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

TL;DR

The paper addresses exact time-dependent density profiles for a tight-binding chain with dephasing noise, formulated via the GKSL equation on the infinite lattice. By mapping to a two-particle Fermi-Hubbard problem with imaginary interaction and exploiting Bethe ansatz on the infinite lattice, it derives a double-contour integral Green's function that yields closed-form density expressions for domain-wall and alternating initial conditions and their long-time asymptotics. It demonstrates diffusive scaling for domain-wall transport and a dissipation-driven dynamical transition for the alternating case, with leading behavior matching SEP and computable corrections, thereby providing precise benchmarks for open quantum systems and clarifying when SEP-like dynamics emerges. The work also highlights the impact of initial-state symmetry on relaxation and lays out a pathway to study fluctuations beyond the current results, potentially extending to other Bethe-ansatz solvable GKSL models. The results offer exact, tunable insights into diffusion, dissipation, and non-equilibrium relaxation in interacting quantum systems.

Abstract

We theoretically investigate the many-body dynamics of a tight-binding chain with dephasing noise on the infinite interval. We obtain the exact solution of an 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 the 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 over-damped 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.

Figures (6)

• Figure 1: Schematic illustrations of the initial states. The blue circles represent fermions and the numbers below the lattice denote the site numbers. (a) Domain wall state $\vert \mathrm{DW}\rangle := \prod_{j\leq 0}a^\dagger_j \vert 0\rangle$. The fermions occupy the left half of the system. (b) Alternating state $\vert \mathrm{ALT}\rangle := \prod_{j\in \mathbb{Z}} a^\dagger_{2j} \vert 0\rangle$. The fermions occupy every other site.
• Figure 2: Density profile $\langle n_x\rangle_t$ with rescaled coordinate $X =x/\sqrt{\tau}$. The colored markers show the exact solution \ref{['eq:density_dw']} for $\gamma =1$, while the dased line does the leading term of the asymptotic expansion \ref{['eq:asymptotics_dw']}.
• Figure 3: Time evolution of $\langle n_x \rangle$ for the alternating initial condition. The colored lines show the exact solution, Eq. \ref{['eq:density_alt']}. The dashed lines show the asymptotic form, Eq. \ref{['eq:density_asymptotics_alt']}.
• Figure 4: Correction $\Delta \langle n_x\rangle_t$ for $\gamma=0.5,1$, and $1.5$. The colored lines show the exact expression of $\Delta \langle n_x\rangle_t$ at $\tau =1000$. The dashed lines show the asymptotic form, Eq. \ref{['eq:density_correction']}.
• Figure 5: Schematic illustration of the contour in the asymptotic analysis of Eq. \ref{['eq:contour_dw']}. The arrowed line shows the contour, while the dashed line represents the unit circle for clarity.