Disordered 2d quasiparticles in class D: Dirac fermions with random mass, and dirty superconductors

Abstract

Disordered noninteracting quasiparticles that are governed by a Majorana-type Hamiltonian -prominent examples are dirty superconductors with broken time-reversal and spin-rotation symmetry, or the fermionic representation of the 2d Ising model with fluctuating bond strengths -are called class D. In two dimensions, weakly disordered systems of this kind may possess a metallic phase beyond the insulating phases expected for strong disorder. We show that the 2d metal phase emanates from the free Majorana fermion point, in the direction of the RG trajectory of a perturbed WZW model. To establish this result, we develop a supersymmetric extension of the method of nonabelian bosonization. On the metallic side of the metal-insulator transition, the density of states becomes nonvanishing at zero energy, by a mechanism akin to dynamical mass generation. This feature is explored in a model of N species of disordered Dirac fermions, via the mapping on a nonlinear sigma model, which encapsulates a Z 2 spin degree of freedom. We compute the density of states in a finite system, and obtain agreement with the random-matrix prediction for class D, in the ergodic limit. Vortex disorder, which is a relevant perturbation at the free-fermion point, changes the density of states at low energy and suppresses the local Z 2 degree of freedom, thereby leading to a different symmetry class, BD.1 The equation (2) fixes an orthogonal Lie algebra in even dimension, so(2N ), which is denoted by D N in Cartan's table. Hence the name "D".2 Throughout this paper, supergroups such as GL(n|n) and OSp(2n|2n) are understood to be defined over the complex number field, C, unless specified otherwise. In places where this fact is of particular importance, we will switch to the notation GL C (n|n) and OSp C (2n|2n).

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

“…The free-fermion theory is conformal, with Virasoro central charge c = −2; θ is clearly a redundant perturbation in this case, as it does not appear in the action (a similar argument appeared in Ref. [18]). By the same argument as before, the conformal invariance with c = −2 should hold for all n, and also for all g 2 σ and θ, though the action is no longer non-interacting in general.…”

Quantum Hall transitions: An exact theory based on conformal restriction

“…[71][72][73] A generic model in this class can have an insulating, a thermal quantum Hall, and a metallic state. The metallic state at weak disorder can be described by a nonlinear sigma model.…”

“…70 The same Gaussian model field theory results from linearization of a different nonlinear sigma model that describes thermal transport of quasiparticles in disordered superconductors in class D in 2D. [71][72][73] The perturbative renormalization group flow in this model is towards weak coupling, and in a finite system of size L one can linearize the nonlinear sigma model to obtain the Gaussian model with a coupling constant of order (ln L) −1 . In this limit quasiparticle transport is essentially classical with thermal conductivities (divided by temperature, and in the corresponding units) growing logarithmically with length scale, σ xx ∼ ln L, while σ xy is arbitrary.…”

Absence of a metallic phase in random-bond Ising models in two dimensions: Applications to disordered superconductors and paired quantum Hall states

“…when det U = 1) corresponds to random matrices in class D, while the case with that quadruplet, det U = −1, corresponds to those in what has been termed class B [9]. These random matrix ensembles are of matrices in the Lie algebras of SO(2N ), SO(2N + 1) (for some N ), respectively [6,9]. Matrices found in class B possess at least one, and typically only one, exact zero eigenvalue.…”

“…An important point that arises along the way is that the distinctions between ensembles D, B, and BD, introduced in Ref. [9], are not important for local properties, such as these correlations. In Sec.…”