Krylov space approach to Singular Value Decomposition in non-Hermitian systems
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| author | Nandy, Pratik Pathak, Tanay Xian, Zhuo-Yu Erdmenger, Johanna |
| author_facet | Nandy, Pratik Pathak, Tanay Xian, Zhuo-Yu Erdmenger, Johanna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09309 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Krylov space approach to Singular Value Decomposition in non-Hermitian systems Nandy, Pratik Pathak, Tanay Xian, Zhuo-Yu Erdmenger, Johanna Quantum Physics Disordered Systems and Neural Networks Statistical Mechanics High Energy Physics - Theory 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. |
| title | Krylov space approach to Singular Value Decomposition in non-Hermitian systems |
| topic | Quantum Physics Disordered Systems and Neural Networks Statistical Mechanics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2411.09309 |