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arXiv:0805.4484 · Physical Review A

p band in a rotating optical lattice

R. O. Umucalılar, Mehmet Özgür Oktel

Published Sep 4, 2008• 10 citations
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Abstract

We investigate the effects of rotation on the excited bands of a tight-binding lattice, focusing particularly on the first excited $(p)$ band. Both the on-site energies and the hopping between lattice sites are modified by the effective magnetic field created by rotation, causing a nontrivial splitting and magnetic fine structure of the $p$ band. We show that Peierls substitution can be modified to describe $p$ band under rotation, and use this method to derive an effective Hamiltonian. We compare the spectrum of the effective Hamiltonian with a first-principles calculation of the magnetic band structure and find excellent agreement, confirming the validity of our approach. We also discuss the on-site interaction terms for bosons and argue that many-particle phenomena in a rotating $p$ band can be investigated starting from this effective Hamiltonian.

DOI
10.1103/physreva.78.033602

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We investigate the effects of rotation on the excited bands of a tight-binding lattice, focusing particularly on the first excited (p)(p)(p) band. Both the on-site energies and the hopping between lattice sites are modified by the effective magnetic field created by rotation, causing a nontrivial splitting and magnetic fine structure of the ppp band. We show that Peierls substitution can be modified to describe ppp band under rotation, and use this method to derive an effective Hamiltonian. We compare the spectrum of the effective Hamiltonian with a first-principles calculation of the magnetic band structure and find excellent agreement, confirming the validity of our approach. We also discuss the on-site interaction terms for bosons and argue that many-particle phenomena in a rotating ppp band can be investigated starting from this effective Hamiltonian.
We study the behavior of interacting ultracold bosons in optical lattices in synthetic magnetic fields with wide range of in-cell fluxes $\ensuremath{\alpha}=p/q$. The problem is similar to the one of an electron moving in a tight-binding scheme in the magnetic field and becomes difficult to tackle for a growing number of magnetic subbands, qqq. To overcome this, we focus on the interplay of the width, shape, and number of the subbands on the formation of the coherent state of cold bosons. Using the quantum rotor approach, which goes beyond the mean-field approximation, we are able to pinpoint the elements of the band structure, which are the most significant in a proper theoretical description of the synthetic magnetic field in a bosonic lattice system. As a result, we propose a method of reconstruction of the Hofstadter butterfly spectrum by replacing the magnetic subbands with renormalized bands of a square lattice. This allows us to effectively investigate the properties of the studied system for a wide range of magnetic fluxes and their impact on the Mott-insulator--superfluid transition.
We study interacting bosons in a two-dimensional square bipartite optical lattice. By focusing on the regime where the first three excited bands are nearly degenerate (i.e., the first-excited ppp bands in one sublattice are nearly degenerate with the sss band of the other sublattice), we derive a multi-orbital tight-binding model which captures the most relevant features of the band structure. In addition, we also derive a corresponding generalized Bose-Hubbard model and solve it numerically under different situations, both with and without a confining trap. It is especially found that the hybridization between sublattices can strongly influence the phase diagrams and, in a trap, enable even appearances of condensed phases intersecting the same Mott insulating plateaus.
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
We develop a theory of weakly interacting fermionic atoms in shaken optical lattices based on the orbital mixing in the presence of time-periodic modulations. Specifically, we focus on fermionic atoms in a circularly shaken square lattice with near-resonance frequencies, i.e., tuned close to the energy separation between the sss band and the ppp bands. First, we derive a time-independent four-band effective Hamiltonian in the noninteracting limit. Diagonalization of the effective Hamiltonian yields a quasienergy spectrum consistent with the full numerical Floquet solution that includes all higher bands. In particular, we find that the hybridized sss band develops multiple minima and therefore nontrivial Fermi surfaces at different fillings. We then obtain the effective interactions for atoms in the hybridized sss band analytically and show that they acquire momentum dependence on the Fermi surface even though the bare interaction is contactlike. We apply the theory to find the phase diagram of fermions with weak attractive interactions and demonstrate that the pairing symmetry is s+ds+ds+d wave. Our theory is valid for a range of shaking frequencies near resonance, and it can be generalized to other phases of interacting fermions in shaken lattices.
We study cold atoms in an optical lattice with synthetic spin-orbit coupling in the Mott-insulator regime. We calculate the parameters of the corresponding tight-binding model using Peierls substitution and "localized Wannier states method" and derive the low-energy spin Hamiltonian for bosons and fermions. The spin Hamiltonian is a combination of Heisenberg model, quantum compass model and Dzyaloshinskii-Moriya interactions and it has a rich classical phase diagram with collinear, spiral and vortex phases. We discuss the state of the art of experiments to realize and detect magnetic orderings in strongly correlated optical lattices.
We consider the Bose-Hubbard model in a two-dimensional rotating optical lattice and investigate the consequences of the effective magnetic field created by rotation. Using a Gutzwiller-type variational wave function, we find an analytical expression for the Mott insulator (MI)--superfluid (SF) transition boundary in terms of the maximum eigenvalue of the Hofstadter butterfly. The dependence of phase boundary on the effective magnetic field is complex, reflecting the self-similar properties of the single particle energy spectrum. Finally, we argue that fractional quantum Hall phases exist close to the MI-SF transition boundaries, including MI states with particle densities greater than one.