Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2409.09411 · Journal of the Physical Society of Japan

Nonequilibrium Steady States in the Weakly-Coupled XXZ Model

Shimpei Senda, Tatsuro Sunami, Yuto Matsumoto, Ayumu Sugita

Published Sep 18, 2024• 0 citations
arXivHTMLPDF

Abstract

We study nonequilibrium steady states (NESSs) in the weakly-coupled XXZ model in contact with two heat baths at different temperatures. We show that the density matrix can be represented using only projection operators specified by the quantum numbers of the quantum group $U_q(sl_2)$ in a very good approximation. By using this property, we numerically calculate physical quantities such as temperature profile, energy current, and correlation functions, for the spin chain consisting of several hundred spins. We analytically derive the exact density matrix in the limit $q \rightarrow 0$.

DOI
10.7566/jpsj.93.104003

Related papers

Perturbative Analysis of Nonequilibrium Steady States in Quantum Systems

Ayumu Sugita

We study the nonequilibrium steady state (NESS) in a quantum system in contact with two heat baths at different temperatures. We use a time-independent perturbative expansion with respect to the coupling with the two heat baths to obtain the density matrix for the NESS. In particular, we show an explicit representation of the density matrix for the reflection symmetric and weakly nonequilibrium case. We also calculate the expectation value of the energy current and show that the Kubo formula holds in this case.

arXiv:1203.3817•Mar 16, 2012•1 citation

Origin of Temperature Gradient in Nonequilibrium Steady States in Weakly Coupled Quantum Spin Systems

Toyohiko Ishida, Ayumu Sugita

We study nonequilibrium steady states (NESSs) in quantum spin-1/2 chains in contact with two heat baths at different temperatures.We consider the weak-coupling limit both for spinspin coupling in the system and for system-bath coupling.This setting allows us to treat NESSs with a nonzero temperature gradient analytically.We develop a perturbation theory for this weak-coupling situation and show a simple condition for the existence of nonzero temperature gradient.This condition is independent of the integrability of the system.

arXiv:1704.01731•Journal of the Physical Society of Japan•Jun 27, 2016•1 citation

Integrable non-equilibrium steady state density operators for boundary driven XXZ spin chains: observables and full counting statistics

Tomaž Prosen, Berislav Buča

We will review some known exact solutions for the steady state of the open quantum Heisenberg $XXZ$ spin chain coupled to a pair of baths [Phys. Rev. Lett. 107, 137201 (2011).]. The dynamics is modelled by the Lindblad master equation. We also review how to calculate some relevant physical observables and provide the statistics of spin current assuming the spin chain is weakly coupled to the baths [Phys. Rev. Lett. 112, 067201 (2014).].

arXiv:1501.06156•Jan 25, 2015•0 citations

Spin- 1/2 XXZ chain coupled to two Lindblad baths: Constructing nonequilibrium steady states from equilibrium correlation functions

Tjark Heitmann, Jonas Richter, Fengping Jin +7 more

State-of-the-art approaches to extract transport coefficients of many-body quantum systems broadly fall into two categories: (i) they target the linear-response regime in terms of equilibrium correlation functions of the closed system; or (ii) they consider an open-system situation typically modeled by a Lindblad equation, where a nonequilibrium steady state emerges from driving the system at its boundaries. While quantitative agreement between (i) and (ii) has been found for selected model and parameter choices, also disagreement has been pointed out in the literature. Studying magnetization transport in the spin-$1/2$ XXZ chain, we here demonstrate that at weak driving, the nonequilibrium steady state in an open system, including its buildup in time, can remarkably be constructed just on the basis of correlation functions in the closed system. We numerically illustrate this direct correspondence of closed-system and open-system dynamics, and show that it allows the treatment of comparatively large open systems, usually only accessible to matrix product state simulations. We also point out potential pitfalls when extracting transport coefficients from nonequilibrium steady states in finite systems.

arXiv:2303.00430•Physical review. B./Physical review. B•Nov 27, 2023•6 citations

Non-equilibrium evolution of quantum systems connected to multiple baths

Jinshan Wu

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.

•Jan 1, 2010•0 citations

Exact Nonequilibrium Steady State of a Strongly Driven Open XXZ Chain

Tomaž Prosen

An exact and explicit ladder-tensor-network ansatz is presented for the nonequilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is-apart from a normalization constant-a polynomial of degree 2n - 2 in the coupling constant. Efficient computation of physical observables is facilitated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cosine spin profiles, 1/n(2) scaling of the spin current, and long-range correlations in the steady state. This is a fully nonperturbative extension of a recent result [Phys. Rev. Lett. 106, 217206 (2011)].

arXiv:1106.2978•Physical Review Letters•Sep 19, 2011•290 citations
We study nonequilibrium steady states (NESSs) in the weakly-coupled XXZ model in contact with two heat baths at different temperatures. We show that the density matrix can be represented using only projection operators specified by the quantum numbers of the quantum group Uq(sl2)U_q(sl_2)Uq​(sl2​) in a very good approximation. By using this property, we numerically calculate physical quantities such as temperature profile, energy current, and correlation functions, for the spin chain consisting of several hundred spins. We analytically derive the exact density matrix in the limit q→0q \rightarrow 0q→0.
We study the nonequilibrium steady state (NESS) in a quantum system in contact with two heat baths at different temperatures. We use a time-independent perturbative expansion with respect to the coupling with the two heat baths to obtain the density matrix for the NESS. In particular, we show an explicit representation of the density matrix for the reflection symmetric and weakly nonequilibrium case. We also calculate the expectation value of the energy current and show that the Kubo formula holds in this case.
We study nonequilibrium steady states (NESSs) in quantum spin-1/2 chains in contact with two heat baths at different temperatures.We consider the weak-coupling limit both for spinspin coupling in the system and for system-bath coupling.This setting allows us to treat NESSs with a nonzero temperature gradient analytically.We develop a perturbation theory for this weak-coupling situation and show a simple condition for the existence of nonzero temperature gradient.This condition is independent of the integrability of the system.
We will review some known exact solutions for the steady state of the open quantum Heisenberg XXZXXZXXZ spin chain coupled to a pair of baths [Phys. Rev. Lett. 107, 137201 (2011).]. The dynamics is modelled by the Lindblad master equation. We also review how to calculate some relevant physical observables and provide the statistics of spin current assuming the spin chain is weakly coupled to the baths [Phys. Rev. Lett. 112, 067201 (2014).].
State-of-the-art approaches to extract transport coefficients of many-body quantum systems broadly fall into two categories: (i) they target the linear-response regime in terms of equilibrium correlation functions of the closed system; or (ii) they consider an open-system situation typically modeled by a Lindblad equation, where a nonequilibrium steady state emerges from driving the system at its boundaries. While quantitative agreement between (i) and (ii) has been found for selected model and parameter choices, also disagreement has been pointed out in the literature. Studying magnetization transport in the spin-1/21/21/2 XXZ chain, we here demonstrate that at weak driving, the nonequilibrium steady state in an open system, including its buildup in time, can remarkably be constructed just on the basis of correlation functions in the closed system. We numerically illustrate this direct correspondence of closed-system and open-system dynamics, and show that it allows the treatment of comparatively large open systems, usually only accessible to matrix product state simulations. We also point out potential pitfalls when extracting transport coefficients from nonequilibrium steady states in finite systems.
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 exact and explicit ladder-tensor-network ansatz is presented for the nonequilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is-apart from a normalization constant-a polynomial of degree 2n - 2 in the coupling constant. Efficient computation of physical observables is facilitated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cosine spin profiles, 1/n(2) scaling of the spin current, and long-range correlations in the steady state. This is a fully nonperturbative extension of a recent result [Phys. Rev. Lett. 106, 217206 (2011)].