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arXiv:1104.1105 · Physical Review A

Spotlighting quantum critical points via quantum correlations at finite temperatures

T. Werlang, G A P Ribeiro, Gustavo Rigolin

Published Jun 27, 2011• 81 citations
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Abstract

We extend the program initiated by T. Werlang et al. [Phys. Rev. Lett. 105, 095702 (2010)] in several directions. Firstly, we investigate how useful quantum correlations, such as entanglement and quantum discord, are in the detection of critical points of quantum phase transitions when the system is at finite temperatures. For that purpose we study several thermalized spin models in the thermodynamic limit, namely, the $\mathit{XXZ}$ model, the $\mathit{XY}$ model, and the Ising model, all of which with an external magnetic field. We compare the ability of quantum discord, entanglement, and some thermodynamic quantities to spotlight the quantum critical points for several different temperatures. Secondly, for some models we go beyond nearest neighbors and also study the behavior of entanglement and quantum discord for second nearest neighbors around the critical point at finite temperature. Finally, we furnish a more quantitative description of how good all these quantities are in spotlighting critical points of quantum phase transitions at finite $T$, bridging the gap between experimental data and those theoretical descriptions solely based on the unattainable absolute zero assumption.

DOI
10.1103/physreva.83.062334

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We extend the program initiated by T. Werlang et al. [Phys. Rev. Lett. 105, 095702 (2010)] in several directions. Firstly, we investigate how useful quantum correlations, such as entanglement and quantum discord, are in the detection of critical points of quantum phase transitions when the system is at finite temperatures. For that purpose we study several thermalized spin models in the thermodynamic limit, namely, the XXZ\mathit{XXZ}XXZ model, the XY\mathit{XY}XY model, and the Ising model, all of which with an external magnetic field. We compare the ability of quantum discord, entanglement, and some thermodynamic quantities to spotlight the quantum critical points for several different temperatures. Secondly, for some models we go beyond nearest neighbors and also study the behavior of entanglement and quantum discord for second nearest neighbors around the critical point at finite temperature. Finally, we furnish a more quantitative description of how good all these quantities are in spotlighting critical points of quantum phase transitions at finite TTT, bridging the gap between experimental data and those theoretical descriptions solely based on the unattainable absolute zero assumption.
We review the main results and ideas showing that quantum correlations at finite temperatures (T), in particular quantum discord, are useful tools in characterizing quantum phase transitions (QPT) that only occur, in principle, at the unattainable absolute zero temperature. We first review some interesting results about the behavior of thermal quantum discord for small spin-1/2 chains and show that they already give us important hints of the infinite chain behavior. We then study in detail and in the thermodynamic limit (infinite chains) the thermal quantum correlations for the XXZ and XY models, where one can clearly appreciate that the behavior of thermal quantum discord at finite T is a useful tool to spotlight the critical point of a QPT.
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 with 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.
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.
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.
We comparatively study three of the most useful quantum information tools to detect quantum critical points (QCPs) when only finite temperature data are available. We investigate quantitatively how the quantum discord, the quantum-teleportation-based QCP detectors, and the quantum coherence spectrum pinpoint the QCPs of several spin-1/21/21/2 chains. We work in the thermodynamic limit (infinite number of spins) and with the spin chains in equilibrium with a thermal reservoir at temperature TTT. The models here studied are the XXZ model with and without an external longitudinal magnetic field, the Ising transverse model, and the XY model subjected to an external transverse magnetic field.
In quantum statistical mechanics, finite-temperature phase transitions are typically governed by classical field theories. In this context, the role of quantum correlations is unclear: recent contributions have shown how entanglement is typically very short-ranged, and thus uninformative about long-ranged critical correlations. In this work, we show the existence of finite-temperature phase transitions where a broader form of quantum correlation than entanglement, the entropic quantum discord, can display genuine signatures of critical behavior. We consider integrable bosonic field theories in both two- and three-dimensional lattices, and show how the two-mode Gaussian discord decays algebraically with the distance even in cases where the entanglement negativity vanishes beyond nearest-neighbor separations. Systematically approaching the zero-temperature limit allows us to connect discord to entanglement, drawing a generic picture of quantum correlations and critical behavior that naturally describes the transition between entangled and discordant quantum matter.