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International Journal of Bifurcation and Chaos

FINITE TEMPERATURE APPROACH TO QUANTUM PHASE TRANSITIONS

Angelo Plastino, Evaldo M. F. Curado

Published Feb 1, 2010• 2 citations
Publisher

Abstract

We show how ordinary, finite-temperature tools of Statistical Mechanics can be used to detect quantum phase transitions in many-body systems. In particular, statistical correlations exhibit a quite idiosyncratic behavior at the critical interaction-strengths. Moreover, what one may call "soft quantum phase-transitions" (crossings of excited states) are characterized by the rather special way of vanishing that these correlations adopt.

DOI
10.1142/s0218127410025855

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We show how ordinary, finite-temperature tools of Statistical Mechanics can be used to detect quantum phase transitions in many-body systems. In particular, statistical correlations exhibit a quite idiosyncratic behavior at the critical interaction-strengths. Moreover, what one may call "soft quantum phase-transitions" (crossings of excited states) are characterized by the rather special way of vanishing that these correlations adopt.
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.
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.
We face the problem of detecting and featuring footprints of quantum criticality in the finite-temperature behavior of quantum many-body systems. Our strategy is that of comparing the phase diagram of a system displaying a T=0T=0T=0 quantum phase transition with that of its classical limit, in order to single out the genuinely quantum effects. To this aim, we consider the one-dimensional Ising model in a transverse field: while the quantum S=1∕2S=1∕2S=1∕2 Ising chain is exactly solvable and extensively studied, results for the classical limit $(S\ensuremath{\rightarrow}\ensuremath{\infty})$ of such model are lacking, and we supply them here. They are obtained numerically, via the transfer-matrix method, and their asymptotic low-temperature behavior is also derived analytically by self-consistent spin-wave theory. We draw the classical phase diagram according to the same procedure followed in the quantum analysis, and the two phase diagrams are found unexpectedly similar: Three regimes are detected also in the classical case, each characterized by a functional dependence of the correlation length on temperature and field analogous to that of the quantum model. What discriminates the classical from the quantum case are the different values of the exponents entering such dependencies, a consequence of the different nature of zero-temperature quantum fluctuations with respect to the thermal ones.
We study the quantum phase transition in the one-dimensional Ising model at zero temperature. As the strength g of the transverse field is changed, the system undergoes a quantum phase transition from an ordered to a disordered phase, characterized by spontaneous magnetization in the z-direction. We apply the standard technique of finite-size scaling in statistical mechanics to detect the nature of the quantum phase transition in the system. PACS numbers: 05.50.+q, 64.60.Cn, 64.60.Fr Keywords: Finite-size scaling, Quantum phase transition, Ising model
We review the non-zero temperature relaxational dynamics of quantum systems near a zero temperature, second-order phase transition. We begin with the quantum Ising chain, for which universal and exact results for the relaxation rates can be obtained in all the distinct limiting regimes of the phase diagram. Next, we describe the crossovers in the electron spectral function near a transition involving a change in the pairing symmetry of BCS superconductors in two dimensions. Finally, we consider dynamic spin models which may provide a mean-field description of magnetic ordering transitions in the heavy fermion compounds.
We review the non-zero temperature relaxational dynamics of quantum systems near a zero temperature, second-order phase transition. We begin with the quantum Ising chain, for which universal and exact results for the relaxation rates can be obtained in all the distinct limiting regimes of the phase diagram. Next, we describe the crossovers in the electron spectral function near a transition involving a change in the pairing symmetry of BCS superconductors in two dimensions. Finally, we consider dynamic spin models which may provide a mean-field description of magnetic ordering transitions in the heavy fermion compounds.