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arXiv:cond-mat/0008119 · Physical review. B, Condensed matter

Various delocalized phases of a disordered fermion model on a two-dimensional square lattice with π -flux

Takahiro Fukui

Published Aug 7, 2001• 0 citations
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Abstract

A disordered fermion model defined on a two-dimensional square lattice with $\ensuremath{\pi}$ flux is studied. Time-reversal symmetry and one of four possible chiral symmetries are imposed. The continuum limit of the model near zero energy yields Dirac fermions with chiral symmetry. It is shown that one of them has $\mathrm{Sp}(n)\ifmmode\times\else\texttimes\fi{}\mathrm{Sp}(n)$ symmetry whereas others have U$(2n)$ symmetry, both of which are broken to Sp$(n),$ and the fluctuation around a saddle point is described, respectively, by the Sp${(n)}_{2}$ Wess-Zumino-Witten model and U$(2n)$/Sp$(n)$ nonlinear sigma model.

DOI
10.1103/physrevb.64.085108

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A disordered fermion model defined on a two-dimensional square lattice with \ensuremathπ\ensuremath{\pi}\ensuremathπ flux is studied. Time-reversal symmetry and one of four possible chiral symmetries are imposed. The continuum limit of the model near zero energy yields Dirac fermions with chiral symmetry. It is shown that one of them has Sp(n)\ifmmode×\else\texttimes\fiSp(n)\mathrm{Sp}(n)\ifmmode\times\else\texttimes\fi{}\mathrm{Sp}(n)Sp(n)\ifmmode×\else\texttimes\fiSp(n) symmetry whereas others have U(2n)(2n)(2n) symmetry, both of which are broken to Sp(n),(n),(n), and the fluctuation around a saddle point is described, respectively, by the Sp(n)2{(n)}_{2}(n)2​ Wess-Zumino-Witten model and U(2n)(2n)(2n)/Sp(n)(n)(n) nonlinear sigma model.
Anderson localization is studied for two flavors of massless Dirac fermions in two-dimensional 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 two-dimensional lattice with a linear energy dispersion such as the honeycomb lattice (graphene) or the square lattice with $\ensuremath{\pi}$ flux per plaquette. The latter TRS is realized by the surface states of three-dimensional Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-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 four 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 three-dimensional Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-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-$\ensuremath{\sigma}$ model (NL$\ensuremath{\sigma}$M) with a Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-topological term. We thereby complete the derivation of topological or Wess-Zumino-Novikov-Witten terms in the NL$\ensuremath{\sigma}$M description of disordered fermionic models in all ten symmetry classes relevant to Anderson localization in two spatial dimensions.
Gade [Nucl. Phys. B 398,398,398, 499 (1993)] has shown that the density of states for a particle hopping on a two-dimensional bipartite lattice in the presence of weak disorder and in the absence of time-reversal symmetry (chiral unitary universality class) is anomalous in the vicinity of the band center $\ensuremath{\varepsilon}=0$ whenever the disorder preserves the sublattice symmetry. More precisely, using a nonlinear $\ensuremath{\sigma}$ model that encodes the sublattice (chiral) symmetry and the absence of time-reversal symmetry she argues that the disorder average density of states diverges as $|\ensuremath{\varepsilon}{|}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}c|\mathrm{ln}\ensuremath{\varepsilon}{|}^{\ensuremath{\kappa}})$ with c some nonuniversal positive constant and $\ensuremath{\kappa}=1/2$ a universal exponent. Her analysis has been extended to the case when time-reversal symmetry is present (chiral orthogonal universality class) for which the same exponent $\ensuremath{\kappa}=1/2$ was predicted. Motrunich et al. [Phys. Rev. B 65,65,65, 064206 (2002)] have argued that the exponent $\ensuremath{\kappa}=1/2$ does not apply to the density of states in the chiral orthogonal universality class. They predict that $\ensuremath{\kappa}=2/3$ instead. We confirm the analysis of Motrunich et al. within a field theory for two flavors of Dirac fermions subjected to two types of weak uncorrelated random potentials: a purely imaginary vector potential and a complex valued mass potential. This model is the naive continuum limit of a model describing a particle hopping on a square lattice in the background of a $\ensuremath{\pi}$-flux phase and subjected to weak disorder that preserves the sublattice symmetry and time-reversal invariance. By commonly held universality arguments, this model is believed to belong to the chiral orthogonal universality class. Our calculation relies in an essential way on the existence of infinitely many local composite operators with negative anomalous scaling dimensions.
A lattice model for disordered d-wave superconductors with spin-rotational and time-reversal symmetries is reconsidered. Near the band center, the lattice model can be described by Dirac fermions with several species. Each fermion yields a Wess-Zumino-Witten (WZW) term for an effective action of the Goldstone mode, which is known to play a crucial role in critical phenomena in two dimensions. The WZW terms cancel out each other because of the fourfold symmetry of the model, which suggests that the quasiparticle states are localized. If the lattice model has, however, symmetry breaking terms which generate mass for any species of the Dirac fermions, remaining a WZW term which avoids the cancellation can derive the system to a delocalized strong-coupling fixed point.
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The low-energy limits of models with disorder are frequently described by $\ensuremath{\sigma}$ models. In two dimensions, most $\ensuremath{\sigma}$ models admit either a Wess-Zumino-Witten term or a $\ensuremath{\theta}$ term. When such a term is present the model can have a stable critical point with gapless excitations. We describe how such a critical point appears, in particular in two-dimensional superconductors with disorder. The presence of such terms is required by the underlying (anomalous) symmetries of the original electron model. This indicates that the usual symmetry classes of disordered systems in two dimensions can be further refined. Conversely, our results also indicate that models previously thought to be in different universality classes are in fact the same once the appropriate extra terms are included.