Bethe-ansatz diagonalization of steady state of boundary driven integrable spin chains

Abstract

We find that the density operator of non-equilibrium steady state (NESS) of XXZ spin chain with strong ``sink and source" boundary dissipation, can be described in terms of quasiparticles, with renormalized -- dissipatively dressed -- dispersion relation. The spectrum of the NESS is then fully accounted for by Bethe ansatz equations for an associated coherent system. The dissipative dressing generates an extra singularity in the dispersion relation, which strongly modifies the NESS spectrum with respect to the spectrum of the corresponding coherent model. In particular, this leads to a dissipation-assisted entropy reduction, due to the suppression -- in the NESS spectrum -- of plain wave-type Bethe states in favor of Bethe states localized at the boundaries.

Figures (5)

• Figure 1: Schematic picture of the dissipative setup. The boundary spins are fixed by dissipation, while the internal spins follow an effective dynamics consisting of fast coherent dynamics (\ref{['XXZ']}) and slow relaxation dynamics (\ref{['ClassicalNESS']}) towards the NESS. For our "sink and source" model (\ref{['LME']}) $\vec{n}_r=(0,0,1)$ and $\vec{n}_l=-\vec{n}_r$.
• Figure 2: Surfaces $\epsilon(\mathop{\rm Re} u,\mathop{\rm Im} u)$ (left panel) and $\tilde{\epsilon}(\mathop{\rm Re} u,\mathop{\rm Im} u)$ (right panel) for isotropic case $\Delta=1$ showing singularities at $u=\pm i/2$ and at $u=\pm i/2$, $u=\pm 3i/2$, respectively.
• Figure 3: Left panel: quasiparticle energies $\epsilon(u_\alpha)$ (blue joined points) and $\tilde{\epsilon}(u_\alpha)$ (red joined points) in the XXX model with $N=18$ spin, in the block with one magnon $M=1$. The state $\alpha=1$ is a localized Bethe state with $u_1\simeq 3i/2 +i e^{-24.5}$. Right panel: coefficients $A_k$ of the normalized localized Bethe state $\ket{\alpha=1} = \sum_{k=1}^N A_k\ \sigma_k^{-} \ket{\uparrow\cdots\uparrow}$ (black empty circles). The dashed red line is the fit $A_k = 1.7 \times 2^{-k}$. The green joined points are the coefficients $\mathop{\rm Re} A_k$ and $\mathop{\rm Im} A_k$ for the plain-wave like Bethe state with $u_4\simeq 1.78139$.
• Figure 4: Von Neumann entropy $S(\rho) = -\operatorname{tr}\!{(\rho \log \rho)}$ per spin versus the system size $N$ for $\rho=\rho_\mathrm{Gibbs}$ (blue points) and $\rho=\rho_{\rm NESS}$ (red points), illustrating the dissipation assisted entropy reduction. Left upper panel corresponds to the isotropic Heisenberg model $\Delta=1$, red dashed line is a fit given by $2/(3 N^{0.7})$. Left bottom panel shows a nonintegrable case in which a staggered magnetic field $(-1)^j h\sigma^z_j$, with $h=1.5$, has been added. Right panels correspond to the anisotropic Heisenberg model in the easy plane $\Delta<1$ and easy axis $\Delta>1$ regime.
• Figure 5: Distribution of the eigenvalues $E$ of $-\log \rho_\mathrm{Gibbs}$ (left panel) and $\tilde{E}$ of $-\log \rho_\mathrm{NESS}$ (right panel) for the XXX model with $N=13$. In both panels, the yellow and blue histograms correspond, respectively, to the integrable and nonintegrable cases of Fig. \ref{['FigEntropy']} with $\Delta=1$. Distinct asymmetricity of $P(\tilde{E})$ (right Panel, yellow area) for the integrable case is due to the dissipative dressing effect.