Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry

We study the entanglement properties of a higher-integer-spin Affleck-Kennedy-Lieb-Tasaki model with quantum group symmetry in the periodic boundary condition. We exactly calculate the finite size correction terms of the entanglement entropies from the double scaling limit. We also evaluate the geom...

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2012
Main Authors: Arita, C., Motegi, K.
Format: Article
Language:English
Published: Інститут математики НАН України 2012
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/148659
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Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry / C. Arita, K. Motegi // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 41 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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author Arita, C.
Motegi, K.
author_facet Arita, C.
Motegi, K.
citation_txt Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry / C. Arita, K. Motegi // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 41 назв. — англ.
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container_title Symmetry, Integrability and Geometry: Methods and Applications
description We study the entanglement properties of a higher-integer-spin Affleck-Kennedy-Lieb-Tasaki model with quantum group symmetry in the periodic boundary condition. We exactly calculate the finite size correction terms of the entanglement entropies from the double scaling limit. We also evaluate the geometric entanglement, which serves as another measure for entanglement. We find the geometric entanglement reaches its maximum at the isotropic point, and decreases with the increase of the anisotropy. This behavior is similar to that of the entanglement entropies.
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 081, 18 pages Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry Chikashi ARITA † and Kohei MOTEGI ‡ † Institut de Physique Théorique CEA, F-91191 Gif-sur-Yvette, France E-mail: chikashi.arita@cea.fr ‡ Okayama Institute for Quantum Physics, Kyoyama 1-9-1, Okayama 700-0015, Japan E-mail: motegi@gokutan.c.u-tokyo.ac.jp Received July 06, 2012, in final form October 23, 2012; Published online October 27, 2012 http://dx.doi.org/10.3842/SIGMA.2012.081 Abstract. We study the entanglement properties of a higher-integer-spin Affleck–Kennedy– Lieb–Tasaki model with quantum group symmetry in the periodic boundary condition. We exactly calculate the finite size correction terms of the entanglement entropies from the double scaling limit. We also evaluate the geometric entanglement, which serves as another measure for entanglement. We find the geometric entanglement reaches its maximum at the isotropic point, and decreases with the increase of the anisotropy. This behavior is similar to that of the entanglement entropies. Key words: valence-bond-solid state; entanglement; quantum group 2010 Mathematics Subject Classification: 17B37; 81V70; 82B23 1 Introduction Quantum entanglement is a fundamental feature in quantum mechanics, and is a primary re- source in quantum communication and quantum computation [6, 12, 23, 37]. Entanglement has become an important tool to characterize quantum many-body systems (see [2] for example for a review). In one dimensional spin systems, typical quantifications of quantum entanglement are the Rényi entropy SR(L, `) and von Neumann entropy SvN(L, `) of a subsystem A with ` sites and environment B with L− ` sites (see Fig. 1) SR(L, `) = log Tr(ρ(L, `))α 1− α , SvN(L, `) = lim α→1 SR(L, `). Here the reduced density matrix ρ(L, `) is obtained from the density matrix of a ground state |Ψ〉 by tracing out all spin degrees of freedom in the environment B ρ(L, `) = TrB |Ψ〉〈Ψ| 〈Ψ|Ψ〉 . (1) The entanglement spectrum, i.e. the set of the eigenvalues of the reduced density matrix, deter- mines the entanglement entropies. For one-dimensional gapless spin chains, the generic behavior of the entanglement entropies has been analyzed [7] by use of the conformal field theory. The entanglement entropies scale logarithmically with the size of the subsystem, the prefactor essen- tially given by the central charge of the corresponding conformal field theory. On the other hand, gapful chains have been analyzed by investigating particular models. One of the most important models is the Affleck–Kennedy–Lieb–Tasaki (AKLT) model [1] which was introduced to understand the massive behavior of integer spin chains [13, 14]. The entanglement mailto:chikashi.arita@cea.fr mailto:motegi@gokutan.c.u-tokyo.ac.jp http://dx.doi.org/10.3842/SIGMA.2012.081 2 C. Arita and K. Motegi 1 2 ` L A B 1 2 ` 2` N` Figure 1. Schematic pictures of the entanglement entropies (left) and the geometric entanglement (right) for the q-AKLT model with S = 3. entropies of the isotropic AKLT models have been investigated by examing the exact valence- bond-solid (VBS) ground state [9, 16, 17, 18, 22, 28, 31, 40]. For gapped systems which have finite correlation lengths, the entanglement entropies saturate at certain values when the size of the subsystems exceed certain lengths. The saturated values of higher rank and higher spin AKLT models are larger than the spin-1 AKLT model. Recently Santos et al. found surprisingly simple and useful formula for calculating the reduced density matrix for matrix product ground states [32, 33]. They applied it to the AKLT model of spin-1 and general integer spin S with quantum group symmetry (q-AKLT model) [3, 5, 10, 19, 20, 24, 35], and another massive Klümper–Schadschneider–Zittartz model [21] to study anisotropic effect. In this article, we study the entanglement properties of the q-AKLT model, following the results of [32, 33] and giving remarks and additional results. The more precise definition of the q-AKLT model on an L-site chain with the periodic boundary condition is as follows H = ∑ k∈ZL 2S∑ J=S+1 CJ(k, k + 1)(πJ)k,k+1, (2) where CJ(k, k+1) > 0, and (πJ)k,k+1, which acts on the k-th and (k+1)-th sites, is the Uq(su(2)) projection operator from VS ⊗ VS to VJ , where Vj is the (2j + 1)-dimensional highest weight representation of the quantum group Uq(su(2)) [8, 15]. The valence-bond-solid (VBS) ground state of this hamiltonian H has a matrix product form [3, 24], which generalize the isotropic higher-integer-spin [4, 11, 36] and spin-1 q-deformed AKLT models [5, 19, 35]. We check that the entanglement spectra for ` = 1 calculated from the formula of the reduced density matrix [32, 33] reproduce the one point functions originally derived in [3]. We achieve the finite size corrections of the entanglement entropies from the double scaling limit, which requires the second order term of the perturbation of the entanglement spectrum. We exactly calculate the finite size correction term of the von Neumann entanglement entropy SvN(`). Besides the entanglement entropies which characterize the bipartite entanglement, we also study the geometric entanglement, which is another kind of measure for entanglement, see Fig. 1. The geometric entanglement has been proposed as a measure for multipartite entanglement. It has been used to study quantum phase transitions [25, 26, 27, 28, 29, 30, 34, 38, 39], and has been measured experimentally recently [41]. Systems near criticality exhibit logarithmic divergences as the entanglement entropies. On the other hand, only a few analytic results are known for gapped systems. The geometric entanglement defined below can be regarded as the actual distance between the ground state of the system and the nearest fully separable state in the Hilbert space. We divide the L-site chain into N parties (L = N`). Consider a pure quantum state of N parties |Ψ〉 ∈ H = ⊗Ni=1H [i], where H [i] is the space of the ith party. The entanglement can be Entanglement Properties of a Higher-Integer-Spin AKLT Model 3 quantified by maximizing the fidelity |Λ| between the quantum state |Ψ〉 and all the possible separable and normalized states of N parties |Φ〉 = ⊗Ni=1|φ[i]〉, |φ[i]〉 ∈ H [i], |Λmax| = max |Φ〉 |Λ|, Λ = 〈Φ|Ψ〉√ 〈Ψ|Ψ〉 . (3) The logarithm of |Λmax| is taken E(Ψ) = −Log |Λmax|2, such that its value becomes zero when |Ψ〉 is separable or positive otherwise. The geometric entanglement per block is defined as the above quantity per party E(`) = − lim N→∞ E(Ψ) N , well defined in the thermodynamic limit. We evaluate the geometric entanglement for the spin S q-deformed VBS state |Ψ〉. We obtain the expression of the geometric entanglement for `→∞ and its finite size corrections with help of numerical calculations. For the evaluation of the entanglement entropies and the geometric entanglement, the spectral structure of the transfer matrix of the q-VBS state in the matrix product representation [3, 24] will be helpful. This article is organized as follows. In Section 2, we briefly review the matrix product repre- sentation [3, 24] of the VBS ground state of the q-AKLT model, which helps us for evaluating the entanglement entropies and the geometric entanglement. In Section 3, the finite-size correction terms of the entanglement entropies from the double scaling limit are calculated by perturba- tive analysis. We emphasize that the double scaling limits of the entanglement entropies and the leading term of the finite-size correction of the entanglement spectrum have been originally obtained by Santos et al. [32]. But we make Section 3 partially overlap their results so that this article can be self-contained and easy to read. In Section 4, we investigate the geometric entanglement with help of numerical calculations. Section 5 is devoted to the summary of this article. 2 q-VBS state In this section, we briefly review the matrix product representation of the higher-integer-spin q- VBS ground state and the spectral structure of the transfer matrix of the q-AKLT model [3, 24]. We use the following notations. For a real number c we define its q analogue as [c] = qc − q−c q − q−1 . We also define the q-shifted factorial and the q-shifted binomial for n ∈ Z≥0 as [n]! =  n∏ i=1 [i], n ∈ N, 1, n = 0, [ n k ] =  [n]! [n]![n− k]! , k = 0, . . . , n, 0, otherwise. The q-VBS state [3, 24], which is the exact ground state of the q-AKLT model (2), is expressed in the following matrix product form |Ψ〉 = Tr[g1 ? g2 ? · · · ? gL−1 ? gL], 4 C. Arita and K. Motegi where gk is an (S + 1)× (S + 1) vector-valued matrix acting on the k-th site whose element is given by gk(a, b) = (−1)S−iq(a+b−S)(S+1)/2 √[ S a ] [ S b ] [S − a+ b]![S + a− b]! |S; b− a〉k =: hab|S; b− a〉k, (0 ≤ a, b ≤ S). The symbol ? denotes the product A ? B = {∑ y |α〉xy ⊗ |β〉yz } xz for vector-valued matrices A = {|α〉xy}xy and B = {|β〉xy}xy. We define g†k by replacing each ket vector in the matrix gk by its corresponding bra vector: g†k(a, b) = hab k〈S; b− a|. Let us set an (S + 1)2 dimensional vector space as W = ⊕ 0≤a,b≤S C|ab〉〉, where {|ab〉〉 | a, b = 0, . . . , S} is an orthonormal basis. We define an (S+1)2×(S+1)2 matrix G acting on the space W as G = g† ⊗ g, 〈〈ab|G|cd〉〉 = g†(a, c)g(b, d) = δa−c,b−dhachbd, which plays the role of a transfer matrix. In [3], the spectral structure of the G matrix was clarified, i.e. the eigenvalues of G are given as λn = (−1)n([S]!)2 [ 2S + 1 S − n ] , n = 0, 1, . . . , S, with the degree of degeneracy 2n+ 1, and thus the squared norm of the ground state is given as 〈Ψ|Ψ〉 = TrGL = ∑ 0≤n≤S (2n+ 1)λLn . (4) The matrix G has the following block diagonal structure since 〈〈ab|G|cd〉〉 = 0 for a−b 6= c−d: G = ⊕ −S≤j≤S G(j), G(j) ∈ EndWj , Wj = min(S,S−j)⊕ i=max(0,−j) C|i, i+ j〉〉. The size of each block G(j) is (S − |j|+ 1)× (S − |j|+ 1). Each element of G(j) is 〈〈a, a+ j|G(j)|c, c+ j〉〉 = (−1)jq(a+c+j−S)(S+1) [S − a+ c]! [S + a− c]! × √[ S a ] [ S a+ j ] [ S c ] [ S c+ j ] . We construct intertwiners among the 2S + 1 blocks G(j) (j = −S, . . . , S). This helps us to construct eigenvectors of each block from another block with a smaller size. Let us define a family of linear operators {Ij}1≤|j|≤S as Ij : Wj →Wj−1 (j > 0), Entanglement Properties of a Higher-Integer-Spin AKLT Model 5 〈〈a, a+ j − 1|Ij |c, c+ j〉〉 =  q−a √ [a+ j] [S − a− j + 1] [j] [S − j + 1] , c = a, −q1−a−j √ [a] [S − a+ 1] [j] [S − j + 1] , c = a− 1, 0, otherwise, Wj →Wj+1 (j < 0), 〈〈a− j − 1, a|Ij |c− j, c〉〉 = 〈〈a, a− j − 1|I−j |c, c− j〉〉. By direct calculation, one finds that the matrix Ij enjoys the intertwining relation IjG (j) = G(j−1)Ij (1 ≤ j ≤ S), IjG (j) = G(j+1)Ij (−S ≤ j ≤ −1). Each block G(j) has a simple (nondegenerated) spectrum SpecG(j) = {λ`}|j|≤`≤S , and the corresponding eigenvectors are given by |λn〉〉j =  ∑ 0≤i≤S−n q(n+1)i √ [S − n]! [i+ n]! [S − i]! [S]![n]![S − i− n]![i]! |i, i+ n〉〉, n = j ≥ 0, ∑ 0≤i≤S−n q(n+1)i √ [S − n]! [i+ n]! [S − i]! [S]![n]![S − i− n]![i]! |i+ n, i〉〉, −n = j < 0, Ij+1Ij+2 · · · In|λn〉〉n, 1 ≤ j + 1 ≤ n ≤ S, Ij−1Ij−2 · · · I−n|λn〉〉−n, 1 < −j + 1 ≤ n ≤ S. (5) The `th-power of the G matrix is formally expanded as G` = ⊕ −S≤j≤S ∑ |j|≤n≤S λ`n j〈〈λn|λn〉〉j |λn〉〉j j〈〈λn|. 3 Finite size correction of the entanglement entropies In this section, we examine the finite-size correction of the entanglement entropies by studying the reduced density matrix. Recently, the following simple formula for the reduced density matrix (1) was found [33] ρ(L, `) = K(L− `)K(`) TrGL , (6) where the “K matrix” is defined as K(`) =MG` with a linear map M M (|ab〉〉〈〈cd|) = |ac〉〉〈〈bd|. (7) The reduced density matrix (6) is an (S + 1)2 × (S + 1)2 matrix, from which the rank of the density matrix is equal to or smaller than (S + 1)2. We study the reduced density matrix by combining (6) and the spectral structure of the transfer matrix G reviewed in the last section. Here we introduce some notations and make some general remarks. We define Kn = ∑ −n≤j≤n 1 j〈〈λn|λn〉〉j M (|λn〉〉j j〈〈λn|) , 6 C. Arita and K. Motegi so that the K matrix and the reduced density matrix are written as K(`) = ∑ 0≤n≤S λ`nKn, ρ(L, `) = 1∑ 0≤n≤S (2n+ 1)λLn ∑ 0≤n≤S 0≤n′≤S λL−`n′ λ`nKn′Kn. (8) One observes that Kn, K(`) and ρ(L, `) enjoy the same block diagonal structure as G: Kn = ⊕ −S≤j≤S K(j) n , K(`) = ⊕ −S≤j≤S K(j)(`), ρ(L, `) = ⊕ −S≤j≤S ρ(j)(L, `), K(j) n ,K(j)(`), ρ(j)(L, `) ∈ EndWj , since 〈〈ab|Kn|cd〉〉 = 0 for a−b 6= c−d. Note thatM (7) does not always map a matrix acting on a sector Wj to a matrix acting on the same sector. The spectrum of ρ(L, `) is, of course, given by the union of the spectra of ρ(j)(L, `)’s. Due to the symmetry 〈〈ab|ρ(L, `)|cd〉〉 = 〈〈ba|ρ(L, `)|dc〉〉, we have the degeneracy Spec ρ(j)(L, `) = Spec ρ(−j)(L, `). 3.1 Double scaling limit We first review the double scaling limit [32, 33] ρ = lim `→∞ lim L→∞ ρ(L, `), ρ(i) = lim `→∞ lim L→∞ ρ(i)(L, `). Noting the form (8) and |λn/λ0| < 1 (n = 1, . . . , S), we find the reduced density matrix becomes diagonal ρ = K0K0, 〈〈ab|ρ|cd〉〉 = δacδbd q2(a+b−S) [S + 1]2 (=: δacδbdpab). (9) The entanglement spectrum is, of course, given by the diagonal elements of ρ, i.e. {pab|a, b = 0, 1, . . . , S}.1 We notice that the degree of the degeneracy of the eigenvalue q2k [S+1]2 is S− |k|+ 1. For example, the spectrum for S = 2 is given as Spec ρ(2) : p02 = 1 [3]2 , Spec ρ(1) : p01 = 1 q2[3]2 , p12 = q2 [3]2 , Spec ρ(0) : p00 = 1 q4[3]2 , p11 = 1 [3]2 , p22 = q4 [3]2 , Spec ρ(−1) : p10 = 1 q2[3]2 , p21 = q2 [3]2 , Spec ρ(−2) : p20 = 1 [3]2 . (10) One can calculate P := Tr ρα = ∑ 0≤a≤S 0≤b≤S pαab = ( [α(S + 1)] [α][S + 1]α )2 . (11) 1This notation is different from that in [32]. Entanglement Properties of a Higher-Integer-Spin AKLT Model 7 0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 q S 2Log2 2Log3 2Log4 2Log5 2Log6 S=1 S=2 S=3 S=4 S=5 vN Figure 2. The von Neumann entropy in the double scaling limit SvN (12). Then we achieve the entanglement entropies in the double scaling limit [32, 33] SR = 1 1− α LogP = 2 1− α Log [α(S + 1)] [α][S + 1]α , SvN = 2 Log q ( q + q−1 q − q−1 − (S + 1) qS+1 + q−(S+1) qS+1 − q−(S+1) ) + 2 Log[S + 1], (12) see Fig. 2 for the von Neumann entropy in the double scaling limit. In particular, when q = 1, the spectrum is totally degenerated pab = 1 (S + 1)2 , (13) and the entropies become SR = SvN = 2 Log(S + 1), which agree with the case of the open boundary condition [16, 22, 40]. On the other hand, in the limit q → 0, only one eigenvalue survives p00 = 1, pab = 0 (a + b > 0), and the entropies become zero. 3.2 Finite-size correction We examine the finite-size correction of the entanglement entropies. We first take the limit L→∞ ρ(`) := lim L→∞ ρ(L, `) = K0 ∑ 0≤n≤S Knκ ` n, (14) ρ(j)(`) := lim L→∞ ρ(j)(L, `) = K (j) 0 ∑ 0≤n≤S K(j) n κ`n, (15) with κn = λn λ0 , and then consider the case ` = 1 and the behavior of the entropies for ` → ∞. Fig. 3 provides plots of the spectrum Spec ρ(`) of the reduced density matrix (14), i.e. the union of the spectra Spec ρ(j)(`)’s of (15), and the von Neumann entropy for S = 2 with q = 4/5. For ` = 1 and L→∞ the reduced density matrix becomes ρ(1) = K0(MG)/λ0. 8 C. Arita and K. Motegi 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 1 2 3 4 5 6 7 8 9 10 11 12 ` p (`) ab 1 2 3 4 5 6 7 8 9 10 11 12 1.5 1.6 1.7 1.8 1.9 2 2.1 ` S (`)vN Figure 3. The entanglement spectrum of ρ(`) (left) and the von Neumann entropy (right) for S = 2 with q = 4/5. The lines are drawn for ` ∈ R with replacement κ`n → |κn|` and κ`n → κn|κn|`−1 in (14) or (15). In the left figure, the dashed, dotted and solid lines correspond to Spec ρ(±2)(`), Spec ρ(±1)(`) and Spec ρ(0)(`), respectively. The eigenvalues of ρ(1) become zero except 2S + 1 ones, which is pointed out for S = 1 and 2 in [32, 33] pab(1) =  δa+b,k [S + |k|]![S − |k|]! [2S + 1]! S−|k|∑ c=0 q(S+2)(2c+|k|−S) [ S c ][ S c+ |k| ] , a× b = 0, 0, otherwise. (16) For example, the spectrum {pab(1)}ab of ρ(1) for S = 2 is given as Spec ρ(2)(1) : p02(1) = 1 [5] , Spec ρ(1)(1) : p01(1) = 1 + q8 q2(1 + q4)[5] , p12(1) = 0, Spec ρ(0)(1) : p00(1) = 1− q2 + 2q6 − q10 + q12 q4(1 + q4)[5] , p11(1) = 0, p22(1) = 0, Spec ρ(−1)(1) : p10(1) = 1 + q8 q2 (1 + q4) [5] , p21(1) = 0, Spec ρ(−2)(1) : p20(1) = 1 [5] . Let us show (16). We consider the submatrix ρ(k)(1), k ≥ 0. The case for k < 0 is similar. By direct calculation, we find the matrix elements of (S− k+ 1)× (S− k+ 1) submatrix ρ(k)(1) are given by 〈〈c, c+ k|ρ(k)(1)|c+ j, c+ j + k〉〉 = (−1)jq2c+k−S+(2c+j+k−S)(S+1) [S + k]! [S − k]! [2S + 1]! × √[ S c ] [ S c+ j ] [ S c+ k ] [ S c+ j + k ] . (17) The rank of ρ(k)(1) is 1, since the element of ρ(k)(1) (17) has a form Ac ×Bc+j . Thus, only one eigenvalue of ρ(k)(1) is nonzero, which is given by Tr(k) ρ (k)(1) = S−k∑ c=0 〈〈c, c+ k|ρ(k)(1)|c, c+ k〉〉 Entanglement Properties of a Higher-Integer-Spin AKLT Model 9 0 0.2 0.4 0.6 0.8 1.0 0 0.5 1.0 1.5 2.0 2.5 q Log3 Log5 Log7 Log9 Log11 S=1 S=2 S=3 S=4 S=5 S (1)vN Figure 4. The von Neumann entropy of the one-site subsystem SvN(1). = [S + k]![S − k]! [2S + 1]! S−k∑ c=0 q(S+2)(2c+k−S) [ S c ] [ S c+ k ] , (18) from the fact that the other eigenvalues are all 0. The expression (18) is actually identical to the one point functions derived in [3]. In particular, when q = 1, the non-zero eigenvalues are degenerated as pab(1) = 1 2S+1 (a× b = 0), and we have SR(1) = SvN(1) = Log(2S+ 1) [32]. One observes the monotonicity of the von Neumann entropy SvN(1) while 0 < q < 1, see Fig. 4. We turn to the behavior of entropies for ` → ∞. Noting again the form (8) and |κn| < 1 (n = 1, . . . , S), we find ρ(`) = ρ+ κ`1K0K1 +O ( κ`2 ) , `→∞. We denote the eigenvalue of ρ(`) by pab(`) corresponding to pab (9) when `→∞. Since the density matrix ρ in the double scaling limit is a diagonal matrix, it is not difficult to perform perturbative calculation. Noting |κ1|2 > |κ2| > |κ3| > · · · , we find pab(`) = pab + rabκ ` 1 + tabκ 2` 1 + o ( κ2` 1 ) , `→∞, (19) rab = 〈〈ab|K0K1|ab〉〉, tab = ∑ (c,d)6=(a,b) 0≤c,d≤S 〈〈cd|K0K1|ab〉〉〈〈ab|K0K1|cd〉〉 pab − pcd . Inserting (5) into rab and tab defined above, we have rab = qa+b−S [S + 1] qa+b−3S [3](1− q2a − q2a+2 + q2S+2)(1− q2b − q2b+2 + q2S+2) (q2 − 1)2[S][S + 1][S + 2] , (20) tab = ( q2(a+b−S)[3][2] [S + 2][S + 1][S] )2 × q−4[S − a+ 1][a][S − b+ 1][b]− q4[S − a][a+ 1][S − b][b+ 1] q2 − q−2 . (21) The first-order term (20) has been originally obtained in [32] (see equation (59) of [32] by changing the indices µ = S/2−a, ν = S/2−b and redefining q → q1/2), where the characteristic length is given by ξ = 1 Log([S+2]/[S]) . We also calculated the second-order term (21) which is needed for seeing the finite-size correction of the von Neumann entropy. For example, the spectrum {pab(`)}ab (`→∞) for S = 2 (which is shifted from (10) as (19)) is given as Spec ρ(2)(`) : 1 [3]2 − [2] [3][4] κ`1, 10 C. Arita and K. Motegi Spec ρ(1)(`) : 1 q2[3]2 + (1− q2)[2] q2[3][4] κ`1 + q2[2]2 (1− q4)[4]2 κ2` 1 , q2 [3]2 − (1− q2)[2] [3][4] κ`1 − q2[2]2 (1− q4)[4]2 κ2` 1 , Spec ρ(0)(`) : 1 q4[3]2 + [2] q2[3][4] κ`1 + [2]2 q2(1− q4)[4]2 κ2` 1 , 1 [3]2 + (1− q2)2[2] q2[3][4] κ`1 − [2] [4] κ2` 1 , q4 [3]2 + q2[2] [3][4] κ`1 − q6[2]2 (1− q4)[4]2 κ2` 1 , Spec ρ(−1)(`) : 1 q2[3]2 + (1− q2)[2] q2[3][4] κ`1 + q2[2]2 (1− q4)[4]2 κ2` 1 , q2 [3]2 − (1− q2)[2] [3][4] κ`1 − q2[2]2 (1− q4)[4]2 κ2` 1 , Spec ρ(−2)(`) : 1 [3]2 − [2] [3][4] κ`1, where we omit the symbol +o ( κ2` 1 ) . The Rényi entropy is expressed by pab, rab and tab up to the order of κ2` 1 as SR(`) = 1 1− α Log Tr (ρ(`))α = 1 1− α Log ∑ 0≤a≤S 0≤b≤S (pab(`)) α = 1 1− α Log { P ( 1 +Rκ`1 + Tκ2` 1 ) + o ( κ2` 1 )} = SR + R 1− α κ`1 + 1 1− α ( T − R2 2 ) κ2` 1 + o ( κ2` 1 ) , `→∞, (22) where R = α P ∑ 0≤a≤S 0≤b≤S pα−1 ab rab, T = α P ∑ 0≤a≤S 0≤b≤S ( pα−1 ab tab + α− 1 2 pα−2 ab r2 ab ) , with P defined by (11). By tedious but straightforward calculation, one finds R = α[3] [S][S + 2] ( [S + 2][αS]− [α(S + 2)][S] (q − q−1)[α(S + 1)][α+ 1] )2 , T = α ( [2][3] (q − q−1)2[S][S + 2] )2 [ [2(α− 1)] [2] ( ([(S + 2)(α+ 1)][S]− [S + 2][S(α+ 1)]) [α+ 2][α+ 1][α(S + 1)] )2 + α− 1 2 ( [2(S + 1)]2 [2][S + 1]2 − 2[2(S + 1)][α][(α+ 1)(S + 1)] [S + 1][α(S + 1)][α+ 1] + [2][α][(α+ 2)(S + 1)] [(S + 1)α][α+ 2] )2 ] . Then we find SvN(`) = SvN + ( 4q2 Log q 1− q4 − 1 2 ) κ2` 1 + o ( κ2` 1 ) , `→∞, (23) where the coefficient of κ`1 vanishes. Since the leading order term is κ2` 1 , the characteristic length is 2ξ. We find the coefficient of κ2` 1 depends on the anisotropy parameter q but is independent of the spin value S. Entanglement Properties of a Higher-Integer-Spin AKLT Model 11 As discussed in [32], the perturbation fails for the isotropic case due to the degeneracy (13), but the entanglement spectrum can be written by linear combinations of κn’s and has the same spectral structure for the transfer matrix G. For example, for S = 2, we have Spec ρ(2)(`) : 1 3 ( 1 3 − κ`1 2 + κ`2 6 ) , Spec ρ(1)(`) : 1 3 ( 1 3 − κ`1 2 + κ`2 6 ) , 1 3 ( 1 3 + κ`1 2 − 5κ`2 6 ) , Spec ρ(0)(`) : 1 3 ( 1 3 − κ`1 2 + κ`2 6 ) , 1 3 ( 1 3 + κ`1 2 − 5κ`2 6 ) , 1 3 ( 1 3 + κ`1 + 5κ`2 3 ) , Spec ρ(−1)(`) : 1 3 ( 1 3 − κ`1 2 + κ`2 6 ) , 1 3 ( 1 3 + κ`1 2 − 5κ`2 6 ) , Spec ρ(−2)(`) : 1 3 ( 1 3 − κ`1 2 + κ`2 6 ) , where no higher order term is needed. In [32] the finite-size corrections of the entanglement entropies for q = 1 were calculated as SR(`) = 2 Log(S + 1)− 3 2 ακ2` 1 + o ( κ2` 1 ) , `→∞, SvN(`) = 2 Log(S + 1)− 3 2 κ2` 1 + o ( κ2` 1 ) , `→∞, which agree with the limits q → 1 of (22) and (23). 4 Geometric entanglement In this section, we evaluate the geometric entanglement, which is another kind of measure of entanglement. We divide the chain into N parties (L = N`), and each of the N parties to be contiguous blocks of ` spins S. When N is large enough, the following expression for the fidelity |Λmax| (3) has been shown for PT -symmetric matrix product ground states |Ψ〉 in [25, 26, 29] |Λmax|2 = lim N→∞ |d|2N 〈Ψ|Ψ〉 , (24) where |d|2 is the quantity |d|2 = max {xi}: S∑ i=0 |xi|2=1 〈Aux |G`|Aux 〉, |Aux 〉 = ∑ 0≤a≤S 0≤b≤S xax ∗ b |ab〉〉. (25) Performing the maximization (25), one obtains the fidelity |Λmax| which finally leads to the analytic expression for the geometric entanglement E(`) E(`) = − lim N→∞ Log |Λmax|2 N . For convenience we set xi = rie √ −1θi (ri ≥ 0, θi ∈ R), and write x•i = r•i e √ −1θ•i if the setting {xi = x•i } maximizes 〈Aux |G`|Aux〉. 12 C. Arita and K. Motegi q E(1) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 q E(2) Figure 5. The geometric entanglements for ` = 1 (left) and 2 (right). In each graph, the line corresponds to S = 1 (26), and the markers × (S = 2), � (S = 3) and + (S = 4) are plotted based on numerical calculations. 4.1 Spin-1 We calculate the geometric entanglement for S = 1. By direct calculation, we have 〈Aux |G`|Aux 〉 = ( [3]` q[2] + (−1)` q−1[2] ) |x0|4 + ( 2[3]` [2] + 2(−1)` − 2(−1)` [2] ) |x0|2|x1|2 + ( q[3]` [2] + q−1(−1)` [2] ) |x1|4. Inserting r2 1 = 1− r2 0, we get 〈Aux |G`|Aux 〉 = 1− q 1 + q2 ( [3]` − (−1)` ) ( (1− q)r4 0 + 2qr2 0 ) + q2[3]` 1 + q2 + (−1)` 1 + q2 , where θi’s do not appear. Thus we find 0 < q < 1 : r•0 = 1, |d|2 = [3]` 1 + q2 + q2(−1)` 1 + q2 , q = 1 : |d|2 = 3` 2 + (−1)` 2 , q > 1 : r•0 = 0, |d|2 = q2[3]` 1 + q2 + (−1)` 1 + q2 , (26) where |d|2 = 〈Aux |G`|Aux 〉 is independent of {xi} at the isotropic point q = 1 [25], and the choice of r•0 changes discontinuously at this point. (We will see that this kind of “degeneracy” occurs for the higher spin case.) Inserting these forms and 〈Ψ|Ψ〉 = [3]L + 3(−1)L into (24), we finally achieve the geometric entanglement 0 < q < 1 : E(`) = Log ( 1 + q2 ) − Log ( 1 + q2(−[3])−` ) , q = 1 : E(`) = Log 2− Log ( 1 + (−[3])−` ) , q > 1 : E(`) = Log ( 1 + q2 ) − Log ( q2 + (−[3])−` ) , which generalizes [25]. The entanglement entropy takes its maximum at the isotropic point, decreases with the decrease of the anisotropy parameter q and finally becomes E(`) = 0 at q = 0, see Fig. 5. This behavior of the geometric entanglement is similar to the entanglement entropies. In the limit `→∞, we have E = Log ( 1 + q2 ) . Entanglement Properties of a Higher-Integer-Spin AKLT Model 13 q r 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 i . q ri . 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 6. Numerical calculations of the sets {r•i } which maximize |d|2 for S = 2 and ` = 1 (left) and 2 (right). The markers ©, 4 and × correspond to r•0 , r•1 and r•2 , respectively. 4.2 Spin-2 Let us consider first the isotropic case, where we have 〈Aux |G`|Aux 〉 = 1 3 λ`0 + 1 2 λ`1 + 1 6 λ`2 − (1− (r0 − r2)2)2 − 4r0r 2 1r2(cos(θ0 − 2θ1 + θ2) + 1) 2 ( λ`1 − λ`2 ) = 1 3 λ`0 + 1 2 λ`1 + 1 6 λ`2 − (r2 1 − 2r0r2)2 − 4r0r 2 1r2(cos(θ0 − 2θ1 + θ2)− 1) 2 ( λ`1 − λ`2 ) with λ0 = 40, λ1 = −20 and λ2 = 4. When ` is odd (resp. even), λ`1 < λ`2 (resp. λ`1 > λ`2). Using the first (resp. second) form, we find ` odd : r•0 − r•2 = 0, cos(θ•0 − 2θ•1 + θ•2) = −1, |d|2 = 1 3 λ`0 + 2 3 λ`2, ` even : r•21 − 2r•0r • 2 = 0, cos(θ•0 − 2θ•1 + θ•2) = 1, |d|2 = 1 3 λ`0 + 1 2 λ`1 + 1 6 λ`2. In the anisotropic case, thanks to the form 〈Aux |G`|Aux〉 = 4q2r0r 2 1r2 1 + q4 ( λ`1 − λ`2 ) cos(θ0 − 2θ1 + θ2) + (independent of {θi}), we have ` odd : cos(θ•0 − 2θ•1 + θ•2) = −1, ` even : cos(θ•0 − 2θ•1 + θ•2) = 1, which is the same as for the isotropic case. We use help of numerical calculations (see Fig. 6 for ` = 1 and 2), which indicates that ` odd : r•i = δi0 for q ≤ ∃ q•; r•0 > 0, r•1 = 0, r•2 > 0 for q• < q < 1, ` even : r•i = δi0 always maximizes 〈Aux |G`|Aux〉. One observes that the geometric entanglement with ` odd is not completely monotonic while 0 < q < 1, see Fig. 5. The set {r•0, r•2} is obtained by d dr0 ( 〈Aux |G`|Aux〉 ∣∣ r1=0,r2= √ 1−r20 ,cos(θ0−2θ1+θ2)=1 ) = 0 14 C. Arita and K. Motegi in the case where ` is odd and q• < q < 1 r•0 = q √ −(1− q2)(1 + q4)λ`0 + 2[3]λ`1 − (1 + 3q2 + q4 + q6)λ`2 (1− q2)2(1 + q4)λ`0 + 4q2[3]λ`1 − (1 + 2q2 + 6q4 + 2q6 + q8)λ`2 . (27) The transition point q• is obtained by solving (27) = 1, which approaches 1 as ` → ∞. The set {r•i } for q > 1 is obtained by replacing r0 ↔ r1 and q → 1/q. Under the assumption r•i = δi0, we have E(`) = −Log |d|2 λ`0 = −Log〈Aux |G`|Aux〉|ri=δi0 = Log(1 + q2 + q4)− Log ( 1 + q2(1 + q2 + q4)κ`1 1 + q4 + q8κ`2 1 + q4 ) , for 0 < q < 1 and sufficiently large `. 4.3 Spin-3 For the isotropic case, we have 〈Aux |G`|Aux 〉 = 1 4 λ`0 + 9 20 λ`1 + 1 4 λ`2 + 1 20 λ`3 − A 5 (λ`1 − λ`3), A = [ (r1r2 − 3r0r3)2 + 2 ( r2 2 − √ 3r1r3 )2 + 2 ( r2 1 − √ 3r0r2 )2 − 2r1r2 { 3r0r3(cos(α+ β)− 1) + 2 √ 3r0r1(cosα− 1) + 2 √ 3r2r3(cosβ − 1) }] , where α = θ0 − 2θ1 + θ2 and β = θ1 − 2θ2 + θ3. When ` is even, λ`1 > λ`3. Thus 〈Aux |G`|Aux 〉 is maximized by r•22 − √ 3r•1r • 3 = r•21 − √ 3r•0r • 2 = 0, cosα• = cosβ• = 1, |d|2 = 1 4 λ`0 + 9 20 λ`1 + 1 4 λ`2 + 1 20 λ`3. When ` is odd, the candidates of {θi} that maximize 〈Aux |G`|Aux 〉 for given {ri} are cosα = 3r2 2r 2 3 − 4r2 1r 2 2 − 3r2 0r 2 1 4 √ 3r0r2 1r2 , cosβ = 3r2 0r 2 1 − 4r2 1r 2 2 − 3r2 2r 2 3 4 √ 3r1r2 2r3 , cos(α+ β) = −3r2 0r 2 1 − 4r2 1r 2 2 + 3r2 2r 2 3 6r0r1r2 2r3 , (28) or cosα = −1, cosβ = −1, cos(α+ β) = 1, (29) or cosα = 1, cosβ = −1, cos(α+ β) = −1, (30) or cosα = −1, cosβ = 1, cos(α+ β) = −1. (31) Inserting (28), we get A = 9 4 − ( 3r2 0 + 2r2 1 + r2 2 − 3 2 )2 , and thus we find 3r•20 + 2r•21 + r•22 = 3 2 , |d|2 = 1 4 λ`0 + 1 4 λ`2 + 1 2 λ`3. (32) Entanglement Properties of a Higher-Integer-Spin AKLT Model 15 q ri . 1.00.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 q ri . 1.00.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 Figure 7. Numerical calculations of the sets {r•i } which maximize |d|2 for S = 3 and ` = 1 (left) and 2 (right). The markers ©, 4, × and � correspond to r•0 , r•1 , r•2 and r•3 , respectively. We end up achieving the same value |d|2 (32) for (29)–(31). For example, inserting (29), we get A = 9 4 − ( 3r2 0 + 2r2 1 + r2 2 − 3 2 )2 − (√ 3r0r1 − r1r2 + √ 3r2r3 )2 . The maximization for the anisotropic case with ` odd is more complicated than S = 2, see the numerical result in Fig. 7. We expect that ` odd : there exist q•, q••, q••• and q•••• such that r•i = δi0, 0 < q ≤ q•, r•1 = r•3 = 0, r•0 > 0, r•2 > 0, q• < q ≤ q••, r•i > 0, i = 0, 1, 2, 3, q•• < q < q•••, r•i = δi1, q••• ≤ q < q••••, r•0 = r•2 = 0, r•1 > 0, r•3 > 0, q•••• ≤ q < 1, ` even : r•i = δi0 always maximizes 〈Aux |G`|Aux〉. We also expect that these transition points q•, . . . , q•••• approach 1 as ` → ∞. Under the assumption r•i = δi0, we have E(`) = −Log |d|2 λ`0 = −Log〈Aux |G`|Aux〉|ri=δi0 = Log q3[4]− Log ( 1 + q2[3]2κ`1 [5] + q8κ`2 1− q2 + q4 + q14κ`3 (1− q2 + q4)[5] ) , for 0 < q < 1 and sufficiently large `. 4.4 General case We consider the maximization of 〈Aux |G`|Aux〉 for general S. As we observed in the previous subsections, we expect that, for given q < 1, ` odd: there exists `• such that the set {r•i = δi0} maximizes 〈Aux |G`|Aux〉 when ` > `•, ` even: the set {r•i = δi0} always maximizes 〈Aux |G`|Aux〉. 16 C. Arita and K. Motegi 0 0 q E Log2 Log3 Log4 Log5 Log6 S=1 S=2 S=3 S=4 S=5 0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 Figure 8. The geometric entanglement in the limit `→∞. Since the term λ`0 0〈〈λ0|λ0〉〉0 ∣∣∣∣ S∑ a=0 qar2 a ∣∣∣∣2 dominates in 〈Aux |G`|Aux〉 for ` → ∞, we have r•i → δi0, which supports the above assumption. Inserting {ri = δi0}, we have |d|2 = 〈Aux |G`|Aux 〉 ∣∣∣ ri=δi0 = S∑ k=0 λ`k 0〈〈λk|λk〉〉0 , and find E(`) = −Log ( S∑ k=0 κ`k 0〈〈λk|λk〉〉0 ) = Log(qS [S + 1])− Log ( S∑ k=0 qk(k+1) [2k + 1][S]![S + 1]! [S + k + 1]![S − k]! κ`k ) = Log(qS [S + 1])− q2[3][S] [S + 2] κ`1 + o ( κ`1 ) , `→∞. (33) Here we used the norm (4) of the q-deformed VBS state |Ψ〉 and the norm of the eigenvectors of the transfer matrix [3]. In the limit ` → ∞, we have the geometric entanglement E = Log(qS [S + 1]), which takes the maximum Log(S + 1) at q = 1 and approaches 0 as q → 0, see Fig. 8. The monotonic behavior while 0 < q < 1 is similar to the entanglement entropies. The isotropic point is a special case where the choice ri = δ0i or ri = δSi does not always maximize |d|2 for S ≥ 2 even if ` is large, as we saw for S = 2 and S = 3. Thus the asymptotic form (33) is no longer valid at the isotropic point. 5 Summary and discussion In this article, we studied some entanglement properties of the higher spin q-AKLT model with the periodic boundary condition from the matrix product representation of the q-VBS ground state. We exactly calculated the finite-size correction terms of the entanglement entropies by the perturbative calculation for the spectrum of the reduced density matrix. We found that the first-order correction term of the Rényi entropy vanishes by taking the limit α → 1. 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id nasplib_isofts_kiev_ua-123456789-148659
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1815-0659
language English
last_indexed 2025-12-07T17:40:27Z
publishDate 2012
publisher Інститут математики НАН України
record_format dspace
spelling Arita, C.
Motegi, K.
2019-02-18T17:38:54Z
2019-02-18T17:38:54Z
2012
Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry / C. Arita, K. Motegi // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 41 назв. — англ.
1815-0659
2010 Mathematics Subject Classification: 17B37; 81V70; 82B23
DOI: http://dx.doi.org/10.3842/SIGMA.2012.081
https://nasplib.isofts.kiev.ua/handle/123456789/148659
We study the entanglement properties of a higher-integer-spin Affleck-Kennedy-Lieb-Tasaki model with quantum group symmetry in the periodic boundary condition. We exactly calculate the finite size correction terms of the entanglement entropies from the double scaling limit. We also evaluate the geometric entanglement, which serves as another measure for entanglement. We find the geometric entanglement reaches its maximum at the isotropic point, and decreases with the increase of the anisotropy. This behavior is similar to that of the entanglement entropies.
C. Arita is a JSPS Fellow for Research Abroad
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry
Article
published earlier
spellingShingle Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry
Arita, C.
Motegi, K.
title Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry
title_full Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry
title_fullStr Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry
title_full_unstemmed Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry
title_short Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry
title_sort entanglement properties of a higher-integer-spin aklt model with quantum group symmetry
url https://nasplib.isofts.kiev.ua/handle/123456789/148659
work_keys_str_mv AT aritac entanglementpropertiesofahigherintegerspinakltmodelwithquantumgroupsymmetry
AT motegik entanglementpropertiesofahigherintegerspinakltmodelwithquantumgroupsymmetry