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arXiv:cond-mat/0510830 · Physical Review B

Interaction effects on two-dimensional fermions with random hopping

Matthew S. Foster, Andreas W. W. Ludwig

Published Apr 7, 2006• 30 citations
arXivHTMLPDF

Abstract

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 $BD\mathrm{I}$ [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.

DOI
10.1103/physrevb.73.155104

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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 $BD\mathrm{I}$ [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.
We uncover a disorder-driven instability in the diffusive Fermi liquid phase of a class of many-fermion systems, indicative of a metal-insulator transition of first-order type, which arises solely from the competition between quenched disorder and interparticle interactions. Our result is expected to be relevant for sufficiently strong disorder in d=3d=3d=3 spatial dimensions. Specifically, we study a class of half-filled, Hubbard-like models for spinless fermions with (complex) random hopping and short-ranged interactions on bipartite lattices in $d\ensuremath{\geqslant}2$. In a given realization, the hopping disorder breaks time-reversal invariance but preserves the special ``nesting'' symmetry responsible for the charge density wave instability of the ballistic Fermi liquid. This disorder may arise, e.g., from the application of a random magnetic field to the otherwise clean model. We derive a low-energy effective field theory description for this class of disordered, interacting fermion systems, which takes the form of a Finkel'stein nonlinear sigma model $(\mathrm{FNL}\ensuremath{\sigma}\mathrm{M})$ (A. M. Finkel'stein, Zh. Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]). We analyze the $\mathrm{FNL}\ensuremath{\sigma}\mathrm{M}$ using a perturbative, one-loop renormalization group analysis controlled via an ϵϵϵ expansion in d=2+ϵd=2+ϵd=2+ϵ dimensions. We find that in d=2d=2d=2 dimensions, the interactions destabilize the conducting phase known to exist in the disordered, noninteracting system. The metal-insulator transition that we identify in d>2d>2d>2 dimensions (ϵ>0)(ϵ>0)(ϵ>0) occurs for disorder strengths of order ϵϵϵ, and is therefore perturbatively accessible for ϵ⪡1ϵ⪡1ϵ⪡1. We emphasize that the disordered system has no localized phase in the absence of interactions, so that a localized phase and the transition into it can only appear due to the presence of the interactions.
We study the effects of electron-electron interactions in disordered metals in and close to two dimensions (2D). We consider physical situations in which localization effects are suppressed. The field-theoretical renormalization-group (RG) calculation performed recently by Finkelstein is interpreted and rederived in terms of perturbative results. Surprisingly, except for the density of states, the scaling behavior is independent of the interaction range. We further extend the model to several new universality classes. In the presence of a strong magnetic field the metal is unstable in 2D and undergoes a metal-insulator transition in $d=2+\ensuremath{\epsilon}$. The conductivity exponent, defined by $\ensuremath{\sigma}\ensuremath{\sim}{(n\ensuremath{-}{n}_{c})}^{\ensuremath{\mu}}$, is universal with $\ensuremath{\mu}=1+O(\ensuremath{\epsilon})$ but N(EF)N({E}_{F})N(EF​) depends not only on the range of the interaction but also on its strength for short-ranged interactions. In 2D the conductivity has a universal temperature dependence [$\ensuremath{\delta}\ensuremath{\sigma}(T)={\ensuremath{\sigma}}_{N}(2\ensuremath{-}2\mathrm{ln}2)\mathrm{ln}(T\ensuremath{\tau})$, ${\ensuremath{\sigma}}_{N}=\frac{{e}^{2}}{2{\ensuremath{\pi}}^{2}\ensuremath{\hbar}}$] if the interaction is Coulombic. If magnetic impurities (or strong spin-orbit scattering with a weak magnetic field) are present instead, the noninteracting fixed point is stable for short-ranged interactions ($\ensuremath{\mu}=\frac{1}{2}$). For the Coulomb interaction the interaction is relevant and drives a metal-insulator transition in $d=2+\ensuremath{\epsilon}$ with universal critical properties ($\ensuremath{\mu}=1$). In 2D the conductivity also has a universal temperature dependence [$\ensuremath{\delta}\ensuremath{\sigma}(T)={\ensuremath{\sigma}}_{N}\mathrm{ln}(T\ensuremath{\tau})$]. We also discuss the behavior of the dielectric constant on the insulator side and the frequency (temperature) dependence of the conductivity at criticality. Remarks are made on the relationship of the above to experiments.
An instability of a diffusive Fermi liquid, indicative of a metal-insulator transition (expected to be of first order), arising solely from the competition between quenched disorder and short-ranged interparticle interactions is identified in Hubbard-like models for spinless fermions, subject to (complex) random hopping at half filling on bipartite lattices. The instability, found within a Finkel'stein nonlinear $\ensuremath{\sigma}$ model treatment in d=(2+ϵ)>2d=(2+ϵ)>2d=(2+ϵ)>2 dimensions, originates from an underlying particle-hole-like (so-called chiral) symmetry, shared by both disorder and interactions. In the clean, interacting Fermi liquid this symmetry is responsible for the (completely different) nesting instability.
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.
We show that the crossover from the weak interaction limit towards the strong interaction limit may be accompanied by a delocalization effect in one dimensional disordered quantum models. The spin degrees of freedom are frozen and the spatial wave functions remain symmetric or antisymmetric when the strength U of a short range interaction is varied. The study concerns the excited states for two interacting particles and the ground state for a finite density of carriers. First, for two particles in a chain of length L, we establish a duality transformation mapping the behavior at weak U onto the behavior at strong U. For intermediate U, the mixing of the one body states and the interaction induced delocalization effect are maximum. Furthermore, if L ≈︂ L1 (the one particle localization length), the system becomes weakly chaotic with critical spectral statistics. This weak chaos is related to the multifractality of the interaction matrix. For two particles starting close to each other, localization is reached in two steps. Before the time t1 necessary to propagate over L1 , U de-favors the propagation. On the contrary, U favors a very slow delocalization after t1 , characterized by a log(t) spreading of the center of mass. Similarly, the curvatures of the energy levels with respect to an enclosed magnetic flux decrease as a function of U for L < L1 and increase for L > L1 . The changes of the curvatures can be described by a conductance-like single scaling parameter. Second, using the density renormalization group algorithm, we have studied the ground state energy of a finite density of spinless fermions and its change under twisted boundary conditions. For a large disorder, a charge reorganization is induced by the interaction: When the system becomes instable between the inhomogeneous configuration driven by the random potential (Anderson insulator) and the homogeneous one driven by repulsive interactions (Mott insulator), the ground state sensitivity can be enhanced by orders of magnitude. In contrast, no enhancement occurs at weaker disorder, when there are many particles on a scale L1.
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.