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arXiv:cond-mat/9904229 · Physical review. B, Condensed matter

Low-energy transition in spectral statistics of two-dimensional interacting fermions

Pil Hun Song, Dima L. Shepelyansky

Published Jun 15, 2000• 19 citations
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Abstract

We study the level spacing statistics $P(s)$ and eigenstate properties of spinless fermions with Coulomb interaction on a two-dimensional lattice at constant filling factor and various disorder strengths. For the noninteracting localized phase in the limit of large lattice size, Coulomb interaction induces a transition of $P(s)$ from the Poisson to the Wigner-Dyson distribution, at a critical total energy independent of the number of fermions. This implies the emergence of quantum ergodicity induced by interaction and delocalization in the Hilbert space at zero temperature inside the one-particle insulating phase.

DOI
10.1103/physrevb.61.15546

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The effect of Coulomb and short-range interactions on the spectral properties of two-dimensional disordered systems with two spinless fermions is investigated by numerical scaling techniques. The size independent universality of the critical nearest-level-spacing distribution P(s)P(s)P(s) allows one to find a delocalization transition at a critical disorder Wc{W}_{\mathrm{c}}Wc​ for any nonzero value of the interaction strength. At the critical point the spacings distribution has a small- sss behavior ${P}_{c}(s)\ensuremath{\propto}s$, and a Poisson-like decay at large spacings.
We study the level spacing statistics P(s)P(s)P(s) in many-body Fermi systems and determine a critical two-body interaction strength Uc{U}_{c}Uc​ at which a crossover from Poisson to Wigner-Dyson statistics takes place. Near the Fermi level, the results allow one to find a critical temperature Tch{T}_{\mathrm{ch}}Tch​ above which quantum chaos and thermalization set in.
The spectral properties of a disordered system with few interacting three-dimensional spinless fermions are investigated. We show the existence of a critical spacing distribution which is invariant upon increase of the system size but depends strongly on the number of particles. At the critical point, we report a substantial decrease of the level repulsion as the number of particles increases indicating a decrease of nearest level correlations associated with the sparsity of the Hamiltonian matrix.
We consider a two-dimensional gas of spinless fermions with Coulomb and short-range interactions on a square lattice at T=0T=0T=0. Using the exact-diagonalization technique we study finite clusters up to 16 particles at filling factors $\ensuremath{\nu}=1/2$ and 1/6. By increasing the hopping amplitude we obtain the low-energy spectrum of the system in a wide range from the classical Wigner crystal to an almost free gas of fermions. Particular effort is made to study the mechanism of the structural and insulator-metal transitions. We show that both transitions are determined by the energy band of the defect with the lowest energy in the Wigner crystal.
The emergence of quantum chaos for interacting Fermi systems is investigated by numerical calculation of the level spacing distribution P(s)P(s)P(s) as a function of interaction strength U and the excitation energy $\ensuremath{\epsilon}$ above the Fermi level. As U increases, P(s)P(s)P(s) undergoes a transition from Poissonian (nonchaotic) to Wigner-Dyson (chaotic) statistics and the transition is described by a single scaling parameter given by $Z=(U{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}\ensuremath{-}{u}_{0}){\ensuremath{\epsilon}}^{1/(2\ensuremath{\nu})},$ where u0{u}_{0}u0​ is a constant. While the exponent $\ensuremath{\alpha},$ which determines the global change of the chaos border, is indecisive within a broad range of $0.9\ensuremath{\sim}2.0,$ the finiteness of $\ensuremath{\nu},$ which comes from the increase of the Fock space size with $\ensuremath{\epsilon},$ suggests that the transition becomes sharp as $\ensuremath{\epsilon}$ increases.
We study the level statistics of a nonintegrable one-dimensional interacting fermionic system characterized by the Gaussian orthogonal ensemble distribution. We calculate numerically on a finite size system the level spacing distribution P(s)P(s)P(s) and the Dyson-Mehta ${\ensuremath{\Delta}}_{3}$ correlation. We observe that its low energy spectrum follows rather the Poisson distribution, characteristic of an integrable system, consistent with the fact that the low energy excitations of this system are described by the Luttinger model. We propose this random matrix theory analysis as a probe, but no proof, for the existence and integrability of low energy effective Hamiltonians for strongly correlated systems.