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arXiv:2309.12978 · SciPost Physics Core

Free fermions with dephasing and boundary driving: Bethe Ansatz results

Vincenzo Alba

Published Jan 30, 2025• 7 citations
arXivHTMLPDF

Abstract

By employing the Lindblad equation, we derive the evolution of the two-point correlator for a free-fermion chain of length L L subject to bulk dephasing and boundary losses. We use the Bethe Ansatz to diagonalize the Liouvillian \mathcal{L}^{(2)} ℒ ( 2 ) governing the dynamics of the correlator. The majority of its eigenvalues are complex. Precisely, L(L-1)/2 L ( L − 1 ) / 2 complex eigenvalues do not depend on dephasing, apart from a trivial shift. The remaining complex levels are perturbatively related to the dephasing-independent ones for large L L . The long-time dynamics is governed by a band of real eigenvalues, which contains an extensive number of levels. They give rise to diffusive scaling at intermediate times, when boundaries can be neglected. Moreover, they encode the breaking of diffusion at asymptotically long times. Interestingly, for large loss rate two boundary modes appear in the spectrum. The real eigenvalues correspond to string solutions of the Bethe equations, and can be treated effectively for large chains. This allows us to derive compact formulas for the dynamics of the fermionic density. We check our results against exact diagonalization, finding perfect agreement.

DOI
10.21468/scipostphyscore.8.1.011

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By employing the Lindblad equation, we derive the evolution of the two-point correlator for a free-fermion chain of length L L subject to bulk dephasing and boundary losses. We use the Bethe Ansatz to diagonalize the Liouvillian \mathcal{L}^{(2)} ℒ ( 2 ) governing the dynamics of the correlator. The majority of its eigenvalues are complex. Precisely, L(L-1)/2 L ( L − 1 ) / 2 complex eigenvalues do not depend on dephasing, apart from a trivial shift. The remaining complex levels are perturbatively related to the dephasing-independent ones for large L L . The long-time dynamics is governed by a band of real eigenvalues, which contains an extensive number of levels. They give rise to diffusive scaling at intermediate times, when boundaries can be neglected. Moreover, they encode the breaking of diffusion at asymptotically long times. Interestingly, for large loss rate two boundary modes appear in the spectrum. The real eigenvalues correspond to string solutions of the Bethe equations, and can be treated effectively for large chains. This allows us to derive compact formulas for the dynamics of the fermionic density. We check our results against exact diagonalization, finding perfect agreement.
Solving for the time evolution of a many particle system whose dynamics is governed by Lindblad equation is hard. We extend the use of the transfer matrix approach to a class of Lindblad equations that admit a closed hierarchy of two point correlators. An example that we treat is the X X spin chain, i.e., free fermions, subject to the local onsite dephasing, but can be extended to other Hermitian dissipators, e.g., nonlocal dephasing. We find a simple expression of the Green's function in the Laplace domain. The method can be used to get analytical results in the thermodynamic limit, for instance, to get the evolution of the magnetization density and to explicitly see the crossover between ballistic and diffusive behavior, or to show that the correlations between operators at distance l decay with time as 1 / t ⌈ l / 2 ⌉ + 1 / 2 . It also provides a fast numerical method to determine the evolution of the density with a complexity scaling with the system size more favorably than in previous methods, easily allowing one to study systems with ∼ 10 6 spins.
Abstract We theoretically investigate the many-body dynamics of a tight-binding chain with dephasing noise on an infinite interval. We obtain the exact solution of the average particle-density profile for the domain wall and the alternating initial conditions via the Bethe ansatz, analytically deriving the asymptotic expressions for the long-time dynamics. For the domain wall initial condition, we obtain the scaling form of the average density, elucidating that diffusive transport always emerges in the long-time dynamics if the strength of the dephasing, no matter how small, is positive. For the alternating initial condition, our exact solution leads to the fact that the average density displays oscillatory decay or overdamped decay depending on the strength of the dissipation. Furthermore, we demonstrate that the asymptotic forms approach those of the symmetric simple exclusion process, identifying corrections from it.
We use the Bethe Ansatz technique to study dissipative systems experiencing loss. The method allows us to exactly calculate the Liouvillian spectrum. This opens the possibility of analytically calculating the dynamics of a wide range of experimentally relevant models including cold atoms subjected to one and two body losses, coupled cavity arrays with bosons escaping the cavity, and cavity quantum electrodynamics. As an example of our approach we study the relaxation properties in a boundary driven XXZ spin chain. We exactly calculate the Liouvillian gap and find different relaxation rates with a novel type of dynamical dissipative phase transition. This physically translates into the formation of a stable domain wall in the easy-axis regime despite the presence of loss. Such analytic results have previously been inaccessible for systems of this type.
We construct an exact map between a tight-binding model on any bipartite lattice in the presence of dephasing noise and a Hubbard model with imaginary interaction strength. In one dimension, the exact many-body Liouvillian spectrum can be obtained by application of the Bethe ansatz method. We find that both the nonequilibrium steady state and the leading decay modes describing the relaxation at late times are related to the η-pairing symmetry of the Hubbard model. We show that there is a remarkable relation between the time evolution of an arbitrary k-point correlation function in the dissipative system and k-particle states of the corresponding Hubbard model.
The Lindblad quantum master equation is one of the central approaches to the physics of open quantum systems. In particular, boundary driving enables the study of transport, where a steady state emerges in the long-time limit, which features a constant current and a characteristic density profile. While the Lindblad equation complements other approaches to transport in closed quantum systems, it has become clear that a connection between closed and open systems exists in certain cases. Here, we build on this connection for magnetization transport in the spin-1/2 XXZ chain with and without integrability-breaking perturbations. Specifically, we study the question of whether the time evolution of the open quantum system can be described on the basis of classical correlation functions, as generated by the Hamiltonian equations of motion for real vectors. By comparing the results to exact numerical simulations of the Lindblad equation, we find a good accuracy of such a description for a range of model parameters which is consistent with previous studies on closed systems. While this agreement is an interesting physical observation, it also suggests that classical mechanics can be used to solve the Lindblad equation for comparatively large system sizes which lie outside the possibilities of a quantum mechanical treatment. We also point out counterexamples and limitations for the quantitative extraction of transport coefficients. Remarkably, our classical approach to large open systems allows us to detect superdiffusion at the isotropic point.
We study a model of noninteracting spinless fermions coupled to local dephasing and boundary drive and described within a Lindblad master equation. The model features an interplay between infinite temperature thermalization due to bulk dephasing and a nonequilibrium stationary state due to the boundary drive and dissipation. We revisit the linear and nonlinear transport properties of the model, featuring a crossover from ballistic to diffusive scaling, and compute the spectral and occupation properties encoded in the single-particle Green's functions, that we compute exactly using the Lindblad equations of motion in spite of the interacting nature of the dephasing term. We show that the distribution function in the bulk of the system becomes frequency independent and flat, consistent with infinite temperature thermalization, while near the boundaries it retains strong nonequilibrium features that reflect the continuous injection and depletion of particles due to driving and dissipation.