Approaching the scaling limit of transport through lattices with dephasing

TL;DR

This work develops a generalized Markovian framework for dephasing and relaxation in quadratic fermionic networks, showing that the one-particle correlation matrix $\mathbfcal{C}$ obeys a closed evolution that reduces to a Lyapunov-like form in the absence of dephasing. It provides an efficient, self-consistent method to compute the steady state $\mathbfcal{C}^\infty$ via a non-Hermitian Hamiltonian and a structured decomposition of self-energies, with a detailed cost analysis that enables scaling to $\sim 10^4$ sites. The authors apply the method to a 1D lattice with long-range hopping and onsite dephasing, demonstrating a super-diffusive-to-diffusive transition at $\alpha_c \approx 1.5$ and achieving improved estimates of the critical point and transport exponent using large system sizes. The approach offers a versatile tool for benchmarking and exploring transport in Markovian open quantum systems, including extended reservoirs and quantum networks for machine learning and beyond.

Abstract

We examine the stationary--state equations for lattices with generalized Markovian dephasing and relaxation. When the Hamiltonian is quadratic, the single--particle correlation matrix has a closed system of equations even in the presence of these two processes. The resulting equations have a vectorized form related to, but distinct from, Lyapunov's equation. We present an efficient solution that helps to achieve the scaling limit, e.g., of the current decay with lattice length. As an example, we study the super--diffusive--to--diffusive transition in a lattice with long--range hopping and dephasing. The approach enables calculations with up to $10^4$ sites, representing an increase of $10$ to $40$ times over prior studies. This enables a more precise extraction of the diffusion exponent, enhances agreement with theoretical results, and supports the presence of a phase transition. There is a wide range of problems that have Markovian relaxation, noise, and driving. They include quantum networks for machine--learning--based classification and extended reservoir approaches (ERAs) for transport. The results here will be useful for these classes of problems.

Figures (3)

• Figure 1: Markovian network dynamics. A network of modes undergo Markovian relaxation, represented with arrows and weights $\gamma^{+(-)}_{mm^\prime}$ ($\gamma^{+(-)}_{m}$ for onsite), and dephasing, represented by blurred bonds with strength $\sigma_{mm^\prime}$ or blurred sites with strength $\sigma_m$. These processes drive dynamics and transport. Extended reservoirs are an example of such a network where a subset of modes forms the left reservoir ($\mathcal{L}$) and another the right ($\mathcal{R}$), each with relaxation that maintains a potential or temperature drop. Another example is boundary transport with injection and depletion only at terminal sites.
• Figure 2: Boundary--drive with long--range hopping: (a) A one--dimensional lattice is held at an infinite bias with the left reservoir $\mathcal{L}$ completely filled and the right reservoir $\mathcal{R}$ completely empty. Long--range hopping is given by a power--law, $v_\mathcal{S}/r^{\alpha}$. The injection and depletion rates are $\gamma^{+}$ and $\gamma^{-}$ from $\mathcal{L}$ and to $\mathcal{R}$, respectively. Each site is subject to Markovian dephasing $\sigma$ indicated by the blurred sites. (b) Stationary--state occupation, $n_i$, versus $i$ for various hopping exponents $\alpha$. The solid line ($\alpha=1.51$) shows a perfectly linear density drop, while smaller $\alpha$ (dashed and dotted curves) develop pronounced curvature and boundary layers.
• Figure 3: Super--diffusive--to--diffusive transition: (a) Phase diagram showing the transport exponent $\nu$ as a function of the long--range hopping exponent $\alpha$. The stationary--state resistance scales as $R_{SS} \propto N_\mathcal{S}^\nu$, where $N_\mathcal{S}$ is the system size. For $\alpha \leq \alpha_c = 1.5$, the system exhibits super--diffusive transport with an exponent $\nu = 2 \, \alpha - 2$. For $\alpha > \alpha_c$, a transition to diffusive transport occurs, where $\nu = 1$. The critical value $\alpha_c =1.5$ analytically marks the boundary where the current operator norm bounds coherent transport. (b) The exponent $\nu$ of the system size scaling for the resistance $R_{SS} = J_{SS}^{-1}$ in the stationary--state as a function of the long--range exponent parameter $\alpha$. Orange squares (with gray error bars) and green circles (errors are smaller than the symbol) correspond to the system sizes between 512 and 1024, and between 7500 and 9000 sites, respectively. The orange and green solid lines guide the eye, and the green dot--dashed and orange dashed lines correspond to the linear fit below $\alpha = 1.5$. The black solid line corresponds to the analytical $\nu = 2 \, \alpha - 2$. Results are with $\sigma = 10^3 \, v_\mathcal{S}$ and $v_\mathcal{S}$ sets the frequency scale.