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arXiv:1503.08064 · Physical Review E

One- and two-dimensional quantum models: Quenches and the scaling of irreversible entropy

Shraddha Sharma, Amit Dutta

Published Aug 6, 2015• 23 citations

Abstract

Using the scaling relation of the ground state quantum fidelity, we propose the most generic scaling relations of the irreversible work (the residual energy) of a closed quantum system at absolute zero temperature when one of the parameters of its Hamiltonian is suddenly changed. We consider two extreme limits: the heat susceptibility limit and the thermodynamic limit. It is argued that the irreversible entropy generated for a thermal quench at low enough temperatures when the system is initially in a Gibbs state is likely to show a similar scaling behavior. To illustrate this proposition, we consider zero-temperature and thermal quenches in one-dimensional (1D) and 2D Dirac Hamiltonians where the exact estimation of the irreversible work and the irreversible entropy is possible. Exploiting these exact results, we then establish the following. (i) The irreversible work at zero temperature shows an appropriate scaling in the thermodynamic limit. (ii) The scaling of the irreversible work in the 1D Dirac model at zero temperature shows logarithmic corrections to the scaling, which is a signature of a marginal situation. (iii) Remarkably, the logarithmic corrections do indeed appear in the scaling of the entropy generated if the temperature is low enough while they disappear for high temperatures. For the 2D model, no such logarithmic correction is found to appear.

DOI: 10.1103/physreve.92.022108

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Using the scaling relation of the ground state quantum fidelity, we propose the most generic scaling relations of the irreversible work (the residual energy) of a closed quantum system at absolute zero temperature when one of the parameters of its Hamiltonian is suddenly changed. We consider two extreme limits: the heat susceptibility limit and the thermodynamic limit. It is argued that the irreversible entropy generated for a thermal quench at low enough temperatures when the system is initially in a Gibbs state is likely to show a similar scaling behavior. To illustrate this proposition, we consider zero-temperature and thermal quenches in one-dimensional (1D) and 2D Dirac Hamiltonians where the exact estimation of the irreversible work and the irreversible entropy is possible. Exploiting these exact results, we then establish the following. (i) The irreversible work at zero temperature shows an appropriate scaling in the thermodynamic limit. (ii) The scaling of the irreversible work in the 1D Dirac model at zero temperature shows logarithmic corrections to the scaling, which is a signature of a marginal situation. (iii) Remarkably, the logarithmic corrections do indeed appear in the scaling of the entropy generated if the temperature is low enough while they disappear for high temperatures. For the 2D model, no such logarithmic correction is found to appear.

We compare the irreversible work produced in an infinitesimal sudden quench of a quantum system at zero temperature with its ground state fidelity susceptibility, giving an explicit relation between the two quantities. We find that the former is proportional to the latter but for an extra term appearing in the irreversible work which includes also contributions from the excited states. We calculate explicitly the two quantities in the case of the quantum Ising chain, showing that at criticality they exhibit different scaling behaviors. The irreversible work, rescaled by square of the quench’s amplitude, exhibits a divergence slower than that of the fidelity susceptibility. As a consequence, the two quantities obey also different finite-size scaling relations.

The entropy produced when a system undergoes an infinitesimal quench is directly linked to the work parameter susceptibility, making it sensitive to the existence of a quantum critical point. Its singular behavior at

T=0

, however, disappears as the temperature is raised, hindering its use as a tool for spotting quantum phase transitions. Notwithstanding, the entropy production can be split into classical and quantum components, related with changes in populations and coherences. In this paper we show that these individual contributions can continue to exhibit signatures of the quantum phase transition, even at arbitrarily high temperatures. This is a consequence of their intrinsic connection to the derivatives of the energy eigenvalues and the energy eigenbasis. We illustrate our results in the Landau-Zener model and the prototypical quantum critical system, the

XY

model.

Topologically ordered quantum phases are robust in the sense that perturbations in the Hamiltonian of the system will not change the topological nature of the ground-state wave function. However, in order to exploit topological order for applications such as self-correcting quantum memories and information processing, these states need to be also robust both dynamically and at finite temperature in the presence of an environment. It is well known that systems like the toric code in two spatial dimensions are fragile in temperature. In this paper, we show a completely analytic treatment of the toric code away from equilibrium, after a quantum quench of the system Hamiltonian. We show that, despite being subject to unitary evolution (and at zero temperature), the long-time behavior of the topological entropy is thermal, therefore vanishing. If the quench preserves a local gauge structure, there is a residual long-lived topological entropy. This also is the thermal behavior in presence of such gauge constraints. The result is obtained by studying the time evolution of the topological 2-R\'enyi entropy in a fully analytical, exact way.

We consider critical models in one dimension. We study the ground state in the thermodynamic limit (infinite lattice). We are interested in an entropy of a subsystem. We calculate the entropy of a part of the ground state from a space interval (0,x). At zero temperature it describes the entanglement of the part of the ground state from this interval with the rest of the ground state. We obtain an explicit formula for the entropy of the subsystem at any temperature. At zero temperature our formula reproduces a logarithmic formula, discovered by Vidal, Latorre, Rico, and Kitaev for spin chains. We prove our formula by means of conformal field theory and the second law of thermodynamics. Our formula is universal. We illustrate it for a Bose gas with a delta interaction and for the Hubbard model.

The irreversible work during a driving protocol constitutes one of the most widely studied measures in nonequilibrium thermodynamics, as it constitutes a proxy for entropy production. In quantum systems, it has been shown that the irreversible work has an additional, genuinely quantum mechanical contribution, due to coherence produced by the driving protocol. The goal of this paper is to explore this contribution in systems that undergo a quantum phase transition. Substantial effort has been dedicated in recent years to understanding the role of quantum criticality in work protocols. However, practically nothing is known about how coherence contributes to it. To shed light on this issue, we study the entropy production in infinitesimal quenches of the one-dimensional XY model. We find that coherence plays a significant role in the entropy production and can even account for most of it in certain situations. Moreover, at low temperatures, the coherence presents a finite cusp at the critical point, whereas the entropy production diverges logarithmically. For high temperatures, however, the coherence presents a kink at the critical point indicating the quantum phase transition that occurs only at T=0. Alternatively, if the quench is performed in the anisotropy parameter, then we find that there are situations where all of the entropy produced is due to quantum coherences.

We consider the dynamics of an isolated quantum many-body system after a sudden change of one control parameter, focusing on the statistics of the work done. We establish a connection between the generating function of the distribution of the work and the partition function of a classical system in a film geometry. Using this connection, we first show that the scaling of the fidelity susceptibilities close to a quantum phase transition can be understood in terms of the critical behavior of the excess entropy and specific heat in the classical model. Remarkably, we show that the statistics of the work close to the threshold and to criticality is connected to the so-called critical Casimir free energy which is responsible for the interaction between the boundaries of the classical system. On the basis of this relation, we highlight the emerging universal features of the statistics of the work. Our findings are exemplified for a global quench of the transverse Ising chain at zero temperature.