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arXiv:1210.7696 · Physical Review Letters

Fractional Topological States of Dipolar Fermions in One-Dimensional Optical Superlattices

Zhihao Xu, Linhu Li, Shu Chen

Published May 21, 2013• 58 citations
arXivHTMLPDF

Abstract

We study the properties of dipolar fermions trapped in one-dimensional bichromatic optical lattices and show the existence of fractional topological states in the presence of strong dipole-dipole interactions. We find some interesting connections between fractional topological states in one-dimensional superlattices and the fractional quantum Hall states: (i) the one-dimensional fractional topological states for systems at filling factor ν=1/p have p-fold degeneracy, (ii) the quasihole excitations fulfill the same counting rule as that of fractional quantum Hall states, and (iii) the total Chern number of p-fold degenerate states is a nonzero integer. The existence of crystalline order in our system is also consistent with the thin-torus limit of the fractional quantum Hall state on a torus. The possible experimental realization in cold atomic systems offers a new platform for the study of fractional topological phases in one-dimensional superlattice systems.

DOI
10.1103/physrevlett.110.215301

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We study the properties of dipolar fermions trapped in one-dimensional bichromatic optical lattices and show the existence of fractional topological states in the presence of strong dipole-dipole interactions. We find some interesting connections between fractional topological states in one-dimensional superlattices and the fractional quantum Hall states: (i) the one-dimensional fractional topological states for systems at filling factor ν=1/p have p-fold degeneracy, (ii) the quasihole excitations fulfill the same counting rule as that of fractional quantum Hall states, and (iii) the total Chern number of p-fold degenerate states is a nonzero integer. The existence of crystalline order in our system is also consistent with the thin-torus limit of the fractional quantum Hall state on a torus. The possible experimental realization in cold atomic systems offers a new platform for the study of fractional topological phases in one-dimensional superlattice systems.
Extensive research interest has been attracted by the fact that recent experiments in cold atoms simulate the chiral edge states of (1 + 1)-dimensional quantum Hall systems with laser-assisted transitions between different atomic nuclear spin states in a one-dimensional optical lattice. Within such an optical lattice, we study the phase diagrams of ν = 1 / 3 filling hard-core fermions and ν = 1 / 2 hard-core bosons with internal spin states in the presence of dipole-dipolar interactions. Our results show that the interaction drives the fermionic (bosonic) systems from the normal (superfluid) phase into the fractional topological phase with possible crystalline orderings. The topological nature of the latter phase is identified by the numerical evaluation of the many-body Chern number. Remarkably, we find that a considerable spectrum gap stabilizes the fractional topological phase with a moderate interaction strength, while, for the normal/superfluid phase, the Thouless charge pumping fluctuates strongly for small spin systems but approaches its universal value 1/3 for fermionic and 1/2 for bosonic systems with large spins.
We present exactly solvable examples showing that topological Mott insulators can emerge from topologically trivial states due to strong interactions between atoms for atomic mixtures trapped in one-dimensional optical superlattice systems. The topological Mott insulating state is characterized by a nonzero Chern number and appears in the strongly interacting limit as long as the total band filling factor is an integer that is not sensitive to the filling of each component. The topological nature of the Mott phase can be revealed by observing the density profile of the trapped system. Our results can be also generalized to multicomponent atomic systems.
We study fermions and hard-core bosons with long-range dipolar interactions at fractional fillings in a topological checkerboard lattice with short-range hoppings up to next-next-nearest neighbors [T. Neupert, L. Santos, C. Chamon, and C. Mudry, Phys. Rev. Lett. 106, 236804 (2011)]. We consider the case that the dipoles are aligned in the perpendicular direction by an external field without the complication of anisotropic interaction. Using exact diagonalization, we find clear signatures of fractional quantum Hall (FQH) states at filling factors 1/3 and 1/5 for fermions (1/2 and 1/4 for bosons) in the lowest Chern band with a robust spectrum gap at moderate dipolar interaction strength. The robustness of these FQH states against a long-range interaction tail and band flatness is investigated. When the dipolar interaction decreases, the fermionic FQH states turn into normal states, and the bosonic 1/4 FQH state turns into a superfluid state. The bosonic 1/2 FQH state survives even in the absence of the dipolar interaction, but vanishes when the hard core becomes a soft core with a critical on-site repulsion. In the thin torus limit, the static density structure factors indicate that the FQH state turns into a commensurate charge density wave state.
Twisted bilayer materials provide a versatile platform for realizing correlated topological states. Motivated by the recent experimental realization of atomic Bose-Einstein condensates in twisted bilayer optical lattices, we investigate the emergence of fractional topological phases in such systems. For experimentally realistic parameters, the single-particle spectrum hosts a pair of quasi-degenerate nearly flat moiré bands. We show using self-consistent Hartree-Fock calculations that atomic interactions spontaneously break time-reversal symmetry, lift the quasi-degeneracy, and generate an isolated nearly flat band with a nonzero Chern number. At fractional filling, exact diagonalization of the Hamiltonian projected to this band reveals a threefold degenerate ground-state manifold and a characteristic particle entanglement spectrum, providing strong evidence for a fractional Chern insulator. The resulting many-body gap is of order 10−3ER10^{-3}E_R10−3ER​, corresponding to a nanokelvin energy scale accessible to state-of-the-art cold-atom experiments. Our scheme requires neither spin-orbit coupling, as in transition metal dichalcogenides, nor a finite magnetic field, as in existing twisted-bilayer-graphene experiments, providing a highly tunable route to strongly correlated topological phases in twisted bilayer optical lattices.
We analytically and numerically analyze the one-dimensional "thin-torus" limit of Fractional Topological Insulators in a series of simple models exhibiting exactly flat bands with local hopping. These models are the one-dimensional limit of two dimensional Chern Insulators, and the Hubbard-type interactions projected into their lowest band take particularly simple forms. By exactly solving the many-body interacting spectrum of these models, we show that, just like in the Fractional Quantum Hall effect, the zero modes of the thin-torus limit are CDW states of occupation numbers satisfying generalized Pauli principles. As opposed to the FQH where the thin-torus CDW appear in orbital space, in the thin-torus FCI states, the CDW states are in real-space. We show the counting of the quasihole excitations in the energy spectrum cannot distinguish between a CDW state and a FQH state. However, by exactly computing the entanglement spectrum for the thin-torus states, we show that it can qualitatively and quantitatively distinguish between a CDW and a fractional topological state such as the FCI. We then discover a previously unknown separation of energy scales of the full FQH energy spectrum in the thin torus limit and find that Chern insulator models exhibiting strong isotropic FCI states have a similar structure in their thin-torus limit spectrum. We close by numerically computing the evolution of energy and entanglement spectra from the thin-torus to the isotropic limit. Our results can also be interpreted as an analysis of one-body, 1-dimensional topological insulators stabilized by inversion symmetry in the presence of interactions.
Recent theoretical works have demonstrated various robust Abelian and non-Abelian fractional topological phases in lattice models with topological flat bands carrying Chern number C=1C=1C=1. Here we study hard-core bosons and interacting fermions in a three-band triangular-lattice model with the lowest topological flat band of Chern number C=2C=2C=2. We find convincing numerical evidence of bosonic fractional quantum Hall effect at the $\ensuremath{\nu}=1/3$ filling characterized by threefold quasidegeneracy of ground states on a torus, a fractional Chern number for each ground state, a robust spectrum gap, and a gap in quasihole excitation spectrum. We also observe numerical evidence of a robust fermionic fractional quantum Hall effect for spinless fermions at the $\ensuremath{\nu}=1/5$ filling with short-range interactions.