidity is suppressedto zero. We will compute the retarded correlation function ofthe charge density, χ(k,ω), where k is the wavevector and ω isfrequency; the dynamic conductivity, σ(ω), is related to χ by theKubo formula, σ(ω) = limk→0(−iω/k2)χ(k,ω). It can be shownthat all CFTs in d = 1 have the following universal χ:

χ(k,ω) =4e2

hK

vk2

(v2k2 − (ω+ iη)2), σ(ω) =

4e2

h

vK

(−iω);

d = 1, all hω/kBT , (18)

where e and h are fundamental constants that normalize theresponse, K is a dimensionless universal number that characterizesthe CFT, v is the velocity of ‘light’ in the CFT and η is a positiveinfinitesimal. This result holds both at T =0 and also at T> 0. Notethat this is a response characteristic of freely propagating excitationsat velocity v. Thus, despite the complexity of the excitations ofthe CFT, there is collisionless transport of charge in d = 1 atT > 0. Of course, in reality, there are always corrections to theCFT hamiltonian in any experimental system, and these will inducecollisions—these issues are discussed in ref. 62.

It was argued in ref. 61 that the behaviour of CFTs inhigher spatial dimensions is qualitatively different. Despite theabsence of particle-like excitations of the critical ground state,the central characteristic of the transport is a crossover fromcollisionless to collision-dominated transport. At high frequenciesor low temperatures, we can obtain a collisionless responseanalogous to equation (18), using similar arguments on the basis




of conformal invariance,

χ(k,ω) =4e2

hK

k2√v2k2 − (ω+ iη)2

, σ(ω) =4e2

hK;

d = 2, hω� kBT , (19)

where again K is a universal number. However, it was argued61 thatphase-randomizing collisions were intrinsically present in CFTsin d = 2 (unlike d = 1), and these would lead to relaxation ofperturbations to local equilibrium and the consequent emergenceof hydrodynamic behaviour. So at low frequencies, we have insteadan Einstein relation that determines the conductivity with

χ(k,ω) = 4e2χc

Dk2

Dk2 − iω, σ(ω) = 4e2χcD =

4e2

hΘ1Θ2;

d = 2, hω� kBT , (20)

where χc is the compressibility and D is the charge diffusionconstant of the CFT, and these obey χc = Θ1kBT/(hv)2 andD = Θ2hv2/(kBT), where Θ1,2 are universal numbers. A largenumber of papers in the literature, particularly those on criticalpoints in quantum Hall systems, have used the collisionless methodof equation (19) to compute the conductivity. However, the correctd.c. limit is given by equation (20); given the distinct physicalinterpretation of the two regimes, we expect that K 6= Θ1Θ2, andso a different conductivity in this limit. This has been shown in aresummed perturbation expansion for a number of CFTs63.

Much to the surprise of condensed-matter physicists64,remarkable progress has become possible recently on the abovequestions as a result of advances in string theory. The AdS/CFTcorrespondence connects the T > 0 properties of a CFT to adual theory of quantum gravity of a black hole in a negativelycurved ‘anti-de Sitter’ universe65–69: the temperature and entropyof the CFT determine the Hawking temperature and Bekensteinentropy of the black hole. This connection has been especiallypowerful for a particular CFT, leading to the first exact resultsfor a quantum-critical point in 2 + 1 dimensions. The theory is aYang–Mills non-abelian gauge theory with an SU(N) gauge group(similar to quantum chromodynamics) andN =8 supersymmetry.In 2 + 1 dimensions, the Yang–Mills coupling g flows under therenormalization group to a strong-coupling fixed point g = gc: thus,such a theory is generically quantum critical at low energies, andin fact realizes a critical spin liquid similar to those consideredin Section IIIB. The CFT describing this fixed point is dualto a particular phase of M theory, and this connection enablescomputation, in the limit of large N , of the full χ(k,ω)of a particular conserved density. These results contain70 thecollisionless to hydrodynamic crossover postulated above betweenexactly the forms for χ(k,ω) in equations (19) and (20), includingthe exact values70,71 of the numbers K , Θ1, Θ2. A curious featureis that for this solvable theory, K = Θ1Θ2, and this was traced toa special electromagnetic self-duality property in M theory, whichis not present in other CFTs. It is worth noting that these arethe first exact results for the conductivity of an interacting many-body system (there are also conjectured exact results for the spindiffusivity of one-dimensional antiferromagnets with a spin gap72,in a phase similar to Section IIB, whose dynamics was observedin ref. 12).

A. HYDRODYNAMICS

The transport results for the superfluid–insulator transitionpresented so far have an important limitation: they are valid onlyat the specific particle densities of the insulators, where thereare a rational number of particles per unit cell, for example,

T

g µ

Insulator x = 1/8

Superconductor

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Figure 10 Superfluid–insulator transition applied to the cuprate superconductors.The parameter g is as in Fig. 9, and that figure applies only exactly at hole densityx= 1/8. The figure shows how the quantum critical point (red circle) controls thephase diagram at general chemical potential µ and hole density x. We hypothesizethat the density modulation in the insulator is similar to the bond-centredmodulations in the underdoped superconductor observed by Kohsaka et al.15, asindicated by the inset picture of their observations (reprinted with permission fromref. 15). Possible supersolid phases are not shown.

the insulators in Figs 6–8. Whereas the particle density is indeedpinned at such commensurate values in the insulator at T = 0,there is no pinning of the density in the superfluid state, oranywhere in the phase diagram at T > 0. Even at small variationsin density from the commensurate value in these states, we expectstrong modifications of the transport results above. This is becauseequations (19) and (20) relied crucially61 on the lack of mixingbetween the conserved momentum current (in which the particleand hole excitations are transported in the same direction) andthe electrical current (which is not conserved because of collisionsbetween particles and holes moving in opposite directions). Whenparticle–hole symmetry is broken by a variation in the density,the mixing between these currents is non-zero, and a new analysisis necessary.

As a particular motivation for such an analysis, Fig. 10shows a hypothetical phase diagram for the hole-doped cupratesuperconductors as a function of a chemical potential, µ, whichcontrols the hole density. We also have a tuning parameter, g , whichcontrols the proximity to an insulating state whose features havepresumably been measured in refs 15 and 57. We are interestedin using the quantum critical point shown in Fig. 10 to describetransport in the analogue of the quantum critical region ofFig. 9 when extended to generic densities. Furthermore, it is alsointeresting to add impurities and a magnetic field, B, to be able todescribe various magnetotransport experiments73.

Further motivation for the above analysis arises from potentialapplications to graphene74. The electronic excitations of graphenehave a Dirac-like spectrum, and after including their Coulombinteractions, we obtain a system with many similarities to quantumcritical states75. At zero bias, we expect the conductivity ofpure graphene to be described by equations (19) and (20) (withlogarithmic corrections from the marginally irrelevant Coulombinteractions63), but we are interested in the bias dependence of theconductivity, and also in the influence of impurity scattering.

A general theory of quantum critical transport with all ofthe above perturbations has not been achieved. However, in thelow-frequency, collision-dominated regime of equation (20), it hasrecently been shown76 that a nearly complete description can beobtained by using general hydrodynamic arguments on the basis




of the positivity of entropy production and relaxation to localequilibrium. The results were verified in an explicit solution of thehydrodynamics of a particular CFT solvable through the AdS/CFTcorrespondence: after the addition of a chemical potential and amagnetic field, the CFT maps onto a black hole with electric andmagnetic charges.

The complete hydrodynamic analysis can be found in ref. 76,along with a comparison to experimental measurements ofthe Nernst effect in the cuprate superconductors73; reasonableagreement is obtained, with only two free parameters. The analysisis intricate, but is mainly a straightforward adaption of the classicprocedure outlined by Kadanoff and Martin77 to the relativisticfield theories that describe quantum critical points. The steps areas follows. (1) Identify the conserved quantities, which are theenergy–momentum tensor, Tµν, and the particle number current,Jµ. (2) Obtain the real time equations of motion, which express theconservation laws:

∂νTµν

= FµνJν, ∂µJµ = 0;

here, Fµν represents the externally applied electric and magneticfields that can change the net momentum or energy of the system,and we have not written a term describing momentum relaxationby impurities. (3) Identify the state variables that characterize thelocal thermodynamic state—we choose these to be the density,ρ, the temperature T , and an average velocity uµ. (4) ExpressTµν and Jµ in terms of the state variables and their spatial andtemporal gradients; here, we use the properties of the observablesunder a boost by the velocity uµ, and thermodynamic quantitiessuch as the energy density, ε, and the pressure, P, which aredetermined from T and ρ by the equation of state of the CFT. Wealso introduce transport coefficients associated with the gradientterms. (5) Express the equations of motion in terms of the statevariables, and ensure that the entropy production rate is positive78.This is a key step that ensures relaxation to local equilibrium,and leads to important constraints on the transport coefficients.In d = 2, it was found that situations with the velocity uµ space-time independent are characterized by only a single independenttransport coefficient76. This we choose to be the longitudinalconductivity at B = 0. (6) Solve the initial value problem for thestate variables using the linearized equations of motion. (7) Finally,translate this solution to the linear response functions, as describedin ref. 77.

As an example of the above analysis, we highlight an importantresult—the modification of the conductivity in equation (20) by anon-zero density difference, ρ, from that of the Mott insulator atB = 0. We found that a non-zero ρ introduces a further componentof the current that cannot relax by collisions among the excitations,and which requires the presence of impurities to yield a finiteconductivity:

σ(ω) =4e2

hΘ1Θ2 +

4e2ρ2v2

(ε+P)

1

(−iω+1/τimp).

Here, the first term is as in equation (20), arising from the collisionsbetween the particle and hole excitations of the underlying CFT,although the values of Θ1 and Θ2 can change as a function ofρ. The additive Drude-like contribution arises from the excessparticles or holes induced by the non-zero ρ, and 1/τimp is theimpurity scattering rate (which can also be computed from theCFT73,79). Note that in the ‘non-relativistic limit’, achieved whenρ is so large that we are well away from the critical point,the energy and pressure are dominated by the rest mass of theparticles ε+ P ≈ |ρ|mv2, and then the second term reduces to theconventional Drude form. The above theory can also be extendedto include the Hall response in a magnetic field, B,76,80; in particular

we obtain the d.c. Hall conductivity, σxy ∝ ρ/B. This predicts anegative Hall resistance in the cuprates at hole densities smallerthan that of the insulator at x = 1/8, as has been observed in recentexperiments on the cuprates81.

VI. CONCLUSIONS

We have described progress in understanding new types ofentanglement in many-electron systems. Entanglement is a keyingredient for quantum computing, and the critical spin liquidsdescribed here are probably the states with the most complexentanglement known. This progress has been made possible byadvances on several distinct fronts. On the experimental side, new‘correlated electron’ materials have been discovered, such as thosedescribed in refs 13,14. There have been advances in experimentaltechniques, such as scanning tunnelling microscopy in ref. 15 andneutron scattering in ref. 12. And a new frontier has openedin the study of the correlated phase of cold atoms in opticallattices52. On the theoretical side, large-scale numerical studiesof new types of quantum magnet have become possible by newalgorithms, including the first studies of phases with no magneticorder in S = 1/2 square-lattice antiferromagnets19–21. An importantfrontier is the extension of such methods to a wider class offrustrated quantum systems, which have so far proved numericallyintractable because of the ‘sign’ problem of quantum MonteCarlo. In field-theoretical studies of novel quantum phases, newconnections have emerged to those studied in particle physics andstring theory64,70,76, and this promises an exciting cross-fertilizationbetween the two fields.

doi:10.1038/nphys894

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AcknowledgementsI thank S. Bais, S. Hartnoll, R. Kaul, R. Melko, M. Metlitski, M. Muller and A. Sandvik for valuablecomments. This work was supported by NSF Grant No. DMR-0537077.

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