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arXiv:cond-mat/9508091 · Physical review. B, Condensed matter

Dirac fermions with disorder in two dimensions: Exact results

K. Ziegler

Published Apr 15, 1996• 16 citations
arXivHTMLPDF

Abstract

Dirac fermions on a two-dimensional lattice with disorder are considered. The Dirac mass, which controls the gap between the two bands of the fermions, is subject to random fluctuations. Another type of disorder is discussed presented by a random vector potential. It is shown that the imaginary part of the one-particle Green's function can be written as the imaginary part of another Green's function, which has only poles on the lower half plane. Therefore, it is possible to perform a Cauchy integration for a Lorentzian distribution in analogy with the Lloyd model. The results are compared with calculations performed in the continuum limit based on renormalization group and bosonization methods. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.9653

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Dirac fermions on a two-dimensional lattice with disorder are considered. The Dirac mass, which controls the gap between the two bands of the fermions, is subject to random fluctuations. Another type of disorder is discussed presented by a random vector potential. It is shown that the imaginary part of the one-particle Green's function can be written as the imaginary part of another Green's function, which has only poles on the lower half plane. Therefore, it is possible to perform a Cauchy integration for a Lorentzian distribution in analogy with the Lloyd model. The results are compared with calculations performed in the continuum limit based on renormalization group and bosonization methods. \textcopyright{} 1996 The American Physical Society.
We study the effects of generic short-ranged interactions on a system of two-dimensional (2D) Dirac fermions subject to a special kind of static disorder, often referred to as ``chiral.'' The noninteracting system is a member of the disorder class BDIBD\mathrm{I}BDI [M. R. Zirnbauer, J. Math. Phys. 37, 4986 (1996)]. It emerges, for example, as a low-energy description of a time-reversal invariant tight-binding model of spinless fermions on a honeycomb lattice, subject to random hopping, and possessing particle-hole symmetry. It is known that, in the absence of interactions, this disordered system is special in that it does not localize in 2D, but possesses extended states and a finite conductivity at zero energy, as well as a strongly divergent low-energy density of states. In the context of the hopping model, the short-range interactions that we consider are particle-hole symmetric density-density interactions. Using a perturbative one-loop renormalization group analysis, we show that the same mechanism responsible for the divergence of the density of states in the noninteracting system leads to an instability, in which the interactions are driven strongly relevant by the disorder. This result should be contrasted with the limit of clean Dirac fermions in 2D, which is stable against the inclusion of weak short-ranged interactions. Our work suggests a mechanism wherein a clean system, initially insensitive to interaction effects, can be made unstable to interactions upon the inclusion of weak static disorder. We dub this mechanism a ``disorder-driven Mott transition.'' Our result for 2D fermions also contrasts sharply with known results in one dimension, where a similar delocalized phase has been shown to be robust against the inclusion of weak interaction effects.
Quasi-one-dimensional spin-Peierls and spin-ladder systems are characterized by a gap in the spin-excitation spectrum, which can be modeled at low energies by that of Dirac fermions with a mass. In the presence of disorder, these systems can still be described by a Dirac fermion model, but with a random mass. Some peculiar properties, like the Dyson singularity in the density of states, are well known and attributed to creation of low-energy states due to the disorder. We take one step further and study single-particle correlations by means of Berezinskii's diagram technique. We find that, at low energy $\ensuremath{\epsilon}$, the single-particle Green function decays in real space like $G(x,\ensuremath{\epsilon})\ensuremath{\propto}{(1/x)}^{3/2}$. It follows that at these energies the correlations in the disordered system are strong --- even stronger than in the pure system without the gap. At distances $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}\ensuremath{\propto}{\mathrm{ln}}^{2}(1/\ensuremath{\epsilon})$, the Green function crosses over to a standard exponential decay. The scale ${L}_{\ensuremath{\epsilon}}$ is different from the Thouless localization length ${\ensuremath{\lambda}}_{\ensuremath{\epsilon}}\ensuremath{\propto}\mathrm{ln}(1/\ensuremath{\epsilon})$. Additionally, we study the effects of boundaries on the local density of states. We find that the latter is logarithmically (in the energy) enhanced close to the boundary. This enhancement decays into the bulk as 1/x1/\sqrt{x}1/x​ saturating at $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}.$ We also discuss some implications of these results for the spin systems and their relation to the investigations based on the real-space renormalization group approach.
A Dirac fermion model with random axial vector potential is proposed. At a special strength of randomness, the symmetry of the action is enhanced, which is due to the gauge symmetry in the manner of Nishimori. Some exact scaling exponents of single-particle Green functions are computed. The relationship with the XY\mathrm{XY}XY gauge glass model is discussed.
We present a nonperturbative calculation of all multifractal scaling exponents at strong disorder for critical wave functions of Dirac fermions interacting with a non-Abelian random vector potential in two dimensions. The results, valid for an arbitrary number of fermionic flavors, are obtained by deriving from conformal field theory an effective Gaussian model for the wave function amplitudes and mapping it to the thermodynamics of a single particle in a random potential. Our spectrum confirms that the wave functions remain delocalized in the presence of strong disorder.
Two-dimensional (2D) disordered superconductor (SC) in class D exhibits a disorder-induced quantum multicritical phenomenon among diffusive thermal metal (DTM), topological superconductor (TS), and conventional localized (AI) phases. To characterize the quantum tricritical point where these three phases meet, we carry out a two-loop renormalization group (RG) analysis for 2D Dirac fermion with random mass in terms of the $\ensuremath{\epsilon}$-expansion in the spatial dimension $d=2\ensuremath{-}\ensuremath{\epsilon}$. In two dimensions $(\ensuremath{\epsilon}=0)$, the random mass is marginally irrelevant around a clean-limit fixed point of the gapless Dirac fermion, while there exists an IR unstable fixed point at finite disorder strength that corresponds to the tricritical point. The critical exponent, dynamical exponent, and scaling dimension of the (uniform) mass term are evaluated around the tricritical point by the two-loop RG analysis. Using a mapping between an effective theory for the 2D random-mass Dirac fermion and the (1+1)(1+1)(1+1)-dimensional Gross-Neveu model, we further deduce the four-loop evaluation of the critical exponent, and the scaling dimension of the uniform mass around the tricritical point. Both the two-loop and four-loop results suggest that criticalities of a AI-DTM transition line as well as TS-DTM transition line are controlled by other saddle-point fixed point(s) at finite uniform mass.
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.