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arXiv:1712.03523 · Physical review. B./Physical review. B

Disorder perturbed flat bands: Level density and inverse participation ratio

Pragya Shukla

Published Aug 31, 2018• 33 citations
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Abstract

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.

DOI
10.1103/physrevb.98.054206

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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.
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 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.
Johri and Bhatt found singular behavior near the band edge in the density of states as well as in the inverse participation ratio of the Anderson model. These singularities mark a transition to an energy range dominated by resonant states. We study the interacting case using an ensemble of two-site Anderson-Hubbard systems. We find the ensemble-averaged density of states and generalized inverse participation ratio have more structure than in the non-interacting case because there are more transitions and in particular the transitions depend on the ground state. Nonetheless, there are regions of sharp decline in the generalized inverse participation ratio associated with specific density of state features. Moreover these features move closer to the Fermi level with the addition of interactions making them more experimentally accessible. Unfortunately resonances unique to interacting systems cannot be specifically identified. Author Keywords: Correlated electrons, Disorder, Localization
We test the usefulness of a generalized inverse participation ratio (GIPR) as a measure of Anderson localization. The GIPR differs from the usual inverse participation ratio in that it is constructed from the local density of states rather than the single-electron wave functions. This makes it suitable for application to many-body systems. We benchmark the GIPR by performing a finite-size scaling analysis of a disordered, noninteracting, three-dimensional tight-binding lattice. We find values for the critical disorder and critical exponents that are in agreement with published values.
We study the effect of infinitesimal onsite disorder on ddd-dimensional all bands flat lattices. The lattices are generated from diagonal Hamiltonians by a sequence of (d+1)(d+1)(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 WWW is only affecting the overall scale of the effective Hamiltonian. Upon variation of the manifold angles for d=1d=1d=1 and d=2d=2d=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=3d=3d=3 we identify a nonperturbative metal-insulator transition upon varying all bands' flat manifold angles.
We use the density-matrix renormalization group method to investigate the superconductor-insulator transition (SIT) in disordered quasi-one-dimensional systems. Focusing on the case of an interacting spinful Hamiltonian at quarter filling, we contrast the differences arising in the SIT when the parent noninteracting model features flat or dispersive bands. Furthermore, we unveil the critical disorder amplitude that triggers insulating behavior by comparing disorder distributions that preserve or not SU(2) symmetry. While scaling analysis suggests the transition be of a Berezinskii-Kosterlitz-Thouless type for all models (two lattices and two disorder types), only in the flat-band model with Zeeman-like disorder is the critical disorder nonvanishing. In this sense, the flat-band structure does strengthen superconductivity in the presence of attractive interactions. For both flat- and dispersive-band models, (i) in the presence of SU(2)-symmetric random chemical potentials, the disorder-induced transition is from superconductor to insulator of singlet pairs; (ii) for the Zeeman-type disorder, the transition is from superconductor to insulator of unpaired fermions. In all cases, our numerical results suggest no intermediate disorder-driven metallic phase.