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arXiv:1807.02436 · Physical review. B./Physical review. B

Disorder perturbed flat bands. II. Search for criticality

Pragya Shukla

Published Nov 7, 2018• 19 citations

Abstract

We present a common mathematical formulation of the level statistics of a disordered tight-binding lattice, with one or many flat bands in the clean limit, in which system-specific details enter through a single parameter. The formulation, applicable to both single- as well as many-particle flat bands, indicates the possibility of two different types of critical statistics: one in weak disorder regime (below a system-specific disorder strength) and insensitive of the disorder strength, another in strong disorder regime and occurs at specific critical disorder strengths. The single-parametric dependence, however, relates the statistics in the two regimes (notwithstanding different scattering conditions therein). This also helps in revealing an underlying universality of the statistics in weakly disordered flat bands, shared by a wide range of other complex systems irrespective of the origin of their complexity.

DOI: 10.1103/physrevb.98.184202

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We present a common mathematical formulation of the level statistics of a disordered tight-binding lattice, with one or many flat bands in the clean limit, in which system-specific details enter through a single parameter. The formulation, applicable to both single- as well as many-particle flat bands, indicates the possibility of two different types of critical statistics: one in weak disorder regime (below a system-specific disorder strength) and insensitive of the disorder strength, another in strong disorder regime and occurs at specific critical disorder strengths. The single-parametric dependence, however, relates the statistics in the two regimes (notwithstanding different scattering conditions therein). This also helps in revealing an underlying universality of the statistics in weakly disordered flat bands, shared by a wide range of other complex systems irrespective of the origin of their complexity.

We consider the effect of disorder on the tight-binding Hamiltonians with a flat band and derive a common mathematical formulation of the average density of states and inverse participation ratio applicable for a wide range of them. The system information in the formulation appears through a single parameter which plays an important role in search of the critical points for disorder driven transitions in flat bands [P. Shukla, arXiv:1807.02436]. In a weak disorder regime, the formulation indicates an insensitivity of the statistical measures to disorder strength, thus confirming the numerical results obtained by our as well as previous studies.

The level statistics of a tight-binding Hamiltonian describing strongly interacting particles in a disordered system is investigated. It is found to be characterized by two transitions as functions of interaction strength. The first is from Poisson statistics to Wigner (GOE) statistics as interactions between the particles are turned on, and the second is back to Poisson statistics as the interactions become stronger. The dependences of both transitions on the interaction strength, filling factor, size of the sample, disorder and range of interactions are considered. Possible experimental consequences are discussed.

We study the effect of infinitesimal onsite disorder on

d

-dimensional all bands flat lattices. The lattices are generated from diagonal Hamiltonians by a sequence of

(d+1)

local unitary transformations parametrized by angles ${\ensuremath{\theta}}_{i}$. Without loss of generality, we consider the case of two flat bands separated by a finite gap, $\mathrm{\ensuremath{\Delta}}$. The perturbed states originating from the flat bands are described by an effective tight-binding network with finite on- and off-diagonal disorder strength which depends on the manifold angles ${\ensuremath{\theta}}_{i}$. The original infinitesimal on-site disorder strength

W

is only affecting the overall scale of the effective Hamiltonian. Upon variation of the manifold angles for

d=1

and

d=2

we find that localization persists for any choice of local unitaries, and the localization length can be maximized for specific values of ${\ensuremath{\theta}}_{i}$. Instead, in

d=3

we identify a nonperturbative metal-insulator transition upon varying all bands' flat manifold angles.

We consider the effect of weak disorder on eigenstates in a special class of tight-binding models. Models in this class have short-range hopping on periodic lattices; their defining feature is that the clean systems have some energy bands that are dispersionless throughout the Brillouin zone. We show that states derived from these flat bands are generically critical in the presence of weak disorder, being neither Anderson localized nor spatially extended. Further, we establish a mapping between this localization problem and the one of resonances in random impedance networks, which previous work has suggested are also critical. Our conclusions are illustrated using numerical results for a two-dimensional lattice, known as the square lattice with crossings or the planar pyrochlore lattice.