Eur. Phys. J. Special Topics 227, 421–444 (2018)
https://doi.org/10.1140/epjst/e2018-00100-9

Regular Article

Strongly correlated non-equilibrium steady states with currents – quantum and classical picture

¹ Department of Medical Physics and Biophysics, University of Split School of Medicine, 21000 Split, Croatia
² Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK
³ Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

^a e-mail: tomaz.prosen@fmf.uni-lj.si

Received: 21 October 2017
Received in final form: 14 March 2018
Published online: 28 September 2018

Abstract

In this minireview we will discuss recent progress in the analytical study of current-carrying non-equilibrium steady states (NESS) that can be constructed in terms of a matrix product ansatz. We will focus on one-dimensional exactly solvable strongly correlated cases, and will study both quantum models, and classical models which are deterministic in the bulk. The only source of classical stochasticity in the time-evolution will come from the boundaries of the system. Physically, these boundaries may be understood as Markovian baths, which drive the current through the system. The examples studied include the open XXZ Heisenberg spin chain, the open Hubbard model, and a classical integrable reversible cellular automaton, namely the Rule 54 of A. Bobenko et al. [A. Bobenko et al., Commun. Math. Phys. 158, 127 (1993)] with stochastic boundaries. The quantum NESS can be at least partially understood through the Yang–Baxter integrability structure of the underlying integrable bulk Hamiltonian, whereas for the Rule 54 model NESS seems to come from a seemingly unrelated integrability theory. In both the quantum and the classical case, the underlying matrix product ansatz defining the NESS also allows for construction of novel conservation laws of the bulk models themselves. In the classical case, a modification of the matrix product ansatz also allows for construction of states beyond the steady state (i.e., some of the decay modes – Liouvillian eigenvectors of the model). We hope that this article will help further the quest to unite different perspectives of integrability of NESS (of both quantum and classical models) into a single unified framework.