Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0311302 · Journal of the Physical Society of Japan

Density of States of Disordered Dirac Particles: Infinitely Many Operators with Negative Scaling Dimensions and Freezing Transitions

Shinsei Ryu, Christopher Mudry, Akira Furusaki

Published Jan 3, 2003• 0 citations
arXivHTMLPDF

Abstract

The global density of states (GDOS) close to the band center $ε=0$ for a particle hopping on a square lattice and subjected to disorder that preserves the bipartite symmetry of the lattice is computed using field theoretical methods. The GDOS diverges like $ |ε|^{-1}\exp (-c|\lnε|^κ ) $ with $κ=2/3$ in agreement with a prediction by Motrunich \textit{et al.} and in disagreement with an older prediction by Gade ($κ=1/2$).

DOI
10.1143/jpsjs.72sa.219

Related papers

Density of states for the π -flux state with bipartite real random hopping only: A weak disorder approach

Christopher Mudry, Shinsei Ryu, Akira Furusaki

Gade [Nucl. Phys. B $398,$ 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 $65,$ 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.

arXiv:cond-mat/0207723•Physical review. B, Condensed matter•Feb 28, 2003•64 citations

Near Critical States of Random Dirac Fermions

Yoshifumi Morita, Yasuhiro Hatsugai

Two-dimensional random Dirac fermions are studied numerically. They are realized on a square lattice by the $\ensuremath{\pi}$-flux model with random hopping. It preserves a symmetry denoted by ${H,\ensuremath{\gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ in an effective field theory. Although it belongs to the orthogonal ensemble, the zero-energy states do not localize but become critical. The density of states vanishes as $\ensuremath{\sim}{E}^{\ensuremath{\alpha}}$ and the exponent $\ensuremath{\alpha}$ changes with strength of the randomness (the critical line). Rapid enhancement of the Thouless number is observed near the zero energy. The level-spacing distribution is also investigated, which is consistent with the existence of the critical states at $E\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$.

arXiv:cond-mat/9705192•Physical Review Letters•Nov 10, 1997•49 citations

Random hopping fermions on bipartite lattices: density of states, inverse participation ratios, and their correlations in a strong disorder regime

Hiroki Yamada, Takahiro Fukui

arXiv:cond-mat/0302376•Nuclear Physics B•Dec 22, 2003•12 citations

Particle-hole symmetric localization in optical lattices using time modulated random on-site potentials

Yue Zou, Ryan Barnett, Gil Refael

The random hopping models exhibit many fascinating features, such as diverging localization length and density of states as energy approaches the band center due to its particle-hole symmetry. Nevertheless, such models are yet to be realized experimentally because the particle-hole symmetry is easily destroyed by diagonal disorder. Here we propose that a pure random hopping model can be effectively realized in ultracold atoms by modulating a disordered onsite potential in particular frequency ranges. This idea is motivated by the recent development of the phenomena called ``dynamical localization'' or ``coherent destruction of tunneling.'' Investigating the application of this idea in one dimension, we find that if the oscillation frequency of the disorder potential is gradually increased from zero to infinity, one can tune a noninteracting system from an Anderson insulator to a random hopping model with diverging localization length at the band center, and eventually to a uniform-hopping tight-binding model.

arXiv:1007.2245•Physical Review B•Dec 15, 2010•2 citations

Disordered critical wave functions of two-dimensional Dirac fermions on a lattice

Yasuhiro Hatsugai, Yoshifumi Morita, Xiao-Gang Wen +1 more

Physica B Condensed Matter•Jun 1, 1998•0 citations

Interaction effects on two-dimensional fermions with random hopping

Matthew S. Foster, Andreas W. W. Ludwig

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.

arXiv:cond-mat/0510830•Physical Review B•Apr 7, 2006•30 citations
The global density of states (GDOS) close to the band center ε=0ε=0ε=0 for a particle hopping on a square lattice and subjected to disorder that preserves the bipartite symmetry of the lattice is computed using field theoretical methods. The GDOS diverges like ∣ε∣−1exp⁡(−c∣ln⁡ε∣κ)|ε|^{-1}\exp (-c|\lnε|^κ )∣ε∣−1exp(−c∣lnε∣κ) with κ=2/3κ=2/3κ=2/3 in agreement with a prediction by Motrunich \textit{et al.} and in disagreement with an older prediction by Gade (κ=1/2κ=1/2κ=1/2).
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.
Two-dimensional random Dirac fermions are studied numerically. They are realized on a square lattice by the $\ensuremath{\pi}$-flux model with random hopping. It preserves a symmetry denoted by ${H,\ensuremath{\gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ in an effective field theory. Although it belongs to the orthogonal ensemble, the zero-energy states do not localize but become critical. The density of states vanishes as $\ensuremath{\sim}{E}^{\ensuremath{\alpha}}$ and the exponent $\ensuremath{\alpha}$ changes with strength of the randomness (the critical line). Rapid enhancement of the Thouless number is observed near the zero energy. The level-spacing distribution is also investigated, which is consistent with the existence of the critical states at E=0E\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0E=0.
The random hopping models exhibit many fascinating features, such as diverging localization length and density of states as energy approaches the band center due to its particle-hole symmetry. Nevertheless, such models are yet to be realized experimentally because the particle-hole symmetry is easily destroyed by diagonal disorder. Here we propose that a pure random hopping model can be effectively realized in ultracold atoms by modulating a disordered onsite potential in particular frequency ranges. This idea is motivated by the recent development of the phenomena called ``dynamical localization'' or ``coherent destruction of tunneling.'' Investigating the application of this idea in one dimension, we find that if the oscillation frequency of the disorder potential is gradually increased from zero to infinity, one can tune a noninteracting system from an Anderson insulator to a random hopping model with diverging localization length at the band center, and eventually to a uniform-hopping tight-binding model.
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.