Quantum Correlations in Spin Chains at Finite Temperatures and Quantum Phase Transitions

Abstract

We compute the quantum correlation (quantum discord (QD)) and the entanglement (EoF) between nearest neighbor qubits (spin-1/2) in an infinite chain described by the Heisenberg model (XXZ Hamiltonian) at finite temperatures. The chain is in the thermodynamic limit and thermalized with a reservoir at temperature T (canonical ensemble). We show that QD, in contrast to EoF and other thermodynamic quantities, spotlight the critical points associated to quantum phase transitions (QPT) for this model even at finite T . This remarkable property of QD may have important implications for experimental characterization of QPTs when one is unable to reach temperatures below which a QPT can be seen. Quantum phase transition (QPT) is a purely quantum process [1] occurring at absolute zero temperature (T = 0), where no thermal fluctuations exist and hence no classical phase transition is allowed to occur. QPT is caused by changing the system's Hamiltonian, such as an external magnetic field or the coupling constant. These quantities are generally known as the tuning parameter. As one changes the Hamiltonian one may reach a special point (critical point) where the ground state of the system suffers an abrupt change mapped to a macroscopic change in the system's properties. This change of phase is solely due to quantum fluctuations, which exist at T = 0 due to the Heisenberg uncertainty principle. This whole process is called QPT. The paramagnetic-ferromagnetic transition in some metals [2], the superconductor-insulator transition [3], and superfluid-Mott insulator transition [4] are remarkable examples of this sort of phase transition.In principle QPTs occur at T = 0, which is unattainable experimentally due to the third law of thermodynamics. Hence, one must work at very small T , as close as possible to the absolute zero, in order to detect a QPT. More precisely, one needs to work at regimes in which thermal fluctuations are insufficient to drive the system from its ground to excited states. In this scenario quantum fluctuations dominate and one is able to measure a QPT.So far the theoretical tools available to determine the critical points (CP) for a given Hamiltonian assume T = 0. For spin chains, for instance, the CPs are determined studying, as one varies the tuning parameter, the behavior of either its magnetization, or bipartite [5] and multipartite [6] entanglement, or its quantum correlation (QC) [7]. By investigating the extremal values of these quantities as well as the behavior of their first and second order derivatives one is able to spotlight the CP. However, the T = 0 assumption limits a direct connection between these theoretical "CP detectors" and experiment. Indeed, if thermal fluctuations are not small enough excited states become relevant and the tools developed so far cannot be employed to clearly indicate the CP.In this Letter we remove this limitation and present a theoretical tool that is able to clearly detect CPs for QPTs at finite T . We show that the behavior of strictly QCs [8] at finite T , as gi...

Spotlighting quantum critical points via quantum correlations at finite temperatures

“…2a and 2b we plot, respectively, TQD and EoF as a function of ∆ for some values of T > 0 and h = 0. As discussed in our previous work [12], and illustrated in Fig. 2, for h = 0 the first-order derivative of TQD is discontinuous at the CP ∆ = 1 not only at T = 0, but also at T > 0.…”

“…It is worth mentioning that the cusp-like behavior in Fig. 1 between the two CPs is due to an exchange in the set of projectors that minimizes the quantum conditional entropy [12], Eq. (A.2), and so far could not be associated with any known QPT for this model.…”

“…Our goal in this work is to extend the study initiated in [12] in at least two directions. We want, on one hand, to study different models to see whether or not TQD is still a good CP detector at finite T .…”

“…Recently, however, we have studied, in the thermodynamic limit, the XXZ Hamiltonian with no magnetic field and shown that quantum discord (QD) is able to detect CPs at finite T while other thermodynamic quantities and entanglement fail in this task [12]. For that model we have shown that the thermal quantum discord (TQD), i.e., QD computed for a system described by the canonical ensemble, possessed a characteristic behavior at the CP robust enough to survive an increasing T and, therefore, spotlight a CP of a QPT at finite T .…”

Phase transitions and thermal entanglement of the distorted Ising–Heisenberg spin chain: topology of multiple-spin exchange interactions in spin ladders

“…Quantum spin chains as the most well-studied quantum models of statistical mechanics exhibit a wide variety of attractive phenomena at zero temperature such as the quantum phase transitions, which are driven by the change of the external parameters at absolute zero temperature [1][2][3], and also at low temperatures [4] that are controlled by the behavior of their ground states and low lying excited states. Entanglement [5,6] is one of the most interesting of such quantum phenomena, as it is a fundamental resource in performing most tasks in quantum computation and quantum information processing [7][8][9][10].…”