arXiv:cond-mat/9808139 · Physical Review Letters

Critical Spectral Statistics in Two-Dimensional Interacting Disordered Systems

Emilio Cuevas

Published Jul 5, 1999• 43 citations

Abstract

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 $P(s)$ allows one to find a delocalization transition at a critical disorder ${W}_{\mathrm{c}}$ for any nonzero value of the interaction strength. At the critical point the spacings distribution has a small- $s$ behavior ${P}_{c}(s)\ensuremath{\propto}s$, and a Poisson-like decay at large spacings.

DOI: 10.1103/physrevlett.83.140

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arXiv:cond-mat/9808139 · Physical Review Letters

Critical Spectral Statistics in Two-Dimensional Interacting Disordered Systems

Emilio Cuevas

Published Jul 5, 1999• 43 citations

arXiv HTML PDF

Abstract

P(s)

allows one to find a delocalization transition at a critical disorder

{W}_{\mathrm{c}}

for any nonzero value of the interaction strength. At the critical point the spacings distribution has a small-

s

behavior

{P}_{c}(s)\ensuremath{\propto}s

, and a Poisson-like decay at large spacings.

DOI: 10.1103/physrevlett.83.140

Related papers

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

Pil Hun Song, Dima L. Shepelyansky

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.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.

arXiv:cond-mat/9904229Physical review. B, Condensed matterJun 15, 200019 citations

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Philippe Jacquod

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 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.

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Many-body localization transition in a lattice model of interacting fermions: Statistics of renormalized hoppings in configuration space

Cécile Monthus, Thomas Garel

We consider the one-dimensional lattice model of interacting fermions with disorder studied previously by Oganesyan and Huse [Phys. Rev. B 75, 155111 (2007)]. To characterize a possible many-body localization transition as a function of the disorder strength $W$ , we use an exact renormalization procedure in configuration space that generalizes the Aoki real-space renormalization procedure for Anderson localization one-particle models [H. Aoki, J. Phys. C 13, 3369 (1980)]. We focus on the statistical properties of the renormalized hopping ${V}_{L}$ between two configurations separated by a distance $L$ in configuration space (distance being defined as the minimal number of elementary moves to go from one configuration to the other). Our numerical results point toward the existence of a many-body localization transition at a finite disorder strength ${W}_{c}$ . In the localized phase $W>{W}_{c}$ , the typical renormalized hopping ${V}_{L}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{L}}}$ decays exponentially in $L$ as $(\text{ln}\text{ }{V}_{L}^{typ})\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}$ and the localization length diverges as ${\ensuremath{\xi}}_{loc}(W)\ensuremath{\sim}{(W\ensuremath{-}{W}_{c})}^{\ensuremath{-}{\ensuremath{\nu}}_{loc}}$ with a critical exponent of order ${\ensuremath{\nu}}_{loc}\ensuremath{\simeq}0.45$. In the delocalized phase $W<{W}_{c}$ , the renormalized hopping remains a finite random variable as $L\ensuremath{\rightarrow}\ensuremath{\infty}$ and the typical asymptotic value ${V}_{\ensuremath{\infty}}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{\ensuremath{\infty}}}}$ presents an essential singularity $(\text{ln}\text{ }{V}_{\ensuremath{\infty}}^{typ})\ensuremath{\sim}\ensuremath{-}{({W}_{c}\ensuremath{-}W)}^{\ensuremath{-}\ensuremath{\kappa}}$ with an exponent of order $\ensuremath{\kappa}\ensuremath{\sim}1.4$. Finally, we show that this analysis in configuration space is compatible with the localization properties of the simplest two-point correlation function in real space.We consider the one-dimensional lattice model of interacting fermions with disorder studied previously by Oganesyan and Huse [Phys. Rev. B 75, 155111 (2007)]. To characterize a possible many-body localization transition as a function of the disorder strength $W$, we use an exact renormalization procedure in configuration space that generalizes the Aoki real-space renormalization procedure for Anderson localization one-particle models [H. Aoki, J. Phys. C 13, 3369 (1980)]. We focus on the statistical properties of the renormalized hopping ${V}_{L}$ between two configurations separated by a distance $L$ in configuration space (distance being defined as the minimal number of elementary moves to go from one configuration to the other). Our numerical results point toward the existence of a many-body localization transition at a finite disorder strength ${W}_{c}$. In the localized phase $W>{W}_{c}$, the typical renormalized hopping ${V}_{L}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{L}}}$ decays exponentially in $L$ as $(\text{ln}\text{ }{V}_{L}^{typ})\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}$ and the localization length diverges as ${\ensuremath{\xi}}_{loc}(W)\ensuremath{\sim}{(W\ensuremath{-}{W}_{c})}^{\ensuremath{-}{\ensuremath{\nu}}_{loc}}$ with a critical exponent of order ${\ensuremath{\nu}}_{loc}\ensuremath{\simeq}0.45$. In the delocalized phase $W<{W}_{c}$, the renormalized hopping remains a finite random variable as $L\ensuremath{\rightarrow}\ensuremath{\infty}$ and the typical asymptotic value ${V}_{\ensuremath{\infty}}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{\ensuremath{\infty}}}}$ presents an essential singularity $(\text{ln}\text{ }{V}_{\ensuremath{\infty}}^{typ})\ensuremath{\sim}\ensuremath{-}{({W}_{c}\ensuremath{-}W)}^{\ensuremath{-}\ensuremath{\kappa}}$ with an exponent of order $\ensuremath{\kappa}\ensuremath{\sim}1.4$. Finally, we show that this analysis in configuration space is compatible with the localization properties of the simplest two-point correlation function in real space.

arXiv:1001.2984Physical Review BApr 21, 2010134 citations

Many-Body Delocalization in Strongly Disordered System with Long-Range Interactions: Finite Size Scaling

Alexander L. Burin

Many-body localization in a disordered system of interacting spins coupled by the long-range interaction $1/R^α$ is investigated combining analytical theory considering resonant interactions and a finite size scaling of exact numerical solutions with a number of spins $N$ . The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for $d$ -dimensional system including the absence of localization in the infinite system at $α<2d$ and a universal scaling of a critical energy disordering $W_{c} \propto N^{\frac{2d-α}{d}}$ . %The finite size effect on the interaction stimulated delocalization of energy in the ensemble of interacting two level systems in amorphous solids at low temperature is discussed.Many-body localization in a disordered system of interacting spins coupled by the long-range interaction $1/R^α$ is investigated combining analytical theory considering resonant interactions and a finite size scaling of exact numerical solutions with a number of spins $N$. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for $d$-dimensional system including the absence of localization in the infinite system at $α<2d$ and a universal scaling of a critical energy disordering $W_{c} \propto N^{\frac{2d-α}{d}}$. %The finite size effect on the interaction stimulated delocalization of energy in the ensemble of interacting two level systems in amorphous solids at low temperature is discussed.

arXiv:1409.7990Sep 29, 20141 citation

P(s)

allows one to find a delocalization transition at a critical disorder

{W}_{\mathrm{c}}

for any nonzero value of the interaction strength. At the critical point the spacings distribution has a small-

s

behavior

{P}_{c}(s)\ensuremath{\propto}s

, and a Poisson-like decay at large spacings.

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.

d

is the spatial dimensionality. We refine earlier arguments and show that the system undergoes a localization transition as a function of the rescaled disorder ${W}^{*}=W/{L}^{2d\ensuremath{-}\ensuremath{\alpha}}lnL$, where

W

is the disorder strength and

L

the system size. This transition has much in common with that on random regular graphs. We further perform a detailed analysis of the inverse participation ratio (IPR) of many-body wave functions, exploring how ergodic behavior in the delocalized phase switches to fractal one at the critical point and on the localized side of the transition. Our analytical results for the scaling of the critical disorder

W

with the system size

L

and for the scaling of IPR in the delocalized and localized phases are supported and corroborated by exact diagonalization of spin chains.

We suggest that if a localized phase at nonzero temperature

T>0

exists for strongly disordered and weakly interacting electrons, as recently argued, it will also occur when both disorder and interactions are strong and

T

is very high. We show that in this high-

T

regime, the localization transition may be studied numerically through exact diagonalization of small systems. We obtain spectra for one-dimensional lattice models of interacting spinless fermions in a random potential. As expected, the spectral statistics of finite-size samples cross over from those of orthogonal random matrices in the diffusive regime at weak random potential to Poisson statistics in the localized regime at strong randomness. However, these data show deviations from simple one-parameter finite-size scaling: the apparent mobility edge ``drifts'' as the system's size is increased. Based on spectral statistics alone, we have thus been unable to make a strong numerical case for the presence of a many-body localized phase at nonzero

T

W

, we use an exact renormalization procedure in configuration space that generalizes the Aoki real-space renormalization procedure for Anderson localization one-particle models [H. Aoki, J. Phys. C 13, 3369 (1980)]. We focus on the statistical properties of the renormalized hopping

{V}_{L}

between two configurations separated by a distance

L

in configuration space (distance being defined as the minimal number of elementary moves to go from one configuration to the other). Our numerical results point toward the existence of a many-body localization transition at a finite disorder strength

{W}_{c}

. In the localized phase

W>{W}_{c}

, the typical renormalized hopping ${V}_{L}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{L}}}$ decays exponentially in

L

as $(\text{ln}\text{ }{V}_{L}^{typ})\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}$ and the localization length diverges as ${\ensuremath{\xi}}_{loc}(W)\ensuremath{\sim}{(W\ensuremath{-}{W}_{c})}^{\ensuremath{-}{\ensuremath{\nu}}_{loc}}$ with a critical exponent of order ${\ensuremath{\nu}}_{loc}\ensuremath{\simeq}0.45$. In the delocalized phase

W<{W}_{c}

, the renormalized hopping remains a finite random variable as $L\ensuremath{\rightarrow}\ensuremath{\infty}$ and the typical asymptotic value ${V}_{\ensuremath{\infty}}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{\ensuremath{\infty}}}}$ presents an essential singularity $(\text{ln}\text{ }{V}_{\ensuremath{\infty}}^{typ})\ensuremath{\sim}\ensuremath{-}{({W}_{c}\ensuremath{-}W)}^{\ensuremath{-}\ensuremath{\kappa}}$ with an exponent of order $\ensuremath{\kappa}\ensuremath{\sim}1.4$. Finally, we show that this analysis in configuration space is compatible with the localization properties of the simplest two-point correlation function in real space.

Many-body localization in a disordered system of interacting spins coupled by the long-range interaction

1/R^α

is investigated combining analytical theory considering resonant interactions and a finite size scaling of exact numerical solutions with a number of spins

N

. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for

d

-dimensional system including the absence of localization in the infinite system at

α<2d

and a universal scaling of a critical energy disordering

W_{c} \propto N^{\frac{2d-α}{d}}

. %The finite size effect on the interaction stimulated delocalization of energy in the ensemble of interacting two level systems in amorphous solids at low temperature is discussed.