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arXiv:1910.04777 · SciPost Physics

Landau theory for non-equilibrium steady states

Camille Aron, Claudio Chamon

Published May 8, 2020• 15 citations
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Abstract

We examine how non-equilibrium steady states close to a continuous phase transition can still be described by a Landau potential if one forgoes the assumption of analyticity. In a system simultaneously coupled to several baths at different temperatures, the non-analytic potential arises from the different density of states of the baths. In periodically driven-dissipative systems, the role of multiple baths is played by a single bath transferring energy at different harmonics of the driving frequency. The mean-field critical exponents become dependent on the low-energy features of the two most singular baths. We propose an extension beyond mean field.

DOI
10.21468/scipostphys.8.5.074

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We examine how non-equilibrium steady states close to a continuous phase transition can still be described by a Landau potential if one forgoes the assumption of analyticity. In a system simultaneously coupled to several baths at different temperatures, the non-analytic potential arises from the different density of states of the baths. In periodically driven-dissipative systems, the role of multiple baths is played by a single bath transferring energy at different harmonics of the driving frequency. The mean-field critical exponents become dependent on the low-energy features of the two most singular baths. We propose an extension beyond mean field.
We study the fate of the Ising model and its universal properties when driven by a rapid periodic drive and weakly coupled to a bath at equilibrium. The far-from-equilibrium steady-state regime is accessed by means of a Floquet mean-field approach. We show that, depending on the details of the bath, the drive can strongly renormalize the critical temperature to higher temperatures, modify the critical exponents, or even change the nature of the phase transition from second to first order after the emergence of a tricritical point. Moreover, by judiciously selecting the frequency of the field and by engineering the spectrum of the bath, one can drive a ferromagnetic Hamiltonian to an antiferromagnetically ordered phase and vice versa.
In this thesis we develop a systematic formalism that allows the calculation of transport properties for small quantum systems coupled to multiple baths with different temperatures and/or chemical potentials. Our approach is based on a generalization of the projector operator technique, previously used to study the evolution towards equilibrium of systems weakly coupled to a single bath. Applying this technique to a system weakly coupled to multiple baths, we find a dynamical equation for the reduced density matrix of the system (the reduced density matrix is obtained by tracing out from the total density matrix the baths’ degrees of freedom). Integration of this dynamical equation gives the time evolution of the biased system. After a time interval long enough compared to the characteristic time-scales of the problem, the system arrives to a non-equilibrium steady-state. The density matrix in this non-equilibrium steady state can also be calculated directly, avoiding the time-integration of the dynamical equation. In the second part of the thesis we study non-equilibrium steady states and heat transport through short spin chains coupled to two thermal baths with different temperature. Two new definitions for the local temperature in nonequilibrium are proposed, based on local spin and energy at different sites. The definition based on the local energy seems to be more reliable. We find that for small biases, the heat and spin currents increase proportionally to the bias, however non-linear behavior is observed for large biases. The temperature profile of the chain depends strongly on the parameters of the system.
An external force dynamically drives an isolated mean-field Hamiltonian system to a long-lasting quasistationary state, whose lifetime increases with population of the system. For second order phase transitions in quasistationary states, two non-classical critical exponents have been reported individually by using a linear and a nonlinear response theories in a toy model. We provide a simple way to compute the critical exponents all at once, which is an analog of the Landau theory. The present theory extends universality class of the non-classical exponents to spatially periodic one-dimensional systems, and shows that the exponents satisfy a classical scaling relation inevitably by using a key scaling of momentum.
In this paper we address the question how the Kibble-Zurek mechanism, which describes the formation of topological defects in quantum systems subjected to a quench across a critical point, is generalized to the same scenario but for driven-dissipative quantum critical systems. In these out of equilibrium systems, the critical behavior is manifested in the steady states rather than in the ground states as in quantum critical models of closed systems. To give the generalization we need to establish what is meant by adiabaticity in open quantum systems. Another most crucial concept that we clarify is the characterization of non-adiabatic excitations. It is clear what these are for a Hamiltonian systems, but question is more subtle for driven-dissipative systems. In particular, the important observation is that the instantaneous steady states serve as reference states. From these the excitations can then be extracted by comparing the distance from the evolved state to the instantaneous steady state. With these issues resolved we demonstrate the applicability of the generalized Kibble-Zurek mechanism to an open Landau-Zener problem and show universal Kibble-Zurek scaling for an open transverse Ising model. Thus, our results support any assumption that non-equilibrium quantum critical behavior can be understood from universal features.
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By considering a solvable driven-dissipative quantum model, we demonstrate that continuous phase transitions in dissipative systems may occur without an accompanying symmetry breaking. As such, the underlying mechanism for this type of transition is qualitatively different from that of continuous equilibrium phase transitions. In our model, the transition is solely a result of the interplay between Hamiltonian and dissipative dynamics and manifests as a nonanalyticity in the steady state ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{ss}}$ in the thermodynamic limit. Based on knowledge from critical classical models we suggest that this behavior derives from a rounding of a first-order phase transition into a continuous one due to large environment-induced fluctuations. Despite being conceptually different from the traditional continuous transitions, we show that expectations of local observables can still be characterized by a set of critical exponents.