Non-linear sigma models for non-Hermitian random matrices in symmetry classes AI$^{\dagger}$ and AII$^{\dagger}$

• URL: http://arxiv.org/abs/2410.24043v1
• Date: Thu, 31 Oct 2024 15:38:13 GMT
• Title: Non-linear sigma models for non-Hermitian random matrices in symmetry classes AI$^{\dagger}$ and AII$^{\dagger}$
• Authors: Anish Kulkarni, Kohei Kawabata, Shinsei Ryu,
• Abstract summary: chaotic open quantum systems exhibit universal bulk spectral correlations on the basis of time-reversal symmetry$dagger$. We analytically study the spectral correlations of non-Hermitian random matrices in the presence of TRS$dagger$ with signs $+1$ and $-1$, corresponding to symmetry classes AI$dagger$ and AII$dagger$, respectively.
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• Abstract: Symmetry of non-Hermitian matrices underpins many physical phenomena. In particular, chaotic open quantum systems exhibit universal bulk spectral correlations classified on the basis of time-reversal symmetry$^{\dagger}$ (TRS$^{\dagger}$), coinciding with those of non-Hermitian random matrices in the same symmetry class. Here, we analytically study the spectral correlations of non-Hermitian random matrices in the presence of TRS$^{\dagger}$ with signs $+1$ and $-1$, corresponding to symmetry classes AI$^{\dagger}$ and AII$^{\dagger}$, respectively. Using the fermionic replica non-linear sigma model approach, we derive $n$-fold integral expressions for the $n$th moment of the one-point and two-point characteristic polynomials. Performing the replica limit $n\to 0$, we qualitatively reproduce the density of states and level-level correlations of non-Hermitian random matrices with TRS$^{\dagger}$.

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