Krylov space approach to Singular Value Decomposition in non-Hermitian systems

TL;DR

This paper introduces a Krylov space method using SVD for non-Hermitian systems, enabling analysis of singular values and complexity without complex eigenvalues, with applications to random matrices and models like SYK.

Contribution

It presents a novel tridiagonalization approach leveraging SVD for non-Hermitian matrices, analyzing Krylov complexity and spectral features in chaotic and integrable systems.

Findings

Complexity peaks in chaotic cases due to singular value repulsion.

Analytical computation of Krylov complexity for certain non-Hermitian classes.

Distinct spectral features differentiate chaotic and integrable non-Hermitian systems.

Abstract

We propose a tridiagonalization approach for non-Hermitian random matrices and Hamiltonians using singular value decomposition (SVD). This technique leverages the real and non-negative nature of singular values, bypassing the complex eigenvalues typically found in non-Hermitian systems. We analyze the tridiagonal elements, namely the Lanczos coefficients and the associated Krylov (spread) complexity, appropriately defined through the SVD, across several examples, including Ginibre ensembles and the non-Hermitian Sachdev-Ye-Kitaev model. We demonstrate that in chaotic cases, the complexity exhibits a distinct peak due to the repulsion between singular values, a feature absent in integrable cases. Using our approach, we analytically compute the Krylov complexity for two-dimensional non-Hermitian random matrices within a subset of non-Hermitian symmetry classes including time-reversal, time-reversal$^{\dagger}$, chiral, and sublattice symmetry.