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Europhysics Letters (EPL)

Crossover Phenomena in the Excitation Statistics of Disordered Many-Particle Systems

Richard Berkovits

Published Mar 20, 1994• 19 citations
Publisher

Abstract

The energy level separation distribution of a fermionic disordered many-particle system is studied. For a non-interacting system, a crossover is observed from a Wigner (GOE) distribution for the first and second excitations to a different form which depends on the number of particles for low-lying excitations and then to a Poisson distribution for high excitations. This behaviour also holds for weakly interacting systems, while for stronger interactions a transition of the high excitations to a Wigner distribution is observed.

DOI
10.1209/0295-5075/25/9/008

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The distribution P1{P}_{1}P1​ of the first many-body excitation energy of a weakly and moderately interacting electron gas in a finite conductor (in the diffusive regime) is calculated. As the interaction is increased, P1{P}_{1}P1​ crosses over from Wigner-Dyson to Poisson distributions. We characterize this transition through the inverse participation ratio in Hilbert space, and examine its manifestation in a projected two-dimensional space.
The development of quantum chaos in finite interacting Fermi systems is considered. At sufficiently high excitation energy the direct two-particle interaction may mix into an eigenstate the exponentially large number of simple Slater determinant states. Nevertheless, the transition from Poisson to Wigner-Dyson statistics of energy levels is governed by the effective high-order interaction between states very distant in the Fock space. The concrete form of the transition depends on the way one chooses to work out the problem of factorial divergence of the number of Feynman diagrams. In the proposed scheme the change of statistics has a form of narrow phase transition and may happen even below the direct interaction threshold.
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Using numerical diagonalization we study the crossover among different random matrix ensembles (Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE)) realized in two different microscopic models.The specific diagnostic tool used to study the crossovers is the level spacing distribution.The first model is a one-dimensional lattice model of interacting hard-core bosons (or equivalently spin 1/2 objects) and the other a higher dimensional model of non-interacting particles with disorder and spin-orbit coupling.We find that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place.This exponent is independent of microscopic details of the perturbation.We also find that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution.We also conjecture that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system.
Statistical properties of the low‐lying states of the entanglement spectrum of a one‐dimensional interacting disordered system are studied in order to understand the localized to extended transition as function of interaction strength and excitation number expected from the many‐body localization transition. It is shown that such a transition is observed in the statistics of the level‐spacing of the entanglement spectrum. For an intermediate range of excitation numbers and strong interaction strength where the entanglement spectrum shows a clear extended behavior, a signature of non‐ergodic behavior also emerges. We interpret this as evidence for a non‐ergodic‐extended phase suggested by previous studies.