Dynamics of operator size distribution in q-local quantum Brownian SYK and spin models
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866913803741954048 |
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| author | Xu, Shenglong |
| author_facet | Xu, Shenglong |
| contents | We study operator dynamics in Brownian quantum many-body models with $q$-local interactions. The operator dynamics are characterized by the time-dependent size distribution, for which we derive an exact master equation in both the Brownian Majorana Sachdev-Ye-Kitaev (SYK) model and the spin model for general $q$. This equation can be solved numerically for large systems. Additionally, we obtain the analytical size distribution in the large $N$ limit for arbitrary initial conditions and $q$. The distributions for both models take the same form, related to the $χ$-squared distribution by a change of variable, and strongly depend on the initial condition. For small initial sizes, the operator dynamics are characterized by a broad distribution that narrows as the initial size increases. When the initial operator size is below $q-2$ for the Majorana model or $q-1$ for the spin model, the distribution diverges in the small size limit at all times. The mean size of all operators, which can be directly measured by the out-of-time ordered correlator, grows exponentially during the early time. In the late time regime, the mean size for a single Majorana or Pauli operator for all $q$ decays exponentially as $t e^{-t}$, much slower than all other operators, which decay as $e^{-t}$. At finite $N$, the size distribution exhibits modulo-dependent branching within a symmetry sector for the $q \geq 8$ Majorana model and the $q \geq 4$ spin model. Our results reveal universal features of operator dynamics in $q$-local quantum many-body systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11737 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamics of operator size distribution in q-local quantum Brownian SYK and spin models Xu, Shenglong Quantum Physics Statistical Mechanics High Energy Physics - Theory We study operator dynamics in Brownian quantum many-body models with $q$-local interactions. The operator dynamics are characterized by the time-dependent size distribution, for which we derive an exact master equation in both the Brownian Majorana Sachdev-Ye-Kitaev (SYK) model and the spin model for general $q$. This equation can be solved numerically for large systems. Additionally, we obtain the analytical size distribution in the large $N$ limit for arbitrary initial conditions and $q$. The distributions for both models take the same form, related to the $χ$-squared distribution by a change of variable, and strongly depend on the initial condition. For small initial sizes, the operator dynamics are characterized by a broad distribution that narrows as the initial size increases. When the initial operator size is below $q-2$ for the Majorana model or $q-1$ for the spin model, the distribution diverges in the small size limit at all times. The mean size of all operators, which can be directly measured by the out-of-time ordered correlator, grows exponentially during the early time. In the late time regime, the mean size for a single Majorana or Pauli operator for all $q$ decays exponentially as $t e^{-t}$, much slower than all other operators, which decay as $e^{-t}$. At finite $N$, the size distribution exhibits modulo-dependent branching within a symmetry sector for the $q \geq 8$ Majorana model and the $q \geq 4$ spin model. Our results reveal universal features of operator dynamics in $q$-local quantum many-body systems. |
| title | Dynamics of operator size distribution in q-local quantum Brownian SYK and spin models |
| topic | Quantum Physics Statistical Mechanics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2408.11737 |