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\pi-flux Dirac bosons and Bogoliubov edge states of Bose-Einstein condensed atoms

Zhi-Fang Xu, Li You, Andreas Hemmerich, W. Vincent Liu

Published Feb 15, 2016• 0 citations
OpenAlex

Abstract

Topological edge states and Dirac fermions are fascinating phenomena known for fermionic systems. Whether similar phenomena can occur for interacting bosons is an open question in fundamental science. The recent observation of $^{87}$Rb chiral $p_x\pm ip_y$ condensates in a checkerboard optical lattice further raises the question whether a bosonic superfluid can be topological. Here we show that Dirac bosons experiencing a $\pi$ Berry flux emerge in the elementary excitation spectra of this experimentally realized Bose system. Tuning a population bias between the observed $p_x+ip_y$ and $p_x-ip_y$ condensate components opens an energy gap in the bulk, which results in Chern invariant and topologically protected Bogoliubov edge modes. Different from previous topological scenarios that employ the concept of engineered spin-orbit coupling or the equivalent in single-particle Bloch bands, spontaneous chiral orbital order driven by many-body interaction is a crucial ingredient in our finding.

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Topological edge states and Dirac fermions are fascinating phenomena known for fermionic systems. Whether similar phenomena can occur for interacting bosons is an open question in fundamental science. The recent observation of 87^{87}87Rb chiral px±ipyp_x\pm ip_ypx​±ipy​ condensates in a checkerboard optical lattice further raises the question whether a bosonic superfluid can be topological. Here we show that Dirac bosons experiencing a π\piπ Berry flux emerge in the elementary excitation spectra of this experimentally realized Bose system. Tuning a population bias between the observed px+ipyp_x+ip_ypx​+ipy​ and px−ipyp_x-ip_ypx​−ipy​ condensate components opens an energy gap in the bulk, which results in Chern invariant and topologically protected Bogoliubov edge modes. Different from previous topological scenarios that employ the concept of engineered spin-orbit coupling or the equivalent in single-particle Bloch bands, spontaneous chiral orbital order driven by many-body interaction is a crucial ingredient in our finding.
We study the topological properties of elementary excitations in a staggered p_{x}±ip_{y} Bose-Einstein condensate realized in recent orbital optical lattice experiments. The condensate wave function may be viewed as a configuration space variant of the famous p_{x}+ip_{y} momentum space order parameter of strontium ruthenate superconductors. We show that its elementary excitation spectrum possesses Dirac bosons with π Berry flux. Remarkably, if we induce a population imbalance between the p_{x}+ip_{y} and p_{x}-ip_{y} condensate components, a gap opens up in the excitation spectrum resulting in a nonzero Chern invariant and topologically protected edge excitation modes. We give a detailed description of how our proposal can be implemented with standard experimental technology.
We study topologically nontrivial excitations of a weakly interacting, spin-orbit coupled Bose-Einstein condensate in a two-dimensional square optical lattice, a system recently realized in experiment [Phys. Rev. Lett. 121, 150401 (2018)]. We focus on situations where the system is not subjected to a Zeeman field and thus does not exhibit nontrivial single-particle band topology. Of special interest then is the role of particle interaction as well as its interplay with the symmetry properties of the system in producing topologically nontrivial excitations. We find that the noninteracting system possesses a rich set of symmetries, including the PT\mathcal{PT}PT symmetry, the modified dihedral point-group symmetry ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{4}$ and the nonsymmorphic symmetry. These combined symmetries ensure the existence of pairs of degenerate Dirac points at the edge of Brillouin zone for the single-particle energy bands. In the presence of particle interaction and with sufficient spin-orbit coupling, the atoms condense in a ground state with net magnetization which spontaneously breaks the PT\mathcal{PT}PT and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{4}$ symmetry. We demonstrate that this symmetry breaking leads to a gap opening at the Dirac point for the Bogoliubov spectrum and consequentially topologically nontrivial excitations. We confirm the nontrivial topology by calculating the Chern numbers of the lowest excitation bands and show that gapless edge states form at the interface of systems characterized by different values of the Chern number.
Abstract Topological superfluidity is an important concept in electronic materials as well as ultracold atomic gases1. However, although progress has been made by hybridizing superconductors with topological substrates, the search for a material—natural or artificial—that intrinsically exhibits topological superfluidity has been ongoing since the discovery of the superfluid 3He-A phase2. Here we report evidence for a globally chiral atomic superfluid, induced by interaction-driven time-reversal symmetry breaking in the second Bloch band of an optical lattice with hexagonal boron nitride geometry. This realizes a long-lived Bose–Einstein condensate of 87Rb atoms beyond present limits to orbitally featureless scenarios in the lowest Bloch band. Time-of-flight and band mapping measurements reveal that the local phases and orbital rotations of atoms are spontaneously ordered into a vortex array, showing evidence of the emergence of global angular momentum across the entire lattice. A phenomenological effective model is used to capture the dynamics of Bogoliubov quasi-particle excitations above the ground state, which are shown to exhibit a topological band structure. The observed bosonic phase is expected to exhibit phenomena that are conceptually distinct from, but related to, the quantum anomalous Hall effect3–7 in electronic condensed matter.
We show that Bose–Einstein condensates in optical lattices with broken time-reversal symmetry can support chiral edge modes originating from nontrivial bulk excitation band topology. To be specific, we analyze a Bose–Hubbard extension of the Haldane model, which can be realized with recently developed techniques of periodically modulating honeycomb optical lattices. The topological properties of Bloch bands known for the noninteracting case are shown to be smoothly carried over to Bogoliubov excitation bands for the interacting case. We show that the parameter ranges that display topological bands enlarge with increasing the Hubbard interaction or the particle density. In the presence of sharp boundaries, chiral edge modes appear in the gap between topological excitation bands. We demonstrate that by coherently transferring a portion of a condensate into an edge mode, a density wave is formed along the edge owing to an interference with the background condensate. This offers a unique method of detecting an edge mode through a macroscopic quantum phenomenon.
Recent advanced experimental implementations of optical lattices with highly tunable geometry open up new regimes for exploring quantum many-body states of matter that had not been accessible previously. Here we report that a topological fermionic superfluid with higher Chern number emerges spontaneously from sss-wave spin-singlet pairing in an orbital optical lattice when its geometry is tuned to explicitly break reflection symmetry. Qualitatively distinct from the conventional scheme that relies on higher partial-wave pairing, the crucial ingredient of our model is topology originating from mixing higher Wannier orbitals. It leads to unexpected changes in the topological band structure at the single-particle level, i.e., the bands are transformed from possessing two flux-$\ensuremath{\pi}$ Dirac points into a single quadratic touching point with flux $2\ensuremath{\pi}$. Based on such engineered single-particle bands, spin-singlet pairing of ultracold fermions arising from standard sss-wave attractive interaction is found to induce higher Chern number (Chern number of 2) and topologically protected chiral edge modes, all occurring at a higher critical temperatures in relative scales, potentially circumventing one of the major obstacle for its realization in ultracold gases.
Topological states of matter are peculiar quantum phases showing different edge and bulk transport properties connected by the bulk-boundary correspondence. While noninteracting fermionic topological insulators are well established by now and have been classified according to a tenfold scheme, the possible realization of topological states for bosons has not been explored much yet. Furthermore, the role of interactions is far from being understood. Here, we show that a topological state of matter exclusively driven by interactions may occur in the p band of a Lieb optical lattice filled with ultracold bosons. The single-particle spectrum of the system displays a remarkable parabolic band-touching point, with both bands exhibiting non-negative curvature. Although the system is neither topological at the single-particle level nor for the interacting ground state, on-site interactions induce an anomalous Hall effect for the excitations, carrying a nonzero Chern number. Our work introduces an experimentally realistic strategy for the formation of interaction-driven topological states of bosons.