Integrability and complexity in quantum spin chains
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913235355041792 |
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| author | Craps, Ben De Clerck, Marine Evnin, Oleg Hacker, Philip |
| author_facet | Craps, Ben De Clerck, Marine Evnin, Oleg Hacker, Philip |
| contents | There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation between integrability and reduced complexity in practice has remained elusive. We provide a connection of this sort by constructing a specific matrix in terms of the eigenvectors of a given quantum Hamiltonian. The null eigenvalues of this matrix are in one-to-one correspondence with conserved quantities that have simple locality properties (a hallmark of integrability). The typical magnitude of the eigenvalues, on the other hand, controls an explicit bound on Nielsen's complexity of the quantum evolution operator, defined in terms of the same locality specifications. We demonstrate how this connection works in a few concrete examples of quantum spin chains that possess diverse arrays of highly structured conservation laws mandated by integrability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_00037 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Integrability and complexity in quantum spin chains Craps, Ben De Clerck, Marine Evnin, Oleg Hacker, Philip Quantum Physics Statistical Mechanics High Energy Physics - Theory Mathematical Physics Exactly Solvable and Integrable Systems There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation between integrability and reduced complexity in practice has remained elusive. We provide a connection of this sort by constructing a specific matrix in terms of the eigenvectors of a given quantum Hamiltonian. The null eigenvalues of this matrix are in one-to-one correspondence with conserved quantities that have simple locality properties (a hallmark of integrability). The typical magnitude of the eigenvalues, on the other hand, controls an explicit bound on Nielsen's complexity of the quantum evolution operator, defined in terms of the same locality specifications. We demonstrate how this connection works in a few concrete examples of quantum spin chains that possess diverse arrays of highly structured conservation laws mandated by integrability. |
| title | Integrability and complexity in quantum spin chains |
| topic | Quantum Physics Statistical Mechanics High Energy Physics - Theory Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2305.00037 |