Statistics-encoded tensor network approach in disordered quantum many-body spin chains

Abstract

Simulating the dynamics of quantum many-body systems with disorder is a fundamental challenge. In this work, we propose a general approach -- the statistics-encoded tensor network (SeTN) -- to study such systems. By encoding disorder into an auxiliary layer and averaging separately, SeTN restores translational invariance, enabling a well-defined transfer matrix formulation. We derive a universal criterion, $n \gg α^2 t^2$, linking discretization $n$, disorder strength $α$, and evolution duration $t$. This sets the resolution required for faithful disorder averaging and shows that encoding is most efficient in the weak-disorder, typically chaotic regime. Applied to the disordered transverse-field Ising model, SeTN shows that the spectral form factor is governed by the leading transfer-matrix eigenvalue, in contrast to the kicked Ising model. SeTN thus provides a novel framework for probing the disorder-driven dynamical phenomena in many-body quantum systems.

Figures (3)

• Figure 1: (Color online) (a) Tensor network (TN) representation of the Trotterized evolvation operator. The tensor encoding the disorder is decomposed into a vector (blue circle) and a Kronecker delta tensor (black point). (b) A general two-layer TN representation of the disorder-averaged transfer matrix $\mathrm{T}$. The orange-shaded region indicates the disorder average. (c) Diagrammatic illustration of the SeTN compression. $S^{(n)}$ is the singular values obtained via corresponding singular value decomposition.
• Figure 2: (Color online) (a) Blue dots: squared relative singular values from SeTN compression under various parameters (Num.). Dashed lines: analytic predictions from perturbation theory (Ana.), with coefficients fitted to numerical data. (b) Blue dots: first seven coefficients $c_i$ obtained by fitting the data in (a). These are further fitted to exponential (green dashed), Gaussian (purple dashed), and gamma-like (red dot-dashed) functions. (c) Spectral form factor of the disordered TFIM with system size $L = 4$, computed using four methods: exact diagonalizing 1000 disorder realizations (avg #1000, green dotted; shaded band indicates the standard error), numerical integration (Exact, blue dashed), SeTN (red solid), and perturbative results (Pert., gray dot-dashed). The mean deviation between SeTN and the exact result is approximately 0.021. Parameters: $J = b = 1$, Trotter step $\tau = 0.005$, averaging over $M = 10^6$ realizations; singular values below $10^{-10}$ are truncated.
• Figure 3: (Color online) (a) SFF of the disordered TFIM with sizes $L = 9$ to $16$ (top to bottom), computed by ED using 2440 independent disorder realizations per size. Standard errors are below 4% at all times, and are therefore not visible on the logarithmic scale. Shaded regions (i) and (ii) indicate the first and second rebounce windows. (b) Magnitudes of the leading and subleading eigenvalues of the SeTN-derived transfer matrix as functions of time, obtained using Krylov (blue crosses) and non-Hermitian DMRG (red circles). Horizontal bars (i) and (ii) match the time windows in (a), highlighting the near-degeneracy of dominant eigenvalues. Black lines represent the fitted coefficient extracted by fitting the data in panel (a) to the form $\lambda(t)^L$, and the shaded region indicates the 80% confidence interval. Parameters: Trotter step $\tau = 0.05$, $J = b = 1, \alpha=0.5$.