PHYSICAL REVIEW E 88, 032110 (2013)

Quantum fidelity for degenerate ground states in quantum phase transitions

Yao Heng SuCentre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, China and

School of Science, Xi’an Polytechnic University, Xi’an 710048, China

Bing-Quan Hu, Sheng-Hao Li, and Sam Young Cho*

Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, China(Received 30 December 2012; revised manuscript received 1 July 2013; published 6 September 2013)

Spontaneous symmetry breaking in quantum phase transitions leads to a system having degenerate groundstates in its broken-symmetry phase. In order to detect all possible degenerate ground states for a broken-symmetryphase, we introduce a quantum fidelity defined as an overlap measurement between a system ground state andan arbitrary reference state. If a system has N -fold degenerate ground states in a broken-symmetry phase, thequantum fidelity is shown to have N different values with respect to an arbitrarily chosen reference state. Thequantum fidelity then exhibits an N -multiple bifurcation as an indicator of a quantum phase transition withoutknowing any detailed broken symmetry between a broken-symmetry phase and a symmetry phase as a systemparameter crosses its critical value (i.e., a multiple bifurcation point). Each order parameter, characterizing abroken-symmetry phase from each degenerate ground state reveals an N -multiple bifurcation. Furthermore,it is shown that it is possible to specify how each order parameter calculated from degenerate ground statestransforms under a subgroup of a symmetry group of the Hamiltonian. Examples are given through study of thequantum q-state Potts models with a transverse magnetic field by employing tensor network algorithms based oninfinite-size lattices. For any q, a general relation between the local order parameters is found to clearly show thesubgroup of the Zq symmetry group. In addition, we systematically discuss criticality in the q-state Potts model.

DOI: 10.1103/PhysRevE.88.032110 PACS number(s): 05.30.Rt, 03.67.−a, 05.50.+q, 75.40.Cx

I. INTRODUCTION

Quantum phase transitions (QPTs) have attracted substan-tial attention in connection with quantum information [1,2].Compared to local order parameters in the conventionalLandau-Ginzburg-Wilson paradigm based on the spontaneoussymmetry-breaking mechanism, quantum entanglement, i.e., apurely quantum correlation being absent in classical systems,can be used as an indicator of QPTs driven by quantumfluctuations in quantum many-body systems [3]. Quantumfidelity, based on the basic notions of quantum mechanicson quantum measurement, has also provided an another wayto characterize QPTs [4–7]. In the past few years, variousquantum fidelity approaches such as fidelity per lattice site(FLS) [4], reduced fidelity [8], fidelity susceptibility [9],density-functional fidelity [9], and operator fidelity [10], havebeen suggested and implemented to explore QPTs.

Fidelity is a measure of similarity between two quantumstates by defining an overlap function between them. The factthat ground states in different phases should be orthogonal dueto their distinguishability in the thermodynamic limit allows afidelity between quantum many-body states in different phasessignaling QPTs, because an abrupt change of the fidelityis expected across a critical point in the thermodynamiclimit [4–10]. Thus, the fidelity has great advantages tocharacterize QPTs in a variety of quantum lattice systems.This results because the ground state of a system undergoes adrastic change in its structure at a critical point, regardless ofwhat type of internal orders are present in quantum many-bodystates. The ground-state FLS especially has been demonstrated

*sycho@cqu.edu.cn

to capture drastic changes in the ground-state wave functionsaround a critical point, even for those which cannot be de-scribed in the framework of Landau-Ginzburg-Wilson theorysuch as a Beresinskii-Kosterlitz-Thouless (BKT) transtion andtopological QPTs in quantum lattice many-body systems [11].

Even though these latest advances in understanding QPTshave been achieved, directly understanding degenerate groundstates originating from spontaneous symmetry breaking asthe heart of the Landau-Ginzburg-Wilson theory still remainslargely unexplored. Recent development of tensor network(TN) algorithms [12–15] in quantum lattice systems have madeit possible to directly investigate degenerate ground stateswith a randomly chosen initial state subject to an imaginarytime evolution. Based on the tensor network algorithm, Zhaoet al. [16] detected doubly degenerate ground states by meansof the FLS bifurcations in the quantum Ising model andspin-1/2 XYX model with transverse magnetic field. Also,in various other spin-lattice models [17], doubly degenerateground states have been detected with the FLS bifurcations. Inthe quantum three-state Potts model [18], the existence of thethreefold degenerate ground state has been inferred from themass distribution function with a bifurcation of FLS. However,the three ground states were not explicitly distinguished fromthe bifurcation of FLS. It is not still clear how to explicitlyidentify all degenerate ground states when a system has morethan threefold degenerate ground states in its broken phase.

In this paper, we investigate how to explicitly detect N -folddegenerate ground states calculated from a tensor networkalgorithm. When a system has more than twofold ground-statedegeneracy, the degenerate ground states are orthogonal toone another in the thermodynamic limit. A quantum fidelitydefined by an overlap function between degenerate groundstates then may not distinguish all the degenerate ground states

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SU, HU, LI, AND CHO PHYSICAL REVIEW E 88, 032110 (2013)

properly. In order to detect more than threefold degenerateground states, in contrast to the quantum fidelity defined asthe overlap function between ground-state wave functions inRef. [16], we introduce a quantum fidelity between degenerateground states and an arbitrary reference state. In fact, ourquantum fidelity corresponds to a projection of each groundstate onto the chosen reference state. Straightforwardly, fora broken-symmetry phase, the number of different projectionmagnitudes denotes the ground-state degeneracy. At a criticalpoint, the different projection magnitudes collapse to oneprojection magnitude. The critical point corresponds to thecollapsed point of the quantum fidelities, which can be called amultiple bifurcation point. With such a property of the quantumfidelity, the different projection magnitudes of the groundstates starting from the collapsed point can be described asa multiple bifurcation of the quantum fidelity.

As a prototypical example, we explore the ground-statewave functions in the q-state Potts model [19,20] in transversemagnetic fields. By employing the infinite matrix product state(iMPS) with the time-evolving block decimation (iTEBD)method [14], we calculate the ground states of the model.Due to the broken Zq symmetry, the q-fold degenerate groundstates in the broken-symmetry phase are obtained by meansof the quantum fidelity with q branches. A continuous (dis-continuous) QPT for q � 4 (q > 4) has been manifested by acontinuous (discontinuous) fidelity function across the criticalpoint. The multiple bifurcation points are shown to correspondto the critical points. Also, we discuss a multiple bifurcation oflocal order parameters and its characteristic properties for thebroken-symmetry phase. We find a general relation betweenthe order parameters from each of the degenerate ground states.The general relation shows clearly how the order parametersfrom each of the degenerate ground states transform under thesubgroup of the symmetry group Zq . In addition, for q = 4,we calculate the critical exponents, which agree well with theirexact values. From the finite-entanglement scalings of the vonNeumann entropy and the correlation length, the cental chargesare calculated to classify the universal classes for each q-statePotts model.

This paper is organized as follows. In Sec. II, we brieflyexplain the iMPS representation and the iTEBD method inone-dimensional quantum lattice systems. In Sec. III, theq-state Potts model is introduced. Section IV discusses how todetect a degenerate ground state by using a quantum fidelitybetween the degenerate ground states and a reference state. Byusing the quantum fidelity per lattice site, the quantum phasetransitions are discussed based on its multiple bifurcationsand multiple bifurcation points indicating quantum criticalpoints in Sec. V. In Sec. VI, to complete the description of theordered phases, we discuss the magnetizations given from thedegenerate ground states and obtain a general relation betweenthem. Section VII presents the critical exponents for q = 4 andthe central charges for q = 3 and q = 4. We also discuss thequantum phase transitions from the von Neumann entropies.Finally, our summary is given in Sec. VIII.

II. NUMERICAL METHOD: IMPS APPROACH

Recently, significant progress has been made in numericalstudies based on TN representations for the investigation of

quantum phase transitions [12–14,21], which offers a newperspective from quantum entanglement and fidelity, thusproviding a deeper understanding on characterizing criticalphenomena in finite and infinite spin-lattice systems. Actually,a wave function represented in TNs allows us to performthe classical simulation of quantum many-body systems. Inone-dimensional spin systems especially, a wave function forinfinite-size lattices can be described by use of the iMPSrepresentation [14], which has been successfully applied toinvestigate the properties of ground-state wave functions invarious infinite spin-lattice systems [8,11].

For an infinite one-dimensional lattice system, a state canbe written as [14]

|�〉 =∑{S}

∑{α}

· · · λ[i]αi

�[i]αi ,si ,αi+1

λ[i+1]αi+1

�[i+1]αi+1,si+1,αi+2

× λ[i+2]αi+2

· · · | · · · Si−1SiSi+1 · · ·〉, (1)

where |Si〉 denotes a basis of the local Hilbert space atthe site i, the elements of a diagonal matrix λ[i]

αiare the

Schmidt decomposition coefficients of the bipartition betweenthe semi-infinite chains L(−∞, . . . ,i) and R(i + 1, . . . ,∞),and �

[i]αi ,Si ,αi+1

are a three-index tensor. The physical indices Si

take the value 1, . . . ,d with the local Hilbert space dimensiond at the site i. The bond indices αi take the value 1, . . . ,χ

with the truncation dimension of the local Hilbert space atthe site i. The bond indices connect the tensors � in thenearest-neighbor sites. Such a representation in Eq. (1) iscalled the iMPS representation. If a system Hamiltonian hasa translational invariance, one can introduce a translationalinvariant iMPS representation for a state. Practically, forinstance, for a two-site translational invariance, the state canbe reexpressed in terms of only the three-index tensors �A(B)

and the two diagonal matrices λA(B) for the even (odd) sites,where {�,λ} are in the canonical form, i.e.,

|�〉 =∑{S}

∑{l,r}

· · · λA�AλB�BλA · · · | · · · Si−1SiSi+1 · · ·〉,

(2)

where l and r are the left and right bond indices, respectively.Once a random initial state |�(0)〉 is prepared in the

iMPS representation, one may employ the iTEBD algorithmto calculate a ground-state wave function numerically. Forinstance, if a system Hamiltonian is translational invariant andthe interaction between spins consists of the nearest-neighborinteractions, i.e., the Hamiltonian can be expressed by H =∑

i h[i,i+1], where h[i,i+1] is the nearest-neighbor two-body

Hamiltonian density, a ground-state wave function of thesystem can be expressed in the form in Eq. (2). The imaginarytime evolution of the prepared initial state |�(0)〉, i.e.,

|�(τ )〉 = exp[−Hτ ]|�(0)〉|| exp[−Hτ ]|�(0)〉|| , (3)

leads to a ground state of the system for a large-enough τ .By using the Suzuki-Trotter decomposition [22], actually, theimaginary time evolution operator U = exp[−Hτ ] can bereduced to a product of two-site evolution operators U (i,i + 1)that only acts on two successive sites i and i + 1. For thenumerical imaginary time evolution operation, the continuoustime evolution can be approximately realized by a sequence of

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the time slice evolution gates U (i,i + 1) = exp[−h[i,i+1]δτ ]for the imaginary time slice δτ = τ/n � 1. A time-sliceevolution gate operation contracts �A, �B , one λA, two λB ,and the evolution operator U (i,i + 1) = exp[−h[i,i+1]δτ ]. Inorder to recover the evolved state in the iMPS representation,a singular value decomposition (SVD) is performed and the χ

largest singular values are obtained. From the SVD, the newtensors �A, �B , and λA are generated. The latter is used toupdate the tensors λA as the new one for all other sites. Similarcontraction on the new tensors �A, �B , two new λA, one λB ,and the evolution operator U (i + 1,i + 2) = exp[−h[i+1,i+2]],and its SVD produce the updated �A, �B , and λB for all othersites. After the time-slice evolution, then, all the tensors �A,�B , λA, and λB are updated. This procedure is repeatedlyperformed until the system energy converges to a ground-stateenergy that yields a ground-state wave function in the iMPSrepresentation. The normalization of the ground-state wavefunction is guaranteed by requiring the norm 〈�|�〉 = 1.Finally, one can determine how many ground states exist fora fixed system parameters with different initial states. Once asystem undergoes a spontaneous symmetry breaking, the iMPSalgorithm can automatically produce degenerate ground stateswith randomly chosen initial states for the broken-symmetryphase. It has been manifested by successfully detecting doublydegenerate ground states from Z2 broken-symmetry phases invarious spin systems such as quantum Ising model, spin-1/2XYX with transverse magnetic field, and a one-dimensionalspin model with competing two-spin and three-spin interac-tions [16,17]. However, it has not been discussed yet how todetect an N -fold degenerate ground state. In the followingsections, we discuss this issue clearly by introducing theq-state Potts model that may have a q-fold degenerate groundstate in the broken-symmetry phases.

III. QUANTUM q-STATE POTTS MODEL

In the lattice statistical mechanics, the quantum q-statePotts model is one of the most important models as ageneralization of the Ising model (q = 2) [20,23]. The q-statePotts model has the very intriguing critical behavior that hasbecome an important testing platform for different numericaland analytical methods and approaches in studying criticalphenomena [20,24–28]. As is well-known in Ref. [20], thequantum q-state Potts model exhibits a continuous quantumphase transition for q � 4 and a first-order (discontinuous)phase transition for q > 4 at the critical point.

We consider the q-state quantum Potts model in a transversemagnetic field λ on an infinite-size lattice,

Hq = −∞∑

j=1

q−1∑p=1

(Mx, p(j )Mx, q−p(j + 1) + λMz(j )), (4)

where λ is the transverse magnetic field and Mx/z,p(i) withp ∈ [1,q − 1] are the q-state Potts spin matrices at the latticesite j . The q-state Potts spin matrices are given as

Mx,1 =(

0 Iq−1

1 0

)and Mz =

(q − 1 0

0 −Iq−1

),

where Iq−1 is the (q − 1) × (q − 1) identity matrix andMx,p = (Mx,1)p. As is known, the model Hamiltonian has

a Zq symmetry [20]. If the ground state of the Hamiltoniandoes not preserve the Zq symmetry, the system undergoesZq symmetry breaking, i.e., a QPT. Specifically, when themagnetic field changes across λc = 1, the q-state Potts modelundergoes a QPT between an ordered phase and a disorderedphase. The QPT is originated from the Zq symmetry breaking,which results in the emergence of long-range order and the q

degenerate ground states in the ordered phase.

IV. DEGENERATE GROUND STATESAND QUANTUM FIDELITY

From a tensor network approach with an infinite latticesystem, once one obtains a ground state |ψ (n)〉 with then-th random initial state, one can define a quantum fidelityF (|ψ (n)〉,|φ〉) = |〈ψ (n)|φ〉| between the ground state and achosen reference state |φ〉. If F (|ψ (n)〉,|φ〉) has only oneconstant value with random initial states, the system has onlyone ground state. If F (|ψ (n)〉,|φ〉) has N projection valueswith random initial states, the system must have N degenerateground states. To distinguish the degenerate ground states, weemploy the ground-state FLS in Ref. [4] as

ln d(|ψ (n)〉,|φ〉) ≡ limL→∞

ln F (|ψ (n)〉,|φ〉)L

, (5)

where L is the system size. The FLS is well defined in thethermodynamic limit, even if F becomes trivially zero. Fromthe fidelity F , the FLS has several properties as (i) d(|ψ (n)〉 =|φ〉) = 1 and (ii) its range 0 � d(|ψ (n)〉,|φ〉) � 1. Within theiMPS approach, the FLS d(|ψ (n)〉,|φ〉) is given by the largesteigenvalue μ0 of the transfer matrix T up to the correctionsthat decay exponentially in the linear system size L. Then,for the infinite-size system, d(|ψ (n)〉,|φ〉) = μ

(n)0 . Actually, in

order to study quantum critical phenomena in quantum latticesystems, Zhou and Barjaktarevic defined the FLS from thefidelity between ground states [4,15]. The Zhou-BarjaktarevicFLS has been manifested as a model-independent and universalindicator that successfully detects quantum phase transitionpoints, including the KT transition and the topological phasetransition [11]. The FLS introduced in Eq. (5) for this study isa simple extension of the Zhou-Barjaktarevic FLS. However,the FLS in Eq. (5) allows us to detect all possible degenerateground states for a broken-symmetry phase.

In order to show clearly how to detect degenerate groundstates based on Eq. (5), as an example, we consider the four-state Potts model. In Fig. 1, we plot the FLS d for random initialstates. Here, λ = 0.8 and 1.2 are chosen. For the quantumfidelity, the numerical arbitrary reference state |φ〉 is chosenrandomly.

Due to the broken Z4 symmetry for q = 4, the systemhas four degenerate ground states for the broken-symmetryphase λ < λc = 1. Figure 1 shows clearly that, for λ = 0.8(λ < λc), there exist four different values of the FLS, while,for λ = 1.2 (λ > λc), there exists only one value of the FLS.From each value of the FLS for λ = 0.8, we label the fourdegenerate ground states as |ψ1〉, |ψ2〉, |ψ3〉, and |ψ4〉. Forλ = 1.2, the ground state is denoted by |ϕ〉. Actually, we havechosen more than 200 random initial states. The probabilityP (n) that the system is in each ground state for λ < λc isfound to be P (n) 1/4 in the broken-symmetry phase. Then,

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FIG. 1. (Color online) Ground-state quantum FLS d for the four-state Potts model with an arbitrary reference state as a function ofrandom initial state trials. Here |φ〉 is an arbitrary reference state, thenumerical ground states |ψ〉 are in the broken-symmetry phase withλ = 0.8, and |ϕ〉 is in the symmetry phase with λ = 1.2, respectively.It is clearly shown that there exist four degenerate ground states(black rectangle) which are labeled by |ψ1〉, |ψ2〉, |ψ3〉, and |ψ4〉.In the broken-symmetry phase for λ = 1.2, only one ground state(red circle) exists. The dotted lines are a guided for the eyes. If thenumber of random initial state trials increases, the probability thatthe system is in each degenerate ground state approaches 1/4 in thebroken-symmetry phase for the four-state Potts model.

for the q-state Potts model, with a large number of randominitial state trials, one may detect the q degenerate groundstates with the probability P (n → ∞) = 1/q, finding eachdegenerate ground state in the broken Zq symmetry phase.Consequently, all degenerate states for a broken-symmetryphase can be detected by using the quantum fidelity with anarbitrary reference state in Eq. (5).

As we discussed, one may choose the reference state as oneof the degenerate ground states. However, the reference statechosen from the degenerate ground states cannot distinguishall other ground states. In order to show this clearly, in Fig. 2,we plot the quantum fidelity with the reference state fromone of the degenerate ground states. In Fig. 2, the referencestate of the quantum fidelity is chosen as one of the groundstates for the broken-symmetry phase in (a) |�1〉, (b) |�2〉,(c) |�3〉, and (d) |�4〉, and as the ground state |ϕ〉 for thesymmetry phase in (e). In Figs. 2(a)–2(d), it is clearly shownthat, for λ = 0.8 (λ < λc), there exist only two different valuesof the FLS, i.e., d(|ψm〉,|ψm〉) = 1 and d(|ψm〉,|ψm=m′ 〉) =const = 1(m,m′ ∈ [1,q]) for the broken-symmetry phase. Thisresult implies that, if one of the ground states for thebroken-symmetry phase is chosen as the reference state, thequantum fidelity cannot distinguish other degenerate groundstates for the broken-symmetry phase. For the symmetryphase, any reference state chosen in the ground states ofthe broken-symmetry phase gives d(|ψm〉,|ϕ〉) = const. In thiscase, as discussed in Ref. [18], one may use a probability massdistribution function to infer how many degenerate groundstates are in the broken-symmetry phase. Figure 2(e) showsclearly that the reference state chosen as the ground state inthe symmetry phase cannot distinguish the degenerate groundstates for the broken-symmetry phase, i.e., d(|ϕ〉,|ϕ〉) = 1 andd(|ψm〉,|ϕ〉) = const = 1. As a result, it is shown that thedegenerate states can be distinguished by a reference state

FIG. 2. (Color online) Ground-state quantum FLS d for the four-state Potts model as a function of random initial state trials. Thereference states are chosen in (a) |�1〉, (b) |�2〉, (c) |�3〉, (d) |�4〉,and (e) |ϕ〉. It is clearly shown that, if the reference state is chosen asone of the degenerate ground states in the broken-symmetry phase,the other three ground states has the same value of the FLS in (a)–(d).If the reference state is chosen as the ground state in the symmetryphase, the FLSs for the four degenerate ground states has the samevalue in the broken-symmetry phase in (e).

chosen as arbitrary states except for the degenerate groundstates in the broken-symmetry phase and the ground state inthe symmetry phase.

V. QUANTUM FIDELITY PER LATTICE SITEFOR PHASE TRANSITIONS

In the view of a ground-state degeneracy, the degenerateground states in the broken-symmetry phase exist untilthe system reaches its critical point. Once one can detectdegenerate ground states, one may also detect a quantumphase transition by the quantum fidelity in Eq. (5). Detecteddegenerate ground-state wave functions may allow us also toinvestigate directly a property of quantum phases even forunexplored broken-symmetry phases. In this section, then, wewill discuss the quantum phase transitions for the q-state Pottsmodel. In the next section, we will discuss the relation betweenthe system symmetry and the order parameter directly from theground states.

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FIG. 3. (Color online) Ground-state quantum FLS d for (a) three-,(b) four-, and (c) five-state Potts models as a function of the transversemagnetic field λ with the truncation dimension χ = 32. In the broken-symmetry phase, the q branches of the FLS correspond to the numbersof the q degenerate ground states. As the magnetic field crossesthe critical point λc, the FLSs show the multiple bifurcations withthree, four, and five branches, respectively, in the broken-symmetryphase. Note that, for the three- and four-state Potts models, theFLSs show their continuous behaviors, while, for the five-state Pottsmodel, the FLS shows a discontinuous behavior. These continuousand discontinuous behaviors indicate the continuous phase transitionsfor q = 3 and 4, and the discontinuous phase transition for q = 5. Thenumerical multiple bifurcations are given as λc = 1.0004 for q = 3and 4 and λc = 1 for q = 5.

From the detected degenerate ground states, in Fig. 3, weplot the FLSs as a function of the transverse magnetic field λ

for q = 3, 4, and 5. Here, the truncation dimension is chosen asχ = 32. Figure 3 shows clearly that, as the transverse magneticfield decreases, the FLSs in the symmetry phase branch offthe q FLSs. The branch points are estimated numerically asλc = 1.0004 for the three- and four-state Potts models. Also,for the five-state Potts model, the multiple bifurcation pointexists exactly at λc = 1. These results show that the branchpoints agree well the exact critical point λc = 1. The branchpoints then correspond to the critical point. Such branchingbehavior of the FLS can be called multiple bifurcation and abranch point can be called the multiple bifurcation point. In

fact, the numerical reference states were chosen randomlyfor each q. Any randomly chosen reference state, exceptfor the system ground states, gives the same number of theground states and the critical point even though the valuesof the quantum fidelity differ. Consequently, it is shownthat the FLS from the quantum fidelity between degenerateground states and a reference state can detect a criticalpoint.

As is known, for the q-state Potts model, the continuous(discontinuous) phase transitions occurs for q � 4 (q > 4).Figure 3(a) for q = 3 and 3(b) q = 4 show that the FLSsare a continuous function for the quantum phase transition.While Fig. 3(c) q = 5 shows that the FLS is a discontinuousfunction for the quantum phase transition. The continuous(discontinuous) behaviors at the critical points then implythat a continuous (discontinuous) phase transition occurs.As a result, the FLS in Eq. (5) can clarify continuous(discontinuous) quantum phase transitions.

VI. SYMMETRY AND ORDER PARAMETER

The existence of a degenerate ground state means thateach of the degenerate ground states possesses its own orderdescribed by each corresponding order parameter. Each orderparameter, which is nonzero value only in an ordered phase,should distinguish an ordered phase from a disordered phase[29]. To complete the description of an ordered phase, it isnecessary to specify how each order parameter from each of thedegenerate ground states transforms under a symmetry groupG that is possessed by the Hamiltonian because each orderparameter is invariant under only a subgroup of the symmetrygroup G although the Hamiltonian remains invariant underthe full symmetry group G [2]. In this section, we then discusslocal magnetizations obtained from each of the degenerateground states.

Let us first discuss the local magnetizations for the three-state Potts model, i.e., q = 3. In Fig. 4, we plot the magnetiza-tions [Fig. 4(a)] 〈Mx,1〉 and [Fig. 4(b)] 〈Mx,2〉 as a function ofthe traverse magnetic field λ. The magnetizations disappear tozero at the numerical critical point λc = 1.0004. Moreover,all the magnetizations show that the phase transition is acontinuous phase transition. For the broken-symmetry phaseλ < λc, each magnetization is calculated from each of the threedegenerate ground states, which is denoted by |ψm〉. Note thatall the absolute values of the magnetizations 〈ψm|Mx,p|ψm〉 ≡〈Mx,p〉m are the same values at a given magnetic field. Fur-thermore, for a given magnetic field, the magnetizations in thecomplex magnetization plane seem to have a relation betweenthem under a rotation, which is characterized by the valueω3 = exp[2πi/3]. Then, in Figs. 4(a) and 4(b), it is observedthat, for a given magnetic field λ < λc, there are the rela-tions between the magnetizations as 〈Mx,1〉1 = ω−1

3 〈Mx,1〉2 =ω−2

3 〈Mx,1〉3 and 〈Mx,2〉1 = ω−23 〈Mx,2〉2 = ω−4

3 〈Mx,2〉3. More-over, for each ground-state wave function, the magnetizationshave the relations 〈Mx,1〉1 = 〈Mx,2〉1, 〈Mx,1〉2 = ω−1

3 〈Mx,2〉2,and 〈Mx,1〉3 = ω−2

3 〈Mx,2〉3. These results show that, in thecomplex magnetization plane, the rotations between themagnetizations for a given magnetic field are determined bythe characteristic rotation angles θ = 0, 2π/3, and 4π/3, i.e.,〈Mx,p〉m = g3〈Mx,p′ 〉m′ with g3 ∈ {I,ω3,ω

23}. Here, we have

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(a)

(b)

FIG. 4. (Color online) Magnetizations (a) 〈Mx,1〉 and (b) 〈Mx,2〉as a function of the transverse magnetic field λ for the three-state Pottsmodel. For the broken-symmetry phase λ < λc, each magnetizationis given for each of the three degenerate ground states denotedby |ψm〉 (m ∈ [1,q]. The numerical critical point locates at λc =1.0004.

chosen the |ψ1〉 that gives a real value of the magnetizations,i.e., 〈Mx,1〉1 and 〈Mx,2〉1 are real. Moreover, the degenerateground states give the same values for the z-componentmagnetizations, i.e., 〈Mz〉1 = 〈Mz〉2 = 〈Mz〉3.

Next, we consider the magnetizations for the four-statePotts model, i.e., q = 4. In Fig. 5, we plot the magnetizations(a) 〈Mx,1〉, (b) 〈Mx,2〉, and (c) 〈Mx,3〉 as a function of thetraverse magnetic field λ. The numerical critical point isλc = 1.0004. All the magnetizations show that the phasetransition is a continuous phase transition. Similarly to the caseq = 3, all the absolute values of the magnetizations have thesame values at a given magnetic field and the magnetizationsin the complex magnetization plane have a relation betweenthem under a rotation, which is characterized by the valueω4 = exp[2πi/4]. In Fig. 5, we observe that, for a givenmagnetic field λ < λc, there are the relations between themagnetizations as 〈Mx,1〉1 = ω−1

4 〈Mx,1〉2 = ω−24 〈Mx,1〉3 =

ω−34 〈Mx,1〉4 from (a), 〈Mx,2〉1 = ω−2

4 〈Mx,2〉2 = ω−44 〈Mx,2〉3 =

ω−64 〈Mx,2〉4 from (b), and 〈Mx,3〉1 = ω−3

4 〈Mx,3〉2 =ω−6

4 〈Mx,3〉3 = ω−94 〈Mx,3〉4 from (c). Also, for each ground-

state wave function, the magnetizations have the relationsas 〈Mx,1〉1 = 〈Mx,2〉1 = 〈Mx,3〉1, 〈Mx,1〉2 = ω−1

4 〈Mx,2〉2 =ω−2

4 〈Mx,3〉2, 〈Mx,1〉3 = ω−24 〈Mx,2〉3 = ω−4

4 〈Mx,3〉3, and〈Mx,1〉4 = ω−3

4 〈Mx,2〉4 = ω−64 〈Mx,3〉4. These results show

that, in the complex magnetization plane, the rotationsbetween the magnetizations for a given magnetic field aredetermined by the characteristic rotation angles θ = 0,2π/4, 4π/4, and 6π/4, i.e., 〈Mx,p〉m = g4〈Mx,p′ 〉m′ withg4 ∈ {I,ω4,ω

24,ω

34}. Moreover, the degenerate ground states

give the same values for the z-component magnetizations,i.e., 〈Mz〉1 = 〈Mz〉2 = 〈Mz〉3 = 〈Mz〉4.

From the discussions for the three-state and four-state Pottsmodels, one may refer to a general relation between themagnetizations for any q-state Potts model. Actually, for anyq, we find the relations

ωp(1−m)q 〈Mx,p〉m = ωp′(1−m′)

q 〈Mx,p′ 〉m′ , (6a)

〈Mz〉m = 〈Mz〉m′ , (6b)

where ωq = exp[2πi/q]. When one calculate the magnetiza-tions of the operator Mx,p with different wave functions, the

relations in Eq. (6) reduce to 〈Mx,p〉m = ωp(m−m′)q 〈Mx,p〉m′ that

satisfies the relations of the magnetizations in Figs. 4 and 5.Moreover, if one choose a wave function, the magnetizationsof the operators Mx,1, . . . ,Mx,q−2, and Mx,q−1 have the

relations 〈Mx,p〉m = ω(p′−p)(1−m)q 〈Mx,p′ 〉m reduced from the

relations in Eq. (6). Furthermore, these results show that, inthe complex magnetization plane, the rotations between themagnetizations for a given magnetic field are determined by thecharacteristic rotation angles θ = 0, 2π/q, 4π/q, 6π/q, . . . ,

and 2(q − 1)π/q. Thus, Eq. (6) can be rewritten as

〈Mx,p〉m = gq〈Mx,p′ 〉m′ , (7a)

gq ∈ {I,ωq,ω

2q, . . . ,ω

q−1q

}. (7b)

Equation (7) then shows clearly that the q-state Potts modelhas the discrete symmetry group Zq consisting of q elements.For q = 2, the q-state Potts model becomes the quantumIsing model that has a doubly degenerate ground state dueto a Z2 symmetry. From Eq. (7), one can easily confirm〈Mx〉1 = −〈Mx〉2 because of g2 ∈ {I,ω2} with ω2 = exp[πi].In addition, the q-state Potts Hamiltonian in Eq. (4) is invariantwith respect to the q transformations, i.e., UmHqU

†m = Hq

for m ∈ [1,q], which implies that the system has the q-folddegenerate ground states for the broken-symmetry phaseaccording to the spontaneous symmetry-breaking mechanism.The transformations are given as

Um :

{Mx,p → (

ωpq

)m−1Mx,p

Mz → Mz

, (8)

where ωq = exp[2πi/q]. Although the q-state Potts Hamil-tonian remains invariant under the full q transformations,for the Zq broken-symmetry phase, the q order states de-scribed by the q degenerate ground states are invariant underonly the subgroup of the Zq symmetry group. Obviously,Eqs. (6) and (7) show the relations between the orderparameters of the q equivalent ordered states under the q

transformations. As a consequence, it is shown that, fromthe degenerate ground states, one can determine the orderparameters as the magnetizations and their specification ofhow the order parameters transform under the symmetrygroup Zq .

To make clearer the general relation of the magnetizationsin Eqs. (6) and (7), let us consider the five-state Potts model.In Fig. 6, we plot the magnetizations (a) 〈Mx,1〉, (b) 〈Mx,2〉,(c) 〈Mx,3〉, and (d) 〈Mx,4〉 as a function of the traverse magneticfield λ. The numerical critical point is obtained as the exactvalue λc = 1. In addtion, all the magnetizations shows that thephase transition is a discontinuous phase transition. Similarlyto the cases q = 3 and q = 4, all the magnetizations have thesame values at a given magnetic field, and the magnetizations

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FIG. 5. (Color online) Magnetizations (a) 〈Mx,1〉, (b) 〈Mx,2〉, and (c) 〈Mx,3〉 as a function of the transverse magnetic field λ for the four-statePotts model. For the broken-symmetry phase, the magnetizations are given from each of the four degenerate ground states. The critical pointis estimated numerically as λc = 1.0004.

in the complex magnetization plane have a relation betweenthem under a rotation, which is characterized by the valueω5 = exp[2πi/5]. From Figs. 6(a)–6(d), the relations betweenthe magnetizations agree with those in Eqs. (6) and (7).

VII. CRITICAL EXPONENTS, CENTRAL CHARGE,AND UNIVERSALITY

As is known, for q � 4, the quantum phase transitions area continuous phase transition. If q > 4, discontinuous (firstorder) phase transitions occur in the q-state Potts model. In

this sense, as is known, the critical q is qc = 4. In this section,we will study the critical exponents of qc = 4 based on theiMPS ground-state wave functions. Actually, all the degenerate ground states for the broken-symmetry phase givethe same exponents, as it should be, although we present oneof the numerical results from the fourfold degenerate groundstates. In addition, in order to calculate a central charge, wediscuss the von Neumann entropy for q = 3, 4, and 5. In theTable I, for the four-state Potts model, the critical exponentsand the critical charge from the our iMPS results are comparedwith their exact values.

(a) (b)

(c) (d)

FIG. 6. (Color online) Magnetizations (a) 〈Mx,1〉, (b) 〈Mx,2〉, (c) 〈Mx,3〉, and (d) 〈Mx,4〉 as a function of the transverse magnetic field λ forthe five-state Potts model. For the broken-symmetry phase, the magnetizations are given from each of the five degenerate ground states. Thenumerical critical point is the exact value λc = 1.

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TABLE I. Critical exponents and central charge for the four-statePotts model.

Exponents α β γ δ ν η c

Exacta 2/3 1/12 7/6 15 2/3 1/4 1iMPS 0.0843 1.0718 0.6300 0.2510 0.9803

aThe exact values of the critical exponents and central charge aretaken from Refs. [20,35].

A. Critical exponents

In our iMPS approach, we obtain the fourfold degenerateground states at zero temperature. The critical exponentsα for specific heat and δ for the field dependence of themagnetization at the critical temperature cannot be calculated.In the following, we discuss the four exponents, i.e., β, γ ,η, and ν. Thus, we start with the magnetization near thecritical point λc = 1.0004. In Fig. 7, the magnetization 〈Mx〉[Fig. 7(a)] and the susceptibility ∂λ〈Mx〉 [Fig. 7(b)] are plottedas a function of |λ − λc| in the log-log plot. It is shownclearly that both the magnetization and the susceptibilitycan be described by power laws, i.e., 〈Mx〉 ∝ |λ − λc|β and∂λ〈Mx〉 ∝ |λ − λc|−γ with their characteristic exponents β

and γ , respectively. Thus, the fitting function is chosen tobe ln〈Mx〉 = β ln |λ − λc| + β0 for the magnetization andln ∂λ〈Mx〉 = −γ ln |λ − λc| + γ0 for the susceptibility withthe fitting constants β0 and γ0, respectively. From the numer-ical fittings, we obtain β = 0.0843 and β0 = 0.0394 for themagnetization [Fig. 7(a)] and γ = 1.0718 and γ0 = −3.0308for the susceptibility [Fig. 7(b)]. The fitted critical exponentsβ = 0.0843 and γ = 1.0718 are quite close to the exactvalues [20] β = 1/12 (=0.0833) and γ = 7/6 (=1.1667),respectively.

In Fig. 8, we plot the correlation length ξ as a functionof |λ − λc| [Fig. 8(a)] and the correlation 〈Mx(i)Mx(j )〉 asa function of the lattice site distance |i − j | [Fig. 8(b)] atthe critical point λc = 1.0004. It is shown clearly that boththe correlation length and the correlation at the critical pointλc can be described by power laws, i.e., ξ ∝ |λ − λc|−ν and〈Mx(i)Mx(j )〉 ∝ |λ − λc|−η with their characteristic expo-nents ν and η, respectively. We choose the fitting functionsas ln ξ = −ν ln |λ − λc| + ν0 for the correlation length andln〈Mx(i)Mx(j )〉 = −η ln |i − j | + η0 for the correlation at the

FIG. 7. (Color online) (a) Magnetization 〈Mx〉 and (b) suscep-tibility ∂λ〈Mx〉 as a function of |λ − λc| near the critical pointλc = 1.0004 for four-state Potts model.

FIG. 8. (Color online) (a) Correlation length ξ as a function of|λ − λc| for the four-state Potts model. (b) Correlation 〈Mx(i)Mx(j )〉as a function of the lattice site distance |i − j | at the critical pointλc = 1.0004.

critical point with the fitting constants ν0 and η0, respectively.From the numerical fittings, we obtain (a) ν = 0.6300 andν0 = −1.2153 for the correlation length and (b) η = 0.2510and η0 = −3.5644 for the correlation. The fitted criticalexponents ν = 0.6300 and η = 0.2510 are quite close to theexact values [20] ν = 2/3 and η = 1/4, respectively.

B. von Neumann entropy and central charge

In our iMPS approach, the von Neumann entropy S

can be directly evaluated by the elements of the diago-nal matrix λ[i]

αithat are the Schmidt decomposition coef-

ficients of the bipartition between the semi-infinite chainsL(−∞, . . . ,i) and R(i + 1, . . . ,∞). This implies that Eq. (1)can be rewritten by |�〉 = ∑χ

α=1 λα|ψLα 〉|ψR

α 〉, where |ψLα 〉

and |ψRα 〉 are the Schmidt bases for the semi-infinite chains

L(−∞, . . . ,i) and R(i + 1, . . . ,∞), respectively. For thebipartition, then, the von Neumann entropy S can be definedas [30] S = −Tr[�L log �L] = −Tr[�R log �R], where �L =TrR � and �R = TrL � are the reduced density matrices of thesubsystems L and R, respectively, with the density matrix� = |�〉〈�|. For the semi-infinite chains L and R in the iMPSrepresentation, the von Neumann entropy S calculated byS = −∑χ

α=1 λ2α log2 λ2

α . Actually, the von Neumann entropyas one of the quantum entanglement measures have beenproposed as a general indicator to determine and characterizequantum phase transitions [31,32]. Also, the logarithmicscaling of the von Neumann entropy was conformed to exhibitconformal invariance and the scaling is governed by a universalfactor [5,33,34], i.e., a central charge c of the associatedconformal field theory. In the iMPS approach, for a continuousphase transition, the diverging entanglement at a quantumcritical point gives simple scaling relations [33] for (i) the vonNeumann entropy S and (ii) a correlation length ξv with respectto the truncation dimension χ as S ∼ cκ

6 log2 χ and ξv ∼ Aχ κ ,where c is a central charge, κ is a so-called finite-entanglementscaling exponent, and A is a constant. By using the relations, acentral charge can be obtained numerically at a critical point.

In Fig. 9(a), we plot the von Neumann entropies as afunction of the transverse magnetic field λ for the three-, four-,and five-state Potts models with the truncation dimension χ =32. In the entropies, there are singular points at λc = 1.0004for q = 3 and 4, and λc = 1 for q = 5, which are consistent

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FIG. 9. (Color online) (a) Ground-state von Neumann entropiesfor the three-, four-, and five-state Potts models as a function ofthe transverse magnetic field λ. (b) Von Neumann entropies S and(c) correlation lengths ξv for q = 3 and q = 4 with respect to thetruncation dimension χ at the exact critical point λc = 1. Here thelogarithm is taken as base 2 in (b).

with the critical points from the multiple bifurcation points inFig. 3 and from the magnetizations as the order parametersin Figs. 4–6. It is shown clearly that the von Neumannentropies capture the phase transitions. The (dis-)continuityof the von Neumann entropies for q = 3 and q = 4 (q = 5)indicates a (dis-)continuous phase transition between thebroken-symmetry phases and the symmetry phases. However,the von Neumann entropies for the different degenerate groundstates give the same values, which implies that the vonNeumann entropy cannot distinguish the different degenerateground states in the broken-symmetry phases.

For q = 3 and q = 4, in Figs. 9(b) and 9(c), we plot[Fig. 9(b)] the von Neumann entropy and [Fig. 9(c)] thecorrelation length ξv as a function of the truncation dimensionχ at the critical points λc = 1. Here, the truncation dimensionsare taken as χ = 8, 16, 25, 32, 50, and 64. It is shown thatboth the von Neumann entropy S and the correlation length

ξv diverge as the truncation dimension χ increases. In orderto obtain the central charges, we use the numerical fittingfunctions, i.e., Sq(χ ) = aq + bq log2 χ and ξv,q(χ ) = Aqχ

κq .Numerically, the constants of the von Neumann entropiesare fitted as a3 = −0.1215 and b3 = 0.2000 for q = 3 anda4 = −0.1830 and b4 = 0.2016 for q = 4. The power-lawfittings on the correlation lengths ξv,q give the fitting constantsas A3 = 0.2060 and κ3 = 1.4775 for q = 3 and A4 = 0.2243and κ4 = 1.2340 for q = 4. As a result, from the κq and bq , thecentral charges are obtained as c3 = 0.8121 for q = 3 and c4 =0.9803 for q = 4, which are quite close to the exact values [35]c = 4/5 for q = 3 and c = 1 for q = 4, respectively.

VIII. SUMMARY

We have investigated how to identify ground-state degener-acy by using quantum fidelity. To do this, we have introducedthe quantum fidelity between the degenerate ground states andan arbitrary reference state. As an example we employed theiMPS with the iTEBD to study the q-state Potts model.

We have obtained the q-fold degenerate ground statesexplicitly. The distinguished degenerate ground-state wavefunctions naturally permits the detection of a quantum phasetransition that is indicated by a multiple bifurcation point of thequantum fidelity per lattice site. Furthermore, as the completedescription of an order phase, we have discussed to how toclassify the order parameters calculated from the degenerateground states transforming under the subgroup of a symmetrygroup of the Hamiltonian. A general relation between themagnetizations calculated from the degenerate ground statesis obtained to show the spontaneous symmetry breaking of theZq symmetry group in the q-state Potts model. In addition,the critical exponents and the central charges are directlycalculated from the degenerate ground states, which showsthat the iMPS results are very close to the exact values.

ACKNOWLEDGMENTS

Y.H.S. thanks Bo Li for helpful discussions about thecritical exponents. We thank Huan-Qiang Zhou for helpfuldiscussions to inspire and encourage us to complete this workand Jon Links for useful comments and suggestions on ourmanuscript. The work was supported by the National NaturalScience Foundation of China (Grant No. 11374379).

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