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arXiv:2308.16440 · Physical Review Letters

Localizing Transitions via Interaction-Induced Flat Bands

Alireza Parhizkar, Victor M. Galitski

Published Oct 16, 2024• 2 citations
arXivHTMLPDF

Abstract

This Letter presents a theory of interaction-induced band-flattening in strongly correlated electron systems. We begin by illustrating an inherent connection between flat bands and index theorems, and presenting a generic prescription for constructing flat bands by periodically repeating local Hamiltonians with topological zero modes. Specifically, we demonstrate that a Dirac particle in an external, spatially periodic magnetic field can be cast in this form. We derive a condition on the field to produce perfectly flat bands and provide an exact analytical solution for the flat band wave functions. Furthermore, we explore an interacting model of Dirac fermions in a spatially inhomogeneous field. We show that certain Hubbard-Stratonovich configurations exist that "rectify" the field configuration, inducing band flattening. We present an explicit model where this localization scenario is energetically favorable-specifically in Dirac systems with nearly flat bands, where the energy cost of rectifying textures is quadratic in the order parameter, whereas the energy gain from flattening is linear. In conclusion, we discuss alternative symmetry-breaking channels, especially superconductivity, and propose that these interaction-induced band-flattening scenarios represent a generic nonperturbative mechanism for spontaneous symmetry breaking, pertinent to many strongly correlated electron systems.

DOI
10.1103/physrevlett.133.166502

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This Letter presents a theory of interaction-induced band-flattening in strongly correlated electron systems. We begin by illustrating an inherent connection between flat bands and index theorems, and presenting a generic prescription for constructing flat bands by periodically repeating local Hamiltonians with topological zero modes. Specifically, we demonstrate that a Dirac particle in an external, spatially periodic magnetic field can be cast in this form. We derive a condition on the field to produce perfectly flat bands and provide an exact analytical solution for the flat band wave functions. Furthermore, we explore an interacting model of Dirac fermions in a spatially inhomogeneous field. We show that certain Hubbard-Stratonovich configurations exist that "rectify" the field configuration, inducing band flattening. We present an explicit model where this localization scenario is energetically favorable-specifically in Dirac systems with nearly flat bands, where the energy cost of rectifying textures is quadratic in the order parameter, whereas the energy gain from flattening is linear. In conclusion, we discuss alternative symmetry-breaking channels, especially superconductivity, and propose that these interaction-induced band-flattening scenarios represent a generic nonperturbative mechanism for spontaneous symmetry breaking, pertinent to many strongly correlated electron systems.
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In a partially filled flat Bloch band electrons do not have a well defined Fermi surface and hence the low-energy theory is not a Fermi liquid. Nevertheless, under the influence of an attractive interaction, a superconductor well described by the Bardeen-Cooper-Schrieffer (BCS) wave function can arise. Here we study the low-energy effective Hamiltonian of a generic Hubbard model with a flat band. We obtain an effective Hamiltonian for the flat band physics by eliminating higher-lying bands via the perturbative Schrieffer-Wolff transformation. At first order in the interaction energy we recover the usual procedure of projecting the interaction term onto the flat band Wannier functions. We show that the BCS wave function is the exact ground state of the projected interaction Hamiltonian, if a simple uniform pairing condition on the single-particle states is satisfied, and that the compressibility is diverging as a consequence of an emergent SU(2)\text{SU}(2)SU(2) symmetry. This symmetry is broken by second-order interband transitions resulting in a finite compressibility, which we illustrate for a one-dimensional ladder with two perfectly flat bands. These results motivate a further approximation leading to an effective ferromagnetic Heisenberg model. The gauge-invariant result for the superfluid weight of a flat band can be obtained from the ferromagnetic Heisenberg model only if the maximally localized Wannier functions in the Marzari-Vanderbilt sense are used. Finally, we prove an important inequality $D\ensuremath{\ge}{\mathcal{W}}^{2}$ between the Drude weight DDD and the winding number W\mathcal{W}W, which guarantees ballistic transport for topologically nontrivial flat bands in one dimension.
In a flat band superconductor, bosonic excitations disperse while unpaired electrons are immobile. To study this strongly interacting system, we construct a family of multi-band Hubbard models with exact eta-pairing ground states in all space groups. We analytically compute their many-body excitations and find that the Cooper pair bound states obey an effective single-boson hopping Hamiltonian written in terms of the non-interacting wavefunctions. These bound states possess a unique zero-energy excitation whose quadratic dispersion is determined by the minimal quantum metric. The rest of the bound state spectrum is classified by topological quantum chemistry, which we use to identify Cooper pairs with Weyl nodes, higher angular momentum pairing, and fragile topology. Broadening the band dispersion, we find the strongest pairing occurs at half filling and not at the highest density of states. Our analytical results prove that the elementary fermionic and bosonic excitations, corresponding respectively to pairing and coherence, are governed by distinct properties of the flat band, rigorously establishing its unique phenomenology. Superconductivity in flat bands is a central feature of quantum geometric effects, supported by mean-field theory and numerics. Here, the authors exhibit a large family of flat band Hubbard models whose ground states, Bogoliubov-de-Gennes particles, and Cooper pairs are exactly solvable with analytically calculable dispersions.
We investigate superconducting transport in the DC field induced Wannier-Stark flat bands in the presence of interactions. Flat bands offer the possibility of unconventional high temperature superconductivity, where the superfluid weight, Ds{D}_{s}Ds​, is enhanced by the density overlap of the localized states. However, construction of flat bands typically requires very precise tuning of Hamiltonian parameters. To overcome this difficulty, we propose a feasible alternative to realize flat bands by applying a DC field in a commensurate lattice direction. We systematically characterize the superconducting behavior on these flat bands by studying the effect of the DC field and attractive Hubbard interaction strengths on the wave function, correlation length, pairing order parameter, and the superfluid weight Ds{D}_{s}Ds​. Our main result is that the superfluid weight is dramatically enhanced at an optimal value of the interaction strength and weak DC fields.
Band flattening has been identified as key ingredient to correlation phenomena in Moiré materials and beyond. Here, we examine strongly repulsive fermions on a ladder -- a minimal platform for unconventional ddd-wave pairing -- and show that flattening of the lower band through an additional diagonal hopping term produces non-Fermi liquid behavior, evidenced by the violation of Luttinger's theorem, as well as axial ddd-wave pairing correlations. Alternatively, plaquette ring exchange can also generate pairing, albeit with a distinct diagonal ddd-wave pairing symmetry. Hence, our finding showcases a competition of different unconventional pairing channels, and demonstrates via a simple model how band geometry can induce fermionic pairing. This offers broadly relevant insights for correlated flat-band systems, ranging from ultracold atoms to strongly interacting electrons in solids.
A Moiré system is formed when two periodic structures have a slightly mismatched period, resulting in unusual strongly correlated states in the presence of particle-particle interactions. The periodic structures can arise from the intrinsic crystalline order and periodic external field. We investigate a one-dimensional Hubbard model with periodic on-site potential of period n 0 , which is commensurate to the lattice constant. For large n 0 , the exact solution demonstrates that there is a midgap flat band with zero energy in the absence of Hubbard interaction. Each Moiré unit cell contributes two zero energy levels to the flat band. In the presence of Hubbard interaction, the midgap physics is demonstrated to be well described by a uniform Hubbard chain in which the effective hopping and on-site interaction strength can be controlled by the amplitude and period of the external field. Numerical simulations are performed to demonstrate the correlated behaviors in the finite-sized Moiré Hubbard system, including the existence of an η -pairing state and bound pair oscillation. This finding provides a method to enhance the correlated effect by a spatially periodic external field.