arXiv:cond-mat/0404474 · Physical Review B

Two-level correlation function of critical random-matrix ensembles

Emilio Cuevas

Published Jan 24, 2005• 8 citations

Abstract

The two-level correlation function ${R}_{d,\ensuremath{\beta}}(s)$ of $d$-dimensional disordered models ($d=1$, 2, and 3) with long-range random-hopping amplitudes is investigated numerically at criticality. We focus on models with orthogonal $(\ensuremath{\beta}=1)$ or unitary $(\ensuremath{\beta}=2)$ symmetry in the strong $({b}^{d}⪡1)$ coupling regime, where the parameter ${b}^{\ensuremath{-}d}$ plays the role of the coupling constant of the model. It is found that ${R}_{d,\ensuremath{\beta}}(s)$ is of the form ${R}_{d,\ensuremath{\beta}}(s)=1+\ensuremath{\delta}(s)\ensuremath{-}{F}_{\ensuremath{\beta}}({s}^{\ensuremath{\beta}}∕{b}^{d\ensuremath{\beta}})$, where ${F}_{1}(x)=\mathrm{erfc}({a}_{d,\ensuremath{\beta}}x)$ and ${F}_{2}(x)=\mathrm{exp}(\ensuremath{-}{a}_{d,\ensuremath{\beta}}{x}^{2})$, with ${a}_{d,\ensuremath{\beta}}$ being a numerical coefficient depending on the dimensionality and the universality class. Finally, the level number variance and the spectral compressibility are also considerded.

DOI: 10.1103/physrevb.71.024205

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arXiv:cond-mat/0404474 · Physical Review B

Two-level correlation function of critical random-matrix ensembles

Emilio Cuevas

Published Jan 24, 2005• 8 citations

arXiv HTML PDF

Abstract

The two-level correlation function

{R}_{d,\ensuremath{\beta}}(s)

d

-dimensional disordered models (

d=1

, 2, and 3) with long-range random-hopping amplitudes is investigated numerically at criticality. We focus on models with orthogonal

(\ensuremath{\beta}=1)

or unitary

(\ensuremath{\beta}=2)

symmetry in the strong

({b}^{d}⪡1)

coupling regime, where the parameter

{b}^{\ensuremath{-}d}

plays the role of the coupling constant of the model. It is found that

{R}_{d,\ensuremath{\beta}}(s)

is of the form

{R}_{d,\ensuremath{\beta}}(s)=1+\ensuremath{\delta}(s)\ensuremath{-}{F}_{\ensuremath{\beta}}({s}^{\ensuremath{\beta}}∕{b}^{d\ensuremath{\beta}})

, where

{F}_{1}(x)=\mathrm{erfc}({a}_{d,\ensuremath{\beta}}x)

and

{F}_{2}(x)=\mathrm{exp}(\ensuremath{-}{a}_{d,\ensuremath{\beta}}{x}^{2})

, with

{a}_{d,\ensuremath{\beta}}

being a numerical coefficient depending on the dimensionality and the universality class. Finally, the level number variance and the spectral compressibility are also considerded.

DOI: 10.1103/physrevb.71.024205

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Abstract Some properties of d‐dimensional disordered models with long‐range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension d2 (for d = 2) and the nearest level spacing distribution Pc(s) (for d = 3) in both the weak (bd ≫ 1) and the strong (bd ≪ 1) coupling regime, where the parameter b− d plays the role of the coupling constant of the model. It is found that (i) the extrapolated values of d2 are of the form d2 = cdbd in the strong coupling limit and d2 = d − ad/bd in the case of weak coupling, and (ii) Pc(s) has the asymptotic form Pc(s) ∼ exp (−Adsα) for s ≫ 1, with the critical exponent α = 2 − ad/bd for bd ≫ 1 and α = 1 + cdbd for bd ≪ 1. In these cases the numerical coefficients Ad, ad and cd depend only on the dimensionality. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)Abstract Some properties of d‐dimensional disordered models with long‐range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension d2 (for d = 2) and the nearest level spacing distribution Pc(s) (for d = 3) in both the weak (bd ≫ 1) and the strong (bd ≪ 1) coupling regime, where the parameter b− d plays the role of the coupling constant of the model. It is found that (i) the extrapolated values of d2 are of the form d2 = cdbd in the strong coupling limit and d2 = d − ad/bd in the case of weak coupling, and (ii) Pc(s) has the asymptotic form Pc(s) ∼ exp (−Adsα) for s ≫ 1, with the critical exponent α = 2 − ad/bd for bd ≫ 1 and α = 1 + cdbd for bd ≪ 1. In these cases the numerical coefficients Ad, ad and cd depend only on the dimensionality. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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arXiv:cond-mat/0607269Physical Review BOct 4, 20066 citations

The two-level correlation function

{R}_{d,\ensuremath{\beta}}(s)

d

-dimensional disordered models (

d=1

, 2, and 3) with long-range random-hopping amplitudes is investigated numerically at criticality. We focus on models with orthogonal

(\ensuremath{\beta}=1)

or unitary

(\ensuremath{\beta}=2)

symmetry in the strong

({b}^{d}⪡1)

coupling regime, where the parameter

{b}^{\ensuremath{-}d}

plays the role of the coupling constant of the model. It is found that

{R}_{d,\ensuremath{\beta}}(s)

is of the form

{R}_{d,\ensuremath{\beta}}(s)=1+\ensuremath{\delta}(s)\ensuremath{-}{F}_{\ensuremath{\beta}}({s}^{\ensuremath{\beta}}∕{b}^{d\ensuremath{\beta}})

, where

{F}_{1}(x)=\mathrm{erfc}({a}_{d,\ensuremath{\beta}}x)

and

{F}_{2}(x)=\mathrm{exp}(\ensuremath{-}{a}_{d,\ensuremath{\beta}}{x}^{2})

, with

{a}_{d,\ensuremath{\beta}}

being a numerical coefficient depending on the dimensionality and the universality class. Finally, the level number variance and the spectral compressibility are also considerded.

Some properties of

d

-dimensional disordered models with long-range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension

d_2

(for

d=2

) and the nearest level spacing distribution

P_c(s)

(for

d=3

) in both the weak (

b^d \gg 1

) and the strong (

b^d \ll 1

) coupling regime, where the parameter

b^{-d}

plays the role of the coupling constant of the model. It is found that (i) the extrapolated values of

d_2

are of the form

d_2=c_db^d

in the strong coupling limit and

d_2=d-a_d/b^d

in the case of weak coupling, and (ii)

P_ (s)

has the asymptotic form

P_c(s)\sim\exp (-A_ds^{\alpha})

for $s\gg $, with the critical exponent

\alpha=2-a_d/b^d

for

b^d \gg 1

and

\alpha=1+c_d b^d

for

b^d \ll 1

. In these cases the numerical coefficients

A_d

a_d

and

c_d

depend only on the dimensionality.

V

, and

{A}_{1},{A}_{2}

, and

{A}_{3}

are coefficients of order unity which depend on the shape of the system. The energy-level correlation function, which we study, allows for a direct experimental observation, for example, by measuring the correlations of the ac conductance of the system at different frequencies.

1/r.

By performing an exact diagonalization scheme on finite chains, we compute the participation ratio of all energy eigenstates. Employing a finite-size scaling analysis, we report on the relevant scaling exponents characterizing this delocalization transition as well as the level-spacing distribution at the critical point $\ensuremath{\alpha}=1.$ The random hopping amplitudes are taken from both uniform and random sign distributions. We show that these models display similar critical behavior in the vicinity of $\ensuremath{\alpha}=1.$ However, the random sign model exhibits an asymptotic delocalization in the limit of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\infty}$ and the universal scaling behavior in this regime is also reported.

We study by the strong disorder renormalization group (RG) method the low-energy properties of the one-dimensional Hubbard model with random-hopping matrix-elements

{t}_{min}<t<{t}_{max}

, and with random onsite Coulomb repulsion terms $0\ensuremath{\leqslant}{U}_{min}<U<{U}_{max}$. There are two critical phases, corresponding to an infinite randomness spin random singlet for strong interactions

({U}_{min}>{t}_{max})

and to an orbital infinite randomness fixed point for vanishing interactions $({U}_{max}∕{t}_{max}\ensuremath{\rightarrow}0)$. To each critical infinite randomness fixed point is connected a Griffiths phase, the correlation length, and dynamical exponent of which have well defined asymptotic dependences on the corresponding quantum control parameter. The theoretical predictions for the scaling in the vicinity of the critical points compare well to numerical RG simulations.