Bell-type correlation at quantum phase transitions in spin-1 chain

Quantum correlation, that has no classical counterparts, plays a pivotal role in studying the many-body systems. Investigation of such a complex massive system through the correlation is a major research topic in the statistical and the condensed matter physics as they provide useful view for the quantification of physical properties in a large scaled systems De Chiara and Sanpera (2018). Especially, the trait of quantum phase and its phase transition driven by the quantum fluctuation in correlation have been attracted much interests throughout the many years as they are closely related to macroscopic quantum phenomena such as superconductivities and topological state of a matter. They are analyzed through the various measures that has been rooted at the quantum information theory.

Quantum entanglement is one of the most crucial concepts that has been extensively studied in the last two decades to comprehend the quantum phase transitions and the exotic quantum properties in condensed matter systems Amico et al. (2008). Concurrence, one of the most used measures for two-qubit entanglement, is exploited to estimate the quantum criticality and, further, its scaling behaviour is also analyzed near the critical point of the one-dimensional XY model Osterloh et al. (2002); Osborne and Nielsen (2002). Entanglement entropy of a block in various many-body systems also has been discussed during the analysis using the conformal field theory Eisert et al. (2010). The entanglement entropy demonstrates logarithmic divergence at critical points, whereas it saturates for noncritical systems associated with the area law Eisert et al. (2010); Vidal et al. (2003). Subsequently, entanglement instabilities demonstrates the origin of the quantum criticality precisely and their analytic dynamics as the interaction parameters are modified Son et al. (2009).

It has been well known that all pure entangled states for two-qubit systems violate CHSH-Bell inequality Capasso et al. (1973). On the other hands, the non-local character of mixed entangled states are not easily quantifiable since it does not necessarily violate Bell inequalities Werner (1989). It means that quantum entanglement is not necessarily equivalent to the concept of Bell non-locality Werner (1989); Augusiak et al. (2015). Thus, it can be said that the concept of entanglement is closely related to the non-locality while they addresses the different physical quantities in general.

Unlikely to the many-body entanglement, that has been subjected to extensive investigation, the Bell non-locality in many-body systems has been attracted less attention relatively so far. It is because generalization of non-locality is needed to consider all the possible local measurements and locate their non-trivial combinations of the joint probabilities. It is only recent that the non-localities in many-body systems get to make an active progress. At first, Bell correlations for many-body systems are applied to two-site reduced density states which are generated from the ground state in one-dimensional (1D) spin-1/2 XY model Batle and Casas (2010) and spin-1/2 XXZ model Justino and de Oliveira (2012). As a result, it has been known that the Bell correlation demonstrates the non-analyticity at the critical point while, due to the monogamy characteristics of the correlation, any bipartite Bell inequality is not violated by the translational invariant many-body systems de Oliveira et al. (2013).

Differently from the bi-partite quantum non-locality, when multipartite correlation is considered, the violation of local realism is possible to be observed, especially in the 1D spin-1/2 model of the nearest-neighbor interactions Sun et al. (2014a, b). It is also quite recent that non-locality in the Lipkin-Meshkov-Glick (LMG) model can be found through multipartite Mermin inequalities Bao et al. (2020). The violation is manifested when the external magnetic field is adjusted. Furthermore, multipartite non-locality, that takes into account one- and two-body correlations imposing on the permutational symmetry, has been found in the Dicke state as well as the ground state of the isotopic LMG model Tura et al. (2014, 2015). The violation of the permutational invariant Bell inequality has been experimentally tested by the spin squeezed state that is realized by the large number of atoms in the Bose-Einstein condensate state Schmied et al. (2016). It is also found that the maximal violation is observed at the quantum criticality in the Ising model of infinite-range interactions Piga et al. (2019).

It is in two-dimensional local Hilbert space that the most Bell correlations are elucidated in aforementioned studies. In other words, Bell correlations in d-dimensional systems of many particles have been rarely investigated, whereas the other measures of quantum information theory, such as fidelity and quantum coherence, are used to characterize the quantum phase transitions (QPTs) many times in spin-1 chains or more Tzeng and Yang (2008); Goli et al. (2013); Malvezzi et al. (2016). In this work, we use a generalized high-order Bell correlation obtained from high-dimensional spin measurements which is proposed by D. Collins et al Collins et al. (2002) and W. Son et al Son et al. (2006); Bae and Son (2018). Especially, we analyze its Bell correlation in the spin-1 chain to identify its exotic quantum phase near the point of QPTs.

In particular, we choose 1D spin-1 XXZ model with the onsite anisotropy, that is one of the most studied spin-1 chain model. Higher-spin XXZ system have richer quantum phases in various different parameter regimes and certain parts of the system are need to be depicted with more complicated physics like symmetry-protected topological phases Kjäll et al. (2013). Since the ground state of spin-1 chains are not possible to be obtained analytically, we utilize numerical techniques, the density-matrix renormalization group (DMRG) method, for the ground state of spin-1 system which are described in matrix product state (MPS) representations. As a result, we found various QPTs that is accurately indicated by analyzing the derivatives of the CGLMP-SLK correlation : FM-to-XY, Haldane-to-AFM, Haldane-to-large-D, and XY1-to-XY2 quantum phase transitions. The first-order QPTs (FM-to-XY transitions) shows discontinuity of the CGLMP-SLK correlation. The Ising-type (Haldane-to-AFM and XY1-to-XY2) QPTs occur where the second derivative of the correlation vanishes. The Haldane-to-large-D transition, also referred to as the Gaussian phase transition, emerges when $\partial\langle\hat{\mathcal{B}}_{r}\rangle/\partial D=0$. The criticality of the correlation is caused by the fact that both the derivatives of the spin-spin correlations $C^{(1)}_{r}$ and $C^{(2)}_{r}$ are simultaneously zero in the criticality. Moreover, we discuss whether violation of the CGLMP-SLK inequality can be detected even in three-dimensional systems.

The outline of this paper is given as following: In Sec.II, we starts with the system hamiltonian of the 1D spin-1 XXZ model with the onsite anisotropy. There, we briefly explains its phases at the different parameter regime and their criticalities at the QPTs. In sec.III, we presents the Bell correlation in the CGLMP-Bell inequality from the generalized SLK-Mermin-Bell formalism described in an operator form of nonlocal order parameter. In Sec.IV, we introduce the matrix product state representations for the ground state of the infinite-size system along with the iDMRG method. Then, using the state obtained, we present our results for the description of Bell correlations in overall parameter space and analysis in the vicinity of the criticality in Sec.V, and we conclude this paper in Sec.VI.

We now start with the description of the system to be analyzed. The Hamiltonian for the one-dimensional spin-1 XXZ model with the on-site anisotropy is of the form

where $\hat{S}^{a}_{j}$ for $a=x,y,z$ denotes the spin-1 operators at site $j$. In the spin-z basis, the spin-1 operators are given by

The summation in Eq.(1) illustrates the one-dimensional spin-1 XXZ chain with the exchange anisotropy $J_{z}$ and we set $J=1$ for the unit energy scale.

The second sum in Eq.(1) describes an on-site anisotropy where the constant D means the uniaxial single-site anisotropy. It has been known that this model can be in the various phases including the antiferromagnetic (AFM) phase, the ferromagnetic (FM) phase, XY phase, the Haldane phase, and the large-D phase. Additionally, the ground state of the system undergoes four types of the QPTs, as varying the parameters $J_{z}$ and $D$. The ground-state phase diagram of the spin-1 chain Eq.(1) is investigated in the earlier study Chen et al. (2003). Here we now briefly demonstrate the characteristics of the different phases and QPTs in the given parameter space.

Bell-type inequalities examine the correlation between two parties of a composite quantum state and determine whether a given quantum state is nonlocal through the violation of local realistic bounds. Clauser-Horne-Shimony-Holt (CHSH) version of Bell inequality is given for the simplest symmetric states where two dichotomic measurements are performed for each of the parties Clauser et al. (1969). If the CHSH correlation admits the local hidden variable model, its absolute value does not exceed the local realistic bound 2. Taking pure entangled states into account, one can find violation of the CHSH inequality and its maximal violation $2\sqrt{2}$ derived by Tsirelson Cirel’son (1980). For the case of two-qubit mixed states, entangled ones do not always violates this inequality Werner (1989).

When it comes to the systems of arbitrary dimension $d$, the inequality derived by Collinset. al. provides a generalized version of CHSH inequality Collins et al. (2002). Further generalization of CHSH for the multiparty $d$-dimesional systems has been made by Son et. al. which is also consistent to the CGLMP Son et al. (2006); Bae and Son (2018). In these scenarios, all the parties are permitted to perform two different measurements at each site with $d$-outcome measurements.

From here, the correlation on the left hand side of the inequality for $d\times d$ system is said to be CGLMP correlation as it reduced into the one found by CGMLP Collins et al. (2002). Our description is made not by set of probabilities but by the form of correlation functions. It can be proved that the representations are equivalent as long as an appropriate collection of the correlations is made through the right choices of weighting coefficients Bae and Son (2018).

In the scenario of the CGLMP inequality, there are two observers ($j=1,2$) and each of them has two measurement choices $V_{j}\in\{A_{j},B_{j}\}$. The measurement operators for the $j$-th party are given as

where $d$ is the index of the number of possible outcomes, $\omega^{\alpha}$ is the eigenvalue, and $|\alpha\rangle_{V_{j}}$ is the corresponding eigenvector. One observer chooses the Fourier transformed basis and the other observer takes the inverse Fourier basis. The Fourier transformed basis is given as

where $\omega=\exp(2\pi i/d)$ and $\phi_{V_{j}}$ denotes a phase shift for a measurement $\hat{V}_{j}$. Here, the basis $|\beta\rangle$ is nothing but the computational basis like the spin-$z$ basis for $d=2$.

As to be maximal tests, four different measurement choices can be made through four different phase shifts that are $\phi_{A_{1}}=0$, $\phi_{B_{1}}=1/2$,$\phi_{A_{2}}=-1/4$, and $\phi_{B_{2}}=1/4$. Based upon this expression, CGLMP correlation can be described by the expectation value of the following composite operator,

when the weight function becomes $f_{n}\equiv\frac{1/2}{d-1}\omega^{\frac{n}{4}}\sec(n\pi/2d)$. As it is addressed in Bae and Son (2018), the weight $f_{n}$ can be specified for the convex sum of the probability distributions suggested by Collins et. al. Collins et al. (2002). Bell correlation Eq.(III) for the case of $d=2$ is obviously equivalent to one in the CHSH inequality. Local realistic bounds of the CGLMP correlation for $d=3$ are given by $-4\leq\langle\hat{\mathcal{B}}\rangle\leq 2$, which can be derived from $f_{n}$ Bae and Son (2018).

It is convenient to investigate the structure of the operator $\langle\hat{\mathcal{B}}\rangle$ in Eq.(III) by introducing the $n$-level lowering operator $\hat{J}^{n}_{j}\equiv\sum^{d-1}_{\beta=n}|\beta\rangle_{j}\langle\beta-n|$ for $j$-th party. From Eq.(5) and Eq.(6), one can derive the identities such that

Thus, the operator $\hat{\mathcal{B}}$ can be simplified as

This operator holds for the local basis of dimension $d$. In this work, we analyze the spin-1 chain where the dimension of local basis is set to be three, i.e., $d=3$. When the set of states $\{|\beta\rangle\;|\;\beta=0,1,2\}$ is regarded as the computational basis, the matrix representation of $\hat{J}$ and $\hat{J}^{2}$ is given by

where $\hat{J}\equiv\hat{S}^{-}/\sqrt{2}=(\hat{S}^{x}-i\hat{S}^{y})/\sqrt{2}$ and $\hat{J}^{2}\equiv\hat{J}\cdot\hat{J}$. The operators shift the higher spin state into the lower spin state in one step and two steps respectively. The choice of measurements $A_{i}$ and $B_{i}$ in Eq.(III) using the corresponding $\phi_{A_{i}}$ and $\phi_{B_{i}}$ lead to the annihilation operators of higher order and they identifies all the matrix elements of the given state. It can be found that the convex sum of the correlation function provides the symmetric selection of relevant probabilities for the correlations in the parties as it is addressed in Bae and Son (2018).

In the next step, it is required to investigate the correlation of ground state for the physical properties of many-body systems in Eq. (1). We utilize a matrix product state (MPS) representation, which helps to identify the Schmidt coefficients of the many-body pure state as they are in the product forms of the matrices Pérez-García et al. (2007); Schollwöck (2011); Orús (2014). A quantum state $|\Psi\rangle$ on the 1D systems of length $N$ can be written in canonical MPS form Vidal (2003):

where $\Gamma^{[j]}_{s_{j}}$ is $\chi_{j-1}\times\chi_{j}$ matrices and $|s_{j}\rangle$ is the $j$th-site local basis of dimension $d$ with the physical indices $s_{j}=0,\ldots,d-1$. When the elements of matrices $\Gamma^{[j]}_{s_{j}}$ for any $s_{j}$ are expressed as $\Gamma^{[j]}_{s_{j};\mu_{j-1}\mu_{j}}$, we call the subscript $\mu$ the bond index and its dimension $\chi$ the bond dimension. The matrices $\Gamma^{[1]}$ and $\Gamma^{[N]}$ at the ends ($j=1,N$) become vectors whose dimensions are $1\times\chi_{1}$ and $\chi_{N-1}\times 1$, respectively. In the case of the periodic boundary conditions, these matrices are of dimensions $\chi_{N}\times\chi_{1}$ and $\chi_{N-1}\times\chi_{N}$ and there is an additional $\Lambda^{[N]}$ connected to them by summing over $\chi_{N}$ Verstraete et al. (2004). The matrices $\Lambda^{[j]}$ are real, diagonal, and of dimension $\chi_{j}\times\chi_{j}$. These matrices should satisfy the relation such that

which stems from the normalization condition $\langle\psi|\psi\rangle=1$. In this paper, we are interested in the 1D infinite chains. The infinite matrix product states (iMPS) can be represented by

where the superscript $[j]$ on each matrices is omitted since the boundary effect in the thermodynamic limit is insignificant and it is invariant under the translational symmetry.

The diagonal elements $\lambda_{\mu_{j}}$ in $\Lambda^{[j]}$ are nothing but the Schmidt coefficients and signify the entanglement spectrum of two partitions where the one consist of spins from $1$ to $j$ and the other ranges from $j+1$ to $N$. The Schmidt state of Eq.(11) can be written as

where the nonnegative $\lambda_{\mu_{j}}$ satisfy the property $\sum_{\mu_{j}}\lambda^{2}_{\mu_{j}}=1$. The Schmidt bases of the left partition and the right partition are given as

It is notable that the matrix $\Lambda^{[j]}$ signifies the bipartite entanglement of two subsystems $L$ and $R$. This is because the entanglement entropy is expressed as $S(|\Psi\rangle)=-\sum_{\mu_{j}}\lambda^{2}_{\mu_{j}}\log\lambda^{2}_{\mu_{j}}$ from its definition.

For the 1D gapped systems, the ground state can be efficiently approximated by this MPS representation Verstraete and Cirac (2006) and the entanglement entropy of the ground state of a bipartite system $L$ and $R$ is finite according to the area law Hastings (2007). When the system lies in the vicinity of the quantum criticality, the entanglement entropy for the 1D infinite systems diverges logarithmically Vidal et al. (2003). That is, a lack amount of entanglement that comes from a small bond dimension $\chi$ causes the inefficiency in the MPS representation to describe the ground state at the criticality. Nevertheless, the scaling properties of finite entanglement at the quantum criticality is suggested based on the conformal field theory Pollmann et al. (2009).

In order to calculate the CGLMP correlation in spin chains which describes the two-body correlation among the many particles, the matrix product states (MPS) representation is applied for the depiction of the ground state and the matrix product operator (MPO) representation for the operator $\hat{\mathcal{B}}$. Together with the MPS representation for the ground state, the operator $\hat{\mathcal{B}}_{i,i+r}$ between parties $i$ and $i+r$ can be described by the MPO representation that reads

where

and $\mathds{1}$ denotes the identity matrix of dimension $4\times 4$ as shown in Fig. 1(a). Here, $\hat{\mathcal{J}}^{L}$ and $\hat{\mathcal{J}}^{R}$ can be obtained by using the method according to Fröwis et al. Fröwis et al. (2010) and schematic representation of the operator expectation in MPS form is depicted in Fig.1.

At first, let us present the CGLMP correlation $\langle\hat{\mathcal{B}}_{i,i+r}\rangle$ that is obtained by the expectation value of the operator $\hat{\mathcal{B}}$ for the ground state of Eq.(1) in the thermodynamic limit. The function is plotted as varying the parameters $J_{z}$ and $D$ for the spins at different sites distanced $r$. Here, the class of matrices $\Gamma_{s_{j}}$ and $\Lambda$ for the ground state can be obtained from the iDMRG method mentioned in Sec.IV. The CGLMP correlation $\langle\hat{\mathcal{B}}_{i,i+r}\rangle$ is calculated through the summation of all tensors $\Gamma$ and $\Lambda$ with $\hat{\mathcal{J}}^{L}_{i}$ and $\hat{\mathcal{J}}^{R}_{i+r}$. A graphical description for the evaluation of $\langle\hat{\mathcal{B}}_{i,i+r}\rangle$ is given in Fig.1. In the figures, the identity matrix $\mathds{1}^{L}$ comes from the overlap $\langle\alpha^{\prime}_{i-1}|\alpha_{i-1}\rangle=\delta^{L}_{\alpha^{\prime},\alpha}$ in Eq.(12) and Eq.(15) and the same thing is obtained for $\mathds{1}^{R}$.

As a result, we here have obtained the characteristics of the CGLMP correlation $\langle\hat{\mathcal{B}}\rangle$ for the particular set of fixed parameters $D=0$, $J_{z}=1$ and $J_{z}=-1$. In the parameter region, various phases and QPTs have been identified. The criticalities can be clearly found through the non-analyticity of the correlation function in the first and the second order. Moreover the contribution of higher moment correlation for the criticalities is possible to be discriminated. However, we found that it is not possible to observe the violation of the CGLMP-type inequality in the ground state of Hamiltonian Eq.(1). They are presented in Fig.2 (a), Fig.3 (a) and Fig. 4 (a). Such non-violation of the CGMLP criteria is consistent to the cases of CHSH in the earlier studies of 1D spin-1/2 XY chain Batle and Casas (2010) and XXZ chain Justino and de Oliveira (2012).

The reduced density matrices for two sites in the model of translational symmetry are not possible to violate bipartite Bell inequality. Nevertheless, the CGLMP correlation $\langle\hat{\mathcal{B}}_{r}\rangle$ can be applied to elucidate the physical properties for the various quantum phases in the system of Eq.(1). One of the rationales for the versatile quantification is that Eq.(9) for $d=3$ is nothing but the linear combination of spin-spin correlation functions in the higher moments where the $n$-level lowering operator $\hat{J}^{n}_{j}$ is regarded as the spin-1 lowering operator $(\hat{S}^{-})^{n}_{j}$ in this model. It is, thus, given by

where the transverse spin-spin correlations $C^{(1)}_{r}$ and $C^{(2)}_{r}$ are given by

In Eq.(18), we drop the index $i$ because these correlation functions depend not on the site $i$ but on the relative distance $r$ between two spins due to the translational symmetry in the thermodynamic limit. Based upon the MPS representation, we discuss below how the CGLMP correlation can be used as an indicator of the QPTs by identifying the extremum and inflection points Wolf et al. (2006). It is also discussed how the Bell-type correlations varies as the distance $r$ changed, especially at the point of QPT, in the following subsections.

We have investigated the CGLMP correlations in the 1D XXZ model with onsite anisotropy especially near the quantum phase transitions. In the case of spin-1, the CGLMP correlation obtained from local measurements in the Fourier bases can be interestingly interpreted as a linear combination of the first-order and second-order transverse spin-spin correlations. By analyzing the derivatives of the CGLMP correlation, one can accurately indicate various QPTs: FM-to-XY, Haldane-to-AFM, Haldane-to-large-D, and XY1-to-XY2 quantum phase transitions. The first-order QPTs (FM-to-XY transitions) shows discontinuity of the CGLMP correlation. The Ising-type (Haldane-to-AFM and XY1-to-XY2) QPTs occur where the second derivative of the CGLMP correlation vanishes. The Haldane-to-large-D transition, also referred to as the Gaussian phase transition, emerges when $\partial\langle\hat{\mathcal{B}}_{r}\rangle/\partial D=0$. The criticality of the CGLMP correlation is caused by the fact that both the derivatives of the spin-spin correlations $C^{(1)}_{r}$ and $C^{(2)}_{r}$ are simultaneously zero in the criticality. Moreover, decaying behaviors of the CGLMP correlation with respect to the distance $r$ in each phase can be predicted from the field-theoretical approach Schulz (1986). However, the BKT-type QPT at $J_{z}=0$ cannot be indicated by finding critical points of the derivatives of the CGLMP correlation. This is contrast to the CHSH correlation in spin-1/2 XXZ chain Justino and de Oliveira (2012).

Another crucial result is that there is unfortunately nonviolation of CGLMP inequality in all phases of spin-1 chain, that is, the CGLMP correlations for the ground states of spin-1 chain do not surpass the local realistic bounds. Even though we do not optimize the measurements for the ground states in each phase, this nonviolation coincides with the one in the 1D spin-1/2 XY chain Batle and Casas (2010) and XXZ chain Justino and de Oliveira (2012) invariant under translational symmetry. The reduced density matrices on two sites which display mixed entangled states are not sufficient to violate CGLMP inequality in the spin-1 chain, although many-body ground states are pure and highly entangled in this model. As the CHSH correlation has monogamy trade-off relation de Oliveira et al. (2013), monogamy of CGLMP correlation can be a candidate to explain the non-violation of CGLMP inequality in the spin-1 chain. Therefore, in order to detect nonlocality in this model, one should consider multipartite Bell correlations for arbitrary dimensions. This gets a clue from the result that multipartite Bell correlations for spin-1/2 invariant under the permutational symmetry Tura et al. (2014) do not satisfy the local hidden variable model in the Ising model with long-range interactions Schmied et al. (2016); Piga et al. (2019). We leave that part as a future investigation that is quite consistent to our current approach demonstrated in this work.