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arXiv:cond-mat/9809029 · Physical Review Letters

Comment on “Critical Behavior in Disordered Quantum Systems Modified by Broken Time-Reversal Symmetry”

Martin Janßen, Boris Shapiro, Imre Varga

Published Oct 5, 1998• 2 citations
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Abstract

In a recent Letter [Phys. Rev. Lett. 80, 1003 (1998)] Hussein and Pato employed the maximum entropy principle (MEP) in order to derive interpolating ensembles between any pair of universality classes in random matrix theory. They apply their formalism also to the transition from random matrix to Poisson statistics of spectra that is observed for the case of the Anderson-type metal-insulator transition. We point out the problems with the latter procedure.

DOI
10.1103/physrevlett.81.3048

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In a recent Letter [Phys. Rev. Lett. 80, 1003 (1998)] Hussein and Pato employed the maximum entropy principle (MEP) in order to derive interpolating ensembles between any pair of universality classes in random matrix theory. They apply their formalism also to the transition from random matrix to Poisson statistics of spectra that is observed for the case of the Anderson-type metal-insulator transition. We point out the problems with the latter procedure.
It is shown that the maximum entropy principle provides a natural basis for construction of an interpolating ensemble between any pair of the universality classes of random matrix theory. As an application, the Gaussian unitary ensemble to Gaussian orthogonal ensemble transition is worked out in detail. Transitions from these two ensembles to a Poisson distribution, of relevance to the metal-insulator transition in disordered metals, is also discussed in detail and the ``universal critical ensemble'' was found to depend on the time-reversal symmetry.
We study a physical system consisting of noninteracting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotation invariance. This system belongs to class D within the recent classification scheme of random matrix ensembles, and its phase diagram contains three different phases: metallic and two distinct localized phases with different quantized thermal Hall conductances. We find that critical exponents describing different transitions (insulator-to-insulator and insulator-to-metal) are identical within the error of numerical calculations and also find that critical disorder of the insulator-to-metal transition is energy-independent.
We present a complete analytical and numerical solution of the typical medium theory (TMT) for the Anderson metal-insulator transition. This approach self-consistently calculates the typical amplitude of the electronic wave functions, thus representing the conceptually simplest order-parameter theory for the Anderson transition. We identify all possible universality classes for the critical behavior, which can be found within such a mean-field approach. This provides insights into how interaction-induced renormalizations of the disorder potential may produce qualitative modifications of the critical behavior. We also formulate a simplified description of the leading critical behavior, thus obtaining an effective Landau theory for Anderson localization.
We reconsider the random bond antiferromagnetic spin-1/2 chain for weak disorder and demonstrate the existence of crossover length scale x_W that diverges with decreasing strength of the disorder. Recent DMRG calculations [Phys. Rev. Lett. 89, 127202 (2002); cond-mat/0111027] claimed to have found evidence for a non-universal behavior in this model and found no indications of a universal infinite randomness fixed point (IRFP) scenario. We show that these data are not in the asymptotic regime since the system sizes that have been considered are of the same order of magnitude or much smaller than the crossover length x_W. We give a scaling form for the xx-spin-correlation function that takes this crossover length into account and that is compatible with the IRFP scenario and a random singlet phase.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.