Kavli Affiliate: Cheng Peng
| First 5 Authors: Tingfei Li, Cheng Peng, Jianghui Yu, ,
| Summary:
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.
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