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arXiv:1210.4555 · Journal of Statistical Mechanics Theory and Experiment

Alternation of sign of magnetization current in driven XXZ chains with twisted XY boundary gradients

Vladislav Popkov

Published Dec 14, 2012• 29 citations
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Abstract

We investigate an open XXZ spin 1/2 chain driven out of equilibrium by coupling with boundary reservoirs targeting different spin orientations in the XY-plane. Symmetries of the model are revealed which appear to be different for spin chains of odd and even sizes. As a result, the spin current is found to alternate with chain length, ruling out the possibility of ballistic transport. Heat transport is switched off completely by virtue of another global symmetry. Further, we investigate the model numerically and analytically. At strong coupling, we find an exact nonequilibrium steady state using perturbation theory. The state is determined by solving secular conditions which guarantee self-consistency of the perturbative expansion. We find nontrivial dependence of the magnetization current on the spin chain anisotropy Δ in the critical region |Δ| < 1, and a phenomenon of tripling of the twisting angle along the chain for narrow lacunae of Δ.

DOI
10.1088/1742-5468/2012/12/p12015

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We investigate an open XXZ spin 1/2 chain driven out of equilibrium by coupling with boundary reservoirs targeting different spin orientations in the XY-plane. Symmetries of the model are revealed which appear to be different for spin chains of odd and even sizes. As a result, the spin current is found to alternate with chain length, ruling out the possibility of ballistic transport. Heat transport is switched off completely by virtue of another global symmetry. Further, we investigate the model numerically and analytically. At strong coupling, we find an exact nonequilibrium steady state using perturbation theory. The state is determined by solving secular conditions which guarantee self-consistency of the perturbative expansion. We find nontrivial dependence of the magnetization current on the spin chain anisotropy Δ in the critical region |Δ| < 1, and a phenomenon of tripling of the twisting angle along the chain for narrow lacunae of Δ.
The spin 1/2 X X Z chain driven out of equilibrium by coupling with boundary reservoirs targeting perpendicular spin orientations in the X Y plane is investigated. The existence of an anomaly in the nonequilibrium steady state (NESS) at the isotropic point Δ = 1 is demonstrated in both the weak coupling and weak driving limits. The nature of the anomaly is studied analytically by calculating exact NESSs for small system sizes, and investigating steady currents. The spin current at the points Δ =± 1 has a singularity which leads to a current discontinuity when either driving or coupling vanishes, and the current of energy develops a twin peak anomaly. The character of the singularity is shown to depend qualitatively on whether the system size is even or odd.
We consider an open Heisenberg quantum spin chain, coupled at the ends to boundary reservoirs polarized in different directions, which sets up a twisting gradient across the chain. We derive a matrix product ansatz for the non-equilibrium density matrix. For the isotropic chain we calculate the exact magnetization profiles and magnetization currents in the nonequilibrium steady steady state of a chain with NNN sites. The magnetization profiles are harmonic functions with a frequency proportional to the twisting angle θ\thetaθ. The currents of the magnetization components lying in the twisting plane and in the orthogonal direction behave qualitatively differently: In-plane steady state currents scale as 1/N21/N^{2}1/N2 for fixed and sufficiently large boundary coupling, and vanish as the coupling increases, while the transversal current increases with the coupling and saturates to 2θ/N2\theta/N2θ/N.
The behavior of the magnetic currents in one-dimensional Heisenberg XXZ spin chains kept out of equilibrium by boundary driving fields is investigated. In particular, the dependence of the spin currents on the anisotropy parameter Δ and on the boundary fields is studied both analytically and numerically in the framework of the Lindblad master equation formalism. We show that the spin current can be maximized with appropriate choices of the boundary fields, and for odd system sizes, N, we demonstrate the existence of additional symmetries that cause the current to be an odd function of Δ. From direct numerical integrations of the quantum master equation, we find that for an arbitrary N the current J(z)(N) vanishes for Δ=0, while for Δ negative it alternates its sign with the system size. In the gapless critical region |Δ|<1, the scaling of the current is shown to be J(z)(N)∼1/N while in the gapped region |Δ|>1 we find that J(z)(N)∼exp(-αN). A simple mean-field approach, which predicts rather well the values of J(z)(N) for the gapped region and the values of the absolute current maxima in the critical region, is developed. The existence of two different stationary solutions for the mean-field density matrix in the whole parameter range is also demonstrated.
We consider an open isotropic Heisenberg quantum spin chain, coupled at the ends to boundary reservoirs polarized in different directions, which sets up a twisting gradient across the chain. Using a matrix product ansatz, we calculate the exact magnetization profiles and magnetization currents in the nonequilibrium steady state of a chain with N sites. The magnetization profiles are harmonic functions with a frequency proportional to the twisting angle θ. The currents of the magnetization components lying in the twisting plane and in the orthogonal direction behave qualitatively differently: In-plane steady-state currents scale as 1/N^{2} for fixed and sufficiently large boundary coupling, and vanish as the coupling increases, while the transversal current increases with the coupling and saturates to 2θ/N.
In this paper we investigate the spin currents and the von Neumann entropy (vNE) of a Heisenberg XXZ chain in contact with twisted XY-boundary magnetic reservoirs by means of the Lindblad master equation. Exact solutions for the stationary reduced density matrix are explicitly constructed for chains of small sizes by using a quantum symmetry operation of the system. These solutions are then used to investigate the optimal transport in the chain in terms of the vNE. As a result we show that the maximal spin current always occurs in the proximity of minima of the vNE and for particular choices of parameters (coupling with reservoirs and anisotropy) it can exactly coincide with them. As the coupling is increased, current reversals may occur and in the limit of strong coupling we show that minima of the vNE tend to zero, meaning that the maximal transport is achieved in this case with states that are very close to pure states.
Abstract We investigate the non-equilibrium steady state (NESS) in an open quantum XXZ chain attached at the ends to polarization baths with unequal polarizations. Using the general theory developed in Popkov (2017 Phys. Rev . A 95 052131), we show that in the critical XXZ | Δ | < 1 easy plane case, the steady current in large systems under strong driving shows resonance-like behaviour, by an infinitesimal change of the spin chain anisotropy or other parameters. Alternatively, by fine tuning the system parameters and varying the boundary dissipation strength, we observe a change of the NESS current from diffusive (of order 1 / N , for small dissipation strength) to ballistic regime (of order 1, for large dissipation strength). This drastic change results from an accompanying structural change of the NESS, which becomes a pure spin-helix state characterized by a winding number which is proportional to the system size. We calculate the critical dissipation strength needed to observe this surprising effect.