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arXiv:cond-mat/9612047 · Physical Review Letters

Nonequilibrium Steady State in a Quantum System: One-Dimensional Transverse Ising Model with Energy Current

Tibor Antal A, Zoltàn Ràcz, L. Sasvári

Published Jan 13, 1997• 72 citations
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Abstract

We study the nonequilibrium steady states of an Ising chain in a transverse field $h$ by investigating the effect of a field $\ensuremath{\lambda}$, which drives the current of energy. The zero-temperature, $h$- $\ensuremath{\lambda}$ phase diagram is determined exactly, and it is found that the energy current appears continuously above a threshold $\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c}(h)$. The long-range magnetic order (present for $h<{h}_{c},\ensuremath{\lambda}<{\ensuremath{\lambda}}_{c}$) is destroyed in the current-carrying state where the correlations are characterized by power-law, oscillatory decay. The mechanism which generates the power-law correlations in the current-carrying state is discussed.

DOI
10.1103/physrevlett.78.167

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We study the nonequilibrium steady states of an Ising chain in a transverse field hhh by investigating the effect of a field \ensuremathλ\ensuremath{\lambda}\ensuremathλ, which drives the current of energy. The zero-temperature, hhh- \ensuremathλ\ensuremath{\lambda}\ensuremathλ phase diagram is determined exactly, and it is found that the energy current appears continuously above a threshold \ensuremathλ>\ensuremathλc(h)\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c}(h)\ensuremathλ>\ensuremathλc​(h). The long-range magnetic order (present for h<hc,\ensuremathλ<\ensuremathλch<{h}_{c},\ensuremath{\lambda}<{\ensuremath{\lambda}}_{c}h<hc​,\ensuremathλ<\ensuremathλc​) is destroyed in the current-carrying state where the correlations are characterized by power-law, oscillatory decay. The mechanism which generates the power-law correlations in the current-carrying state is discussed.
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.
The dynamics of quantum anisotropic XY spin chain driven out of equilibrium by a transverse magnetic field, that carry an energy current ( J E ) is investigated. Considering the spin exchange and the flow of energy trapped in magnetic field between two defects in the x y -plane separated by a distance d , we derive an analytical expression of the steady state Green's function of the model using the Grassmann algebra in order to determine the Casimir force between two magnetic defects. As soon as done we find that, cooling the system as well as increasing the anisotropic parameter decrease the energy current and accordingly increase the range of the Casimir force. This symmetry is broken with a significant increase of distance between the defects. A critical temperature is then recorded exhibiting a phase transition according to the anisotropic parameter. An interesting feature of this model is that turning the anisotropic parameter as well as the temperature range will better manipulate the energy flux and control the range of the Casimir force.
The zero-temperature transverse Ising chain carrying an energy flux j(E) is studied with the aim of determining the nonequilibrium distribution functions, P(M(z)) and P(Mx) of its transverse and longitudinal magnetizations, respectively. An exact calculation reveals that P(M(z)) is a Gaussian both at j(E)=0 and at j(E) not equal to 0, and the width of the distribution decreases with increasing energy flux. The distribution of the order-parameter fluctuations, P(Mx), is evaluated numerically for spin chains of up to 20 spins. For the equilibrium case (j(E)=0), we find the expected Gaussian fluctuations away from the critical point, while the critical order-parameter fluctuations are shown to be non-Gaussian with a scaling function Phi(x)=Phi(M(x)/<Mx>)=<Mx>P(Mx) strongly dependent on the boundary conditions. When j(E) not equal to 0, the system displays long-range, oscillating correlations but P(Mx) is a Gaussian nevertheless, and the width of the Gaussian decreases with increasing j(E). In particular, we find that, at critical transverse field, the width has a j(-3/8)(E) asymptotic in the j(E)-->0 limit.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian:
We systematically characterize a previously unidentified dynamical phase boundary alpha*(h/J) in the one-dimensional transverse-field Ising model with power-law decaying interactions ~1/r^alpha. Three main results are established: (i) the dominant oscillation frequency of the mean magnetization satisfies omega = Delta/2pi universally across all interaction ranges alpha tested; (ii) the finite-size scaling of the energy gap changes qualitatively at alpha*(h/J), separating exponential (long-range) from power-law (short-range effective) scaling; (iii) the boundary follows alpha*(h/J) = (c*J/h)^3 + b with c ~ e/2 and b ~ 5/3, and is characterized by the effective crossover condition Jeff(alpha*) ~ (h/J)/sqrt(3) + 1/2, suggesting the crossover is controlled by a balance between collective interaction strength and transverse field energy scale. Results are obtained via exact diagonalization (QuTiP, N=4-12) and symbolic regression (PySR).
To create a more comprehensive understanding of nonequilibrium dynamics of open quantum many-body systems, we visit an exactly solvable example, that is a quenched transverse-field Ising chain coupled to Markovian baths, which act locally on the Jordan-Wigner fermionic space. Performing explicit calculations on the heat transfer, transverse magnetization, and kink density, we find that the imbalance of two opposite damping mechanisms play a crucial role in constructing a nontrivial nonequilibrium steady state accompanied with a dissipative quantum phase transition, also that the competition between unitary drive and decoherence does not necessarily lead to a quasi-stationary state or prethermalization under certain ordinary relaxation.