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arXiv:2107.11365 · Physical review. B./Physical review. B

Metal-insulator transition in infinitesimally weakly disordered flat bands

Tilen Čadež, Yeongjun Kim, Alexei Andreanov, Sergej Flach A

Published Nov 15, 2021• 18 citations
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Abstract

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.

DOI
10.1103/physrevb.104.l180201

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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 \ensuremathθi{\ensuremath{\theta}}_{i}\ensuremathθi​. Without loss of generality, we consider the case of two flat bands separated by a finite gap, \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ. 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 \ensuremathθi{\ensuremath{\theta}}_{i}\ensuremathθi​. 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 \ensuremathθi{\ensuremath{\theta}}_{i}\ensuremathθi​. Instead, in d=3d=3d=3 we identify a nonperturbative metal-insulator transition upon varying all bands' flat manifold angles.
We study the effect of quasiperiodic perturbations on one-dimensional all-bands-flat lattice models. Such networks can be diagonalized by a finite sequence of local unitary transformations parameterized by angles ${\ensuremath{\theta}}_{i}$. Without loss of generality, we focus on the case of two bands with bandgap $\mathrm{\ensuremath{\Delta}}$. Weak perturbations lead to an effective Hamiltonian with both on- and off-diagonal quasiperiodic terms that depend on ${\ensuremath{\theta}}_{i}$. For some angle values, the effective model coincides with the extended Harper model. By varying the parameters of the quasiperiodic potentials, we observe localized insulating states and an entire parameter range hosting critical states with subdiffusive transport. For finite quasiperiodic potential strength, the critical-to-insulating transition becomes energy dependent with what we term fractality edges separating localized from critical states.
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.
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.
Using the determinant quantum Monte Carlo method, we investigate the metal-insulator transition in the interacting disordered Hubbard model of a Lieb lattice in which the system characterizes the flat band centered at the Fermi level. By choosing suitable electron densities, we ensure the weak interaction sign problem to improve the reliability of our results. It is found that disorder and on-site Coulomb repulsive interaction produce interesting effects that induce the metal-insulator transition which is impossible in the half-filled case. The density of states at the Fermi energy is still finite in the thermodynamic limit, suggesting that the system is an Anderson insulator rather than a Mott insulator. Moreover, this doping system is paramagnetic, unlike the half-filled system which is ferrimagnetic.
In recent years, there has been a growing interest in flatband systems which exhibit macroscopic degeneracies. These systems offer a valuable mathematical framework for the extreme sensitivity to perturbations and interactions. This sensitivity unveils a wide variety of exotic and unconventional physical phenomena. Moreover, the progress in their experimental realization contributes to the expanding landscape of exploration in this field. This thesis aims to summarize all the works throughout the Ph.D. program. Firstly, an in-depth exploration was conducted on the impact of weak quasiperiodic perturbations on one-dimensional two-band all-bands-flat lattices. These tight-binding Hamiltonian are diagonalized through a sequence of local unitary transformations. By adjusting the quasiperiodic potential parameters, the key achievement involves finding a critical-to-insulator transition and identifying fractality edges in the flatband systems with quasiperiodic perturbations. Next, the investigation delved into the effects of on-site interactions among hard-core bosons in one- and two-dimensional cross-stitch lattices. One key finding is that groundstate energy primarily depends on compact localized states. Moreover, the presence of barriers of compact localized states trap bosons, leading to the emergence of non-ergodic excitation and Hilbert space fragmentation. Lastly, a compact localized eigenstate of the one-dimensional diamond chain using an electric circuit was successfully generated via local (linear and non-linear) driving. This achievement opens up a versatile circuit platform for the generation of flatbands and holds promise for potential applications in the field of quantum information. I hope these collective efforts have expanded the frontiers of the field and made a meaningful contribution to the scientific community.