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Critical properties in long-range hopping Hamiltonians

Emilio Cuevas

Published Jan 1, 2004• 6 citations
OpenAlex

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 $d_2$ (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.

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Some properties of ddd-dimensional disordered models with long-range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension d2d_2d2​ (for d=2d=2d=2) and the nearest level spacing distribution Pc(s)P_c(s)Pc​(s) (for d=3d=3d=3) in both the weak (bd≫1b^d \gg 1bd≫1) and the strong (bd≪1b^d \ll 1bd≪1) coupling regime, where the parameter b−db^{-d}b−d plays the role of the coupling constant of the model. It is found that (i) the extrapolated values of d2d_2d2​ are of the form d2=cdbdd_2=c_db^dd2​=cd​bd in the strong coupling limit and d2=d−ad/bdd_2=d-a_d/b^dd2​=d−ad​/bd in the case of weak coupling, and (ii) P(s)P_ (s)P(​s) has the asymptotic form Pc(s)∼exp⁡(−Adsα)P_c(s)\sim\exp (-A_ds^{\alpha})Pc​(s)∼exp(−Ad​sα) for s≫s\gg s≫, with the critical exponent α=2−ad/bd\alpha=2-a_d/b^dα=2−ad​/bd for bd≫1b^d \gg 1bd≫1 and α=1+cdbd\alpha=1+c_d b^dα=1+cd​bd for bd≪1b^d \ll 1bd≪1. In these cases the numerical coefficients AdA_dAd​, ada_dad​ and cdc_dcd​ depend only on the dimensionality.
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)
The two-level correlation function ${R}_{d,\ensuremath{\beta}}(s)$ of ddd-dimensional disordered models (d=1d=1d=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 (bd⪡1)({b}^{d}⪡1)(bd⪡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.
Finite-size effects in the generalized fractal dimensions dq{d}_{q}dq​ are investigated numerically. We concentrate on a one-dimensional disordered model with long-range random hopping amplitudes in both the strong- and the weak-coupling regime. At the macroscopic limit, a linear dependence of dq{d}_{q}dq​ on q is found in both regimes for values of $q\ensuremath{\lesssim}{4g}^{\ensuremath{-}1},$ where g is the coupling constant of the model.
The combined effects of disorder and long-range hopping play an important role in determining localization and transport in low-dimensional quantum systems. We study transport in a one-dimensional disordered chain with nearest-neighbor hopping supplemented by a long-range hopping term. The long-range coupling is characterized by site-dependent amplitudes that may be either uniform or random. We investigate the interplay between diagonal disorder and long-range hopping, characterizing the resulting transport efficiency. We find that nearest-neighbor hopping in the presence of uniform long-range hopping amplitudes strongly suppresses delocalization over a range of weak disorder. The disorder-enhanced transport regime, established in previous studies, can be understood here as the recovery of delocalization from this strongly suppressed regime. An analytical solution in the Anderson eigenstate basis reveals the microscopic origin of this behavior. In particular, we show that the low-disorder regime of strongly suppressed delocalization can be eliminated by introducing randomness into the long-range hopping amplitudes, providing an additional mechanism for enhancing transport and qualitatively changing the response of the system to disorder.
We investigate numerically the nature of energy eigenstates in one-dimensional bond-disordered Anderson models with hopping amplitudes decreasing as ${H}_{\mathrm{ij}}\ensuremath{\propto}1/|i\ensuremath{-}j{|}^{\ensuremath{\alpha}}.$ The eigenstates become delocalized whenever the hopping amplitudes decay slower than 1/r.1/r.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.
Critical states are studied by a real space RG in the problem with strong diagonal disorder and long range power law hopping. The RG ow of the distribution of coupling parameters is characterized by a family of non-trivial fix points. We consider the RG flow of the distribution of participation ratios of eigenstates. Scaling of participation ratios is sensitive to the nature of the RG fix point. For some fix points, scaling of participation ratios is characterized by a distribution of exponents, rather than by a single exponent. The RG method can be generalized to treat certain fermionic Hamiltonians with disorder and long range hopping. We derive the RG for a model of interacting two-level systems. Besides couplings, in this problem the RG includes the density of states. The density of states is renormalized so that it develops a singularity near zero energy.