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
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 introduce a lattice model for fermions in a spatially periodic magnetic field that also has spatially periodic hopping amplitudes. We discuss how this model might be realized with cold atoms in an artificial magnetic field on a square optical lattice. When there is an average flux of half a flux quantum per plaquette, the spectrum of low-energy excitations can be described by massless Dirac fermions in which the usually doubly degenerate Dirac cones split into cones with different ``speeds of light.'' These gapless birefringent Dirac fermions arise because of broken chiral symmetry in the kinetic energy term of the effective low-energy Hamiltonian. We characterize the effects of various perturbations to the low-energy spectrum, including staggered potentials, interactions, and domain-wall topological defects.
A time-dependent optical lattice with staggered particle current in the tight-binding regime was considered that can be described by a time-independent effective lattice model with an artificial staggered magnetic field. The low-energy description of a single-component fermion in this lattice at half-filling is provided by two copies of ideal two-dimensional massless Dirac fermions. The Dirac cones are generally anisotropic and can be tuned by the external staggered flux $\ensuremath{\phi}$. For bosons, the staggered flux modifies the single-particle spectrum such that in the weak coupling limit, depending on the flux $\ensuremath{\phi}$, distinct superfluid phases are realized. Their properties are discussed, the nature of the phase transitions between them is established, and Bogoliubov theory is used to determine their excitation spectra. Then the generalized superfluid--Mott-insulator transition is studied in the presence of the staggered flux and the complete phase diagram is established. Finally, the momentum distribution of the distinct superfluid phases is obtained, which provides a clear experimental signature of each phase in ballistic expansion experiments.
Viewed as an ideal quantum simulator, the optical lattice system with trapped ultracold atoms has been developed into a powerful tool to investigate unusual physical properties due to its considerable tunability. In this paper, we propose to construct a feasible two-dimensional optical lattice which can be continuously transformed from a square lattice to a Lieb lattice by tuning laser intensity. In these transformations, accompanied with the crossover between nonrelativistic particles and relativistic Dirac fermions, the topological and transport properties are varied under artificial gauge potentials. It is found that the fractal energy spectrum in an effective magnetic field splits and reassembles from a traditional Hofstadter butterfly to multiple ones with flat band, characterized by different Chern numbers. In a specific case of half a pseudomagnetic flux quantum, the spectrum has a transition from individual Dirac cones to double-layer Dirac cones, then to the Dirac cones with a flat band, the pseudospin of particles changes from 1/2 to 3/2, then to 1. In the structural evolution, a remarkable birefringent tunneling across a potential barrier is demonstrated. The adjustable lattice structure replenished with synthetic gauge fields for cold atoms offers more opportunities to observe intriguing physical properties.
We show that the dynamics of cold bosonic atoms in a two-dimensional square optical lattice produced by a bichromatic light-shift potential is described by a Bose-Hubbard model with an additional effective staggered magnetic field. In addition to the known uniform superfluid and Mott insulating phases, the zero-temperature phase diagram exhibits a novel kind of finite-momentum superfluid phase, characterized by a quantized staggered rotational flux. An extension for fermionic atoms leads to an anisotropic Dirac spectrum, which is relevant to graphene and high-T(c) superconductors.
In this thesis, we discuss quantum phase transitions in low-dimensional optical lattices, namely one- and two-dimensional lattices. The dimensional confinement is realized in experiments by suppressing the hopping in the extra dimensions through a deep potential barrier that prevents the atoms to tunnel across dimensions. The dimensionality considerably affects the physical behavior of the system. For instance, in one dimension, the celebrated Fermi-liquid theory does not describe the many-body behavior of a fermionic system. There are no single particle excitations, but instead collective behavior, and spin and charge degrees of freedom are decoupled . In the first part of this thesis, we consider one-dimensional fermionic gases under the effect of time-dependent driving forces. First, we describe a proposal to realize non-standard Hubbard models that display correlated-hopping terms by modulating interactions through a time-dependent magnetic field. The extra terms describe hopping processes that depend on the particle-density difference in neighboring sites. The effective model is obtained in arbitrary spatial dimensions, but only in one-dimension it is known to be exactly solvable for specific values of the parameters. Moreover, an SU(2) symmetry in the charge sector allows to construct non-trivial superconducting eigenstates (eta-states), discovered by Yang as excited states of the Hubbard model. Then, we study a one-dimensional bipartite optical lattice with fermionic atoms at half-filling. The lattice potential barriers have heights alternating in magnitude. As a consequence, the hopping amplitudes also alternate in magnitude and the elementary unit cell has two sites. This system is known to display Peierls dimerization at half-filling: the fermions form dimers in neighboring sites that inhibit transport and open a gap in the spectrum. After applying shaking on this system we discover that one can dynamically induce a phase transition to an unconventional metallic state with four Fermi-points. The second part of this thesis describes bosonic systems in two-dimensional optical lattices. First, we consider a square optical lattice with alternating deep-shallow wells arranged in a checkerboard pattern. This lattice is realized in the experimental laboratory of Prof. Hemmerich in Hamburg. In the experiment, the visibility is extracted from the time-of-flight imaging to quantify the phase coherence of the bosonic gas. We develop a mean-field theory and determine the phase diagram of this model. We interpret the experimental data by assuming that at large detuning, only the deeper wells are populated and form an imbalanced Mott insulator with empty shallow wells and integer density on the deeper wells. We support this hypothesis by large-scale Gutzwiller calculations. Then, we investigate a system of bosons in the Lieb lattice, which is the lattice geometry of the Cu-O plane in cuprates . We consider configurations of the optical potential where s and p orbitals sitting on neighboring wells become resonant and hybridize. The interest in this problem comes from the possibility to realize time-reversal broken phases, usually known as Varma phases for electronic systems, which have been proposed to explain the pseudo-gap regime of cuprate superconductors. We show the conditions under which the symmetry breaking in a four-band model can lead to states with loop currents. Moreover, the lowest branch of the excitation spectrum has a non-vanishing Chern number and therefore supports topological excitations.
Motivated by the recent progress in engineering artificial non-Abelian gauge fields for ultracold fermions in optical lattices, we investigate the time-reversal-invariant Hofstadter–Hubbard model. We include an additional staggered lattice potential and an artificial Rashba-type spin–orbit coupling term available in experiment. Without interactions, the system can be either a (semi)-metal, a normal or a topological insulator, and we present the non-Abelian generalization of the Hofstadter butterfly. Using a combination of real-space dynamical mean-field theory (RDMFT), analytical arguments, and Monte-Carlo simulations we study the effect of strong on-site interactions. We determine the interacting phase diagram, and discuss a scenario of an interaction-induced transition from a normal to a topological insulator. At half-filling and large interactions, the system is described by a quantum spin Hamiltonian, which exhibits exotic magnetic order due to the interplay of Rashba-type spin–orbit coupling and the artificial time-reversal-invariant magnetic field term. We determine the magnetic phase diagram: both for the itinerant model using RDMFT and for the corresponding spin model in the classical limit using Monte-Carlo simulations.