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arXiv:quant-ph/0203071 · Journal of Physics A Mathematical and General

Ergodic properties of local spectral density for a conservative system of coupled quantum states

Vladimir S. Starovoitov

Published Apr 12, 2002• 1 citation
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Abstract

The shape and the inverse participation ratio (IPR) of local spectral density (LSD) are studied for a generic isolated system of coupled quantum states, the Hamiltonian of which is represented by a band random matrix with disordered leading diagonal. We find for the matrices with arbitrary small band that the lack of ergodicity for LSD can be associated with an exponential increase in IPR with the ratio v /Δ c ( v is the root of the mean square for off-diagonal matrix elements and Δ c is the energy spacing between directly coupled basis states). Criteria specifying transition to localization and ergodicity for LSD are considered.

DOI
10.1088/0305-4470/35/16/303

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Johri and Bhatt found singular behavior near the band edge in the density of states as well as in the inverse participation ratio of the Anderson model. These singularities mark a transition to an energy range dominated by resonant states. We study the interacting case using an ensemble of two-site Anderson-Hubbard systems. We find the ensemble-averaged density of states and generalized inverse participation ratio have more structure than in the non-interacting case because there are more transitions and in particular the transitions depend on the ground state. Nonetheless, there are regions of sharp decline in the generalized inverse participation ratio associated with specific density of state features. Moreover these features move closer to the Fermi level with the addition of interactions making them more experimentally accessible. Unfortunately resonances unique to interacting systems cannot be specifically identified. Author Keywords: Correlated electrons, Disorder, Localization
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The Eigenstate Thermalization Hypothesis explains thermalization in isolated quantum systems through the statistical properties of observables in the energy eigenbasis. We investigate the crossover from integrability to chaos in the spin-1/21/21/2 XXZ chain, establishing a direct correspondence between the spectral correlations of the Hamiltonian and local observables expressed in the energy eigenbasis as a signature of ergodicity breaking. By introducing a local perturbation that drives the system from integrability to chaos, we track the standard ETH indicators and the eigenstate entanglement entropy. We introduce a submatrix-based framework for analyzing local observables in the energy eigenbasis. By extracting real-symmetric blocks along the diagonal of the local observables represented in eigenbasis, we show that these submatrices exhibit both the short-range and long-range spectral features of the Hamiltonian. Remarkably, this correspondence persists even in a partially ergodic regime, indicating that the emergence of chaos is already encoded locally within the observables' matrix structure and that small blocks are sufficient to capture the underlying spectral correlations.