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Physical review. B./Physical review. B

Varied topological properties in a continuously modulated optical square lattice

Yafang Xu, Guojun Jin

Published Aug 31, 2018• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.98.075440

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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.
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
Abstract Topological many-body phases of matter exhibit remarkable electronic properties and ultracold atoms in optical lattices constitute promising candidates to study them in a well-controlled environment. In two-dimensional (2D) electron gases topological phases may emerge in the presence of strong magnetic fields. This situation is not directly applicable to cold atoms because they are charge neutral. Therefore, novel experimental techniques have been developed to engineer novel lattice systems, whose Hamiltonian is formally equivalent to the one of charged particles in magnetic fields. In this Tutorial, we introduce a paradigmatic topological lattice model, the Hofstadter model, and explain how it can be implemented with ultracold atoms using laser-assisted tunneling. The technique is based on imprinting phases on the tunneling matrix elements using additional laser beams. These phases are reminiscent of Aharonov–Bohm phases and can be interpreted as a magnetic flux piercing the lattice unit cell. We present experimental results on the cyclotron-like dynamics of neutral atoms in isolated four-site square plaquettes and discuss the first measurement of a 2D topological invariant, the Chern number, in artificially generated Hofstadter bands. The work presented in this Tutorial was one of the four shortlisted finalists of the 2016 DPG SAMOP dissertation prize.
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.
We investigate the dynamics of neutral atoms in a 2D optical lattice which traps two distinct internal states of the atoms in different columns. Two Raman lasers are used to coherently transfer atoms from one internal state to the other, thereby causing hopping between the different columns. By adjusting the laser parameters appropriately we can induce a non-vanishing phase of particles moving along a closed path on the lattice. This phase is proportional to the enclosed area and we thus simulate a magnetic flux through the lattice. This set-up is described by a Hamiltonian identical to the one for electrons on a lattice subject to a magnetic field and thus allows us to study this equivalent situation under very well defined controllable conditions. We consider the limiting case of huge magnetic fields—which is not experimentally accessible for electrons in metals—where a fractal band structure, the Hofstadter butterfly, characterizes the system.
We present a general framework for engineering two-dimensional subwavelength topological optical lattices using spatially dependent atomic dark states in a Λ -type configuration of the atom-light coupling. By properly designing the spatial profiles of the laser fields inducing coupling between the atomic internal states, we show how to generate subwavelength Kronig-Penney-like geometric scalar potential accompanied by narrow and strong patches of the synthetic magnetic field localized in the same areas as the scalar potential. These sharply peaked magnetic fluxes are compensated by a smooth background magnetic field of opposite sign, resulting in zero net flux per unit cell while still enabling topologically nontrivial band structures. Specifically, for sufficiently narrow peaks, their influence is minimum, and the behavior of the system in a remaining smooth background magnetic field resembles the Landau problem, allowing for the formation of nearly flat energy bands with unit Chern numbers. Numerical analysis confirms the existence of ideal Chern bands and the robustness of the topological phases against nonadiabatic effects and losses. This makes the scheme well suited for simulating quantum Hall systems and fractional Chern insulators in ultracold atomic gases, offering a platform for exploring strongly correlated topological phases with high tunability.
We investigate the properties of the Lieb lattice, that is, a face-centered square lattice, subjected to external gauge fields. We show that an Abelian gauge field leads to a peculiar quantum Hall effect, which is a consequence of the single Dirac cone and the flat band characterizing the energy spectrum. Then we explore the effects of an intrinsic spin-orbit term---a non-Abelian gauge field---and demonstrate the occurrence of the quantum spin Hall effect in this model. Besides, we obtain the relativistic Hamiltonian describing the Lieb lattice at low energy and derive the Landau levels in the presence of external Abelian and non-Abelian gauge fields. Finally, we describe concrete schemes for realizing these gauge fields with cold fermionic atoms trapped in an optical Lieb lattice. In particular, we provide a very efficient method to reproduce the intrinsic (Kane-Mele) spin-orbit term with assisted-tunneling schemes. Consequently, our model could be implemented in order to produce a variety of topological states with cold atoms.