We study the Wigner crystallization on partially filled topological flat bands of kagome, honeycomb and checkerboard lattices. We identify the Wigner crystals (WCs) by analyzing the Cartesian and angular Fourier transform of the pair correlation density of the many-body ground state obtained using exact diagonalization. The crystallization strength, measured by the magnitude of the Fourier peaks, increases with decreasing particle density. The Wigner crystallization observed by us is a robust and general phenomenon, existing in all three lattice models for a broad range of filling factors and interaction parameters. The shape of the resulting WCs is determined by the boundary conditions of the chosen plaquette. It is to a large extent independent on the underlying lattice, including its topology, and follows the behavior of classical point particles.
We study the ground states of cold atoms in the tight-binding bands built from p orbitals on a two dimensional honeycomb optical lattice. The band structure includes two completely flat bands. Exact many-body ground states with on-site repulsion can be found at low particle densities, for both fermions and bosons. We find crystalline order at n=1/6 with a sqrt[3] x sqrt[3] structure breaking a number of discrete lattice symmetries. In fermionic systems, if the repulsion is strong enough, we find the bonding strength becomes dimerized at n=1/2. Experimental signatures of crystalline order can be detected through the noise correlations in time of flight experiments.
We study a tight-binding model on the two-dimensional star lattice (also called the decorated honeycomb lattice) with short-range hoppings modulated by staggered fluxes. We observe a series of topological quantum phase transitions when tuning the hopping parameters, and find nontrivial topological bands with high Chern numbers. We also find in some parameter regions, the system exhibits the interesting topological flat bands with large flatness ratios of the band gaps over the bandwidths. We further illustrate the
1/2 fractional quantum Hall effect of interacting hard-core bosons filled in a lowest topological flat band of the star lattice.
The Lieb lattice possesses three bands and with intrinsic spin orbit coupling
λ, supports topologically non-trivial band insulating phases. At half filling the lower band is fully filled, while the upper band is empty. The chemical potential lies in the flat band (FB) located at the middle of the spectrum, thereby stabilizing a flat band insulator. At this filling, we introduce on-site Hubbard interaction
U on all sites. Within a slave rotor mean field theory we show that, in spite of the singular effect of interaction on the FB, the three bands remain stable up to a fairly large critical correlation strength (
Ucrit), creating a correlated flat band insulator. Beyond
Ucrit, there is a sudden transition to a Mott insulating state, where the FB is destroyed due to complete transfer of spectral weight from the FB to the upper and lower bands. We show that all the correlation driven insulating phases host edge modes with linearly dispersing bands along with a FB passing through the Dirac point, exhibiting that the topological nature of the bulk band structure remains intact in presence of strong correlation. Furthermore, in the limiting case of
U introduced only on one sublattice where
λ=0, we show that the Lieb lattice can support mixed edge modes containing contributions from both spinons and electrons, in contrast to purely spinon edge modes arising in the topological Mott insulator.
We analyze low energy spectral properties of small toroidal configurations of the Kitaev honeycomb spin model in the Abelian topological phase. We begin with a brief classification of honeycomb lattices on a torus. Then, using the Brillouin–Wigner perturbation theory, we explain the low order finite size effects that can occur in these systems and show how they affect their ground state topological degeneracy. Finally, we demonstrate the accuracy of the perturbative method by means of exact diagonalization, and use the insights into the finite size effects to reconstruct the topological degeneracy in a small example system.
We construct quasi-one-dimensional topological and nontopological three-band lattices with a tunable band gap and winding number of the flat band. Using full multiband mean field (MF) and exact density matrix renormalization group (DMRG) calculations, we show explicitly how the band gap affects pairing and superconductivity (SC) in a Hubbard model with attractive interactions. We obtain excellent agreement between MF and DMRG calculations. The SC weight
Ds on the gapped topological, $W\ensuremath{\ne}0$, flat band increases linearly with interaction strength
U for low values and with a slope that depends on the details of the compact localized state at
U=0. As $U\ensuremath{\rightarrow}0$ for the gapped nontopological flat band (
W=0),
Ds decays with a power law faster than quadratically but slower than exponentially. In the gapless case (flat band touching the band above it), we find at low
U (for both
W=0 and $W\ensuremath{\ne}0$) that ${D}_{s}\ensuremath{\propto}{U}^{\ensuremath{\varphi}}$, with $\ensuremath{\varphi}<1$. In other words,
Ds increases faster than linearly for low
U, thus favoring SC at weak interaction more than the gapped case. Our results reestablish that the BCS mean field and quantum metric alone are insufficient to characterize SC at weak coupling.