Scaling symmetry of quantum correlations in quadratic Hamiltonians and its applications
Date: 12 OCTOBER 2022 from 15:00 to 16:00
In quadratic Hamiltonians, various quantum correlation measures such as entanglement entropy, fidelity, and Loschmidt echo possess an inherent scaling symmetry that the Hamiltonian of the system does not have. We exploit this symmetry to address various problems in semi-classical gravity. In the time-independent case, we use this symmetry to establish a one-to-one correspondence between entanglement energy, entropy, and temperature (quantum entanglement mechanics) and the Komar energy, Bekenstein-Hawking entropy, and Hawking temperature of the horizon (black-hole thermodynamics), respectively. This correspondence is also found to exactly capture the Komar relation and Smarr formula for generic 4-D spherically symmetric space-times. The scaling symmetry can be further generalized to time-dependent systems. Such systems may evolve to develop inverted or zero-mode instabilities that potentially apply to various physical phenomena. We show that, asymptotically, in the presence of instabilities, the leading order dynamics of various correlation measures --- entanglement entropy, fidelity, and Loschmidt echo --- are related via simple expressions. We quantify such instabilities in terms of scrambling time and Lyapunov exponents and show that the system mimics classicality under certain conditions. We then discuss its implications for the quantum-classical transition of primordial density perturbations in the early Universe.
[Refs: https://arxiv.org/abs/2205.13338, https://doi.org/10.1103/PhysRevD.103.125008, https://doi.org/10.1103/PhysRevD.102.125025 ]