Ramp from replica trick
Abstract We compute the spectral form factor of the modular Hamiltonian K = −ln ρ A associated to the reduced density matrix of a Haar random state. A ramp is demonstrated and we find an analytic expression for its slope. Our method involves an application of the replica trick, where we first calcul...
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2025-01-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP01(2025)104 |
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| author | Xuchen Cao Thomas Faulkner |
| author_facet | Xuchen Cao Thomas Faulkner |
| author_sort | Xuchen Cao |
| collection | DOAJ |
| description | Abstract We compute the spectral form factor of the modular Hamiltonian K = −ln ρ A associated to the reduced density matrix of a Haar random state. A ramp is demonstrated and we find an analytic expression for its slope. Our method involves an application of the replica trick, where we first calculate the correlator tr ρ A n tr ρ A m $$ \left\langle \textrm{tr}{\rho}_A^n tr{\rho}_A^m\right\rangle $$ at large bond dimension and then analytically continue the indices n, m from integers to arbitrary complex numbers. We use steepest descent methods at large modular times to extract the ramp. The large bond dimension limit of the replicated partition function is dominated by a sum over annular non-crossing permutations. We explored the similarity between our results and calculations of the spectral form factor in low dimensional gravitational theories where the ramp is determined by the double trumpet geometry. We find there is an underlying resemblance in the two calculations, when we interpret the annular non-crossing permutations as representing a discretized version of the double trumpet. Similar results are found for an equilibrated pure state in place of the Haar random state. |
| format | Article |
| id | doaj-art-017dac576df14e02816313ed0a936f73 |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-017dac576df14e02816313ed0a936f732025-02-09T12:07:55ZengSpringerOpenJournal of High Energy Physics1029-84792025-01-012025113510.1007/JHEP01(2025)104Ramp from replica trickXuchen Cao0Thomas Faulkner1Department of Physics, University of IllinoisDepartment of Physics, University of IllinoisAbstract We compute the spectral form factor of the modular Hamiltonian K = −ln ρ A associated to the reduced density matrix of a Haar random state. A ramp is demonstrated and we find an analytic expression for its slope. Our method involves an application of the replica trick, where we first calculate the correlator tr ρ A n tr ρ A m $$ \left\langle \textrm{tr}{\rho}_A^n tr{\rho}_A^m\right\rangle $$ at large bond dimension and then analytically continue the indices n, m from integers to arbitrary complex numbers. We use steepest descent methods at large modular times to extract the ramp. The large bond dimension limit of the replicated partition function is dominated by a sum over annular non-crossing permutations. We explored the similarity between our results and calculations of the spectral form factor in low dimensional gravitational theories where the ramp is determined by the double trumpet geometry. We find there is an underlying resemblance in the two calculations, when we interpret the annular non-crossing permutations as representing a discretized version of the double trumpet. Similar results are found for an equilibrated pure state in place of the Haar random state.https://doi.org/10.1007/JHEP01(2025)104AdS-CFT CorrespondenceModels of Quantum Gravity2D GravityRandom Systems |
| spellingShingle | Xuchen Cao Thomas Faulkner Ramp from replica trick Journal of High Energy Physics AdS-CFT Correspondence Models of Quantum Gravity 2D Gravity Random Systems |
| title | Ramp from replica trick |
| title_full | Ramp from replica trick |
| title_fullStr | Ramp from replica trick |
| title_full_unstemmed | Ramp from replica trick |
| title_short | Ramp from replica trick |
| title_sort | ramp from replica trick |
| topic | AdS-CFT Correspondence Models of Quantum Gravity 2D Gravity Random Systems |
| url | https://doi.org/10.1007/JHEP01(2025)104 |
| work_keys_str_mv | AT xuchencao rampfromreplicatrick AT thomasfaulkner rampfromreplicatrick |