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arXiv:1506.02290 · Physical Review Letters

Landau Levels in Strained Optical Lattices

Binbin Tian, Manuel Endres, David Pekker

Published Dec 1, 2015• 22 citations
arXivHTMLPDF

Abstract

We propose a hexagonal optical lattice system with spatial variations in the hopping matrix elements. Just like in the valley Hall effect in strained graphene, for atoms near the Dirac points the variations in the hopping matrix elements can be described by a pseudomagnetic field and result in the formation of Landau levels. We show that the pseudomagnetic field leads to measurable experimental signatures in momentum resolved Bragg spectroscopy, Bloch oscillations, cyclotron motion, and quantization of in situ densities. Our proposal can be realized by a slight modification of existing experiments. In contrast to previous methods, pseudomagnetic fields are realized in a completely static system avoiding common heating effects and therefore opening the door to studying interaction effects in Landau levels with cold atoms.

DOI
10.1103/physrevlett.115.236803

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We study strongly correlated ground states of dipolar fermions in a honeycomb optical lattice with spatial variations in hopping amplitudes. Similar to strained graphene, such nonuniform hopping amplitudes produce valley-dependent pseudomagnetic fields for fermions near the two Dirac points, resulting in the formation of Landau levels. The dipole moments aligned perpendicular to the honeycomb plane yield a long-range repulsive interaction. By exact diagonalization in the zeroth-Landau-level basis, we show that this repulsive interaction stabilizes a variety of valley-polarized fractional quantum Hall states such as Laughlin and composite-fermion states. The present system thus offers an intriguing platform for emulating fractional quantum Hall physics in a static optical lattice. We calculate the energy gaps above these incompressible states and discuss the temperature scales required for their experimental realization.
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We analyze a recently proposed method to create fractional quantum Hall (FQH) states of atoms confined in optical lattices [A. S\o{}rensen et al., Phys. Rev. Lett. 94, 086803 (2005)]. Extending the previous work, we investigate conditions under which the FQH effect can be achieved for bosons on a lattice with an effective magnetic field and finite on-site interaction. Furthermore, we characterize the ground state in such systems by calculating Chern numbers which can provide direct signatures of topological order and explore regimes where the characterization in terms of wave-function overlap fails. We also discuss various issues which are relevant for the practical realization of such FQH states with ultracold atoms in an optical lattice, including the presence of a long-range dipole interaction which can improve the energy gap and stabilize the ground state. We also investigate a detection technique based on Bragg spectroscopy to probe these systems in an experimental realization.
With the discovery of graphene [1] and topological insulators [2] there has been much recent interest in systems in which low energy excitations can be described using Dirac fermions. A parallel area of interest has been the explorati on of the possibility of generating artificial magnetic fields f or cold atoms confined in an optical lattice. Neutral bosonic co ld atoms cannot couple to a magnetic field directly, so there hav e been numerous proposals [3‐6] of approaches to couple atoms to an artificial magnetic field, several of which have been implemented experimentally [7, 8]. The problem of the spectrum of quantum particles in a uniform magnetic field on a lattice has the well-known Hofstadte r spectrum [9]. Our modification of the proposal by Sorensen et al. [6] leads to an effective Hamiltonian with a tunable Hofstadter-like spectrum that arises from the combination of hopping and an artificial magnetic field with a non-zero average that are both periodically modulated in the x and the y directions. The presence of spatial periodicity in the amplitude as well as the phase of the hopping is the key difference between the model we consider here and previous work on the spectrum of particles in the presence of magnetic fields that are periodic in both the x and y directions [10]. This difference facilitates the unusual Dirac-like spectrum th at we discuss in this Letter. In our effective model, when there is an average of half a flux quantum per plaquette, and at half-filling, the low energy degrees of freedom can be described by a Dirac Hamiltonian with the unusual property that chiral symmetry is broken in the kinetic energy rather than via mass terms. This has the consequence that the doubly degenerate Dirac cone for massless fermions splits into two cones with tunable distinct slopes, analagous to a situation in which there are two speeds of light for fermionic excitations, similar to birefringen ce of light in crystals such as calcite. We discuss the meaning of broken chiral symmetry in our effective model and explore the effects of various perturbations, such as staggered potent ials, domain walls, and interactions between fermions. The approach to obtain an artifical magnetic field for cold atoms in an optical lattice suggested by Sorensen et al. [6] was presented in the context of the Bose-Hubbard model, but ignores interactions and is not specific to bosons. We consider a model of spinless fermions (corresponding to only one available hyperfine state for cold atoms) with Hamiltonian