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arXiv:cond-mat/0407168 · physica status solidi (b)

Critical properties in long‐range hopping Hamiltonians

E. Cuevas

Published Jun 22, 2004• 7 citations
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Abstract

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)

DOI
10.1002/pssb.200404783

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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)
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\gg $, 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.
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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.
In this work, we investigate the critical properties of the Bose-Einstein condensation transition in a model of non-interacting bosons whose one-particle hamiltonian in the tight-binding approximation includes long-range couplings in a linear chain with periodic boundary conditions. The couplings along the chain are described by a kinetic hopping term which is responsible by the quantum mobility of the particles and whose amplitude decays with the distance between the sites as a power-law with a characteristic exponent a. We show that the density of states in the vicinity of the lower band edge of the energy spectrum strongly depends on a. In the re- gime of 1 < α < 2, the density of states vanishes as a power-law, similar to the behavior found for bosons in hypercubic lattices with dimension d = 2/(α -1). We determine the critical properties of the Bose-Einstein condensation transition that takes place in this regime using a hypothesis of a single characteristic length scale and a finite-size scaling analysis. In the regime of 3/2 < α < 2, the critical exponents are quite similar to those that characterize the transition in the corresponding hypercubic lattice. On the other hand, in the regime of 1 < α < 3/2, the exponents obtained from the single length scale hypothesis are quite distinct form those on the hypercubic lattice. This result indicates that the single length scale hypothesis can not correctly capture the critical exponents in the regime where the effective dimensionality of the model is above the upper critical dimension above which the transition has a mean-field critical behavior.