Process-tensor approach to full counting statistics of charge transport in quantum many-body circuits

Abstract

We introduce a numerical tensor-network method to compute the statistics of the charge transferred across an interface partitioning an interacting one-dimensional many-body lattice system with $U(1)$ symmetry. Our approach is based on a matrix-product state representation of the process tensor (also known as influence functional or influence matrix) describing the effect of the bulk system on the degrees of freedom at the interface, allowing us to evaluate a multi-time correlation function that yields the moment-generating function of charge transfer. We develop a scheme to truncate non-Markovian correlations which preserves the proper normalization of the process tensor and ensures the correct physical properties of the generating function. We benchmark our approach by simulating magnetization transport within the Heisenberg spin-$1/2$ XXZ brickwork circuit model at infinite temperature. Our results recover the correct transport exponent describing ballistic, superdiffusive, and diffusive transport in different regimes of the model. We also demonstrate anomalous transport encoded by a self-similar scaling form of the moment-generating function outside of the ballistic regime. In particular, we confirm the breakdown of Kardar-Parisi-Zhang universality in higher-order transport cumulants at the isotropic point. Our work paves the way for process-tensor descriptions of non-Markovian open quantum systems to address current fluctuations in strongly interacting systems far from equilibrium.

Figures (10)

• Figure 1: Local magnetization auto-correlator $\langle Z_0^{}(n) Z_0^{}(0)\rangle$ as function of circuit depth $n$ for various $(\mathcal{J},\mathcal{J}^\prime)$ parameters indicated in the legends that define the two-site gates of the spin-$\frac{1}{2}$ brickwork XXZ circuit model. Coloured lines with markers indicate the correlators obtained using the methodology presented in Sec. \ref{['sec:methodology']}. Left, middle and right panels display the local correlators respectively in the ballistic, super-diffusive and diffusive transport regime of the circuit. Black lines indicate power-law fits to the data, and the respective exponents are displayed in the plot legends. Throughout this work, we use bond dimension $\chi=2^{10}$ in the ballistic regime (left panel) and $\chi=2^{11}$ in the superdiffusive and diffusive regime (middle and right panels, respectively).
• Figure 2: Second cumulant $\kappa_2(n)$ of the magnetization transported across the interface as a function of the circuit depth $n$ for the same choice of the parameters as in Fig. \ref{['fig:szsz']}. Colored lines with markers indicate the numerical data obtained from the moment generating function as described in Sec. \ref{['sec:methodology']} and the black lines indicate power-law fits to the data with the corresponding exponents indicated in the legends. Insets show the local time-dependent exponent for $\mathcal{J}=1/4$ [Eq. \ref{['local_exponent']}], which converges to the expected value (horizontal black line) for ballistic, superdiffusive, or diffusive transport in each of the three regimes.
• Figure 3: Kurtosis $\gamma_4 = \kappa_4/\kappa_2^2$ of the magnetization transported across the interface as a function of the circuit depth $n$, for the same parameters as in Figs. \ref{['fig:szsz']} and \ref{['fig:k2']}. Horizontal dashed lines indicate predictions for the Gaussian distribution, i.e., $\gamma_4=0$. At the isotropic point (middle panel) we additionally display predictions for two non-Gaussian distributions in the KPZ universality class: the Baik-Rains and Tracy-Widom (TW) distributions.
• Figure 4: Sextosis $\gamma_6 = \kappa_6/\kappa_2^3$ of the magnetization transported across the interface as a function of the circuit depth $n$, for the same parameters as in Figs. \ref{['fig:szsz']}--\ref{['fig:g4']}. The Gaussian sextosis of $\gamma_6=0$ is indicated by the horizontal dashed lines.
• Figure 5: Cumulant-generating function $\ln\mathcal{Z}(\lambda,n)$ for the magnetization transported across the interface as a function of the counting field $\lambda$ for various circuit depths $n$ indicated in the colorbars. We use the same parameters and color scheme as all previous figures. In the ballistic transport regime (left panel), the cumulant generating function shows a large-deviation form, $\mathcal{Z}(\lambda,n) = e_{}^{-n \mathcal{F}(\lambda)}$, where the function $\mathcal{F}(\lambda)$ depends on the circuit parameters $(\mathcal{J},\mathcal{J}')$ but not on the circuit depth $n$. In the superdiffusive (middle panel) and diffusive (right panel) regimes, we see scaling collapse when plotting the cumulant generating function as a function of $\lambda n^{1/2z}$, indicating a self-similar form $\mathcal{Z}(\lambda,n)= \mathcal{Z}(\lambda n^{1/2z})$ with the exponents $z=3/2$ and $z=2$, respectively.