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 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.


Introduction
Quantum entanglement is a fundamental feature in quantum mechanics, and is a primary resource 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 S R (L, ) and von Neumann entropy S vN (L, ) of a subsystem A with sites and environment B with L − sites (see The entanglement spectrum, i.e. the set of the eigenvalues of the reduced density matrix, determines 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 essentially 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]. entropies of the isotropic AKLT models have been investigated by examing the exact valencebond-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 where C J (k, k+1) > 0, and (π J ) k,k+1 , which acts on the k-th and (k+1)-th sites, is the U q (su(2)) projection operator from V S ⊗ V S to V J , where V j is the (2j + 1)-dimensional highest weight representation of the quantum group U q (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 S vN ( ). 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 is the space of the ith party. The entanglement can be quantified by maximizing the fidelity |Λ| between the quantum state |Ψ and all the possible separable and normalized states of N parties The logarithm of |Λ max | is taken 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 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 representation [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 perturbative 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.

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 We also define the q-shifted factorial and the q-shifted binomial for n ∈ Z ≥0 as [i], n ∈ N, 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 where g k is an (S + 1) × (S + 1) vector-valued matrix acting on the k-th site whose element is given by 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 g k by its corresponding bra vector: Let us set an (S + 1) 2 dimensional vector space as 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 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 with the degree of degeneracy 2n + 1, and thus the squared norm of the ground state is given as The matrix G has the following block diagonal structure since ab|G|cd = 0 for a−b = c−d: 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 {I j } 1≤|j|≤S as By direct calculation, one finds that the matrix I j enjoys the intertwining relation and the corresponding eigenvectors are given by The th-power of the G matrix is formally expanded as G = −S≤j≤S |j|≤n≤S λ n j λ n |λ n j |λ n j j λ n |.

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, where the "K matrix" is defined as 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 so that the K matrix and the reduced density matrix are written as One observes that K n , K( ) and ρ(L, ) enjoy the same block diagonal structure as G: Note that M (7) does not always map a matrix acting on a sector W j 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

Double scaling limit
We first review the double scaling limit [32,33] Noting the form (8) and |λ n /λ 0 | < 1 (n = 1, . . . , S), we find the reduced density matrix becomes diagonal The entanglement spectrum is, of course, given by the diagonal elements of ρ, i.e. {p ab |a, b = 0, 1, . . . , S}. 1 We notice that the degree of the degeneracy of the eigenvalue q 2k [S+1] 2 is S − |k| + 1. For example, the spectrum for S = 2 is given as Spec ρ (2) : Spec ρ (0) : One can calculate Then we achieve the entanglement entropies in the double scaling limit [32,33] see Fig. 2 for the von Neumann entropy in the double scaling limit. In particular, when q = 1, the spectrum is totally degenerated and the entropies become 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 p 00 = 1, p ab = 0 (a + b > 0), and the entropies become zero.

Finite-size correction
We examine the finite-size correction of the entanglement entropies. We first take the limit with κ n = λn λ 0 , 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 The eigenvalues of ρ(1) become zero except 2S + 1 ones, which is pointed out for S = 1 and 2 in [32,33] p ab (1) = For example, the spectrum {p ab (1)} ab of ρ(1) for S = 2 is given as 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 The rank of ρ (k) (1) is 1, since the element of ρ (k) (1) (17) has a form A c × B c+j . Thus, only one eigenvalue of ρ (k) (1) is nonzero, which is given by c, c + k|ρ (k) (1)|c, c + k 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 p ab (1) = 1 2S+1 (a × b = 0), and we have S R (1) = S vN (1) = Log(2S + 1) [32]. One observes the monotonicity of the von Neumann entropy S vN (1) while 0 < q < 1, see Fig. 4.
We turn to the behavior of entropies for → ∞. Noting again the form (8) and |κ n | < 1 (n = 1, . . . , S), we find We denote the eigenvalue of ρ( ) by p ab ( ) corresponding to p ab (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 |κ 1 | 2 > |κ 2 | > |κ 3 | > · · · , we find p ab ( ) = p ab + r ab κ 1 + t ab κ 2 Inserting (5) into r ab and t ab defined above, we have 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 → q 1/2 ), where the characteristic length is given by ξ = . 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 {p ab ( )} ab ( → ∞) for S = 2 (which is shifted from (10) as (19)) is given as where we omit the symbol +o κ 2 1 . The Rényi entropy is expressed by p ab , r ab and t ab up to the order of κ 2 1 as where R = α P 0≤a≤S 0≤b≤S with P defined by (11). By tedious but straightforward calculation, one finds Then we find where the coefficient of κ 1 vanishes. Since the leading order term is κ 2 1 , the characteristic length is 2ξ. We find the coefficient of κ 2 1 depends on the anisotropy parameter q but is independent of the spin value S.
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 κ n 's and has the same spectral structure for the transfer matrix G. For example, for S = 2, we have Spec ρ (2) ( ) : 1 3 where no higher order term is needed. In [32] the finite-size corrections of the entanglement entropies for q = 1 were calculated as which agree with the limits q → 1 of (22) and (23).

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 |Λ max | (3) has been shown for P T -symmetric matrix product ground states |Ψ in [25,26,29] where |d| 2 is the quantity Performing the maximization (25), one obtains the fidelity |Λ max | which finally leads to the analytic expression for the geometric entanglement E( ) For convenience we set

Spin-1
We calculate the geometric entanglement for S = 1. By direct calculation, we have Inserting where θ i 's do not appear. Thus we find 0 < q < 1 : where |d| 2 = Aux |G | Aux is independent of {x i } 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 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

Spin-2
Let us consider first the isotropic case, where we have with λ 0 = 40, λ 1 = −20 and λ 2 = 4. When is odd (resp. even), λ 1 < λ 2 (resp. λ 1 > λ 2 ). Using the first (resp. second) form, we find odd : In the anisotropic case, thanks to the form 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 : 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 in the case where is odd and q • < q < 1 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 r 0 ↔ r 1 and q → 1/q. Under the assumption r • i = δ i0 , we have for 0 < q < 1 and sufficiently large .
Inserting (28), we get and thus we find We end up achieving the same value |d| 2 (32) for (29)- (31). For example, inserting (29), we get The maximization for the anisotropic case with odd is more complicated than S = 2, see the numerical result in Fig. 7. We expect that We also expect that these transition points q • , . . . , q •••• approach 1 as → ∞. Under the assumption r • i = δ i0 , we have for 0 < q < 1 and sufficiently large .

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 = δ i0 } maximizes Aux |G | Aux when > • , even: the set {r • i = δ i0 } always maximizes Aux |G | Aux .
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(q S [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 r i = δ 0i or r i = δ Si 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.

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. This requires the second-order perturbation of the entanglement spectrum for calculation of the finite-size correction of the von Neumann entropy. It would be interesting to extend the study of entanglement properties to various generalizations, the entanglement entropies with multiple blocks (see [31] for the isotropic spin-1 case), for example. We also investigated the geometric entanglement. The geometric entanglement in the limit → ∞ decreases with the decrease of the anisotropy parameter q while 0 < q < 1. This property is the same as the entanglement entropies. Under an assumption which is based on numerical results, we calculated the finite-size correction of the geometric entanglement.