Global phase diagram of two-dimensional Dirac fermions in random potentials

TL;DR

This paper maps the phase diagram of disordered 2D Dirac fermions across all chiral symmetry classes, revealing critical lines, topological effects, and unifying different symmetry classes within a comprehensive framework.

Contribution

It presents a unified global phase diagram for 2D disordered Dirac fermions across all chiral symmetry classes, including topological terms in the non-linear sigma model.

Findings

Identifies critical lines in the phase diagram for symmetry classes BDI and CII.

Includes symmetry classes AII and D as phase boundaries.

Derives topological terms in the non-linear sigma model for all relevant classes.

Abstract

Anderson localization is studied for two flavors of massless Dirac fermions in 2D space perturbed by static disorder that is invariant under a chiral symmetry (chS) and a time-reversal symmetry (TRS) operation which, when squared, is equal either to plus or minus the identity. The former TRS (symmetry class BDI) can for example be realized when the Dirac fermions emerge from spinless fermions hopping on a 2D lattice with a linear energy dispersion such as the honeycomb lattice (graphene) or the square lattice with $\pi$-flux per plaquette. The latter TRS is realized by the surface states of 3D $\mathbb{Z}_{2}$-topological band insulators in symmetry class CII. In the phase diagram parametrized by the disorder strengths, there is an infrared stable line of critical points for both symmetry classes BDI and CII. Here we discuss a "global phase diagram" in which disordered Dirac fermion systems in all three chiral symmetry classes, AIII, CII, and BDI, occur in 4 quadrants, sharing one corner which represents the clean Dirac fermion limit. This phase diagram also includes symmetry classes AII [e.g., appearing at the surface of a disordered 3D $\mathbb{Z}_2$-topological band insulator in the spin-orbit (symplectic) symmetry class] and D (e.g., the random bond Ising model in two dimensions) as boundaries separating regions of the phase diagram belonging to the three chS classes AIII, BDI, and CII. Moreover, we argue that physics of Anderson localization in the CII phase can be presented in terms of a non-linear-sigma model (NLsM) with a $\mathbb{Z}_{2}$-topological term. We thereby complete the derivation of topological or Wess-Zumino-Novikov-Witten terms in the NLsM description of disordered fermionic models in all 10 symmetry classes relevant to Anderson localization in two spatial dimensions.