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arXiv:2607.05094

Brownian Motion in Orthogonal and Symplectic Groups

Zhiyang Tan, Piet W. Brouwer

quant-phcond-mat.mes-hallmath-phSubmitted Jul 6, 2026• Announced Jul 7, 2026• 0 citations
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Abstract

Matrix Brownian motion provides a powerful framework for studying crossover ensembles in quantum chaos and quantum transport, as well as thermalization and information scrambling in many-body dynamics. Here, we develop a unified diagrammatic framework to characterize Brownian ensembles for orthogonal and symplectic random matrices, which describe systems with particle-hole symmetry. We compute polynomial averages up to fourth order and construct an orthogonally invariant interpolation for the disconnected $\mathrm{SO}^-(q)$ sector of the orthogonal group. We consider applications relating to the fields of quantum information, quantum chaos, and quantum transport.

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13 pages, 5 figures

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Matrix Brownian motion provides a powerful framework for studying crossover ensembles in quantum chaos and quantum transport, as well as thermalization and information scrambling in many-body dynamics. Here, we develop a unified diagrammatic framework to characterize Brownian ensembles for orthogonal and symplectic random matrices, which describe systems with particle-hole symmetry. We compute polynomial averages up to fourth order and construct an orthogonally invariant interpolation for the disconnected SO−(q)\mathrm{SO}^-(q)SO−(q) sector of the orthogonal group. We consider applications relating to the fields of quantum information, quantum chaos, and quantum transport.
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