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arXiv:1504.07232 · Physical Review A

Ferromagnetic-nematic order and strongly correlated phases of fermions in optical flux lattices

Simon C. Davenport, Nigel R. Cooper

Published Aug 10, 2015• 0 citations
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Abstract

We study a model of a two-dimensional ultracold atomic gas subjected to an ``optical flux lattice'': a laser configuration where Raman-dressed atoms experience a strong artificial magnetic field. This leads to a band structure of narrow energy bands with nonzero Chern numbers. We consider the case of two-level (spin-$1/2$) fermionic atoms in this lattice, interacting via a repulsive $s$-wave contact interaction. Atoms restricted to the lowest band are described by an effective model of spinless fermions with interactions that couple states in a momentum-dependent manner across the Brillouin zone; a consequence of the Raman dressing of the two spin states. We present the results of detailed exact diagonalization studies of the many-body states for a range of filling factors, $\ensuremath{\nu}$. First, we present evidence for the existence of a phase with coupled ferromagnetic-nematic ordering, which was previously suggested by a mean-field analysis. Second, we present evidence indicating the presence of a Laughlin-like fractional quantum Hall state occurring at filling factor $\ensuremath{\nu}=1/3$. Finally, we observe a charge-density wave state at $\ensuremath{\nu}=1/2$, which we are able to cleanly distinguish from the Laughlin-like state by its translational symmetry breaking and relatively low participation ratio.

DOI
10.1103/physreva.92.023608

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We study a model of a two-dimensional ultracold atomic gas subjected to an ``optical flux lattice'': a laser configuration where Raman-dressed atoms experience a strong artificial magnetic field. This leads to a band structure of narrow energy bands with nonzero Chern numbers. We consider the case of two-level (spin-1/21/21/2) fermionic atoms in this lattice, interacting via a repulsive sss-wave contact interaction. Atoms restricted to the lowest band are described by an effective model of spinless fermions with interactions that couple states in a momentum-dependent manner across the Brillouin zone; a consequence of the Raman dressing of the two spin states. We present the results of detailed exact diagonalization studies of the many-body states for a range of filling factors, \ensuremathν\ensuremath{\nu}\ensuremathν. First, we present evidence for the existence of a phase with coupled ferromagnetic-nematic ordering, which was previously suggested by a mean-field analysis. Second, we present evidence indicating the presence of a Laughlin-like fractional quantum Hall state occurring at filling factor \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3. Finally, we observe a charge-density wave state at \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2, which we are able to cleanly distinguish from the Laughlin-like state by its translational symmetry breaking and relatively low participation ratio.
Ultracold atoms in Raman-dressed optical lattices allow for effective momentum-dependent interactions among single-species fermions originating from short-range s-wave interactions. These dressed-state interactions combined with the very flat bands encountered in the recently introduced optical flux lattices push the Stoner instability towards weaker repulsive interactions, making it accessible with current experiments. As a consequence of the coupling between spin and orbital degrees of freedom, the magnetic phase features Ising nematic order.
We describe a simple scheme by which "optical flux lattices" can be implemented in ultracold atomic gases using two-photon dressed states. This scheme can be applied, for example, to the ground-state hyperfine levels of commonly used atomic species. The resulting flux lattices simulate a magnetic field with high mean flux density, and have low-energy bands analogous to the lowest Landau level. We show that in practical cases the atomic motion significantly deviates from the adiabatic following of one dressed state, and that this can lead to significant interactions even for fermions occupying a single band. Our scheme allows experiments on cold atomic gases to explore strong correlation phenomena related to the fractional quantum Hall effect, both for fermions and bosons.
We study ultracold color fermions with three internal states, red, green, and blue, with SU(3)\mathrm{SU}(3)SU(3) symmetry in optical lattices, when color-orbit coupling and color-flip fields are present. This system corresponds to a generalization of two internal state fermions with SU(2)\mathrm{SU}(2)SU(2) symmetry in the presence of spin-orbit coupling and spin-flipping Zeeman fields. We investigate the eigenspectrum and Chern numbers to describe different topological phases that emerge in the phase diagrams of color-orbit coupled fermions in optical lattices. We obtain the phases as a function of artificial magnetic, color-orbit, and color-flip fields that can be independently controlled. For fixed artificial magnetic flux ratio, we identify topological quantum phases and phase transitions in the phase diagrams of chemical potential versus color-flip fields or color-orbit coupling, where the chirality and number of midgap edge states changes. The topologically nontrivial phases are classified in three groups: The first group has total nonzero chirality and exhibits only the quantum charge Hall effect; the second group has total nonzero chirality and exhibits both quantum charge and quantum color Hall effects; and the third group has total zero chirality but exhibits the quantum color Hall effect. These phases are generalizations of the quantum Hall and quantum spin Hall phases for charged spin-1/21/21/2 fermions. Lastly, we also describe the color density of states and a staircase structure in the total and color filling factors versus chemical potential for fixed color-orbit coupling, color-flip field, and magnetic flux ratio. We show the existence of incompressible states at rational filling factors precisely given by a gap labelling theorem that relates the filling factors to the magnetic flux ratio and topological quantum numbers.
We study Hubbard models for ultracold bosonic or fermionic atoms loaded into an optical lattice. The atoms carry a high spin F> 1/2, and interact on site via strong repulsive Van der Waals forces. Making convenient rearrangements of the interaction terms, and exploiting their symmetry properties, we derive low-energy effective models with nearest-neighbor interactions, and their properties. We apply our method to F =3/2, and 5/2 fermions on two-dimensional square lattice at quarter, and 1/6 fillings, respectively, and investigate mean-field equations for repulsive couplings. We find for F =3/2 fermions that the plaquette state appearing in the highly symmetric SU (4) case does not require fine tuning, and is stable in an extended region of the phase diagram. This phase competes with an SU (2) flux state, that is always suppressed for repulsive interactions in the absence of external magnetic field. The SU (2) flux state has, however, lower energy than the plaquette phase, and stabilizes in the presence of weak applied magnetic field. For F =5/2 fermions a similar SU (2) plaquette phase is found to be the ground state without external magnetic field.
An interesting route to the realization of topological Chern bands in ultracold atomic gases is through the use of optical flux lattices. These models differ from the tight-binding real-space lattice models of Chern insulators that are conventionally studied in solid-state contexts. Instead, they involve the coherent coupling of internal atomic (spin) states, and can be viewed as tight-binding models in reciprocal space. By changing the form of the coupling and the number NNN of internal spin states, they give rise to Chern bands with controllable Chern number and with nearly flat energy dispersion. We investigate in detail how interactions between bosons occupying these bands can lead to the emergence of fractional quantum Hall states, such as the Laughlin and Moore-Read states. In order to test the experimental realization of these phases, we study their stability with respect to band dispersion and band mixing. We also probe interesting topological phases that emerge in these systems when the Chern number is greater than 1.