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arXiv:1609.06597 · Journal of Physics A Mathematical and Theoretical

On a quantum phase transition in a steady state out of equilibrium

Walter H. Aschbacher

Published Sep 19, 2016• 2 citations
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Abstract

Abstract Within the rigorous axiomatic framework for the description of quantum mechanical systems with a large number of degrees of freedom, we show that the nonequilibrium steady state, constructed in the quasifree fermionic system corresponding to the isotropic XY chain in which a finite sample, coupled to two thermal reservoirs at different temperatures, is exposed to a local external magnetic field, is breaking translation invariance and exhibits a strictly positive entropy production rate. Moreover, we prove that there exists a second-order nonequilibrium quantum phase transition with respect to the strength of the magnetic field as soon as the system is truly out of equilibrium.

DOI
10.1088/1751-8113/49/41/415201

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Using the framework of rigorous algebraic quantum statistical mechanics, we construct the unique nonequilibrium steady state in the isotropic XY chain in which a sample of arbitrary finite size is coupled by a bond coupling perturbation of arbitrary strength to two infinitely extended thermal reservoirs, and we prove that this state is thermodynamically nontrivial. Moreover, extracting the leading second-order contribution to its microscopic entropy production and deriving its entropy production in the van Hove weak coupling regime, we prove that, in the mathematically and physically important XY chain, the van Hove regime reproduces the leading order contribution to the microscopic regime.
We study the current-carrying steady state of a transverse field Ising chain coupled to magnetic thermal reservoirs and obtain the nonequilibrium phase diagram as a function of the magnetization potential of the reservoirs. Upon increasing the magnetization bias we observe a discontinuous jump of the magnetic order parameter that coincides with a divergence of the correlation length. For steady states with a nonvanishing conductance, the entanglement entropy at zero temperature displays a bias dependent logarithmic correction that violates the area law and differs from the well-known equilibrium case. Our findings show that out-of-equilibrium conditions allow for novel critical phenomena not possible at equilibrium.
We obtain the steady-state phase diagram of a transverse-field XY spin chain coupled at its ends to magnetic reservoirs held at different magnetic potentials. In the long-time limit, the magnetization bias across the system generates a current-carrying nonequilibrium steady state. We characterize the different nonequilibrium phases as functions of the chain's parameters and magnetic potentials, in terms of their correlation functions and entanglement content. The mixed-order transition, previously observed for the case of a transverse-field Ising chain, is established to emerge as a generic feature of a wider class of out-of-equilibrium problems. The critical exponents associated with this universality class are determined analytically. Results are also contrasted with those obtained in the limit of Markovian reservoirs. Our findings should prove helpful in establishing the properties of nonequilibrium phases and phase transitions of extended open quantum systems.
We study the current-carrying steady-state of a transverse field Ising chain coupled to magnetic thermal reservoirs and obtain the non-equilibrium phase diagram as a function of the magnetization potential of the reservoirs. Upon increasing the magnetization bias we observe a discontinuous jump of the magnetic order parameter that coincides with a divergence of the correlation length. For steady-states with a non-vanishing conductance, the entanglement entropy at zero temperature displays a bias dependent logarithmic correction that differs from the well-known equilibrium case. Our findings show that out-of-equilibrium conditions allow for novel critical phenomena not possible at equilibrium.
We study nonequilibrium quantum phase transitions in XY spin 1/2 chain using the C∗C^*C∗ algebra. We show that the well-known quantum phase transition at magnetic field h=1h = 1h=1 persists also in the nonequilibrium setting as long as one of the reservoirs is set to absolute zero temperature. In addition, we find nonequilibrium phase transitions associated to imaginary part of the correlation matrix for any two different temperatures of the reservoirs at h=1h = 1h=1 and h=hc≡∣1−γ2∣h = h_{\rm c} \equiv|1-γ^2|h=hc​≡∣1−γ2∣, where γγγ is the anisotropy and hhh the magnetic field strength. In particular, two nonequilibrium quantum phase transitions coexist at h=1h=1h=1. In addition we also study the quantum mutual information in all regimes and find a logarithmic correction of the area law in the nonequilibrium steady state independent of the system parameters. We use these nonequilibrium phase transitions to test the utility of two models of reduced density operator, namely Lindblad mesoreservoir and modified Redfield equation. We show that the nonequilibrium quantum phase transition at h=1h = 1h=1 related to the divergence of magnetic susceptibility is recovered in the mesoreservoir approach, whereas it is not recovered using the Redfield master equation formalism. However none of the reduced density operator approaches could recover all the transitions observed by the C∗C^*C∗ algebra. We also study thermalization properties of the mesoreservoir approach.