Nonequilibrium dynamics and phase transitions in deformed Hamiltonians and the β-ensemble

Das, Adway Kumar (2025) Nonequilibrium dynamics and phase transitions in deformed Hamiltonians and the β-ensemble. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Adway Kumar Das (19RS062)) 19RS062.pdf - Submitted Version Restricted to Repository staff only Download (13MB)

Abstract

In generic dynamical systems, the notion of integrability implies an extensive number of globally conserved quantities. It is usually believed that in integrable quantum systems defined over discrete and finite Hilbert space, there exists similar conservation laws, even in the absence of a semiclassical limit. The perturbation of the underlying global symmetries breaks the integrability and the dynamics is governed by a Hamiltonian belonging to the deformed ensemble. In this thesis, we study the spectral statistics and non-equilibrium dynamics of the deformed Hamiltonians. We analytically compute the level spacing distribution, inverse participation ratio and survival probability of a non-equilibrium initial state in case of two-level systems. Our results from the two-level deformed Hamiltonians are insightful to understand quantum chaos and in a broader context the wave-particle duality of a photon in an interferometer. In higher dimensional Hamiltonians, breaking a global symmetry can have important implications for controlling the fidelity of information transfer in qubit networks. We study random matrix ensembles possessing centrosymmetry and identify different phases, characteristic timescales and equilibrium properties. We also look at the β-ensemble, which governs a disordered single particle system on a one dimensional lattice which can be locally mapped to the Anderson model within approximately decoupled spatial blocks. Despite having uncorrelated short-range hopping in one dimension, the β-ensemble exhibits two second order transitions leading to a nonergodic extended phase with a multiscale energy spectrum where the typical bulk energy states are fractal with no short-range correlations below a critical energy scale. Moreover, a vanishing fraction of localized energy eigenstates coexist with the extended states in the middle of the spectrum without forming a mobility edge. In the zero temperature limit, the β-ensemble reduces to a deterministic Hamiltonian, which has an energycoordinate duality to the quantum Harmonic oscillator. We find that corresponding bulk states have energy dependent multifractality while the energy spectrum has a singular continuous nature.

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