John Chalker

Abstract:

We study the influence of short-range electron-electron interactions on scaling behavior near the integer quantum Hall plateau transitions. Short-range interactions are known to be irrelevant at the renormalization group fixed point which represents the transition in the non-interacting system. We find, nevertheless, that transport properties change discontinuously when interactions are introduced. Most importantly, in the thermodynamic limit the conductivity at finite temperature is zero without interactions, but non-zero in the presence of arbitrarily weak interactions. In addition, scaling as a function of frequency, $\omega$, and temperature, $T$, is determined by the scaling variable $\omega/T^p$ (where $p$ is the exponent for the temperature dependence of the inelastic scattering rate) and not by $\omega/T$, as it would be at a conventional quantum phase transition described by an interacting fixed point. We express the inelastic exponent, $p$, and the thermal exponent, $z_T$, in terms of the scaling dimension, $-\alpha < 0$, of the interaction strength and the dynamical exponent $z$ (which has the value $z=2$), obtaining $p=1+2\alpha/z$ and $z_T=2/p$.

Abstract:

Statistical properties of eigenvectors in non-Hermitian random matrix ensembles are discussed, with an emphasis on correlations between left and right eigenvectors. Two approaches are described. One is an exact calculation for Ginibre's ensemble, in which each matrix element is an independent, identically distributed Gaussian complex random variable. The other is a simpler calculation using $N^{-1}$ as an expansion parameter, where $N$ is the rank of the random matrix: this is applied to Girko's ensemble. Consequences of eigenvector correlations which may be of physical importance in applications are also discussed. It is shown that eigenvalues are much more sensitive to perturbations than in the corresponding Hermitian random matrix ensembles. It is also shown that, in problems with time-evolution governed by a non- Hermitian random matrix, transients are controlled by eigenvector correlations.

Abstract:

We study a delocalization transition for non-interacting quasiparticles moving in two dimensions, which belongs to a new symmetry class. This symmetry class can be realised in a dirty, gapless superconductor in which time reversal symmetry for orbital motion is broken, but spin rotation symmetry is intact. We find a direct transition between two insulating phases with quantized Hall conductances of zero and two for the conserved quasiparticles. The energy of quasiparticles acts as a relevant, symmetry-breaking field at the critical point, which splits the direct transition into two conventional plateau transitions.

Abstract:

We study statistical properties of the eigenvectors of non-Hermitian random matrices, concentrating on Ginibre’s complex Gaussian ensemble, in which the real and imaginary parts of each element of an N × N matrix, J, are independent random variables. Calculating ensemble averages based on the quantity ⟨Lα|Lβ⟩ ⟨Rβ|Rα⟩, where ⟨Lα| and |Rβ⟩ are left and right eigenvectors of J, we show for large N that eigenvectors associated with a pair of eigenvalues are highly correlated if the two eigenvalues lie close in the complex plane. We examine consequences of these correlations that are likely to be important in physical applications. © 1998 The American Physical Society.