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arXiv:2501.07095 · Journal of Statistical Mechanics Theory and Experiment

Exact density profile in a tight-binding chain with dephasing noise

Taiki Ishiyama, Kazuya Fujimoto, Tomohiro Sasamoto

Published Mar 3, 2025• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/adba43

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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 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 full counting statistics (FCS) of current has long provided fundamental insights into nonequilibrium systems. Recently, the FCS in quantum many-body systems has attracted growing attention, driven by rapid experimental progress in measuring current fluctuations. Nevertheless, for diffusive quantum many-body dynamics, the FCS of current has yet to be obtained exactly. In this Letter, we present the first exact solution for the FCS of current in a diffusive quantum many-body system, specifically a tight-binding chain with dephasing noise. By leveraging the system's SU(2) symmetry and a mapping to the one-dimensional Hubbard model, we derive an exact Fredholm determinant representation for the moment generating function of the time-integrated current. Our long-time asymptotic analysis shows that the cumulant generating function, and hence the corresponding large-deviation function, exhibit diffusive scaling for any nonzero dephasing. We compare our theoretical predictions with experimentally measured current variance and find consistent diffusive scaling.
The relaxation dynamics of the one-dimensional totally asymmetric simple exclusion process on a ring is considered in the case of step initial condition. Analyzing the time evolution of the local particle densities and currents by the Bethe ansatz method, we examine their full relaxation dynamics. As a result, we observe peculiar behaviors, such as the emergence of a ripple in the density profile and the existence of the excessive particle currents. Moreover, by making a finite-size scaling analysis of the asymptotic amplitudes of the local densities and currents, we find the scaling exponents with respect to the total number of sites to be -3/2 and -1, respectively.
We study the dynamics of an isotropic spin-12\frac{1}{2}21​ Heisenberg chain starting in a domain-wall initial condition where the spins are initially up on the left half-line and down on the right half-line. We focus on the long-time behavior of the magnetization profile. We perform extensive time-dependent density-matrix renormalization-group simulations (up to t=350t=350t=350) and find that the data are compatible with a diffusive behavior. Subleading corrections decay slowly blurring the emergence of the diffusive behavior. We also compare our results with two alternative scenarios: superdiffusive behavior and enhanced diffusion with a logarithmic correction. We finally discuss the evolution of the entanglement entropy.
The one-dimensional totally asymmetric simple exclusion process, a Markov process describing classical hard-core particles hopping in the same direction, is considered on a periodic lattice of L sites. The relaxation to the nonequilibrium steady state, which occurs on the time scale t∼L^{3/2} for large L, is studied for the half-filled system with N=L/2 particles. Using large L asymptotics of Bethe ansatz formulas for the eigenstates, exact expressions depending explicitly on the rescaled time t/L^{3/2} are obtained for the average and two-point function of the local density, and for the current fluctuations for simple (stationary, flat and step) initial conditions, relating previous results for the infinite system to stationary large deviations. The final formulas have a nice interpretation in terms of a functional integral with the action of a scalar field in a linear potential.
A universal characterization of non-equilibrium steady states in interacting quantum many-body systems remains one of the central challenges of statistical physics. Here, we address this problem for a paradigmatic model of diffusive interacting quantum matter---the boundary-driven XXZ spin chain with bulk dephasing---and derive, directly from its microscopic Lindblad dynamics, an emergent classical Macroscopic Fluctuation Theory (MFT) governing its large-scale fluctuations. Crucially, the resulting hydrodynamics carries a density-dependent diffusivity and mobility as the fingerprint of interactions. This effective description enables the exact computation of the stationary density profile, long-range correlations, and the full counting statistics of the current, in excellent agreement with tensor-network simulations. Our work demonstrates that noisy quantum many-body systems can realize the universality class of genuinely interacting diffusive matter, beyond the constant-diffusivity class of the symmetric simple exclusion process, and establishes MFT as a powerful universal framework for interacting diffusive quantum systems.