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Geometric entanglement for quantum critical spin chains belonging to the Ising and three-state Potts universality classes
Physics Letters A 376 (2012) 2677–2682
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Physics Letters A www.elsevier.com/locate/pla
Geometric entanglement for quantum critical spin chains belonging to the Ising and three-state Potts universality classes Jin-Hua Liu, Hai-Tao Wang, Qian-Qian Shi, Huan-Qiang Zhou ∗ Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 30 March 2012 Received in revised form 8 July 2012 Accepted 17 July 2012 Available online 20 July 2012 Communicated by A.R. Bishop Keywords: Geometric entanglement Critical point Affleck–Ludwig g factor Boundary entropy
a b s t r a c t The leading finite-size correction to the geometric entanglement per lattice site is investigated for the antiferromagnetic–ferromagnetic alternating Heisenberg model, quantum three-state Potts model in a transverse field and a spin-1/2 spin chain with the competing two-spin and three-spin interactions at criticality, belonging to the Ising and three-state Potts universality classes with the central charge c = 1/2 and c = 4/5, respectively. Our results demonstrate that the leading finite-size correction coefficient is essentially the celebrated Affleck–Ludwig boundary entropy corresponding to a conformally invariant boundary condition, which in turn depends on the period of the translation-invariant separable states. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Recently, the geometric entanglement (GE) [1–4], a holistic measure of the multipartite entanglement present in a quantum state wave function, has attracted much attention, due to its role as a detector to locate critical points for quantum many-body lattice systems. In particular, an intriguing connection between the GE per site and the Affleck–Ludwig g factor has been established for a finite-size spin chain with the periodic boundary conditions at criticality. More specifically, the coefficient in the subleading term, which occurs in the finite-size correction to the GE per site, is related to the Affleck–Ludwig boundary entropy, or equivalently, the Affleck–Ludwig g factor. This was first discovered by J.-M. Stéphan, G. Misguich, and F. Alet in Ref. [5], given that the subleading term coefficient had been conjectured to be universal [6]. In order to check whether or not such a connection between the GE per site and the Affleck–Ludwig g factor is universally valid, we have studied three prototypical spin chains belonging to the Ising universality class to see if the same g factor appears [7]. To this end, an efficient method to systematically evaluate the GE per lattice site for quantum many-body lattice systems has been developed in the context of the tensor network algorithms based on matrix product state representations for quantum many-body lattice systems with the periodic boundary conditions [8–12]. Remarkably, it was found that the subleading term coefficient in the GE per lattice site depends on the period of the translation-invariant
*
Corresponding author. E-mail address: [email protected] (H.-Q. Zhou).
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separable states. Actually, for quantum XYX model in an external field, either conformally invariant free or fixed boundary condition appears, depending on the period of the translation-invariant separable states being one- or two-site, respectively. This provides further insights into the connection between the subleading term coefficient in the GE per lattice site and the Affleck–Ludwig boundary entropy. However, two major questions remain to be resolved. The first is whether or not the very same connection between the subleading term coefficient in the GE per lattice site and the Affleck– Ludwig boundary entropy is also valid for quantum critical manybody systems belonging to other universality classes, such as the Potts universality class. The second is whether or not the dependence of the subleading term coefficient in the GE per lattice site on the period of the translation-invariant separable states involved in the definition of the GE is generic, instead of being a peculiar feature for a specific model. In order to address these questions, we investigate the leading finite-size correction to the GE per lattice site for the antiferromagnetic–ferromagnetic (AF–F) alternating Heisenberg model at criticality, belonging to the Ising universality class with the central charge c = 1/2, and quantum three-state Potts model in a transverse magnetic field and a spin-1/2 chain with the competing two-spin and three-spin interactions at criticality, belonging to the three-state Potts universality class with the central charge c = 4/5. Our simulation results presented in this work, together with the previous work [7] on three prototypical critical quantum spin chains belonging to the same Ising universality class, confirm that the leading finite-size correction to the GE per lattice site is universal, in the sense that the subleading term coefficient
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is essentially the celebrated Affleck–Ludwig boundary entropy corresponding to a conformally invariant boundary condition. In addition, the dependence of the subleading term coefficient in the GE per lattice site on the period of the translation-invariant separable states is generic. 2. The geometric entanglement per lattice site and its connection to the Affleck–Ludwig g factor
s = ln( g ). 3. The models
(1)
(2)
H = J AF
where
Λmax = max φ|ψ. |φ
Here, Λmax represents the maximum fidelity between a quantum pure state |ψ and all the possible separable and normalized states |φ of the N parties. Since the contribution to E (ψ) from each party is additive, E (ψ) scales linearly with N, for a multipartite system consisting of N parties. Therefore, it is convenient to define the GE per party as
E N |ψ = N −1 E |ψ .
(3)
For a quantum many-body lattice spin system with the periodic boundary conditions, each lattice site constitutes a party. As such, E N is the GE per lattice site. As argued in Ref. [6], the GE per site, E N , scales as
E N ∼ E∞ +
b N
+O
1 N2
,
2 ln(2)
s.
N /2
σx[2i−1] σx[2i] + σ y[2i−1] σ y[2i] + AF σz[2i−1] σz[2i]
i =1
+ JF
N /2
σx[2i] σx[2i+1] + σ y[2i] σ y[2i+1] + F σz[2i] σz[2i+1] , (7)
i =1
[i ]
where σi (α = x, y , z) are the Pauli spin matrices at site i, J AF and J F are the antiferromagnetic and ferromagnetic couplings, AF and F are the anisotropic constants. Later on, we set J AF = 1, J F = −1, and AF = 1, with F as a control parameter. The second model is quantum three-state Potts model in a transverse magnetic field. The Hamiltonian is written as
H =−
N i =1
(4)
α =1,2
[i ]
[ i +1 ]
M x,α M x,3−α − λ
N
[i ]
Mz ,
(8)
i =1
[i ]
where N is the chain size, and b is the subleading term coefficient. The g factor, also known as the ground-state degeneracy, was introduced by Affleck and Ludwig [13] to characterize conformally invariant boundary conditions. Indeed, following Affleck and Ludwig [13], there is a g function that decreases along a boundary renormalization group flow from a less stable to a more stable critical point in the same bulk universality class. At a critical point, the value of the g function becomes the g factor. To understand the connection between the GE per lattice site and the Affleck–Ludwig g factor, we consider a cylinder of circumference N and height M N with spins at both circular edges. Each edge is supposed to be a state of a one-dimensional quantum chain with the periodic boundary conditions, described by a circle at the spatial direction. As is well known, any state can be projected to a ground state with infinite imaginary time evolution. Thus, the cylinder might represent the overlap between a separable state and the system’s ground state. With the space and imaginary time interchanged, the original edges become two boundary states, corresponding to some conformally invariant boundary conditions. Therefore, the determination of b amounts to calculating a subleading contribution of φbs |G , with |G being the ground state, and |φbs being a boundary state corresponding to a conformally invariant boundary condition. In this geometry, the free energy can be written in the form [13]: F = F bulk + 2F boundary , where F bulk ∼ N M represents the bulk free energy, and F boundary = aN + s + o(1) = − lnz φbs |G represents the boundary free energy. Here, a is the boundary free energy per party, and s, a subleading term in the boundary free energy, is the Affleck–Ludwig boundary entropy. Therefore, there exists a relation between the Affleck–Ludwig boundary entropy s and the subleading term coefficient b [5]:
b=−
(6)
We consider three critical quantum many-body spin chains with the periodic boundary conditions: one belongs to the Ising universality class with the central charge c = 1/2, and the other two belong to the three-state Potts universality class with the central charge c = 4/5. The first model we consider is the AF–F alternating Heisenberg model. The Hamiltonian is given by
Let us first recall the definition of the global GE per party. Mathematically, the global GE between a quantum pure state |ψ and all possible separable and normalized states |φ is defined as
E |ψ = − log2 Λ2max ,
Here, s is defined via the Affleck–Ludwig g factor:
(5)
where λ is the transverse magnetic field and M α (α = x, z) are the Potts matrices:
M x,1 =
Mz =
0 0 1
1 0 0 1 0 0
2 0 0 0 −1 0 0 0 −1
M x,2 =
,
0 0 1 0 0 1
1 0 0
,
.
The third model is a spin-1/2 spin chain with the competing two-spin and three-spin interactions. It is described by the Hamiltonian
H = J2
N
σz[i] σz[i+1] + J 3
i =1
N i =1
σz[i] σz[i+1] σz[i+2] − h
N
σx[i] ,
(9)
i =1
[i ]
where σα (α = x, z) are the Pauli spin-1/2 operators at site i, J 2 and J 3 are the two-spin and three-spin interaction constants, and h is an external field. We choose J 2 = 0.4 and h = 1, with J 3 as a control parameter. 4. Simulation results To simulate a finite-size translation-invariant spin chain with the period boundary conditions, we might resort to tensor network algorithms based on the matrix product state representations [8–12]. Here, we exploit the translation-invariant matrix product state algorithm [10], whose cost does not depend on the chain size N. For a finite-size periodic system, we first take advantage of the von Neumann entropy to locate a quantum critical point. As is well
J.-H. Liu et al. / Physics Letters A 376 (2012) 2677–2682
known, the von Neumann entropy, as a bipartite entanglement measure, characterizes the entanglement between two subsystems when we partition the system into two subsystems with sizes l and N − l. Mathematically, it is defined as S (ρ ) ≡ − tr(ρ log2 ρ ), where ρ denotes the reduced density matrix of a subsystem (of size l) for a given quantum state of the system. Here, we restrict ourselves to study the von Neumann entropy when the system is in a ground state. As such, it takes a maximum value when l = N /2, with N even. The von Neumann entropy for a subsystem consisting of a half chain, which we denote it as S h , exhibits a maximum when the control parameter crosses a pseudo-critical point; the critical point may be determined from an extrapolation to the thermodynamic limit. As is seen from the main panels in Fig. 1, the von Neumann entropy, S h , reaches a maximum value when the control parameter crosses a pseudo-transition point, which is denoted as Fc ( N ) in (a), as λc ( N ) in (b), and as J 3c ( N ) in (c), respectively. In the insets to Fig. 1, the critical points Fc (∞), λc (∞), and J 3c (∞) are, respectively, determined by performing an extrapolation of the pseudo-critical points Fc ( N ), λc ( N ), and J 3c ( N ) with respect to the size N. Specifically, we plot the von Neumann entropy S h as a function of the control parameter F for the AF–F alternating Heisenberg model in Fig. 1(a), with the size ranging from 20 to 100. Actually, the model is in the Haldane phase if F = 1, and in the Néel phase if F 1. Therefore, it undergoes a phase transition, as the control parameter F varies from 1 to ∞. In Fig. 1(a), the von Neumann entropy S h appears to be a maximum at a pseudo-critical point, Fc ( N ) for a given size N. We perform an extrapolation of F ( N ) with respect to the size N. Here, the fitting function takes the form: Fc ( N ) = Fc (∞) + β N −γ , with Fc (∞) = 2.2329, β = 130.35, and γ = 2.1358. Thus, the critical point in the thermodynamic limit is located at Fc (∞) = 2.2329, which is consistent with the known result Fc ∼ 2.32 [14]. Fig. 1(b) shows the von Neumann entropy S h as a function of the control parameter λ for the three-state Potts model, with the size ranging from 20 to 100. The pseudo-critical points λc ( N ) are identified from the maximum values of the von Neumann entropy S h for different values of the size N. By performing an extrapolation of the pseudo-critical points λc ( N ) with respect to the size N, the critical point is determined. Here, the fitting function is λc ( N ) = λc (∞) + β N −γ , with λc (∞) = 1.00097, β = −16.6307, and γ = 2.14455. The quantum critical point we have determined matches very well with the exact value λc = 1. In Fig. 1(c), we plot the von Neumann entropy S h as a function of the control parameter J 3 for a model with the two-spin and three-spin competing interactions, with the size ranging from 30 to 120. By performing an extrapolation of the pseudo-critical points J 3c ( N ) with respect to the size N, the critical point is determined. Here, the fitting function is J 3c ( N ) = J 3c (∞) + β N −γ , with J 3c (∞) = 0.90038, β = 325.48, and γ = 2.8564. For a critical system of the size N, the von Neumann entropy S (l), describing entanglement between two subsystems with sizes l and N − l, follows the universal logarithmic scaling with the subsystem size l: S (l) = c log2 ( N /π sin(π l/ N ))/3 + c 0 , where c is the central charge, and c 0 is a model-dependent constant [15–19]. In Fig. 2, we show the scaling relation between the von Neumann entropy S (l) and T (l) ≡ log2 ( N /π sin(π l/ N )) for three critical quantum models. The scaling of the von Neumann entropy yields the central charge, whose value determines the universality class a system belongs to. In Table 1, the central charge c fit and the constant c 0fit are fitted for three spin chains with different system sizes N. They are consistent with the exact central charge c = 1/2 for the Ising universality class and c = 4/5 for the three-state Potts universality class, as long as the chain size N is large enough. This indicates that the critical points we have determined are accurate.
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Fig. 1. (Color online.) Main: The von Neumann entropy S h for a subsystem consisting of a half chain as a function of the control parameter for three different periodic quantum spin chains: (a) the antiferromagnetic–ferromagnetic (AF–F) alternating Heisenberg model, with F as the control parameter, and the size ranging from 20 to 100; (b) the three-state Potts model, with λ as the control parameter, and the size ranging from 20 to 100; (c) the spin-1/2 chain with two-spin and three-spin competing interactions, with J 3 as the control parameter, and the size ranging from 30 to 120, respectively. Inset: The extrapolation of the pseudo-critical points corresponding to the maximum values of the von Neumann entropy S h with respect to the size N. In (a), the fitting function is Fc ( N ) = Fc (∞) + β N −γ , with Fc (∞) = 2.2329, β = 130.35, and γ = 2.1358. In (b), the fitting function is λc ( N ) = λc (∞) + β N −γ with λc (∞) = 1.00097, β = −16.6307, and γ = 2.14455. In (c), the fitting function is J 3c ( N ) = J 3c (∞) + β N −γ , with J 3c (∞) = 0.90038, β = 325.48, and γ = 2.8564.
Note that the relative fitting errors of the von Neumann entropy in Fig. 2 are all smaller than 3.3 × 10−3 for three models. Now we turn to the leading finite-size correction to the GE per lattice site for three spin chains at their respective critical points, as identified above. Besides the determination of the Affleck– Ludwig g factor for each model, our focus will be on the connection between the period of the translation-invariant separable states and the Affleck–Ludwig g factor. Generically, the imposed
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Table 1 The central charge c fit and the constant c 0fit are fitted from the scaling relation of the von Neumann entropy S (l) with the subsystem size l, for the critical quantum AF–F alternating Heisenberg model with the size N ranging from 20 to 100, the critical three-state Potts model with the size N ranging from 20 to 100, and the critical spin-1/2 chain with two-spin and three-spin competing interactions with the size N ranging from 30 to 150. AF–F alternating Heisenberg model Three-state Potts model
Spin chain with competing interactions
Fig. 2. (Color online.) Main: The scaling of the von Neumann entropy S (l) with the subsystem size l: S (l) = 3c T (l) + c 0 , with T (l) ≡ log2 ( N /π sin(π l/ N )), where c is the central charge, and c 0 is a model-dependent constant for the critical quantum AF–F alternating Heisenberg model, with the system size ranging from 20 to 100 in (a), for the three-state Potts model, with the system size ranging from 20 to 100 in (b), and for the spin-1/2 chain with two-spin and three-spin competing interactions, with the system size ranging from 30 to 150 in (c). Inset: The relative fitting errors S err = ( S fit (l) − S algo (l))/ S algo (l) are always smaller than 1.9 × 10−3 in (a), smaller than 6 × 10−4 in (b), and smaller than 3.3 × 10−3 in (c). Here, S fit (l) is the value extracted from the fit and S algo (l) is the value computed from the algorithm.
period of a translation-invariant separable state acts as a constraint condition. Indeed, for two commensurate periods, the set of all the translation-invariant separable states with the smaller period is a subset of the set of all the translationally invariant separable states with the larger period. As a consequence, the translationinvariant closest separable state could be different in the two sets, depending on whether or not it falls into the subset of the separable states with the smaller period. Therefore, two scenarios are expected: the first is that the GE per lattice site computed from the translation-invariant separable states with the smallest possible period is identical to that from the exact diagonalization
Size N
20
40
60
80
100
c fit
0.5127
0.5058
0.5040
0.5031
0.5025
c 0fit
0.6443
0.6504
0.6524
0.6534
0.6541
Size N
20
40
60
80
100
c fit
0.8031
0.8019
0.8013
0.8010
0.8007
c 0fit
1.0978
1.0982
1.0985
1.0986
1.0989
Size N
30
60
90
120
150
c fit
0.8367
0.8268
0.8226
0.8196
0.8175
c 0fit
0.9580
0.9685
0.9735
0.9768
0.9792
without any translation-invariant assumption, thus the smallest g factor is anticipated to be involved, given that it corresponds to the most stable fixed point in the sense of boundary renormalization group flows (according to the Affleck–Ludwig g theorem). This is exactly what happens in the three-state Potts model, as seen below. The second is that the GE per lattice site computed from the translation-invariant separable states with the smaller period is different from that computed from the translation-invariant separable states with the larger period, while the latter is identical to that from the exact diagonalization without any translationinvariant assumption. As such, different g factors corresponding to different conformally invariant boundary conditions appear in the finite-size correction to the GE per lattice site, with one of them being the smallest. As seen below, the AF–F alternating Heisenberg model and a spin-1/2 spin chain with the competing two-spin and three-spin interactions at criticality fall into this scenario. In Fig. 3, we plot the GE per lattice site E N as a function of the chain size N for the AF–F alternating Heisenberg model at the critical point Fc = 2.232, with the size N ranging from 4 to 100. The GE per lattice site E N is well fitted into the scaling function E N = a + b/ N + f / N 2 , with a = 0.678918, b = −0.010256, and f = −1.324811, and a = 0.078658, b = 1.002489 and f = −0.390788 for (a) and (b), respectively. In (a), the separable states are translation-invariant under two-site shifts. In (b), the separable states are translation-invariant under four-site shifts. The periods of the translation-invariant separable states are chosen to ensure that the translational invariance of the model Hamiltonian under two-site shifts are respected. The exact diagonalization for small sizes up to 22 is also performed to evaluate the GE per lattice site without any translation-invariant assumption, which is denoted by the five point stars in Fig. 3(b). Given the coefficient b, we get the Affleck–Ludwig g factors g FA = 1.0035 and g FA = 0.7064, respectively, for (a) and (b), according to the relation (5) between the Affleck–Ludwig g factor and the coefficient b. The two g factors match very well with the exact g factors g free = 1 and g fixed = √ 2/2, corresponding to the conformally invariant free and fixed boundary conditions in the Ising universality class, respectively. As such, we provide another critical spin chain belonging to the Ising universality class, for which the subleading term coefficient depends on the period of the translation-invariant separable states, given that either conformally invariant free or fixed boundary condition appears for quantum XYX model in an external field [7]. In Fig. 4(a), we plot the GE per lattice site E N as a function of the chain size N for quantum three-state Potts model at the critical point λ = 1, with the size ranging from N = 10 to 200. Here, the separable states are chosen to be translation-invariant under one-site shifts. The reason for such a choice is that, for this model, any other choice of the period of the translation-
J.-H. Liu et al. / Physics Letters A 376 (2012) 2677–2682
Fig. 3. (Color online.) Main: Relation between the GE per lattice site E N and the chain size N for the AF–F alternating Heisenberg model with the periodic boundary conditions. For the size N ranging from 12 to 100, the data are fitted into E N = a + b/ N + f / N 2 with a = 0.678918, b = −0.010256, and f = −1.324811 in (a), and a = 0.078658, b = 1.002489 and f = −0.390788 in (b). The separable states are translation-invariant under two-site shifts in (a), and translation-invariant under four-site shifts in (b), respectively. The five-point stars indicate the GE per lattice site from the exact diagonalization, with the chain size up to 22. The GE per lattice site from the exact diagonalization matches very well with those from the matrix data product state algorithm. Inset: the relative fitting errors εerr = (ε N − ε Nfit )/ε Nfit are always smaller than 3.4 × 10−5 in (a) and (b), where E Nfit is the value extracted from
the fit and E Ndata is our simulation value from the matrix product state algorithm for each N.
invariant separable states results in the same GE per lattice site. This is also consistent with the fact that the GE per lattice site computed from the exact diagonalization without any translationinvariant assumption, as shown by the five-point stars in Fig. 4(a), matches very well with those from the matrix product state algorithm. The finite size scaling for the GE per lattice site E N follows the function E N = a + b/ N + f / N 2 , where the fitting coefficients are a = 0.042783, b = 1.770539, and f = −3.518868. It yields the Affleck–Ludwig g factor g potts = 0.5413. On the other hand, various partition functions, referring to the fixed boundary conditions A, B, and C (corresponding to three possible states of the Potts variable), the mixed boundary conditions A B, AC , and BC , the free boundary condition “free” and a novel boundary condition “new” are systematically investigated in Ref. [20]. As it turns out, four different boundary conditions, labeled as A, A B, free and new, respectively, are conformally invariant, with the g factor being 0.5509, 0.8914, 0.9542, and 1.544, respectively. As such, our result g potts = 0.5413 matches very well with the exact g factor g A = 0.5509 (within a tolerant numerical error). That is, it corresponds to the so-called A-type fixed boundary condition, one of four conformally invariant boundary conditions for the three-state Potts model [20]. In addition, we stress that the Affleck–Ludwig g factor, arising from
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Fig. 4. (Color online.) Main: (a) The scaling of the GE per lattice site E N with the chain size N at the critical point λc = 1 for quantum three-state Potts model. For the size N from 10 to 200, the data are fitted into E N = a + b/ N + f / N 2 with the coefficients a = 0.042783, b = 1.770539, and f = −3.518868. The GE per lattice site from the exact diagonalization, as displayed by the five-pointed stars for the chain size up to 14, matches very well with those from the matrix product state algorithm. (b) The scaling relation between the GE per lattice site E N and the chain size N for a quantum spin chain with competing interactions. For the size N ranging from 9 to 150, the data are fitted into E N = a + b/ N + f / N 2 . We have a = 0.145519, b = 0.044916, and f = −0.526629 in the upper branch, and a = 0.030834, b = 1.776206 and f = −3.487129 in the lower branch. In the upper and lower branches, the separable states are translation-invariant under one-site shifts, and translation-invariant under three-site shifts. Note that the GE per site is the same for the upper and lower branches when the chain size is less than the threshold, which is N = 12. The GE per lattice site from the exact diagonalization, as displayed by the five-pointed stars for the chain size up to 24, matches very well. data Inset: the relative fitting errors εerr = (ε N − ε Nfit )/ε Nfit are less than 5.1 × 10−3 and 4.7 × 10−4 in (a) and (b), respectively, where E Nfit is the value extracted from the fit and E Ndata is our simulation value for each N.
the computation of the ground-state GE for the three-state Potts model, conforms to the previous observation [7] that the smallest g factor g A , which corresponds to the most stable fixed point in the sense of the Affleck–Ludwig g theorem, is involved in the leading finite-size correction coefficient of the GE per lattice site. In Fig. 4(b), we plot the GE per lattice site E N as a function of the chain size N for a spin chain with the competing two-spin and three-spin interactions, where the size N is chosen from 9 to 150. It is shown that the GE per lattice site E N is splitted into two branches for a large enough chain size (N > 12): the upper branch is for the separable states that are translation-invariant under one-site shifts, and the lower branch is for the separable states that are translation-invariant under three-site shifts. We also compute the GE per site by means of the exact diagonalization without any translation-invariant assumption, which is denoted by the fivepoint stars in Fig. 4(b). We find that this only reproduces the lower branch. Thus, the translation-invariant separable states under onesite shifts constitutes a true subset of the translation-invariant separable states under three-site shifts. Therefore, the separable
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states that are translation-invariant under three-site shifts is sufficient for the GE per lattice site to reach the smallest g factor corresponding to the most stable conformally invariant boundary condition. Both of the branches follow the scaling function E N = a + b/ N + f / N 2 with the coefficients a = 0.145519, b = 0.044916, and f = −0.526629 for the upper branch and a = 0.030834, b = 1.776206 and f = −3.487129 for the lower branch. Then we get the Affleck–Ludwig g factors g up = 0.9845 and g down = 0.5403 for the upper and lower branch, respectively. They are consistent with the exact values g free = 0.9542 and g A = 0.5509 [13,21] for conformally invariant free and A-type fixed boundary conditions in the three-state Potts universality class. 5. Summary We have investigated the GE per lattice site for the AF–F alternating Heisenberg model at criticality, belonging to the Ising universality class with the central charge c = 1/2, and quantum three-state Potts model in a transverse magnetic field and a spin1/2 spin chain with the competing two-spin and three-spin interactions at criticality, belonging to the three-state Potts universality class with the central charge c = 4/5. The relevant critical points have been identified by evaluating the von Neumann entropy in the context of the matrix product state representations. Our simulation results, together with the previous work [7], demonstrate that the leading finite-size correction to the GE per lattice site is universal, in the sense that the subleading term coefficient is essentially the celebrated Affleck–Ludwig boundary entropy corresponding to a conformally invariant boundary condition. For all the models investigated, the simulated g factors are comparable with the exact results from conformal field theory, with their relative errors less than 3.1 × 10−2 . In addition, we have focused on the correlation between the period of the translation-invariant separable states and the Affleck– Ludwig g factor. More precisely, the subleading term in the finitesize correction to the GE per lattice site is determined by the Affleck–Ludwig g factor corresponding to one of the boundary conformal field theories compatible with the bulk criticality, regardless of the value of the period. However, the value of the g factor, generically, depends on the period. Specifically, for the AF–F alternating Heisenberg model, either free or fixed boundary condition appears, depending on the period of the translation-invariant separable states: if the period is two-site, it corresponds to the free boundary condition; if the period is four-site, then it corresponds
to the fixed boundary condition. For the spin-1/2 chain with the competing two-spin and three-spin interactions, both conformally invariant, i.e., free and A-type fixed, boundary conditions appear: if the period is one-site, it corresponds to the free boundary condition; if the period is three-site, it corresponds to the A-type fixed boundary condition. Therefore, we conclude that the dependence of the subleading term coefficient on the period of the translationinvariant separable states is generic. Acknowledgements We thank Sam Young Cho and Bing-Quan Hu for enlightening discussions. The work is partially supported by the National Natural Science Foundation of China (Grant No. 11174375), the Innovative Talent Funds for Project 985 (Project No. WLYJSBJRCGR201103), and the Chongqing University Postgraduates Science and Innovation Fund (Project No. 200911C1A0060322). References [1] R.G. Unanyan, C. Ionescu, M. Fleischhauer, Phys. Rev. A 72 (2005) 022326; R. Orús, S. Dusuel, J. Vidal, Phys. Rev. Lett. 101 (2008) 025701; A. Botero, B. Reznik, arXiv:0708.3391; R. Orús, Phys. Rev. Lett. 100 (2008) 130502. [2] R. Orús, T.-C. Wei, Phys. Rev. B 82 (2010) 155120. [3] T.-C. Wei, D. Das, S. Mukhopadyay, S. Vishveshwara, P. Goldbart, Phys. Rev. A 71 (2005) 060305. [4] C.-Y. Huang, F.-L. Lin, Phys. Rev. A 81 (2010) 032304. [5] J.-M. Stéphan, G. Misguich, F. Alet, Phys. Rev. B 82 (2010) 180406R. [6] Q.-Q. Shi, R. Orús, J.O. Fjærestad, H.-Q. Zhou, New J. Phys. 12 (2010) 025008. [7] B.-Q. Hu, X.-J. Liu, J.-H. Liu, H.-Q. Zhou, New J. Phys. 13 (2011) 093041. [8] F. Verstraete, D. Porras, J.I. Cirac, Phys. Rev. Lett. 93 (2004) 227205. [9] P. Pippan, S.R. White, H.G. Evertz, Phys. Rev. B 81 (2010) 081103(R). [10] Q.-Q. Shi, H.-Q. Zhou, J. Phys. A: Math. Theor. 42 (2009) 272002. [11] B. Pirvu, F. Verstraete, G. Vidal, Phys. Rev. B 83 (2011) 125104. [12] D. Rossini, V. Giovannetti, R. Fazio, J. Stat. Mech. (2011) P05021. [13] I. Affleck, A.W.W. Ludwig, Phys. Rev. Lett. 67 (1991) 161. [14] J. Ren, S. Zhu, Eur. Phys. J. D 50 (2008) 103. [15] G. Vidal, et al., Phys. Rev. Lett. 90 (2003) 227902; J.I. Latorre, et al., Quantum Inf. Comput. 4 (48) (2004). [16] V.E. Korepin, Phys. Rev. Lett. 92 (2004) 096402. [17] P. Calabrese, J. Cardy, J. Stat. Mech. P 06002 (2004). [18] C. Holzhey, et al., Nucl. Phys. B 424 (1994) 44. [19] H.-Q. Zhou, T. Barthel, J.O. Fjaerestad, U. Schollwoeck, Phys. Rev. A 74 (2005) 022326. [20] I. Affleck, M. Oshikawa, H. Saleur, J. Phys. A: Math. Gen. 31 (1998) 5827. [21] J.L. Cardy, Nucl. Phys. B 240 (1984) 514; J.L. Cardy, Nucl. Phys. B 324 (1989) 581.
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Physics Letters A www.elsevier.com/locate/pla
Geometric entanglement for quantum critical spin chains belonging to the Ising and three-state Potts universality classes Jin-Hua Liu, Hai-Tao Wang, Qian-Qian Shi, Huan-Qiang Zhou ∗ Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 30 March 2012 Received in revised form 8 July 2012 Accepted 17 July 2012 Available online 20 July 2012 Communicated by A.R. Bishop Keywords: Geometric entanglement Critical point Affleck–Ludwig g factor Boundary entropy
a b s t r a c t The leading finite-size correction to the geometric entanglement per lattice site is investigated for the antiferromagnetic–ferromagnetic alternating Heisenberg model, quantum three-state Potts model in a transverse field and a spin-1/2 spin chain with the competing two-spin and three-spin interactions at criticality, belonging to the Ising and three-state Potts universality classes with the central charge c = 1/2 and c = 4/5, respectively. Our results demonstrate that the leading finite-size correction coefficient is essentially the celebrated Affleck–Ludwig boundary entropy corresponding to a conformally invariant boundary condition, which in turn depends on the period of the translation-invariant separable states. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Recently, the geometric entanglement (GE) [1–4], a holistic measure of the multipartite entanglement present in a quantum state wave function, has attracted much attention, due to its role as a detector to locate critical points for quantum many-body lattice systems. In particular, an intriguing connection between the GE per site and the Affleck–Ludwig g factor has been established for a finite-size spin chain with the periodic boundary conditions at criticality. More specifically, the coefficient in the subleading term, which occurs in the finite-size correction to the GE per site, is related to the Affleck–Ludwig boundary entropy, or equivalently, the Affleck–Ludwig g factor. This was first discovered by J.-M. Stéphan, G. Misguich, and F. Alet in Ref. [5], given that the subleading term coefficient had been conjectured to be universal [6]. In order to check whether or not such a connection between the GE per site and the Affleck–Ludwig g factor is universally valid, we have studied three prototypical spin chains belonging to the Ising universality class to see if the same g factor appears [7]. To this end, an efficient method to systematically evaluate the GE per lattice site for quantum many-body lattice systems has been developed in the context of the tensor network algorithms based on matrix product state representations for quantum many-body lattice systems with the periodic boundary conditions [8–12]. Remarkably, it was found that the subleading term coefficient in the GE per lattice site depends on the period of the translation-invariant
*
Corresponding author. E-mail address: [email protected] (H.-Q. Zhou).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.07.014
separable states. Actually, for quantum XYX model in an external field, either conformally invariant free or fixed boundary condition appears, depending on the period of the translation-invariant separable states being one- or two-site, respectively. This provides further insights into the connection between the subleading term coefficient in the GE per lattice site and the Affleck–Ludwig boundary entropy. However, two major questions remain to be resolved. The first is whether or not the very same connection between the subleading term coefficient in the GE per lattice site and the Affleck– Ludwig boundary entropy is also valid for quantum critical manybody systems belonging to other universality classes, such as the Potts universality class. The second is whether or not the dependence of the subleading term coefficient in the GE per lattice site on the period of the translation-invariant separable states involved in the definition of the GE is generic, instead of being a peculiar feature for a specific model. In order to address these questions, we investigate the leading finite-size correction to the GE per lattice site for the antiferromagnetic–ferromagnetic (AF–F) alternating Heisenberg model at criticality, belonging to the Ising universality class with the central charge c = 1/2, and quantum three-state Potts model in a transverse magnetic field and a spin-1/2 chain with the competing two-spin and three-spin interactions at criticality, belonging to the three-state Potts universality class with the central charge c = 4/5. Our simulation results presented in this work, together with the previous work [7] on three prototypical critical quantum spin chains belonging to the same Ising universality class, confirm that the leading finite-size correction to the GE per lattice site is universal, in the sense that the subleading term coefficient
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is essentially the celebrated Affleck–Ludwig boundary entropy corresponding to a conformally invariant boundary condition. In addition, the dependence of the subleading term coefficient in the GE per lattice site on the period of the translation-invariant separable states is generic. 2. The geometric entanglement per lattice site and its connection to the Affleck–Ludwig g factor
s = ln( g ). 3. The models
(1)
(2)
H = J AF
where
Λmax = max φ|ψ. |φ
Here, Λmax represents the maximum fidelity between a quantum pure state |ψ and all the possible separable and normalized states |φ of the N parties. Since the contribution to E (ψ) from each party is additive, E (ψ) scales linearly with N, for a multipartite system consisting of N parties. Therefore, it is convenient to define the GE per party as
E N |ψ = N −1 E |ψ .
(3)
For a quantum many-body lattice spin system with the periodic boundary conditions, each lattice site constitutes a party. As such, E N is the GE per lattice site. As argued in Ref. [6], the GE per site, E N , scales as
E N ∼ E∞ +
b N
+O
1 N2
,
2 ln(2)
s.
N /2
σx[2i−1] σx[2i] + σ y[2i−1] σ y[2i] + AF σz[2i−1] σz[2i]
i =1
+ JF
N /2
σx[2i] σx[2i+1] + σ y[2i] σ y[2i+1] + F σz[2i] σz[2i+1] , (7)
i =1
[i ]
where σi (α = x, y , z) are the Pauli spin matrices at site i, J AF and J F are the antiferromagnetic and ferromagnetic couplings, AF and F are the anisotropic constants. Later on, we set J AF = 1, J F = −1, and AF = 1, with F as a control parameter. The second model is quantum three-state Potts model in a transverse magnetic field. The Hamiltonian is written as
H =−
N i =1
(4)
α =1,2
[i ]
[ i +1 ]
M x,α M x,3−α − λ
N
[i ]
Mz ,
(8)
i =1
[i ]
where N is the chain size, and b is the subleading term coefficient. The g factor, also known as the ground-state degeneracy, was introduced by Affleck and Ludwig [13] to characterize conformally invariant boundary conditions. Indeed, following Affleck and Ludwig [13], there is a g function that decreases along a boundary renormalization group flow from a less stable to a more stable critical point in the same bulk universality class. At a critical point, the value of the g function becomes the g factor. To understand the connection between the GE per lattice site and the Affleck–Ludwig g factor, we consider a cylinder of circumference N and height M N with spins at both circular edges. Each edge is supposed to be a state of a one-dimensional quantum chain with the periodic boundary conditions, described by a circle at the spatial direction. As is well known, any state can be projected to a ground state with infinite imaginary time evolution. Thus, the cylinder might represent the overlap between a separable state and the system’s ground state. With the space and imaginary time interchanged, the original edges become two boundary states, corresponding to some conformally invariant boundary conditions. Therefore, the determination of b amounts to calculating a subleading contribution of φbs |G , with |G being the ground state, and |φbs being a boundary state corresponding to a conformally invariant boundary condition. In this geometry, the free energy can be written in the form [13]: F = F bulk + 2F boundary , where F bulk ∼ N M represents the bulk free energy, and F boundary = aN + s + o(1) = − lnz φbs |G represents the boundary free energy. Here, a is the boundary free energy per party, and s, a subleading term in the boundary free energy, is the Affleck–Ludwig boundary entropy. Therefore, there exists a relation between the Affleck–Ludwig boundary entropy s and the subleading term coefficient b [5]:
b=−
(6)
We consider three critical quantum many-body spin chains with the periodic boundary conditions: one belongs to the Ising universality class with the central charge c = 1/2, and the other two belong to the three-state Potts universality class with the central charge c = 4/5. The first model we consider is the AF–F alternating Heisenberg model. The Hamiltonian is given by
Let us first recall the definition of the global GE per party. Mathematically, the global GE between a quantum pure state |ψ and all possible separable and normalized states |φ is defined as
E |ψ = − log2 Λ2max ,
Here, s is defined via the Affleck–Ludwig g factor:
(5)
where λ is the transverse magnetic field and M α (α = x, z) are the Potts matrices:
M x,1 =
Mz =
0 0 1
1 0 0 1 0 0
2 0 0 0 −1 0 0 0 −1
M x,2 =
,
0 0 1 0 0 1
1 0 0
,
.
The third model is a spin-1/2 spin chain with the competing two-spin and three-spin interactions. It is described by the Hamiltonian
H = J2
N
σz[i] σz[i+1] + J 3
i =1
N i =1
σz[i] σz[i+1] σz[i+2] − h
N
σx[i] ,
(9)
i =1
[i ]
where σα (α = x, z) are the Pauli spin-1/2 operators at site i, J 2 and J 3 are the two-spin and three-spin interaction constants, and h is an external field. We choose J 2 = 0.4 and h = 1, with J 3 as a control parameter. 4. Simulation results To simulate a finite-size translation-invariant spin chain with the period boundary conditions, we might resort to tensor network algorithms based on the matrix product state representations [8–12]. Here, we exploit the translation-invariant matrix product state algorithm [10], whose cost does not depend on the chain size N. For a finite-size periodic system, we first take advantage of the von Neumann entropy to locate a quantum critical point. As is well
J.-H. Liu et al. / Physics Letters A 376 (2012) 2677–2682
known, the von Neumann entropy, as a bipartite entanglement measure, characterizes the entanglement between two subsystems when we partition the system into two subsystems with sizes l and N − l. Mathematically, it is defined as S (ρ ) ≡ − tr(ρ log2 ρ ), where ρ denotes the reduced density matrix of a subsystem (of size l) for a given quantum state of the system. Here, we restrict ourselves to study the von Neumann entropy when the system is in a ground state. As such, it takes a maximum value when l = N /2, with N even. The von Neumann entropy for a subsystem consisting of a half chain, which we denote it as S h , exhibits a maximum when the control parameter crosses a pseudo-critical point; the critical point may be determined from an extrapolation to the thermodynamic limit. As is seen from the main panels in Fig. 1, the von Neumann entropy, S h , reaches a maximum value when the control parameter crosses a pseudo-transition point, which is denoted as Fc ( N ) in (a), as λc ( N ) in (b), and as J 3c ( N ) in (c), respectively. In the insets to Fig. 1, the critical points Fc (∞), λc (∞), and J 3c (∞) are, respectively, determined by performing an extrapolation of the pseudo-critical points Fc ( N ), λc ( N ), and J 3c ( N ) with respect to the size N. Specifically, we plot the von Neumann entropy S h as a function of the control parameter F for the AF–F alternating Heisenberg model in Fig. 1(a), with the size ranging from 20 to 100. Actually, the model is in the Haldane phase if F = 1, and in the Néel phase if F 1. Therefore, it undergoes a phase transition, as the control parameter F varies from 1 to ∞. In Fig. 1(a), the von Neumann entropy S h appears to be a maximum at a pseudo-critical point, Fc ( N ) for a given size N. We perform an extrapolation of F ( N ) with respect to the size N. Here, the fitting function takes the form: Fc ( N ) = Fc (∞) + β N −γ , with Fc (∞) = 2.2329, β = 130.35, and γ = 2.1358. Thus, the critical point in the thermodynamic limit is located at Fc (∞) = 2.2329, which is consistent with the known result Fc ∼ 2.32 [14]. Fig. 1(b) shows the von Neumann entropy S h as a function of the control parameter λ for the three-state Potts model, with the size ranging from 20 to 100. The pseudo-critical points λc ( N ) are identified from the maximum values of the von Neumann entropy S h for different values of the size N. By performing an extrapolation of the pseudo-critical points λc ( N ) with respect to the size N, the critical point is determined. Here, the fitting function is λc ( N ) = λc (∞) + β N −γ , with λc (∞) = 1.00097, β = −16.6307, and γ = 2.14455. The quantum critical point we have determined matches very well with the exact value λc = 1. In Fig. 1(c), we plot the von Neumann entropy S h as a function of the control parameter J 3 for a model with the two-spin and three-spin competing interactions, with the size ranging from 30 to 120. By performing an extrapolation of the pseudo-critical points J 3c ( N ) with respect to the size N, the critical point is determined. Here, the fitting function is J 3c ( N ) = J 3c (∞) + β N −γ , with J 3c (∞) = 0.90038, β = 325.48, and γ = 2.8564. For a critical system of the size N, the von Neumann entropy S (l), describing entanglement between two subsystems with sizes l and N − l, follows the universal logarithmic scaling with the subsystem size l: S (l) = c log2 ( N /π sin(π l/ N ))/3 + c 0 , where c is the central charge, and c 0 is a model-dependent constant [15–19]. In Fig. 2, we show the scaling relation between the von Neumann entropy S (l) and T (l) ≡ log2 ( N /π sin(π l/ N )) for three critical quantum models. The scaling of the von Neumann entropy yields the central charge, whose value determines the universality class a system belongs to. In Table 1, the central charge c fit and the constant c 0fit are fitted for three spin chains with different system sizes N. They are consistent with the exact central charge c = 1/2 for the Ising universality class and c = 4/5 for the three-state Potts universality class, as long as the chain size N is large enough. This indicates that the critical points we have determined are accurate.
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Fig. 1. (Color online.) Main: The von Neumann entropy S h for a subsystem consisting of a half chain as a function of the control parameter for three different periodic quantum spin chains: (a) the antiferromagnetic–ferromagnetic (AF–F) alternating Heisenberg model, with F as the control parameter, and the size ranging from 20 to 100; (b) the three-state Potts model, with λ as the control parameter, and the size ranging from 20 to 100; (c) the spin-1/2 chain with two-spin and three-spin competing interactions, with J 3 as the control parameter, and the size ranging from 30 to 120, respectively. Inset: The extrapolation of the pseudo-critical points corresponding to the maximum values of the von Neumann entropy S h with respect to the size N. In (a), the fitting function is Fc ( N ) = Fc (∞) + β N −γ , with Fc (∞) = 2.2329, β = 130.35, and γ = 2.1358. In (b), the fitting function is λc ( N ) = λc (∞) + β N −γ with λc (∞) = 1.00097, β = −16.6307, and γ = 2.14455. In (c), the fitting function is J 3c ( N ) = J 3c (∞) + β N −γ , with J 3c (∞) = 0.90038, β = 325.48, and γ = 2.8564.
Note that the relative fitting errors of the von Neumann entropy in Fig. 2 are all smaller than 3.3 × 10−3 for three models. Now we turn to the leading finite-size correction to the GE per lattice site for three spin chains at their respective critical points, as identified above. Besides the determination of the Affleck– Ludwig g factor for each model, our focus will be on the connection between the period of the translation-invariant separable states and the Affleck–Ludwig g factor. Generically, the imposed
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Table 1 The central charge c fit and the constant c 0fit are fitted from the scaling relation of the von Neumann entropy S (l) with the subsystem size l, for the critical quantum AF–F alternating Heisenberg model with the size N ranging from 20 to 100, the critical three-state Potts model with the size N ranging from 20 to 100, and the critical spin-1/2 chain with two-spin and three-spin competing interactions with the size N ranging from 30 to 150. AF–F alternating Heisenberg model Three-state Potts model
Spin chain with competing interactions
Fig. 2. (Color online.) Main: The scaling of the von Neumann entropy S (l) with the subsystem size l: S (l) = 3c T (l) + c 0 , with T (l) ≡ log2 ( N /π sin(π l/ N )), where c is the central charge, and c 0 is a model-dependent constant for the critical quantum AF–F alternating Heisenberg model, with the system size ranging from 20 to 100 in (a), for the three-state Potts model, with the system size ranging from 20 to 100 in (b), and for the spin-1/2 chain with two-spin and three-spin competing interactions, with the system size ranging from 30 to 150 in (c). Inset: The relative fitting errors S err = ( S fit (l) − S algo (l))/ S algo (l) are always smaller than 1.9 × 10−3 in (a), smaller than 6 × 10−4 in (b), and smaller than 3.3 × 10−3 in (c). Here, S fit (l) is the value extracted from the fit and S algo (l) is the value computed from the algorithm.
period of a translation-invariant separable state acts as a constraint condition. Indeed, for two commensurate periods, the set of all the translation-invariant separable states with the smaller period is a subset of the set of all the translationally invariant separable states with the larger period. As a consequence, the translationinvariant closest separable state could be different in the two sets, depending on whether or not it falls into the subset of the separable states with the smaller period. Therefore, two scenarios are expected: the first is that the GE per lattice site computed from the translation-invariant separable states with the smallest possible period is identical to that from the exact diagonalization
Size N
20
40
60
80
100
c fit
0.5127
0.5058
0.5040
0.5031
0.5025
c 0fit
0.6443
0.6504
0.6524
0.6534
0.6541
Size N
20
40
60
80
100
c fit
0.8031
0.8019
0.8013
0.8010
0.8007
c 0fit
1.0978
1.0982
1.0985
1.0986
1.0989
Size N
30
60
90
120
150
c fit
0.8367
0.8268
0.8226
0.8196
0.8175
c 0fit
0.9580
0.9685
0.9735
0.9768
0.9792
without any translation-invariant assumption, thus the smallest g factor is anticipated to be involved, given that it corresponds to the most stable fixed point in the sense of boundary renormalization group flows (according to the Affleck–Ludwig g theorem). This is exactly what happens in the three-state Potts model, as seen below. The second is that the GE per lattice site computed from the translation-invariant separable states with the smaller period is different from that computed from the translation-invariant separable states with the larger period, while the latter is identical to that from the exact diagonalization without any translationinvariant assumption. As such, different g factors corresponding to different conformally invariant boundary conditions appear in the finite-size correction to the GE per lattice site, with one of them being the smallest. As seen below, the AF–F alternating Heisenberg model and a spin-1/2 spin chain with the competing two-spin and three-spin interactions at criticality fall into this scenario. In Fig. 3, we plot the GE per lattice site E N as a function of the chain size N for the AF–F alternating Heisenberg model at the critical point Fc = 2.232, with the size N ranging from 4 to 100. The GE per lattice site E N is well fitted into the scaling function E N = a + b/ N + f / N 2 , with a = 0.678918, b = −0.010256, and f = −1.324811, and a = 0.078658, b = 1.002489 and f = −0.390788 for (a) and (b), respectively. In (a), the separable states are translation-invariant under two-site shifts. In (b), the separable states are translation-invariant under four-site shifts. The periods of the translation-invariant separable states are chosen to ensure that the translational invariance of the model Hamiltonian under two-site shifts are respected. The exact diagonalization for small sizes up to 22 is also performed to evaluate the GE per lattice site without any translation-invariant assumption, which is denoted by the five point stars in Fig. 3(b). Given the coefficient b, we get the Affleck–Ludwig g factors g FA = 1.0035 and g FA = 0.7064, respectively, for (a) and (b), according to the relation (5) between the Affleck–Ludwig g factor and the coefficient b. The two g factors match very well with the exact g factors g free = 1 and g fixed = √ 2/2, corresponding to the conformally invariant free and fixed boundary conditions in the Ising universality class, respectively. As such, we provide another critical spin chain belonging to the Ising universality class, for which the subleading term coefficient depends on the period of the translation-invariant separable states, given that either conformally invariant free or fixed boundary condition appears for quantum XYX model in an external field [7]. In Fig. 4(a), we plot the GE per lattice site E N as a function of the chain size N for quantum three-state Potts model at the critical point λ = 1, with the size ranging from N = 10 to 200. Here, the separable states are chosen to be translation-invariant under one-site shifts. The reason for such a choice is that, for this model, any other choice of the period of the translation-
J.-H. Liu et al. / Physics Letters A 376 (2012) 2677–2682
Fig. 3. (Color online.) Main: Relation between the GE per lattice site E N and the chain size N for the AF–F alternating Heisenberg model with the periodic boundary conditions. For the size N ranging from 12 to 100, the data are fitted into E N = a + b/ N + f / N 2 with a = 0.678918, b = −0.010256, and f = −1.324811 in (a), and a = 0.078658, b = 1.002489 and f = −0.390788 in (b). The separable states are translation-invariant under two-site shifts in (a), and translation-invariant under four-site shifts in (b), respectively. The five-point stars indicate the GE per lattice site from the exact diagonalization, with the chain size up to 22. The GE per lattice site from the exact diagonalization matches very well with those from the matrix data product state algorithm. Inset: the relative fitting errors εerr = (ε N − ε Nfit )/ε Nfit are always smaller than 3.4 × 10−5 in (a) and (b), where E Nfit is the value extracted from
the fit and E Ndata is our simulation value from the matrix product state algorithm for each N.
invariant separable states results in the same GE per lattice site. This is also consistent with the fact that the GE per lattice site computed from the exact diagonalization without any translationinvariant assumption, as shown by the five-point stars in Fig. 4(a), matches very well with those from the matrix product state algorithm. The finite size scaling for the GE per lattice site E N follows the function E N = a + b/ N + f / N 2 , where the fitting coefficients are a = 0.042783, b = 1.770539, and f = −3.518868. It yields the Affleck–Ludwig g factor g potts = 0.5413. On the other hand, various partition functions, referring to the fixed boundary conditions A, B, and C (corresponding to three possible states of the Potts variable), the mixed boundary conditions A B, AC , and BC , the free boundary condition “free” and a novel boundary condition “new” are systematically investigated in Ref. [20]. As it turns out, four different boundary conditions, labeled as A, A B, free and new, respectively, are conformally invariant, with the g factor being 0.5509, 0.8914, 0.9542, and 1.544, respectively. As such, our result g potts = 0.5413 matches very well with the exact g factor g A = 0.5509 (within a tolerant numerical error). That is, it corresponds to the so-called A-type fixed boundary condition, one of four conformally invariant boundary conditions for the three-state Potts model [20]. In addition, we stress that the Affleck–Ludwig g factor, arising from
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Fig. 4. (Color online.) Main: (a) The scaling of the GE per lattice site E N with the chain size N at the critical point λc = 1 for quantum three-state Potts model. For the size N from 10 to 200, the data are fitted into E N = a + b/ N + f / N 2 with the coefficients a = 0.042783, b = 1.770539, and f = −3.518868. The GE per lattice site from the exact diagonalization, as displayed by the five-pointed stars for the chain size up to 14, matches very well with those from the matrix product state algorithm. (b) The scaling relation between the GE per lattice site E N and the chain size N for a quantum spin chain with competing interactions. For the size N ranging from 9 to 150, the data are fitted into E N = a + b/ N + f / N 2 . We have a = 0.145519, b = 0.044916, and f = −0.526629 in the upper branch, and a = 0.030834, b = 1.776206 and f = −3.487129 in the lower branch. In the upper and lower branches, the separable states are translation-invariant under one-site shifts, and translation-invariant under three-site shifts. Note that the GE per site is the same for the upper and lower branches when the chain size is less than the threshold, which is N = 12. The GE per lattice site from the exact diagonalization, as displayed by the five-pointed stars for the chain size up to 24, matches very well. data Inset: the relative fitting errors εerr = (ε N − ε Nfit )/ε Nfit are less than 5.1 × 10−3 and 4.7 × 10−4 in (a) and (b), respectively, where E Nfit is the value extracted from the fit and E Ndata is our simulation value for each N.
the computation of the ground-state GE for the three-state Potts model, conforms to the previous observation [7] that the smallest g factor g A , which corresponds to the most stable fixed point in the sense of the Affleck–Ludwig g theorem, is involved in the leading finite-size correction coefficient of the GE per lattice site. In Fig. 4(b), we plot the GE per lattice site E N as a function of the chain size N for a spin chain with the competing two-spin and three-spin interactions, where the size N is chosen from 9 to 150. It is shown that the GE per lattice site E N is splitted into two branches for a large enough chain size (N > 12): the upper branch is for the separable states that are translation-invariant under one-site shifts, and the lower branch is for the separable states that are translation-invariant under three-site shifts. We also compute the GE per site by means of the exact diagonalization without any translation-invariant assumption, which is denoted by the fivepoint stars in Fig. 4(b). We find that this only reproduces the lower branch. Thus, the translation-invariant separable states under onesite shifts constitutes a true subset of the translation-invariant separable states under three-site shifts. Therefore, the separable
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states that are translation-invariant under three-site shifts is sufficient for the GE per lattice site to reach the smallest g factor corresponding to the most stable conformally invariant boundary condition. Both of the branches follow the scaling function E N = a + b/ N + f / N 2 with the coefficients a = 0.145519, b = 0.044916, and f = −0.526629 for the upper branch and a = 0.030834, b = 1.776206 and f = −3.487129 for the lower branch. Then we get the Affleck–Ludwig g factors g up = 0.9845 and g down = 0.5403 for the upper and lower branch, respectively. They are consistent with the exact values g free = 0.9542 and g A = 0.5509 [13,21] for conformally invariant free and A-type fixed boundary conditions in the three-state Potts universality class. 5. Summary We have investigated the GE per lattice site for the AF–F alternating Heisenberg model at criticality, belonging to the Ising universality class with the central charge c = 1/2, and quantum three-state Potts model in a transverse magnetic field and a spin1/2 spin chain with the competing two-spin and three-spin interactions at criticality, belonging to the three-state Potts universality class with the central charge c = 4/5. The relevant critical points have been identified by evaluating the von Neumann entropy in the context of the matrix product state representations. Our simulation results, together with the previous work [7], demonstrate that the leading finite-size correction to the GE per lattice site is universal, in the sense that the subleading term coefficient is essentially the celebrated Affleck–Ludwig boundary entropy corresponding to a conformally invariant boundary condition. For all the models investigated, the simulated g factors are comparable with the exact results from conformal field theory, with their relative errors less than 3.1 × 10−2 . In addition, we have focused on the correlation between the period of the translation-invariant separable states and the Affleck– Ludwig g factor. More precisely, the subleading term in the finitesize correction to the GE per lattice site is determined by the Affleck–Ludwig g factor corresponding to one of the boundary conformal field theories compatible with the bulk criticality, regardless of the value of the period. However, the value of the g factor, generically, depends on the period. Specifically, for the AF–F alternating Heisenberg model, either free or fixed boundary condition appears, depending on the period of the translation-invariant separable states: if the period is two-site, it corresponds to the free boundary condition; if the period is four-site, then it corresponds
to the fixed boundary condition. For the spin-1/2 chain with the competing two-spin and three-spin interactions, both conformally invariant, i.e., free and A-type fixed, boundary conditions appear: if the period is one-site, it corresponds to the free boundary condition; if the period is three-site, it corresponds to the A-type fixed boundary condition. Therefore, we conclude that the dependence of the subleading term coefficient on the period of the translationinvariant separable states is generic. Acknowledgements We thank Sam Young Cho and Bing-Quan Hu for enlightening discussions. The work is partially supported by the National Natural Science Foundation of China (Grant No. 11174375), the Innovative Talent Funds for Project 985 (Project No. WLYJSBJRCGR201103), and the Chongqing University Postgraduates Science and Innovation Fund (Project No. 200911C1A0060322). References [1] R.G. Unanyan, C. Ionescu, M. Fleischhauer, Phys. Rev. A 72 (2005) 022326; R. Orús, S. Dusuel, J. Vidal, Phys. Rev. Lett. 101 (2008) 025701; A. Botero, B. Reznik, arXiv:0708.3391; R. Orús, Phys. Rev. Lett. 100 (2008) 130502. [2] R. Orús, T.-C. Wei, Phys. Rev. B 82 (2010) 155120. [3] T.-C. Wei, D. Das, S. Mukhopadyay, S. Vishveshwara, P. Goldbart, Phys. Rev. A 71 (2005) 060305. [4] C.-Y. Huang, F.-L. Lin, Phys. Rev. A 81 (2010) 032304. [5] J.-M. Stéphan, G. Misguich, F. Alet, Phys. Rev. B 82 (2010) 180406R. [6] Q.-Q. Shi, R. Orús, J.O. Fjærestad, H.-Q. Zhou, New J. Phys. 12 (2010) 025008. [7] B.-Q. Hu, X.-J. Liu, J.-H. Liu, H.-Q. Zhou, New J. Phys. 13 (2011) 093041. [8] F. Verstraete, D. Porras, J.I. Cirac, Phys. Rev. Lett. 93 (2004) 227205. [9] P. Pippan, S.R. White, H.G. Evertz, Phys. Rev. B 81 (2010) 081103(R). [10] Q.-Q. Shi, H.-Q. Zhou, J. Phys. A: Math. Theor. 42 (2009) 272002. [11] B. Pirvu, F. Verstraete, G. Vidal, Phys. Rev. B 83 (2011) 125104. [12] D. Rossini, V. Giovannetti, R. Fazio, J. Stat. Mech. (2011) P05021. [13] I. Affleck, A.W.W. Ludwig, Phys. Rev. Lett. 67 (1991) 161. [14] J. Ren, S. Zhu, Eur. Phys. J. D 50 (2008) 103. [15] G. Vidal, et al., Phys. Rev. Lett. 90 (2003) 227902; J.I. Latorre, et al., Quantum Inf. Comput. 4 (48) (2004). [16] V.E. Korepin, Phys. Rev. Lett. 92 (2004) 096402. [17] P. Calabrese, J. Cardy, J. Stat. Mech. P 06002 (2004). [18] C. Holzhey, et al., Nucl. Phys. B 424 (1994) 44. [19] H.-Q. Zhou, T. Barthel, J.O. Fjaerestad, U. Schollwoeck, Phys. Rev. A 74 (2005) 022326. [20] I. Affleck, M. Oshikawa, H. Saleur, J. Phys. A: Math. Gen. 31 (1998) 5827. [21] J.L. Cardy, Nucl. Phys. B 240 (1984) 514; J.L. Cardy, Nucl. Phys. B 324 (1989) 581.