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arXiv:2408.09302 · Journal of Physics A Mathematical and Theoretical

Bethe-Ansatz diagonalization of the steady state of boundary driven integrable spin chains

Vladislav Popkov, Xin Zhang, Carlo Presilla A, Tomaž Prosen

Published Oct 22, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract We find that the density operator of the nonequilibrium steady state (NESS) of XXZ spin chains 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 of these quasiparticles. The dissipative dressing generates an extra singularity in the dispersion relation, which significantly changes the NESS spectrum. In particular, it 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.

DOI
10.1088/1751-8121/ae1643

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Abstract We find that the density operator of the nonequilibrium steady state (NESS) of XXZ spin chains 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 of these quasiparticles. The dissipative dressing generates an extra singularity in the dispersion relation, which significantly changes the NESS spectrum. In particular, it 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.
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.
Abstract We construct the nonequilibrium steady state (NESS) density operator of the spin- 1 / 2 XXZ chain with non-diagonal boundary magnetic fields coupled to boundary dissipators. The Markovian boundary dissipation is found with which the NESS density operator is expressed in terms of the product of the Lax operators by relating the dissipation parameters to the boundary parameters of the spin chain. The NESS density operator can be expressed in terms of a non-Hermitian transfer operator (NHTO) which forms a commuting family of quasilocal charges. The optimization of the Mazur bound for the high temperature Drude weight is discussed by using the quasilocal charges and the conventional local charges constructed through the Bethe ansatz.
We construct the nonequilibrium steady state (NESS) density operator of the spin-1/2 XXZ chain with non-diagonal boundary magnetic fields coupled to boundary dissipators. The Markovian boundary dis- sipation is found with which the NESS density operator is expressed in terms of the product of the Lax operators by relating the dissipation parameters to the boundary parameters of the spin chain. The NESS density operator can be expressed in terms of a non-Hermitian transfer operator (NHTO) which forms a commuting family of quasilocal charges. The optimization of the Mazur bound for the high temperature Drude weight is discussed by using the quasilocal charges and the conventional local charges constructed through the Bethe ansatz.
We demonstrate that the exact nonequilibrium steady state of the one-dimensional Heisenberg XXZ spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the nonequilibrium density matrix where the matrices satisfy a quadratic algebra. This algebra turns out to be related to the quantum algebra U(q)[SU(2)]. Coherent state techniques are introduced for the exact solution of the isotropic Heisenberg chain with and without quantum boundary fields and Lindblad terms that correspond to two different completely polarized boundary states. We show that this boundary twist leads to nonvanishing stationary currents of all spin components. Our results suggest that the matrix product ansatz can be extended to more general quantum systems kept far from equilibrium by Lindblad boundary terms.
A new type of quantum transfer matrix, arising as a Cholesky factor for the steady-state density matrix of a dissipative Markovian process associated with the boundary-driven Lindblad equation for the isotropic spin-1/2 Heisenberg (XXX) chain, is presented. The transfer matrix forms a commuting family of non-Hermitian operators depending on the spectral parameter, which is essentially the strength of dissipative coupling at the boundaries. The intertwining of the corresponding Lax and monodromy matrices is performed by an infinitely dimensional Yang–Baxter R-matrix, which we construct explicitly and is essentially different from the standard 4 × 4 XXX R-matrix. We also discuss a possibility to construct Bethe ansatz for the spectrum and eigenstates of the non-equilibrium steady-state density operator. Furthermore, we indicate the existence of a deformed R-matrix in the infinite dimensional auxiliary space for the anisotropic XXZ spin-1/2 chain, which in general provides a sequence of new, possibly quasi-local, conserved quantities of the bulk XXZ dynamics.