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Phase Transitions

Quantum correlations in anisotropic XY-spin chains in a transverse magnetic field

Farshid Mofidnakhaei, F. Khastehdel Fumani, S. Mahdavifar, Javad Vahedi

Published Oct 9, 2018• 16 citations
Publisher

Abstract

The well-known anisotropic XY spin chains in a transverse magnetic field are studied in the context of bipartite entanglement and quantum discord. By using Jordan–Wigner transformation, the model mapped onto a noninteracting spinless fermion model then, by accessing the energy spectrum we have found analytical formulas for concurrence and quantum discord based on reduced density matrix. We used the approach to go beyond the trivial nearest neighbours information, namely the second and third nearest neighbours. It is shown that far distance derivatives of quantum discord are able to capture XY-critical line and furthermore a signature of factorization points is observed.

DOI
10.1080/01411594.2018.1527916

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The well-known anisotropic XY spin chains in a transverse magnetic field are studied in the context of bipartite entanglement and quantum discord. By using Jordan–Wigner transformation, the model mapped onto a noninteracting spinless fermion model then, by accessing the energy spectrum we have found analytical formulas for concurrence and quantum discord based on reduced density matrix. We used the approach to go beyond the trivial nearest neighbours information, namely the second and third nearest neighbours. It is shown that far distance derivatives of quantum discord are able to capture XY-critical line and furthermore a signature of factorization points is observed.
Abstract An exactly solvable cluster spin model with three-spin interaction couplings Jx(for XZX spin components) and J y (for YZY spin components) in the presence of a transverse magnetic field h for a spin chain is investigated. For h = 0, and with only one non-zero interaction strength, the ground state is the cluster state. Through the Jordan–Wigner fermion mapping, the odd sites and the even sites form two separate transverse-XY chains, connected only through the boundary terms. Consequently, all measures of quantum correlations for nearest neighbour spins, the concurrence, the quantum mutual information and the quantum discord are all zero in the ground state. The dynamics is spin conserving for J y = J x , exhibiting a line of critical points for | h / J x | ⩽ 2 , with an uncorrelated direct product ground state for | h / J x | > 2 . There are several quantum critical points in the parameter space, with multi-fold degenerate ground states. The magnetisation and the global entanglement measure exhibit strong singular features for the spin conserving case. The next-neighbour quantum correlation measures are investigated analytically, which exhibit singular features in the vicinity of degeneracy critical points. The Jy - and h- derivatives of the concurrence exhibit singular peak behaviour near the degeneracy critical points, except in the spin conserving case where the derivatives are zero.
We examine the ability of quantum discord (QD) and entanglements (concurrence, EoF and negativity) to detect the critical points associated to quantum phase transitions (QPTs) for XY models, i.e., the isotropic XY model with three-spin interactions at zero temperature, and the anisotropic XY model in a transverse magnetic field h at finite temperatures. For the case of zero temperature, we found that both entanglements and QD can spotlight the critical points of QPTs for these two models. Moreover, QD versus distance M exhibits the long-range behavior of quantum correlation for the anisotropic XY model, while entanglement is short-ranged. For the case of finite temperatures, we found that negativity has the same behaviors with concurrence at or near transition points. Moreover, QD for the anisotropic XY model can increase with temperature even in the absence of a magnetic field.
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We investigate pairwise quantum correlation as measured by the quantum discord as well as its classical counterpart in the thermodynamic limit of anisotropic XY\mathit{XY}XY spin-1/21/21/2 chains in a transverse magnetic field for both zero and finite temperatures. Analytical expressions for both classical and quantum correlations are obtained for spin pairs at any distance. In the case of zero temperature, it is shown that the quantum discord for spin pairs farther than second neighbors is able to characterize a quantum phase transition, even though pairwise entanglement is absent for such distances. For finite temperatures, we show that quantum correlations can be increased with temperature in the presence of a magnetic field. Moreover, in the XX\mathit{XX}XX limit, thermal quantum discord is found to be dominant over classical correlation while the opposite scenario takes place for the transverse field Ising model limit.
One-dimensional spin-12\frac{1}{2}21​ systems are well-known candidates to study the quantum correlations between particles. In condensed matter physics, studies often are restricted to first-neighbor particles. In this work, we consider the one-dimensional XXZXXZXXZ model in a transverse magnetic field (TF) which is not integrable except at specific points. Analytical expressions for quantum correlations (entanglement and quantum discord) between spin pairs at any distance are obtained for both zero and finite temperature by using the analytical approach proposed by Caux et al. [Phys. Rev. B 68, 134431 (2003)]. We compare the efficiency of the quantum discord (QD) with respect to the entanglement in the detection of critical points as the neighboring spin pairs go farther than the next-nearest neighbors. In the absence of the TF and at zero temperature, we show that the QD for spin pairs farther than the second neighbors is able to capture the critical points while the pairwise entanglement is absent. In contrast with the pairwise entanglement, two-site QD is effectively long range in the critical regimes where it decays algebraically with the distance between pairs. We also show that the thermal QD between neighbor spins possesses strong distinctive behavior at the critical point that can be seen at finite temperature and, therefore, spotlights the critical point while the entanglement fails in this task.
Non-classical correlations arise in various quantum mechanical systems. Characterization and quantification of these correlations is an important and active branch of research in the field of quantum information theory. Investigation of non-classical correlations in condensed matter systems gives important insights about the characteristics of these systems. In particular, systems possessing a quantum critical point in their phase diagrams have attracted much attention due to the peculiar behavior of correlations near these points. In this thesis, we have investigated two distinct quantum spin models from the perspective of correlations and, we have discussed the correlation content of an important subclass of bipartite states. We start by an analytical calculation of the quantum discord for a system composed of spin-j and spin-1/2 subsystems possessing rotational symmetry. We have compared our results with the quantum discord of states having similar symmetries and seen that in rotationally invariant states the amount of quantum discord is much higher. Moreover, using the well known entanglement properties of these states, we have compared their quantum discord with entanglement and seen that quantum discord is higher than the entanglement. Next, we have investigated the thermal quantum correlations and entanglement in spin-1 Bose-Hubbard model with two and three particles. We have demonstrated that the energy level crossings in the ground state of the system are signalled by both the behavior of thermal quantum correlations and entanglement. Finally, we have investigated various thermal quantum and total correlations in the anisotropic XY spin-chain with transverse magnetic field. We have shown that the ability of the considered measures to estimate the critical points of this system at finite temperature strongly depends on the anisotropy parameter of the Hamiltonian. Furthermore, we have studied the effect of temperature on long-range correlations of the XY chain.