The mobility edge problem: Continuous symmetry and a conjecture

Wegner, Franz

doi:10.1007/bf01319839

Cited by 666 publications

(611 citation statements)

References 8 publications

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“…This was confirmed by numerical analysis of quasi-one-dimensional (Q1D) systems [2]. Theoretical dependence of ν on dimension d = 2 + ε was found in [3,4] for small ε. Numerically, ν(ε) was studied on bifractals [5].The conductance g was originally chosen as the order parameter in the scaling theory [1]. Soon it became clear, that the absence of self-averaging of g in the critical region must be taken into account [6,7].…”

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confidence: 53%

Critical conductance distribution in various dimensions

Travěnec

Markoš

2002

Phys. Rev. B

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We study numerically the metal -insulator transition in the Anderson model on various lattices with dimension 2 < d ≤ 4 (bifractals and Euclidian lattices). The critical exponent ν and the critical conductance distribution are calculated. We confirm that ν depends only on the spectral dimension. The other parameters -critical disorder, critical conductance distribution and conductance cummulants -depend also on lattice topology. Thus only qualitative comparison with theoretical formulae for dimension dependence of the cummulants is possible.PACS numbers: 71.30.+h, 71.23.An, 72.15.Rn It is commonly accepted, though not proved, that metal -insulator transitions (MIT) can be described by oneparameter scaling theory [1]. The critical exponent ν which describes the divergence of correlation length at MIT depends only on the system dimension for a chosen universality class. Microscopic details of models do not affect it. This was confirmed by numerical analysis of quasi-one-dimensional (Q1D) systems [2]. Theoretical dependence of ν on dimension d = 2 + ε was found in [3,4] for small ε. Numerically, ν(ε) was studied on bifractals [5].The conductance g was originally chosen as the order parameter in the scaling theory [1]. Soon it became clear, that the absence of self-averaging of g in the critical region must be taken into account [6,7]. The shape of the critical conductance distribution P (g) in 3D models was numerically analysed in detail [8,9,10,11,12]. Contrary to the critical exponent, P (g) is not universal. Its shape depends not merely on the dimension [10] and physical symmetry [11] but also on boundary conditions [12] and even anisotropy [13]. Nevertheless, for a given physical model the mean conductance and resistance follow one parameter scaling [14].Analytical theory of MIT is restricted to systems with dimension close to the lower critical dimension: 2 + ε with ε ≪ 1 [15]. In spite of predicted non-universality of higher order conductance cummulants δg n ,(1) the distribution P (g) should be universal in the infinite system size limit [16]. For small ε, the bulk of the distribution is approximately Gaussian near the mean value g . The parameters of Gaussian peak,agree with the estimation of the first cummulants (1). For large g, theory predicts long power-law tailNumerical results for 3D systems show completely different P (g), indicating that theory is not applicable to large ε [8].In this Letter we present the critical exponent and the critical conductance distribution for Anderson model obtained numerically on three bifractal lattices with dimension 2 < d < 3. They all possess the same fractal dimension d f = log 3/ log 2 + 1. Two of these lattices have the same spectral dimension d s . Their critical exponents are identical within error bars. This confirms the universality of MIT. The shape of the P (g), and the value of the critical disorder, depend not only on d s but also on the lattice topology. This novel non-universality of P (g) is found also by numerical analysis of two different 3D...

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confidence: 53%

Critical conductance distribution in various dimensions

Travěnec

Markoš

2002

Phys. Rev. B

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“…N of moduli fields. This allows us to use Random Matrix Theory (RMT) [18][19][20][21] Quantum tunneling to other sites is always present which allows the wavefunction to spread from site to site. Together with the stochastic distribution of sites this ensures the Anderson localization [22] of wavepackets around some vacuum site, at least for all the energy levels up to the disorder strength.…”

Section: A Model Of the Stringy Landscapementioning

confidence: 99%

Why the Universe started from a low entropy state

Holman

Mersini-Houghton

2006

Phys. Rev. D

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We show that the inclusion of backreaction of massive long wavelengths imposes dynamical constraints on the allowed phase space of initial conditions for inflation, which results in a superselection rule for the initial conditions. Only high energy inflation is stable against collapse due to the gravitational instability of massive perturbations. We present arguments to the effect that the initial conditions problem cannot be meaningfully addressed by thermostatistics as far as the gravitational degrees of freedom are concerned. Rather, the choice of the initial conditions for the universe in the phase space and the emergence of an arrow of time have to be treated as a dynamic selection.

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“…Beyond the O(N)-symmetric Heisenberg models ("vector models"), which we have discussed in the previous sections, they correspond to the simplest scalar field theories. There is a wide set of different applications as the metal insulator transition [158] or liquid crystals [159] or strings and random surfaces [160]. The universal behavior of these models in the vicinity of a second order or weak first order phase transition is determined by the symmetries and the field content of the corresponding field theories.…”

Section: Introductionmentioning

confidence: 99%

Non-perturbative renormalization flow in quantum field theory and statistical physics

2002

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We review the use of an exact renormalization group equation in quantum field theory and statistical physics. It describes the dependence of the free energy on an infrared cutoff for the quantum or thermal fluctuations. Non-perturbative solutions follow from approximations to the general form of the coarse-grained free energy or effective average action. They interpolate between the microphysical laws and the complex macroscopic phenomena. Our approach yields a simple unified description for O(N )-symmetric scalar models in two, three or four dimensions, covering in particular the critical phenomena for the second-order phase transitions, including the Kosterlitz-Thouless transition and the critical behavior of polymer chains. We compute the aspects of the critical equation of state which are universal for a large variety of physical systems and establish a direct connection between microphysical and critical quantities for a liquid-gas transition. Universal features of first-order phase transitions are studied in the context of scalar matrix models. We show that the quantitative treatment of coarse graining is essential for a detailed estimate of the nucleation rate. We discuss quantum statistics in thermal equilibrium or thermal quantum field theory with fermions and bosons and we describe the high temperature symmetry restoration in quantum field theories with spontaneous symmetry breaking. In particular we explore chiral symmetry breaking and the high temperature or high density chiral phase transition in quantum chromodynamics using models with effective four-fermion interactions.This work is dedicated to the 60th birthday of Franz Wegner. *

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The mobility edge problem: Continuous symmetry and a conjecture

Cited by 666 publications

References 8 publications

Critical conductance distribution in various dimensions

Critical conductance distribution in various dimensions

Why the Universe started from a low entropy state

Non-perturbative renormalization flow in quantum field theory and statistical physics

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