John Chalker

Abstract:

Statistical lattice ensembles of loops in three or more dimensions typically have phases in which the longest loops fill a finite fraction of the system. In such phases it is natural to ask about the distribution of loop lengths. We show how to calculate moments of these distributions using $CP^{n-1}$ or $RP^{n-1}$ and O(n) $\sigma$ models together with replica techniques. The resulting joint length distribution for macroscopic loops is Poisson-Dirichlet with a parameter $\theta$ fixed by the loop fugacity and by symmetries of the ensemble. We also discuss features of the length distribution for shorter loops, and use numerical simulations to test and illustrate our conclusions.

Abstract:

We develop the theory of electronic Mach-Zehnder interferometers built from quantum Hall edge states at the Landau level filling factor ν=2, which have been investigated in a series of recent experiments and theoretical studies. We show that a detailed treatment of the dephasing and nonequlibrium transport is made possible by using bosonization combined with refermionization to study a model in which interactions between electrons are short range. In particular, this approach allows a nonperturbative treatment of electron tunneling at the quantum point contacts that act as beam splitters. We find an exact analytic expression at an arbitrary tunneling strength for the differential conductance of an interferometer with arms of equal length and obtain numerically exact results for an interferometer with unequal arms. We compare these results with previous perturbative and approximate ones and with observations. © 2013 American Physical Society.

Abstract:

We study the vortex lines that are a feature of many random or disordered three-dimensional systems. These show universal statistical properties on long length scales, and geometrical phase transitions analogous to percolation transitions but in distinct universality classes. The field theories for these problems have not previously been identified, so that while many numerical studies have been performed, a framework for interpreting the results has been lacking. We provide such a framework with mappings to simple supersymmetric models. Our main focus is on vortices in short-range-correlated complex fields, which show a geometrical phase transition that we argue is described by the CP^{k|k} model (essentially the CP^{n-1} model in the replica limit n→1). This can be seen by mapping a lattice version of the problem to a lattice gauge theory. A related field theory with a noncompact gauge field, the 'NCCP^{k|k} model', is a supersymmetric extension of the standard dual theory for the XY transition, and we show that XY duality gives another way to understand the appearance of field theories of this type. The supersymmetric descriptions yield results relevant, for example, to vortices in the XY model and in superfluids, to optical vortices, and to certain models of cosmic strings. A distinct but related field theory, the RP^{2l|2l} model (or the RP^{n-1} model in the limit n→1) describes the unoriented vortices that occur, for instance, in nematic liquid crystals. Finally, we show that in two dimensions, a lattice gauge theory analogous to that discussed in three dimensions gives a simple way to see the known relation between two-dimensional percolation and the CP^{k|k} σ model with a θ term.

Abstract:

We study the vortex lines that are a feature of many random or disordered three-dimensional systems. These show universal statistical properties on long length scales, and geometrical phase transitions analogous to percolation transitions but in distinct universality classes. The field theories for these problems have not previously been identified, so that while many numerical studies have been performed, a framework for interpreting the results has been lacking. We provide such a framework with mappings to simple supersymmetric models. Our main focus is on vortices in short-range correlated complex fields, which show a geometrical phase transition that we argue is described by the CP^{k|k} model (essentially the CP^{n-1} model in the replica limit n\rightarrow 1). This can be seen by mapping a lattice version of the problem to a lattice gauge theory. A related field theory with a noncompact gauge field, the 'NCCP^{k|k} model', is a supersymmetric extension of the standard dual theory for the XY transition, and we show that XY duality gives another way to understand the appearance of field theories of this type. The supersymmetric descriptions yield results relevant, for example, to vortices in the XY model and in superfluids, to optical vortices, and to certain models of cosmic strings. A distinct but related field theory, the RP^{2l|2l} model (or the RP^{n-1} model in the limit n\rightarrow 1) describes the unoriented vortices which occur for instance in nematic liquid crystals. Finally, we show that in two dimensions, a lattice gauge theory analogous to that discussed in three dimensions gives a simple way to see the known relation between two-dimensional percolation and the CP^{k|k} sigma model with a \theta-term.

Abstract:

A highly non-thermal electron distribution is generated when quantum Hall edge states originating from sources at different potentials meet at a quantum point contact. The relaxation of this distribution to a stationary form as a function of distance downstream from the contact has been observed in recent experiments [Phys. Rev. Lett. 105, 056803 (2010)]. Here we present an exact treatment of a minimal model for the system at filling factor \nu=2, with results that account well for the observations.