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

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

2011-11-14

arxiv.org/abs/1111.3249v1

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

26 pages

Scientific paper

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 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 a stable line of critical points in the former case, while there are outgoing RG flows away from an unstable line of critical points in the latter case. 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 (NL$\sigma$M) with a $\mathbb{Z}_{2}$-topological term. We thereby complete the derivation of topological or Wess-Zumino-Novikov-Witten terms in the NL$\sigma$M description of disordered fermionic models in all 10 symmetry classes relevant to Anderson localization in two spatial dimensions.

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