Graph-Theoretic Analysis of $n$-Replica Time Evolution in the Brownian Gaussian Unitary Ensemble

TL;DR

The paper develops a graph-theoretic framework to study the $n$-replica time evolution operator $\mathcal{U}_n(t)=e^{\mathcal{L}_n t}$ for the Brownian Gaussian Unitary Ensemble (BGUE). By analyzing the moments of the generator $\mathcal{L}_n$ in a $D^{2n}$-dimensional Hilbert space, the authors derive explicit representations for $n=2$ and $n=3$ and introduce a systematic graph-denotation method that compresses the problem into a finite-dimensional matrix problem via graph categories. They provide a universal algorithm for arbitrary $n$, illustrate it with $n=4$, and show how the $n$-replica approach facilitates computation of observables such as correlators and spectral form factors, while revealing connections to quantum information concepts like unitary designs and SU$(D)$ representation theory. The work integrates a comprehensive combinatorial organization of contractions with explicit spectra and evolution functions, offering a versatile toolbox for analyzing Brownian disordered systems and their information-theoretic properties. The results underscore how Brownian disorder can be harnessed to study scrambling, design ensembles, and link disordered quantum dynamics to contour-based replica evolution, with potential applications to BGUE, BGOE, and BGSE frameworks.

Abstract

In this paper, we investigate the $n$-replica time evolution operator $\mathcal{U}_n(t)\equiv e^{\mathcal{L}_nt} $ for the Brownian Gaussian Unitary Ensemble (BGUE) using a graph-theoretic approach. We examine the moments of the generating operator $\mathcal{L}_n$, which governs the Euclidean time evolution within an auxiliary $D^{2n}$-dimensional Hilbert space, where $D$ represents the dimension of the Hilbert space for the original system. Explicit representations for the cases of $n = 2$ and $n = 3$ are derived, emphasizing the role of graph categorization in simplifying calculations. Furthermore, we present a general approach to streamline the calculation of time evolution for arbitrary $n$, supported by a detailed example of $n = 4$. Our results demonstrate that the $n$-replica framework not only facilitates the evaluation of various observables but also provides valuable insights into the relationship between Brownian disordered systems and quantum information theory.

Figures (3)

• Figure 1: The four non-trivial cases of $P_{i\bar{j}}\mathsf{F}$.
• Figure 2: The two non-trivial cases of $X_{ij}\mathsf{F}$.
• Figure 3: The pattern coefficient $f_a (t)$ or its log-plot, where we set $D = 10,J=1$. We depict the categories of $p=0,1,2,3$ with red, orange, green and blue color respectively. One can see the three categories with $p=3$ has the energy $0$, so they (blue lines) reminds finite at time $t=\infty$.