Exact Anomalous Current Fluctuations in Quantum Many-Body Dynamics

Abstract

Fluctuations of integrated currents have attracted considerable interest over the past decades in the context of statistical mechanics. Recently, anomalous current fluctuations, characterized by the M-Wright function, were obtained exactly in a classical automaton [$Ž$. Krajnik et al., Phys. Rev. Lett. 128, 160601 (2022)], and previous studies have shown that the anomalous behavior can arise in a variety of classical systems. Despite the rapidly growing interest in such anomalous behaviors, which capture a universal aspect of one-dimensional many-body transport, the exact derivation of the M-Wright function in quantum many-body systems has remained elusive. In this Letter, we present the first exact microscopic derivation of the M-Wright function in quantum many-body dynamics by analyzing the integrated spin current in a one-dimensional Fermi-Hubbard model with infinitely strong repulsive interactions. Our results lay the groundwork for exploring anomalous integrated currents in a broad class of quantum many-body systems.

Figures (4)

• Figure 1: (a) Schematic illustration for main result. The left figure depicts the initial and time evolved configurations of fermions on a one-dimensional lattice. The numbers below the lattice denote labels of sites, and a circle with an up (down) arrow represents a fermion with an up (down) spin. In this figure, an integrated spin current $\Delta S^z_R$ (see Eq. \ref{['eq:def_PS']} for the precise definition) is unity. The right figure shows the probability $\mathbb{P}_S[\Delta S_R^z,t]$ of $\Delta S_R^z$ at time $t$ as a function of a scaled integrated spin current $\mathcal{J}_S = \Delta S_R^z/t^{1/4}$. In this work, we exactly demonstrate that it obeys the M-Wright function $\mathbb{P}_{\rm MW}[\mathcal{J}_S,\sigma]$ in the long time limit. (b) Table for previous results and ours. The symbol of $\checkmark~(\times)$ means presence (absence) of previous works reporting $\mathbb{P}_{\rm MW}[\mathcal{J}_S,\sigma]$ for integrated currents. The theoretical results mean that $\mathbb{P}_{\rm MW}[\mathcal{J}_S,\sigma]$ is derived under some approximations or assumptions.
• Figure 2: Schematic illustration for Eq. \ref{['eq:exactPS']} with $\Delta N_R = 3$ and $n=1$. The three fermions occupying the shadowed cells only contribute to the integrated spin current at time $t$ due to $\mathbb{P}_{C}[ \Delta N_{R},t]$ in Eq. \ref{['eq:exactPS']}. The figure shows the specific spin configuration ($\downarrow, \uparrow, \uparrow$) for the three fermions, and our initial state of Eq. \ref{['eq:initial']} realizes $_{\Delta N_R} C_{n} = 3$ configurations having the same integrated spin current $\Delta S^z_R=1$ with equal probability [other spin configurations are ($\uparrow, \downarrow,\uparrow$) and ($\uparrow, \uparrow,\downarrow$)]. The quantity $_{\Delta N_R} C_{n} / 2^{\Delta N_R} = 3/8$ of Eq. \ref{['eq:exactPS']} reflects this probabilistic fact.
• Figure 3: Numerical verification for the convergence of the scaled probability $t^{1/4} \mathbb{P}_{S}[ t^{1/4}\mathcal{J}_S,t]$ to Eq. \ref{['eq:Ps_limit']}. The numerical method is explained in Sec. \ref{['sec:numerics']} of SM SM and the system size is $2N = 2000$. (Upper panel) Scaled probability at $t=50, 100, 200, 400,$ and $800$. The ordinate and abscissa are the scaled probability $t^{1/4} \mathbb{P}_{S} [ \Delta S_R^z, t]$ and the scaled integrated spin current $\mathcal{J}_S = \Delta S_R^z/t^{1/4}$, respectively. The dashed line represents Eq. \ref{['eq:Ps_limit']}. The inset shows the probability of the main panel with the logarithmic ordinate. (Lower panel) Time evolution of the difference $\mathbb{P}_{S}^{\rm typ} [0] - t^{1/4} \mathbb{P}_{S} [0, t]$. The dashed line represents the line proportional to $t^{-0.22}$.
• Figure 4: Numerical study for the integrated spin current of the $t0$ model in a realistic experimental setup. The numerical method is the same as that of Fig. \ref{['fig3']} and the system size to be $2N = 280$. The ordinate and abscissa are the scaled probability $t^{1/4} \mathbb{P}_{S} [ \Delta S_R^z, t]$ and the scaled integrated spin current $\mathcal{J}_S = \Delta S_R^z/t^{1/4}$, respectively. The inset shows the main panel with the logarithmic ordinate. The dashed and dash-dotted lines represent Eq. \ref{['eq:Ps_limit']} and the Gaussian function $\mathbb{P}_{S}^{\rm typ} [0] e^{-2\mathcal{J}_S^2}$, respectively.