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physica status solidi (b)

Global Quantum Discord in Transverse‐Field Ising Models on N × N Square Lattices

Zhao‐Yu Sun, Mu Zhou, Mei Wang, Jin Fu

Published Mar 18, 2019• 4 citations
Publisher

Abstract

Global quantum discord (GQD) in two‐dimensional (2D) transverse‐field Ising models on N × N square lattices at zero temperature and finite temperatures are studied. First, the GQD defined on the reduced density matrix of a 2 × 2 sub‐square in the center of the N × N lattices, with is studied. It is found that shows little size dependence in the strong‐field and weak‐field limit, and presents dramatic size dependence in the middle‐field region. The results are explained by short‐range correlations in non‐critical regions and long‐range correlations in critical regions of the models, respectively. Then the scaling behavior of the GQD of the ground states for the entire N × N squares is studied. It is found that contains a 1D contribution (corresponding to the side length of the squares) and a 2D contribution (corresponding to the area of the squares). Furthermore, the effect of thermodynamic fluctuations on GQD by investigating the thermal‐state GQD is considered. In high‐temperature regions, GQD is always destroyed by thermodynamic fluctuations for any strength of the magnetic field. However, in low‐temperature regions, some thermal enhancement of GQD under weak fields and some thermal robustness of GQD under strong fields is observed. The results are explained by the low‐lying energy states and the energy gap of the models.

DOI
10.1002/pssb.201800666

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Global quantum discord (GQD) in two‐dimensional (2D) transverse‐field Ising models on N × N square lattices at zero temperature and finite temperatures are studied. First, the GQD defined on the reduced density matrix of a 2 × 2 sub‐square in the center of the N × N lattices, with is studied. It is found that shows little size dependence in the strong‐field and weak‐field limit, and presents dramatic size dependence in the middle‐field region. The results are explained by short‐range correlations in non‐critical regions and long‐range correlations in critical regions of the models, respectively. Then the scaling behavior of the GQD of the ground states for the entire N × N squares is studied. It is found that contains a 1D contribution (corresponding to the side length of the squares) and a 2D contribution (corresponding to the area of the squares). Furthermore, the effect of thermodynamic fluctuations on GQD by investigating the thermal‐state GQD is considered. In high‐temperature regions, GQD is always destroyed by thermodynamic fluctuations for any strength of the magnetic field. However, in low‐temperature regions, some thermal enhancement of GQD under weak fields and some thermal robustness of GQD under strong fields is observed. The results are explained by the low‐lying energy states and the energy gap of the models.
Abstract We study the global quantum discord (GQD) in the Lipkin–Meshkov–Glick (LMG) model at zero and finite temperatures, in which all spins are mutually interacted and introduced in an external magnetic field (denoted by h ). We confirm that the high coordinate number is one of the most distinguishing features of the LMG model, which directly results in the nontrivial behaviors of quantum correlations. We compare the GQD with other quantum correlations measures (such as concurrence, quantum discord, and global entanglement) and find the remarkable difference between them. For instance, we find that GQD spreads in the entire system and captures more information on quantum correlations when comparing with concurrence and quantum (pairwise) discord. We discover that GQD can characterize multipartite correlations in the both broken phase ( h < 1) and the symmetric phase ( h ⩾ 1), while global entanglement and its generalized fail. Moreover, we show that the ground-state GQD can identify second-order quantum phase transitions of the LMG model in the thermodynamic limit. By making the scaling behavior of the GQD in the LMG model analysis, we show that GQD (denoted by G ) scales as G ∼ k ⋅ N + c with k > 0 in the anisotropic cases for any fixed magnetic field. We further show that GQD behaves as G | s n ∼ k ⋅ 1 N + c with k < 0 in the isotropic cases for any Dicke state | s n ⟩. Herein k and c are the fitting parameters. We also find that the thermal stability of the GQD at low temperatures depends on the energy gap. We further reveal that the extraordinary behaviors of the thermal-state GQD in the isotropic LMG model are explained by the contribution theory of the energy levels.
In this paper, we study global quantum discord (GQD) in infinite-size spin chains. For this purpose, in the framework of matrix product states (MPSs), we propose an effective procedure to calculate GQD (denoted as Gn) for consecutive n-site subchains in infinite chains. For a spin-1/2 three-body interaction model, whose ground state can be exactly expressed as MPSs, We use the procedure to study Gn with n up to 242424. Then for a spin-1/2 XXZ chain, we firstly use infinite time-evolving block decimation (iTEBD) algorithm to obtain the approximate wavefunction in the from of MPSs, and then figure out Gn with n up to 181818. In both models, Gn shows an interesting linear growth as the increase of n, that is, Gn = k*n+b. Moreover, in non-critical regions the slope kkk of Gn converges very fast, while in critical regions it converges relatively slow, and the behaviors are explained in a clear physical picture with the short-range and long-range correlations. Based on these results, we propose to use Gn/n to describe the global correlations in infinite chains. Gn/n has twofold physical meanings. Firstly, it can be regarded as "global discord per site", very similar to "energy per site" or "magnetization per site" in quantum magnetic systems. Secondly, Gn/n (when n is large enough) describes the quantum correlation between a single site and an (n-1)-site block. Then we successfully apply our theory to an exactly soluble infinite-size spin XY chain which is beyond the matrix product formula, and the Hamiltonian can reduce to the transverse-field Ising model and the XX model. The relation between GQD and quantum phase transitions in these models is discussed.
We perform an extensive study of the properties of global quantum correlations in finite-size one-dimensional quantum spin models at finite temperature. By adopting a recently proposed measure for global quantum correlations (Rulli and Sarandy 2011 Phys. Rev. A 84 042109), called global discord , we show that critical points can be neatly detected even for many-body systems that are not in their ground state. We consider the transverse Ising model, the cluster-Ising model where three-body couplings compete with an Ising-like interaction, and the nearest-neighbor XX Hamiltonian in transverse magnetic field. These models embody our canonical examples showing the sensitivity of global quantum discord close to criticality. For the Ising model, we find a universal scaling of global discord with the critical exponents pertaining to the Ising universality class.
Using the quantum renormalization group (QRG) method, we investigate the ground state geometric measure of quantum discord (GQD) of the transverse Ising model. The analytical expression of GQD together with the RG recurrence relations are obtained. The critical and scaling behaviors of GQD are investigated. Our results show that with the increasing of RG steps, GQD (first derivative of GQD) exhibits sudden change (dip) in behavior at the critical point, which indicates that the system undergoes phase transitions between the Ising and the paramagnetic phases. The critical exponent that governs the divergences of the first derivative of GQD is calculated. How the critical point of the model is approached as the increasing of the system size is revealed.
We examined decay dynamics of various geometric quantum discords (GQDs) for two interacting qubits described by the Heisenberg XY model and further coupled independently to their respective thermal reservoirs. Compared to the case of noninteracting qubits, our results showed that decay rates of the GQDs can be retarded apparently by properly choosing system parameters of the interaction term. In the long-time limit, the asymptotic values of the GQDs are enhanced evidently by tuning anisotropy of the model and strength of the transverse magnetic field. We further illuminated the relations between different GQDs on characterizing quantum correlations, and observed multiple sudden change behaviors of their dynamics.
We study the scaling of quantum discord (a measure of quantum correlation beyond entanglement) in spin models analytically and systematically. We find that at finite temperature the block scaling of quantum discord satisfies an area law for any two-local Hamiltonian. We show that generically and heuristically the two-site scaling of quantum discord is similar to that of correlation functions. In particular, at zero temperature it decays exponentially and polynomially in gapped and gapless (critical) systems, respectively; at finite temperature it decays exponentially in both gapped and gapless systems. We compute the two-site scaling of quantum discord in the XXZXXZXXZ chain, the XYXYXY chain (in a magnetic field), and the transverse field Ising chain at zero temperature.