Dissipatively dressed quasiparticles in boundary driven integrable spin chains

TL;DR

This work reveals that nonequilibrium steady states of boundary-driven, integrable spin chains can be understood through a renormalized, dissipatively dressed quasiparticle dispersion, connecting the NESS spectrum to the Bethe Ansatz structure of a related coherent system via a dissipation-projected Hamiltonian in the Zeno limit. It provides explicit dressed dispersions for XXX, XXZ, and XYZ models under various boundary drivings, demonstrating an additional singularity in the dispersion that reweights Bethe states and can lead to boundary-localized solutions dominating the NESS. The authors supply a rigorous derivation for the diagonal, $U(1)$-symmetric sink/source case and present substantial analytic and numerical evidence for the chiral invariant subspaces and the XYZ case, suggesting a broad, model-spanning mechanism for dissipative dressing. The results offer a framework for dissipative state engineering in integrable systems and point toward extensions to Floquet-based or higher-dimensional analogs where boundary dissipation can be tuned to realize desired steady states.

Abstract

The nonequilibrium steady state (NESS) of integrable spin chains experiencing strong boundary dissipation is accounted by introducing quasiparticles with a renormalized -- dissipatively dressed -- dispersion relation. This allows us to evaluate the spectrum of the NESS in terms of the Bethe ansatz equations for a related coherent system which has the same set of eigenstates, the so-called dissipation-projected Hamiltonian. We find explicit analytic expressions for the dressed energies of the XXX and XXZ models with effective, i.e., induced by the dissipation, diagonal boundary fields, which are U(1) invariant, as well as the XXZ and XYZ models with effective non-diagonal boundary fields. In all cases, the dissipative dressing generates an extra singularity in the dispersion relation, substantially altering the NESS spectrum with respect to the spectrum of the corresponding coherent model.

Figures (4)

• Figure 1: Schematic picture of the dissipative setup. The boundary spins are fixed by dissipation, while the internal spins follow effective dynamics (\ref{['LMEeff']}), consisting of fast coherent dynamics (\ref{['eq:OpenXYZ']}) and slow relaxation dynamics (\ref{['app:ClassicalNESS']}) towards the NESS.
• Figure 2: Top panel: quasiparticle energies $\epsilon(u_\alpha)$ (blue joined points) and their dissipative dressing $\tilde{\epsilon}(u_\alpha)$ (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 $u_1\simeq 3i/2 +i e^{-24.5}$. Bottom panel: coefficients $\gamma_k$ of the normalized localized Bethe state $\ket{\alpha=1} = \sum_{k=1}^N \gamma_k\ \sigma_k^{-} \ket{0}$ (black empty circles). The dashed red line is the fit $\gamma_k = 1.7 \times 2^{-k}$. The green joined points are the coefficients $\mathop{\rm Re} \gamma_k$ and $\mathop{\rm Im} \gamma_k$ for the plain-wave like Bethe state with $u_4\approx 1.78139$.
• Figure 3: Quasiparticle energies $\epsilon(u_\alpha)$ (blue joined points) and dissipatively dressed ones $\tilde{\epsilon}(u_\alpha)$ (red joined points) in the XXZ model with chiral invariant subspace, in the block with one kink $M=1$. Parameters: $N=18$, $\Delta=0.7$. There are no localized states: all the amplitudes are plain-wave like, data not shown.
• Figure A-1: Brickwall unitary circuit (BUC), original (left) and effective (right). Left panel shows original BUC with reset gates $K$ and $F$ at the edges. Two-body interaction $U$ is given by (\ref{['Ugate']}). Right Panel shows effective BUC, where the first site and the last site are traced out, giving rise to effective Krauss operators $K_{eff}$ and $F_{eff}$.