Exact Current Fluctuations in a Tight-Binding Chain with Dephasing Noise

TL;DR

This work analyzes current fluctuations in a tight-binding chain with dephasing noise under a step density profile. By unveiling a nontrivial omega-dependence of the current's moment generating function and computing exact Green's functions via a Bethe-ansatz treatment of the equivalent Fermi-Hubbard problem on the infinite lattice, the authors derive an exact expression for the variance and its long-time asymptotics. They show that any nonzero dephasing drives the current fluctuations from ballistic to diffusive, with a universal asymptotic form that matches the symmetric exclusion process (SEP) results, and they verify this behavior numerically using GKSL unraveling and determinant formulas. The work also establishes a precise correspondence with SEP for the cumulant generating function, highlighting a deep link between quantum open-system fluctuations and classical stochastic processes. Overall, the paper provides a rigorous quantum-transport analogue of classical fluctuation results and opens pathways to exact higher-order fluctuation analyses in open quantum integrable systems.

Abstract

For a tight-binding chain with dephasing noise on an infinite interval, we exactly calculate the variance of the integrated current for a step initial condition with average densities, $ρ_a$ on the negative axis and $ρ_b$ on the positive axis. Our exact solution reveals that the presence of dephasing, no matter how small, alters the nature of current fluctuations from ballistic to diffusive in the long-time limit. The derivation relies on the Bethe ansatz on the infinite interval and a nontrivial parameter dependence, referred to as the $ω$-dependence, of the moment generating function for the integrated current. Furthermore, we demonstrate that the asymptotic form of the variance and a numerically obtained cumulant generating function coincide with those in the symmetric simple exclusion process.

Figures (6)

• Figure 1: Schematic illustration of the step initial condition for $\rho_a=1$ and $\rho_b=0$. Starting from the step initial condition, we evolve the system according to Eq. \ref{['eq:GKSL']} and investigate the fluctuations of an integrated current $Q_t$ across the bond between site $0$ and site $1$.
• Figure 2: Numerical verification for the asymptotic form in Eq. \ref{['eq:asymptotic_variance_2']}. The dots represent the numerical results for $\gamma=1$ and system size $2L=128$. The dashed line shows Eq. \ref{['eq:asymptotic_variance_2']}.
• Figure 3: Comparison of the cumulant generating function $\chi(\lambda,t)$ for $\rho_a=1$ and $\rho_b=0$. The dots represent the numerical results for our model with $\tau=400$ and system size $2L=256$. The dashed line shows the analytical result in SEP, Eq. \ref{['eq:SEP']}. The inset shows the results with $\tau=20$ and system size $2L=256$.
• Figure S-1: Schematic illustration of the initial condition $\psi^{(2n)}_t(\bm{x};\bm{a}|\bm{y})$.
• Figure S-2: Schematic illustrations of the branch cut of $\alpha(z)$ for (a) $1<\gamma$ and (b) $\gamma \leq 1$. The wavy lines represent the branch cut, where $\beta$ and $\beta'$ are the roots of the quadratic equations, $x^2+2(2\gamma+1)x+1=0$ and $x^2+2(2\gamma-1)x+1=0$, respectively. The dashed lines represent unit circles for clarity.

Theorems & Definitions (8)

• Theorem
• Remark
• Lemma 1
• proof : Proof of Lemma \ref{['lem:reduction']}
• Lemma 2
• proof : Proof of Lemma \ref{['lem:I']}
• Lemma 3
• proof : Proof of Lemma \ref{['lem:scattering']}