aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Energy density of variational states
Leon Balents Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, U.S.A. (Dated: October 16, 2018)We show, in several important and general cases, that a low variational energy density of atrial state is possible even when the trial state represents a different phase from the ground state.Specifically, we ask whether the ground state energy density of a Hamiltonian whose ground stateis in phase A can be approximated to arbitrary accuracy by a wavefunction which represents adifferent phase B. We show this is indeed the case when A has discrete symmetry breaking orderin one dimension or topological order in two dimensions, while B is disordered. We argue that, ifreasonable conditions of physicality are imposed upon the trial wavefunction, then this is not possiblewhen A has discrete symmetry breaking in dimensions greater than one and B is symmetric, or whenA is topologically trivial and B has topological order.
I. INTRODUCTION
Quantum many body theory has revealed the exis-tence of “exotic” ground states phases of matter – whichwe take to mean phases distinguished by means otherthan symmetry breaking. Notable are topological phases,which are fully gapped ground states supporting anyonicexcitations in two dimensions, and embodying a general-ization of this structure in three dimensions. A topolog-ical phase is locally stable to all perturbations, makingit apparently more robust even than a conventional sym-metry breaking state. While the theoretical existenceof a huge family of topological phases is well-establishedthrough exactly soluble models and an extensive formal-ism, these states are so far scarce in the real world.The identification of these exotic phases is hamperedby the dual challenges of calculating ground states ofmany body systems, and of identifying the hidden topo-logical or other structure within those states. Conse-quently, a prominent computational approach is the vari-ational method, in which a trial wavefunction is opti-mized within a given (say) topological class, and the en-ergy of such optimized wavefunctions can be comparedacross classes. A popular example is the use of Gutzwillerprojected free fermion states for quantum spin systems.In this context exotic phases are known as quantum spinliquids . Distinct states can be constructed using the pro-jective symmetry group construction, and the energy ofoptimized states compared. A natural implication, pre-sumed in many studies, is that the true ground stateis in the same phase as the wavefunction with lowest op-timized energy.In this paper, we discuss the reliability of this conclu-sion for systems with different types of ground states. Weconsider local Hamiltonians H for infinite quantum sys-tems of dimensionality d = 1 , ,
3. For finite systems thevariational method is on very firm ground: if a state | ψ i has variational energy E v = h ψ | H | ψ i which is separatedby the ground state by an amount less than the spectralgap, E < E v < E , where E is the exact ground stateenergy and E is the first excited state energy, then | ψ i must have a finite overlap with the exact ground state, and the minimum value of this overlap increases as E v approaches E . However, for infinite systems, there is afundamental difficulty: both the variational energy andthe exact ground state energy are infinite , and generi-cally, E v and E differ by an amount proportional to thesystem volume L d . Standard practice is to compare thevariational energy density (e.g. per site or per unit vol-ume) of different states. The variational principle guar-antees that the variational energy density is bounded be-low by the exact energy density, so one can certainlyuse the variational method to get a good estimate of theground state energy density. But is it actually predictivefor the phase of the ground state?Many studies of strongly correlated problems (e.g.Hubbard models, frustrated quantum magnets) haveshown that states belonging to different phases have veryclose variational energy densities. A common interpre-tation of this observation is that the actual system issomehow indecisive about its true ground state, i.e. thatseveral phases are “close” in phase space, and could be se-lected in the true ground state by small perturbations ofthe original Hamiltonian. This notion is somehow similarto the picture of first order phase transitions, for whichnear the transition the higher (free) energy phase existsas a metastable object even when it is not the global ther-modynamic minimum. According to this line of thinking,distinct variational wavefunctions represent “competingorders” that play a central role in the physics. However,it is not a priori obvious that the phase of the variationalstate has predictive value.This paper addresses this issue at the level of princi-ple. Specifically, we ask whether a wavefunction in the wrong phase can yield an arbitrarily good approximationto the ground state energy density for a given Hamilto-nian. It is natural to suppose that the answer to thisquestion is the same for all non-fine-tuned local Hamil-tonians in a particular dimension, whose ground state isin a given phase. Then we say that a given phase A ina given dimension is variationally robust to phase B, iffor all non-fine-tuned Hamiltonians with a ground statein that phase in that dimension, states which belong to adifferent phase B have variational energy density which isstrictly larger than that of the ground state. Conversely,if variational states in phase B can be found with en-ergy density arbitrarily close to the exact ground statevalue, then we say that the phase A is not variationallyrobust to phase B. Clearly variational methods are morepredictive for states that are variationally robust.Theoretically, it is much easier to show that a state is not variationally robust, simply by providing a construc-tive example of a variational state in the wrong phasewith low energy density. Demonstrating variational ro-bustness is much more difficult, and indeed we do notknow how to prove this rigorously. Part of the difficulty isthat one must specify somehow the allowed set of possiblevariational states. We would like to include only statesthat are characteristic of generic ground state phases oflocal Hamiltonians. This is a small subset of all possiblequantum states. For example, any state which exhibitsa volume law of entanglement entropy cannot be a validground state.With this in mind, we restrict ourselves to gapped phases of bosonic/spin systems, for which it is generallyagreed that ground states can be described by tensor net-work states (TNS), also know as projected entangled pairstates (PEPS). We therefore assume that both the ex-act ground state and the variational state have the TNSform. An important fact is that the variational energydensity, being an expectation value of local operators,is a smooth function of the tensor describing the TNS.Then one sees that the question of variational robustnessis equivalent to the robustness of the phase in questionwith respect to small variations of the TNS tensor. Ifan infinitesimal change of the tensor changes the phaseof the TNS describing the exact ground state, then thephase is not variationally robust. Conversely, a variation-ally robust phase must be preserved by all infinitesimalvariations of the TNS tensor. If the latter condition istrue, the phase is likely to be variationally robust, thoughthis is not a rigorous proof.The TNS description applies only to fully gappedphases. Thus we can consider phases with no order, withdiscrete symmetry breaking, and with topological order.We first consider the situation in which ground state hasno order, and show that broken symmetry states witharbitrarily low variational energy always exist. This isvery straightforward and indeed agrees with a simplisticpicture based on Landau theory. Next, we consider theconverse situation, and ask whether a broken symmetryphase is variationally robust: do low energy variationalstates exist in which the symmetry in question is un-broken ? We show that discrete broken symmetry is not variationally robust in one dimension, but argue that it is robust in two or more dimensions. We then turn totopological order, which exists only in dimensions two orgreater. We show that topological order is not variation-ally robust in two dimensions, i.e. that low energy stateswithout topological order exist even when the groundstate has topological order. This shows that topologicalorder is less robust in the variational sense than discrete symmetry breaking. We do not make general statementsabout topological phases in three dimensions, but do dis-cuss the case of the toric code ( Z topological phase). Fi-nally, we consider the converse question: whether a phasewithout topological order admits low energy variationalstates with topological order. We argue this is not possi-ble, so that topologically trivial phases are variationallyrobust to approximants with topological order.The remainder of this paper discusses these ideas inmore detail. Sec. II discusses symmetry breaking order,and robustness of states with or without such order tothose without/with broken symmetry. ones. Next, inSec. III, we discuss topological order, and the robustnessof states with or without such order. Finally, we concludewith some general comments in Sec. IV. II. DISCRETE SYMMETRY BREAKINGA. Robustness of “disordered” states to symmetrybreaking
Here we consider a Hamiltonian which is invariant un-der a discrete symmetry group G . Suppose there existsa local order parameter φ which is not invariant under G ( φ may be a scalar or have multiple components). Sup-pose that the ground state | i is invariant under G . Weask whether a variational state | φ i exists with non-zero φ and arbitrarily low variational energy density. A sim-plistic Landau theory arguments suggests yes. We writethe energy density E for the state | φ i as a function of φ . Standard Landau reasoning suggests that it takes theform E ( φ ) = E + rφ · φ + · · · , (1)where φ · φ is the quadratic invariant of the order param-eter, and r > φ = 0 and unbroken symmetry. Clearly this predictsthat a broken symmetry state with arbitrarily low energydensity can be found simply by taking the magnitude of φ small but non-zero.Microscopically, we can argue similarly, by the follow-ing construction. We assume the Hamiltonian H has thesymmetry G and ground state | i . Consider a fiduciaryHamiltonian H λ , with H λ = H − λ Z d d x ˆ φ, (2)where the integral over space can be replaced by a sum,as appropriate, and ˆ φ is a local operator with the samesymmetry as the order parameter. The ground state of H λ smoothly goes to that of H as λ →
0, and hence sodoes the energy density. We can consider this state, | λ i as a variational state for H . Then h λ | H | λ i = E ( λ ) + h λ | ˆ φ | λ i , (3)where E ( λ ) is the ground state energy of H λ . By pertur-bation theory (which is valid because, by assumption, H has a gap), we see that the right hand side above is equalto E up to terms of order λ . Hence the state | λ i , whichhas broken symmetry whenever λ = 0, has a variationalenergy density of order λ higher than the exact groundstate.So we conclude that a disordered (or symmetric) phaseis never variationally robust, in any dimension, to asymmetry-broken state. This is not surprising and agreeswith the intuition based on simple Landau theory. B. Robustness of symmetry broken phases toapproximation by symmetric states
Now we consider the reverse question from that of theprevious section. Suppose the ground state of H breaksa discrete symmetry. Can one find a state with unbroken symmetry with arbitrarily low variational energy den-sity? Here Landau theory suggests the answer is no. Thisis because the symmetry broken minima in the Landauenergy functional are separated by a non-zero distancefrom the symmetric point: a small variation of a non-zeroorder parameter φ cannot restore the symmetry. This ar-gument is, however, na¨ıve, and demonstrably incorrect inone dimension, as we now show.
1. Example: Transverse field Ising chain
Let us begin with the archetypal example of symme-try breaking in one dimension, the transverse field Isingmodel, H = X i − σ zi σ zi +1 − hσ xi , (4)where ~σ i are Pauli matrices. The model is exactly sol-uble, and, for | h | <
1, has a ferromagnetic ground statewhich breaks the Ising symmetry, σ zi → − σ zi . Thereis an excitation gap above the ground state, and expo-nentially decaying connected correlations. Consequently,the ground state can be well-approximated by a matrixproduct state, | ψ i = X { σ i = ± } Tr [ M ( σ ) M ( σ ) · · · M ( σ N )] | σ · · · σ N i , (5)where σ i = ± σ zi . A brokensymmetry ground state with positive magnetization isdescribed by a matrix M + ( σ ) which preferentially favors σ = +1 over σ = −
1. One can obtain the other groundstate by transforming the matrix into M − ( σ ) = M + ( − σ ).For a finite system, one expects exponentially weak tun-neling to mix the two symmetry broken states, so that theproper system eigenstates are the Schr¨odinger cat states | M i defined by the enlarged matrix M = M + e iφ M − , (6)where we may choose φ = 0 and φ = π/N to obtainlinearly independent states, even and odd linear combi-nations of the two ferromagnetic states. The energy den-sity of the Schr¨odinger cat states is identical to that ofpure ferromagnetic states up to exponential corrections,and indeed they represents the true eigenstates of largefinite systems. Moreover these states still have brokensymmetry in the sense of off-diagonal long-range order:the correlation functions obey h M | σ zi σ zj | M i → σ for 1 ≪ | i − j | ≪ N. (7)Now we claim that an infinitesimal deformation of thematrix in Eq. (6) leads to a state with unbroken Isingsymmetry. Specifically, consider M ǫ = M + ǫ M ǫ M M − . (8)Provided M ( − σ ) = M ( σ ), the state | M ǫ i retains theIsing symmetry, as can be easily verified. However, when ǫ = 0, the long-range order of the state is destroyed. Thisis true in general, but we illustrate it for transparency forthe simplest case in which M ± are one dimensional, whichcorresponds for ǫ = 0 to a mean field approximation tothe ground state.To see the destruction of long-range order, we can di-rectly calculate the spin-spin correlation function. Fol-lowing standard methods, we have h M ǫ | σ zi σ zj | M ǫ i = 1 Z Tr h T N −| i − j |− ST | i − j |− S i , (9)where the trace is expressed in a doubled Hilbert spacewhich is defined as a direct product of the original matrixproduct one, owing to the product of two wavefunctionsin the expectation value. We have T = X σ M ǫ ( σ ) ⊗ M ǫ ( σ ) , (10) S = X σ σ M ǫ ( σ ) ⊗ M ǫ ( σ ) , (11)and Z = Tr T N . For the one dimensional case, we have M ± ( σ ) = p (1 ± mσ ) /
2, where m is the spontaneousmagnetization when ǫ = 0, and − < m <
1. We take M = M = 1. Expressing the matrices in the directproduct space in terms of two sets of Pauli matrices ~τ and ~τ , we have then T = 2 (cid:2) ( m + + ǫτ x )( m + + ǫτ x ) + m − τ z τ z (cid:3) , (12) S = 2 [ m + m − ( τ z + τ z ) + m − ǫ ( τ z τ x + τ x τ z )] , (13)where m ± = 12 r m ± r − m ! . (14)Expressing Eq. (9) in terms of the eigenstates of T oneobtains h M ǫ | σ zi σ zj | M ǫ i = P a,b t Na (cid:16) t b t a (cid:17) | i − j |− |h a | S | b i| P a t Na , (15)where T | a i = t a | a i and h a | a i = 1, and we can choose aconvention in which t ≥ t ≥ t ≥ t . In the thermody-namic limit, N → ∞ , this is dominated by the term orterms in which t a are maximal. For ǫ = 0, t > t , andthe maximal eigenvalue is unique, and so h M ǫ | σ zi σ zj | M ǫ i ǫ =0 −−−−→ N →∞ X b (cid:18) t b t (cid:19) | i − j |− |h | S | b i| . (16)By Ising symmetry, one readily sees that h | S | i = 0,so all terms in the sum decay exponentially (this breaksdown for ǫ = 0 because the leading eigenvalues are de-generate, t = t in that case), and there is no long-rangeorder. For ǫ ≪
1, we can approximate |h | S | i| ≈ m , t b /t ≈ − ǫ / (1 − √ − m ), so that h M ǫ | σ zi σ zj | M ǫ i <ǫ ≪ −−−−→ N →∞ m e −| i − j | /ξ , (17)with ξ ≈ − √ − m ǫ . (18)So we see that the state | M ǫ i has, for arbitrarily small ǫ ,no long-range order.
2. Domain wall interpretation
For the toy model in the previous subsection, it isstraightforward to interpret the physics of the variationalstate. In general, the two blocks of the double matrix inEqs. (6,8) describe the two physical uniform Ising groundstates. By introducing a small non-zero ǫ in Eq. (8), weintroduce a probability amplitude ǫ for a domain wall toappear in this uniform state. A domain wall is a topo-logical soliton in the Ising ordered phase, which in onedimension is a localized point-like defect separating thetwo Ising domains. It is an excitation of the Ising orderedphase (indeed it is the elementary one). In general, in adiscrete broken symmetry state, a non-zero gap is ex-pected for such an excitation. The state | M ǫ i can beviewed as a state with a superposition of domain walls,whose mean density becomes small for ǫ ≪ ξ above. If the correlation function is measured betweentwo points whose separation is much larger than ξ , the number of domain walls between these two points fluc-tuates randomly (or more properly the wavefunction is asuperposition of components in which this number variessignificantly), leading to contributions to the correlationfunction of opposite sign that cancel and decay exponen-tially with length.In the variational sense, a low density superposition ofdomain walls has a low energy density in the Ising orderedphase, since the energy of each soliton is finite, and thenumber per unit length is 1 /ξ . Hence the variationalenergy density of such a state above the ground state isexpected to be of order ∆ /ξ , where ∆ is the excitationgap for a single soliton. This can be made arbitrarilysmall by increasing ξ .Some comments are in order. The domain wall stateis completely spatially uniform. It is also not a thermalstate. Since it can be expressed as a matrix product state,it clearly obeys the area law for entanglement entropy,unlike a thermal state. Hence it can be a plausible groundstate for some locally interacting quantum system. Infact, it is natural to regard the state as an approxima-tion to the ground state in the vicinity of the quantumcritical point of the Ising chain, on the quantum param-agnetic side. It is a quasi-condensate of the solitons, if weview the latter as particles. Note of course it would be acrude approximation to the actual ground state near thecritical point of the chain, and does not capture the uni-versal physics of the quantum critical regime. It does,however, roughly capture the universal physics of thequantum paramagnetic phase. In this paper, we wish tostress only that in this way one constructs a wavefunctionwhich describes a symmetry unbroken state, which couldbe a good approximation to the ground state of somelocal Hamiltonian, and which approximates the groundstate energy of the ferromagnet arbitrarily well. Thus wehave shown, by contradiction, that the statement that agood variational energy density is a predictor for ferro-magnetism is false in one dimension.From this physical picture, it is clear that the samemust be true for all discrete symmetry breaking states inone dimension. The elementary topological excitationsfor all such states are pointlike solitons, and the prolif-eration of such solitons destroys the broken symmetryphase. Since they are local, these solitons have finite en-ergy, and so states of arbitrarily small energy density canbe constructed by making the solitons dilute.We return to the connection to thermal fluctuations. Itis well-known that one dimensional systems at non-zerotemperature T > not the case at T = 0. The presence of a finite en-ergy gap is sufficient to protect the Ising ordered phaseat T = 0. However, the expectation value of physicalobservables within a matrix product state can generi-cally be expressed as a local classical statistical mechanicsmodel in the same dimension, in which both the physicalspins and the auxiliary ones (matrix degrees of freedom)appear as fluctuating variables. In the physical ferro-magnetic ground state, the corresponding classical sta-tistical mechanics problem has a zero temperature char-acter. Thus is can sustain long-range order. However,when perturbed with non-zero ǫ , the fictitious statisticalmechanics problem develops a non-zero small fictitioustemperature. Consequently, the classical reasoning ap-plies to this fiduciary problem, and we may understandthe destruction of long-range order by non-zero ǫ by thecorresponding destruction at T >
3. Robustness of symmetry broken states in d ≥ We now consider the case of d ≥
2. In analogy tothe approach of Sec. II B 1, we consider a TNS or PEPSrepresentation of the ground state, which generalizes nat-urally the MPS to d >
2. For a spin system, a tensor isdefined for every site containing a spin, with one ten-sor index for each link connecting that site to anotherneighboring site. The link indices, which generalize thematrix indices in the MPS, comprise fiduciary variablesto be summed over in building the wavefunction: | ψ i = X { σ i } X abc ··· [ T abcd ( σ ) T aefg ( σ ) · · · ] | σ · · · σ N i . (19)Here a, b, c, · · · indicate the link indices, which aresummed over D values, defining the “inner dimension”of the TNS, and T abcd ( σ ) is the tensor at site 1 etc. (thechoice of 4 links per site was arbitrary). Each link indexoccurs in just two tensors, and the arrangement of linksdefines the network.It is straightforward to write down tensors correspond-ing to symmetry broken states. For example, the catstate of Eqs. (6) in the simplest case D = 2 is T ( ǫ ) abcd ( σ ) = q aσm a = b = c = d, O ( ǫ ) otherwise , (20)where we take the inner indices a = ±
1, and m > ǫ = 0, this is a perfect cat state,but one can include defects in the cat state (analogous tothe domain walls in 1d) by allowing non-zero off-diagonalentries, ǫ > arbitrary small per-turbations. This can be most simply understood fromthe mapping of TNS matrix elements to classical statis-tical mechanics. Specifically, the spin-spin expectationvalue C ij = h ψ | σ zi σ zj | ψ i / h ψ | ψ i , (21) which diagnoses long-range order ( C ij → C ∞ = 0 as | i − j | → ∞ ) can be expressed via Eq. (19) as C ij = P { σ i } P ab ··· a ′ b ′··· σ i σ j W [ { σ i } , abc . . . , a ′ b ′ c ′ · · · ] P { σ i } P ab ··· a ′ b ′··· W , (22)where W [ { σ i } , abc . . . , a ′ b ′ c ′ · · · ] = [ T abcd ( σ ) T aefg ( σ ) · · · ] × [ T a ′ b ′ c ′ d ′ ( σ ) T a ′ e ′ f ′ g ′ ( σ ) · · · ] (23)gives the weight for the classical statistical mechanicsproblem. It can be visualized as a “bilayer” network oftwo copies of the internal states defining the TNS (oneeach from the bra and the ket) and a single set of spinscorresponding to the physical states. Most importantly,the statistical weight is local and varies smoothly withthe TNS tensors.Hence, we can rely on the well-known fact that in clas-sical statistical mechanics with d ≥
2, a discrete sym-metry breaking state is a stable phase . Hence if a onepoint in parameter space exhibits symmetry breaking,for example the TNS which represents the actual groundstate of the physical hamiltonian H , then generically theneighborhood of this point also exhibits broken symme-try. Thus, unless the original TNS is finely tuned, arbi-trary small variations of the tensor preserve broken sym-metry. This shows that it is not possible to construct astate with low energy density by exploring small varia-tions of a TNS. Symmetry breaking is stable in this sensein d ≥ non-local : domain walls with dimension d −
1. A largedomain of linear size L has a weight which is modified locally on of order L d − tensors, and hence contributesa total weight of order ǫ L d − to the wavefunction, whichvanishes rapidly for large L . Hence large domain wallsare exponentially rare – too rare to modify the long-rangebehavior of correlation functions.This strongly suggests that discrete broken symmetryorder in d ≥ not a small variation of theTNS representation of the ground state, but which nev-ertheless has low energy density. We may imagine twopossibilites. First, there may be a TNS with a tensorwhich is not close to the ground state tensor, but whichnevertheless has low energy. This would appear to bean unlikely accident, and probably it is possible to arguethat it only occurs with fine-tuning: with a small modifi-cation to the Hamiltonian, an accidental degeneracy likethis can be split. The second possibility is that there maybe a state with low energy which does not have a TNSrepresentation.In fact, there definitely are low energy variationalstates which do not have the TNS form. For example, wecan consider a state with one-dimensional domain walls,inspired by the discussion in Secs. II B 1,II B 2.For con-creteness take a square lattice. We suppose the domainwalls are rigid and infinitely long in the y direction, andexist with probability amplitude ǫ on a given x value.Note that the latter assumption violates the locality ofthe TNS form, since the amplitude is not a product oflocal factors, which inevitably would be exponential inthe length. Nevertheless, one can still form a superposi-tion of such domain walls. Essentially this is the sameas taking the MPS state in Eq. (5) and replacing eachsingle spin σ i by an entire column of perfectly correlatedspins in the y direction at fixed x . When ǫ is small, theaverage distance between domain walls is of order 1 /ǫ ,so that the energy density is order ǫ .Clearly, however, this is not truly a disordered state.It possesses long-range correlations along the y direction.It also violates the locality assumptions of the TNS, andconsequently has other pathological properties. For ex-ample, the entanglement entropy of a rectangular regionof width x and height y scales only with x , a sub-area-lawbehavior. We may reject this state as unphysical, or inany case readily diagnose such a state in numerics.We may imagine a variant of this state in which domainwalls are introduced in both directions simultaneously,but still rigidly. In this case the two-spin correlationfunction will decay exponentially in both the x and y directions. However, the long-range order is not fullydestroyed. A four spin correlation function of the form C x,y = h σ , σ x, σ x,y σ ,y i (24)will still not decay even for large x , y . Hence this variantis not truly a disordered state.The above states with completely rigid domain wallsare clearly unsatisfactory. One may wonder, gener-ally, whether there is some less artifical way to write awavefunction which still realizes a state with fluctuatingflipped domains of spins, but weights them differentlythan in a TNS, so that a finite weight is achieved for ar-bitrarily large domains. We have not found a satisfactorygeneral answer to this question. States in which domainwalls are localized are certainly possible, and simply rep-resent additional symmetry breaking on top of the Isingorder. As argued in Sec. II A, this is always possibleat low energy cost. We suspect, without proof, that aweighting of arbitrarily large domains which restores allsymmetries and yields a state which could be a genericground state of some local Hamiltonian is not possible.In summary, the above arguments suggest that discretesymmetry breaking order is robust in d ≥
2, meaningthat when a Hamiltonian exhibits discrete broken sym-metry, any variational state which is a generic groundstate (i.e. away from critical points) of some local Hamil-tonian and which has this symmetry unbroken, must havea variational energy density which is strictly larger thanthat of the ground state.
III. TOPOLOGICAL ORDER IN TWODIMENSIONS
In this section, we consider the variational determi-nation of topological order, in the same sense as dis-cussed above for discrete symmetry breaking order. Weuse topological order in the sense of Wen, to describea fully gapped ground phase of matter at zero tempera-ture, which exhibit ground state degeneracies in the ther-modynamic limit on closed surfaces of non-trivial genus,and for which states within the degenerate subspace areindistinguishable by any local operator. The essentialphysics of topological order is the existence of emergentanyonic excitations, with mutual statistics that cannot beobtained from any finite number of electrons. Topologi-cal order exists only in dimensions greater than or equalto 2. Even ignoring symmetry completely, the concept oftopological order divides phases of matter into distincttopological classes. A topological phase is locally stableto any local perturbation.
A. Variational robustness of topological phases tostates with ”less” topological order
We first consider whether a topological phase is varia-tionally robust to states with lower (or none) topologicalorder. Despite the stability of the actual ground statesto arbitrarily perturbations, we show that Hamiltoniansfor topological phases in two dimensions behave varia-tionally like Hamiltonians for symmetry breaking orderin one dimension. That is, a wavefunction be found in atopological class different (lower, in a sense to be definedlater) from that of the ground state, but with a varia-tional energy density that can be made arbitrarily closeto the exact value.Consider first the simplest example of a topologicalphase: the Z or toric code phase. It is exemplified bythe model of Kitaev, which is exactly soluble: H K = − X p Y i ∈ p σ zi − X s Y i ∈ s σ xi , (25)where the spins live on the bonds i of a 2d lattice, p in-dicates plaquettes and s indicates “stars”, i.e. the setof bonds emanating from a given site. The ground statecan be considered as a uniform sum of all configurationsin the σ xi basis which satisfy the constraint that an evennumber of σ xi = − S T EE = ln 2, as defined in Refs.11 and 12.To get more physical insight, one may represent theconfigurations by coloring the links with σ xi = −
1. Thenthere must be an even number of colored links at everyvertex. This constraint corresponds to the existence of aconserved Z “electric” flux, which can be defined as thenumber of colored links crossing a curve drawn on linksof the dual lattice. This flux can be odd or even, and iszero in the ground state for any contractible closed loop.Excited states may have a non-zero flux, which is dueto an electric charge or e particle inside the loop (whichhave Q i ∈ s σ xi = − s inside the loop). The e particles are topological excitations of the Z phase.There are also “magnetic” m particles, which correspondto defects in which the product of σ zi around a closedloop is equal to − Q i ∈ p σ zi = − e and m particles arebosons, but if one considers “mutual statistics” together,they are relative semions: i.e. adiabatically transportingan e particle around an m particle incurs a change ofthe phase of the state by π . From these two particlesone may also construct a composite e-m particle, whichis a fermion. The e , m , and e-m are the fundamentalanyonic excitations of the Z topological phase, and theirexistence may be regarded as the defining characteristicof the state.The toric code phase is stable to all perturbations. Forexample, the model on the square lattice has been exten-sively studied, and the topological phase has been shownto persist under the generic perturbations H ′ = X i h z σ zi + h x σ xi (26)until the applied fields h x , h z are of order one (beyondwhich a quantum phase transition occurs to a topologi-cally trivial phase). However, we may still ask whether itis possible to obtain a good approximation to the groundstate energy density well within the topological phase bya topologically trivial wavefunction?Following the logic of the previous section, we expectthat the ground state within the Z phase can be approx-imated to arbitrary accuracy by a tensor network state,or Projected Entangled Pair State (PEPS), which is thenatural generalization of an MPS to higher dimensions.Such a representation can be written explicitly for theexactly soluble Kitaev limit. As for an MPS, any PEPSis the ground state of some local Hamiltonian (hence canbe regarded as physical, and for example obeys the arealaw of entanglement entropy), and the variational energy density of a PEPS is a smooth function of the tensor com-ponents. Hence if we can find an infinitesimal change ofthe tensor for a Z state which destroys the topologicalorder, we have found a variational state which satisfiesthe requirements. In fact, such tensors have already beenfound by Chen et al in Ref.13. We do not repeat the de-tails here as they are given very explicitly in the formerpaper. Therefore it is indeeed possible to write a varia-tional state which lacks topological order but has arbi-trarily low energy density for the Z topological phase.The mechanism behind the low energy variationalwavefunctions is very similar to that discussed in the pre-vious section. A suitable deformation of the tensor defin-ing the PEPS state introduces a low density (of order ǫ for an O ( ǫ ) deformation of the tensor) free ends into the ground state wavefunction. Importantly, there are nolong-range correlations between these ends in the varia-tional state. The presence of free ends removes the fluxconservation on long distances and, as shown in Ref., makes the topological entanglement entropy vanish.Physically, the free ends are electric e particles, andthe low density of these particles in the ground state cor-responds to a condensate of electric charge. We may thenask whether the quasiparticles of the toric code persistinto this phase. The e particle no longer exists as a well-defined quasiparticle, like in any condensate, for whichthe particle number of the condensed particle becomesan uncertain variable. The m and e-m particles, more-over, become confined , as their phase is scrambled by thedelocalized background condensate of e particles. Hencenone of the non-trivial anyons persist in the e conden-sate state, so it is topologically trivial, consistent withvanishing topological entanglement entropy.The arbitrarily low energy density of the e condensatestate can be understood simply by the fact that the e par-ticle, being local, has a non-zero but finite energy, andso the energy density of a condensate can be renderedarbitrarily small but rendering the density of the e par-ticles low. This is directly analogous to the low densitysoliton state which destroys symmetry breaking in onedimension.From this reasoning, we can immediately conclude thatthis is a general statement for topological phases in twodimensions: a state with “less” topological order can al-ways be constructed by forming a low density conden-sation of an anyonic excitation of the topological phase.The definition of “less” is in fact just those states whichcan be arrived at by anyon condensation. In general itmay be possible for more than one such state to be con-structed, if several bosonic anyons are available for con-densation.
1. Half-integer spins
An interesting special case of the above discussion isthe situation in which the system contains a half-integerspin per unit cell. Then according to the generalizedLieb-Schulz-Mattis (gLSM) theorem, it is generally notpossible to form a topological trivial state without brokensymmetry or gapless excitations.
Suppose we havesuch a Hamiltonian whose ground state is a topologicalphase with no broken symmetry – a topological spin liq-uid. The arguments of this subsection still apply, so vari-ational wavefunctions can be constructed without topo-logical order and arbitrarily good energy density. How-ever, the gLSM argument implies that these variationalstates must either represent gapless phases or exhibit bro-ken symmetry.Broken symmetry can arise very naturally in the statesconstructed by anyon condensation. This occurs simplyif the anyon which condenses carry symmetry quantumnumbers. For example, in Z gauge theories of spin-1/2spin liquids, the vison (or m particle) carries space groupquantum numbers, transforming under some projectiverepresentation of the symmetry group of the lattice. Thevison condensation then leads directly to valence bondorder. In general we expect that low energy densityvariational states that are topologically trivial and pos-sess symmetry breaking order exist even when the trueground state is a featureless and topological spin liquidstate. B. Robustness of trivial states to topological order
The converse question to the previous subsection iswhether a ground state in a topologically trivial phasecan be approximated well by one with topological order?More generally we can ask whether a phase with a giventopology can be approximated by one with “more” topo-logical order – but we will not attempt to answer thishere. The analogous question in the case of symmetrybreaking order was answered in Sec. II A with a resound-ing yes.However, an attempt to follow the argument ofSec. II A immediately runs into difficulty. There, weshowed how to construct a variational state by introduc-ing a field coupling to the order parameter of the sym-metry breaking. However, for topological order, there isno local order parameter. This construction fails immedi-ately. In general, since we expect that classes of topologi-cal order (including the trivial one) are absolutely stable,there can be no perturbation which defines a fiduciaryHamiltonian with topological order, when the groundstate is trivial. In passing, we note that if the originalground state is critical (i.e. H is gapless), then pertur-bations that produce topological order may be possible.However, gapless states are beyond the considerations ofthis paper.We may turn to the TNS construction for guidance. Asshown in Ref., tensors for states with Z topological or-der must satisfy a symmetry requirement. Any violation,however small, of this requirement destroys the topolog-ical order of the state. Clearly, a generic state without Z topological order has an asymmetric tensor. As a fi-nite object, there are no order of limits questions withrespect to the tensor. It cannot be perturbed infinitesi-mally to restore its symmetry. Hence at least within thisconstruction it appears impossible to approximate a Z topological phase by a trivial one.A physical picture which confirms this notion comesfrom the string-net scheme of Levin and Wen. Theyshowed that topological phases, viewed approaching froma trivial state, arise by the proliferation/condensation of infinitely long extended strings or string-nets. Clearly, ifthe ground state of our Hamiltonian A is in the trivialphase, the strings must be finite there, i.e. there is a non-zero string tension, or energy per length of string. Anyvariational state with infinitely long strings must pay thisstring tension, which is a volume energy since the strings are dense in the topological phase. The situation seemsanalogous to the destruction of symmetry breaking order,which requires condensation of domain walls of dimension d − > d ≥
2. Hence we expect that topologicallytrivial phases are variationally robust to topological ones.
IV. DISCUSSION
We have discussed the question of variational robust-ness of certain phases against others: can a Hamiltonianwith a ground state in phase A be approximated by aphysical wavefunction in another phase B, with arbitrar-ily small energy density? In fact, in surprisingly manycases, the answer is yes. Two cases in which it is notobviously possible are (1) A is a symmetry broken stateand the symmetry is unbroken in B, in dimensions twoand larger, and (2) A is a trivial state and B has topolog-ical order. In case (1), the restriction that the symmetricstate B be physical is non-trivial and crucial.The approach of this paper has been to construct ex-amples of principle, to show that in many cases an ap-proximation is possible, i.e. that phase A is not vari-ationally robust to phase B. The examples were basedon a simple paradigm of defect proliferation. This doesnot mean this is the only way in which a low energyvariational state may manifest. Rather, we intend theconstruction here as a proof of principle, to show that inthese cases the finding of a low energy density variationalstate does not necessarily mean the ground state phasehas been properly identified.Probably the major context in which these results maybe relevant is the study of spin liquid phases of quantummagnets, where variational methods are a dominant ap-proach. The fact that topological spin liquids are notvariationally robust to trivial states suggests that vari-ational methods tend to overestimate the dominance oftopologically trivial but ordered phases. Conversely, thevariational robustness of trivial phases to topological or-der means that a good variational state which has topo-logical order is a strong argument for topological order inthe ground state. It is important to note that the argu-ments expressed in this paper presuppose the existenceof an excitation gap, and this must be checked indepen-dently in a numerical calculation.In practice, there are many means beyond just inspect-ing the energy to evaluate variational states. Many ofthe low energy variational states constructed in this pa-per have emergent long length scales, which could be de-tected by various measurements. Correct properties ofthe ground state might be obtained from a careful studyof the wavefunction on shorter scales.One may envision pursuing these ideas further. Quitelikely some of the statements in this paper could be mademathematically rigorous. The yes/no question of varia-tional robustness formulated here is only crudest type ofstatement one might make about variational states. Itwould be highly desirable to be more quantitative aboutvarious physical properties. For example, one would liketo know how much correlation functions or reduced den-sity matrices over regions of a given size can differ be-tween the ground and variational states, given a partic- ular difference in energy density of the two states. Acknowledgements.—
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