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arXiv:cond-mat/9802226 · Physical Review Letters

Critical Spectral Statistics at the Metal-Insulator Transition in Interacting Fermionic Systems

Philippe Jacquod

Published Aug 31, 1998• 4 citations
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Abstract

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.

DOI
10.1103/physrevlett.81.1913

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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.
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) 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)P(s)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.
The spectral properties of the spinless fermion model with nearest-neighbor repulsive interactions on a one-dimensional lattice are investigated using the Bethe ansatz. Although its bulk quantities are exactly the same as those of the spin-1/2 XXZ chain, the difference in the statistics of particles causes substantial effects on spectral features, such as gapless points of dispersion relations and line shapes of spectral functions. In this Letter, we clarify the origin of the differences in spectral features between fermionic and bosonic systems in terms of Bethe ansatz solutions. We also confirm that the two-string solutions have considerable spectral weights in the high-energy regime.
Using the density-matrix renormalization-group algorithm, we study the model of spinless fermions with nearest-neighbor interaction on a ring in the presence of disorder. We determine the spatial decay of the density induced by a defect (Friedel oscillations), and the phase sensitivity of the ground-state energy \ensuremath{\Delta}E=(-)N{)}^{\mathit{N}})N[E(\ensuremath{\varphi}=0)-E(\ensuremath{\varphi}=\ensuremath{\pi})], where \ensuremath{\varphi}=2\ensuremath{\pi}\ensuremath{\Phi}/${\mathrm{\ensuremath{\Phi}}}_{0}$ (N is the number of fermions, \ensuremath{\Phi} the magnetic flux, and ${\mathrm{\ensuremath{\Phi}}}_{0}$=h/e the flux quantum), for a disordered system versus the system size M. The quantity ln(M\ensuremath{\Delta}E) is found to have a normal distribution to a good approximation. The ``localization length'' decreases (increases) for a repulsive (attractive) interaction. \textcopyright{} 1996 The American Physical Society.
We study a one-dimensional XXZ spin chain in a random field on the metallic side of the many-body localization transition by level statistics. For a fixed interaction, and intermediate disorder below the many-body localization transition, we find that, asymptotically, the number variance grows faster than linear with a disorder-dependent exponent. This is consistent with the existence of an anomalous Thouless energy in the spectrum. In noninteracting disordered metals, this is an energy scale related to the typical time for a particle to diffuse across the sample. In the interacting case, it seems related to a more intricate anomalous diffusion process. This interpretation is not fully consistent with recent claims that for intermediate disorder, level statistics are described by a plasma model with power-law decaying interactions whose number variance grows slower than linear. As disorder is further increased, still on the metallic side, the Thouless energy is gradually washed out. In the range of sizes we can explore, level statistics are scale invariant and approach Poisson statistics at the many-body localization transition. Slightly below the many-body localization transition, spectral correlations, well described by critical statistics, are quantitatively similar to those of a high-dimensional, noninteracting, disordered conductor at the Anderson transition.
We present a theory of the metal-insulator transition in a disordered two-dimensional electron gas. A quantum critical point, separating the metallic phase, which is stabilized by electronic interactions, from the insulating phase, where disorder prevails over the electronic interactions, has been identified. The existence of the quantum critical point leads to a divergence in the density of states of the underlying collective modes at the transition, causing the thermodynamic properties to behave critically as the transition is approached. We show that the interplay of electron-electron interactions and disorder can explain the observed transport properties and the anomalous enhancement of the spin susceptibility near the metal-insulator transition.