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Particle-Hole Symmetric Localization Problems in One and Two Dimensions

Olexei I. Motrunich

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

Particle-hole symmetric localization problems are studied in low dimensions with the emphasis on understanding the origin of the observed singularities in the low-energy density of states. Pure imaginary random hopping and bipartite random hopping problems are considered. In quasi-1D geometry, such systems generically exhibit localized behavior, and are also found to have power-law density of states with nonuniversal dynamical exponents. Griffiths-like mechanism is identified that generates the singularly many low-energy states. Distinct localized phases are separated by delocalized critical points, and at these points the density of states has strong Dyson singularity near the band center. Similar mechanism is argued to be operative in all particle-hole symmetric lo-calized phases in any dimension. Higher-dimensional systems can also exhibit true metallic or critical delocalized phases, and one such system—the 2D bipartite ran-dom hopping problem—is studied in great detail. General such system is known to

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Particle-hole symmetric localization problems are studied in low dimensions with the emphasis on understanding the origin of the observed singularities in the low-energy density of states. Pure imaginary random hopping and bipartite random hopping problems are considered. In quasi-1D geometry, such systems generically exhibit localized behavior, and are also found to have power-law density of states with nonuniversal dynamical exponents. Griffiths-like mechanism is identified that generates the singularly many low-energy states. Distinct localized phases are separated by delocalized critical points, and at these points the density of states has strong Dyson singularity near the band center. Similar mechanism is argued to be operative in all particle-hole symmetric lo-calized phases in any dimension. Higher-dimensional systems can also exhibit true metallic or critical delocalized phases, and one such system—the 2D bipartite ran-dom hopping problem—is studied in great detail. General such system is known to
The one-dimensional (1D) tight-binding model with random nearest-neighbor hopping is known to have a singularity of the density of states and of the localization length at the band center. We study numerically the effects of random long-range (power-law) hopping with an ensemble average magnitude $〈|{t}_{\mathrm{ij}}|〉\ensuremath{\propto}|i\ensuremath{-}j{|}^{\ensuremath{-}\ensuremath{\sigma}}$ in the 1D chain, while maintaining the particle-hole symmetry present in the nearest-neighbor model. We find, in agreement with results of real-space renormalization-group techniques applied to the random XY spin chain with power-law interactions, that there is a change of behavior when the power-law exponent $\ensuremath{\sigma}$ becomes smaller than 2.
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.
We introduce a strong-disorder renormalization group (RG) approach suitable for investigating the quasiparticle excitations of disordered superconductors in which the quasiparticle spin is not conserved. We analyze one-dimensional models with this RG and with elementary transfer matrix methods. We find that such models with broken spin rotation invariance generically lie in one of two topologically distinct localized phases. Close enough to the critical point separating the two phases, the system has a power-law divergent low-energy density of states (with a nonuniversal continuously varying power law) in either phase, due to quantum Griffiths singularities. This critical point belongs to the same infinite-disorder universality class as the one-dimensional particle-hole symmetric Anderson localization problem, while the Griffiths phases in the vicinity of the transition are controlled by lines of strong (but not infinite) disorder fixed points terminating in the critical point.
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.
The low-frequency response of systems near a many-body localization transition can be dominated by rare regions that are locally critical or ``in the other phase.'' It is known that in one dimension, these rare regions can cause the dc conductivity and diffusion constant to vanish even inside the delocalized thermal phase. Here, we present a general analysis of such Griffiths effects in the thermal phase near the many-body localization transition: we consider both one-dimensional and higher-dimensional systems, subject to quenched randomness, and discuss both linear response (including the frequency- and wave-vector-dependent conductivity) and more general dynamics. In all the regimes we consider, we identify observables that are dominated by rare-region effects. In some cases (one-dimensional systems and Floquet systems with no extensive conserved quantities), essentially all long-time local observables are dominated by rare-region effects; in others, generic observables are instead dominated by hydrodynamic long-time tails throughout the thermal phase, and one must look at specific probes, such as spin echo, to see Griffiths behavior.
We consider ddd-dimensional systems which are localized in the absence of interactions, but whose single-particle localization length diverges near a discrete set of (single-particle) energies, with critical exponent $\ensuremath{\nu}$. This class includes disordered systems with intrinsic or symmetry protected topological bands, such as disordered integer quantum Hall insulators. We show that such marginally localized systems exhibit anomalous properties intermediate between localized and extended, including vanishing dc conductivity but subdiffusive dynamics, and fractal entanglement (an entanglement entropy with a scaling intermediate between area and volume law). We investigate the stability of marginal localization in the presence of interactions, and argue that arbitrarily weak short-range interactions trigger delocalization for partially filled bands at nonzero energy density if $\ensuremath{\nu}\ensuremath{\ge}1/d$. We use the Harris-Chayes bound $\ensuremath{\nu}\ensuremath{\ge}2/d$ to conclude that marginal localization is generically unstable in the presence of interactions. Our results suggest the impossibility of stabilizing quantized Hall conductance at nonzero energy density.