Random matrix theory universality of current operators in spin-$S$ Heisenberg chains

TL;DR

The paper investigates whether microcanonically truncated observables in chaotic quantum many-body systems exhibit random-matrix universality described by a unitarily invariant ensemble (UIE) rather than a strict Gaussian unitary ensemble. Using quantum-typicality-based traces and Chebyshev expansions, it analyzes the spin current in translationally invariant Heisenberg chains with $S=\tfrac{1}{2},1,\tfrac{3}{2}$ and demonstrates that, within a sufficiently narrow energy window $ΔE ≤ ΔE_U$, the free cumulants satisfy $Δ_k(δE) ∝ δE^{k-1}$, aligning with UIE predictions in chaotic regimes. In contrast, integrable settings do not show this scaling, indicating the UIE description is tied to chaos. Overall, the results support the universality of observable statistics in chaotic quantum many-body systems and extend RMT concepts to dynamical observables like current operators, beyond spectral properties.

Abstract

Quantum chaotic systems exhibit certain universal statistical properties that closely resemble predictions from random matrix theory (RMT). With respect to observables, it has recently been conjectured that, when truncated to a sufficiently narrow energy window, their statistical properties can be described by an unitarily invariant ensemble, and testable criteria have been introduced, which are based on the scaling behavior of free cumulants. In this paper, we investigate the conjecture numerically in translationally invariant Heisenberg spin chains with spin quantum number $S =\frac{1}{2},1,\frac{3}{2}$. Combining a quantum-typicality-based numerical method with the exploitation of the system's symmetries, we study the spin current operator and find clear evidence of consistency with the proposed criteria in chaotic cases. Our findings further support the conjecture of the existence of RMT universality as manifest in the observable properties in quantum chaotic systems.

Figures (3)

• Figure 1: Numerical results for the XXZ model for spin quantum number $S = 1$. Even cumulants $\Delta_k$ for $k=2,4,6$ as a function of $\delta E$ for the total spin current operator $J_S$ for parameter sets [(a)(b)(c)] $\Delta = 0.5, \Delta^\prime = 0.5$; [(d)(e)(f)] $\Delta = 1.5, \Delta^\prime = 0.5$ and [(g)(h)(i)] $\Delta = 1.5, \Delta^\prime = 0.0$, for system sizes $L=16, 18, 20$. As a guide to the eye, the dashed lines and vertical dotted lines indicate the scaling $\Delta_{k}\propto\delta E^{k-1}$ and an approximate location of $\Delta E_U$, respectively.
• Figure 2: Similar data as the one in Fig. \ref{['Fig-Spin1']} but for spin quantum number $S = \frac{3}{2}$.
• Figure 3: Similar data s the one in Fig. \ref{['Fig-Spin1']} but for spin quantum number $S = \frac{1}{2}$.