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arXiv:1602.04555 · Physical Review Letters

π -Flux Dirac Bosons and Topological Edge Excitations in a Bosonic Chiral p -Wave Superfluid

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

Published Aug 15, 2016• 36 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.117.085301

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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.
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 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.
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.
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.
Spin and orbital are two basic degrees of freedom that play significant roles in exploring exotic quantum phases in optical lattices with synthetic spin-orbit coupling (SOC) and high orbital bands, respectively. Here, we combine these two crucial ingredients for the first time by proposing a completely new orbital optical Raman lattice scheme to explore exotic high-orbital Bose condensates with Raman-induced SOC in a square lattice. We find that both the SOC and p-orbital interactions influence the condensed state of bosons. Their interplay results in two novel high-orbital many-body quantum phases: the uniform angular momentum superfluid phase, which exhibits a global topological chiral orbital current characterized by a uniform Chern number, and the two-dimensional topological spin-orbital supersolid phase, which is characterized by the spin and orbital angular momentum density wave patterns and topological excitations with opposite Chern numbers, respectively protecting the chiral and antichiral edge modes in the neighboring supersolid clusters. Our scheme may open a new avenue for exploring exotic SOC and high-orbital physics in optical lattices, and is expected to advance the experimental realization of novel supersolids in higher dimensions.
We discuss topological phase transitions in ultracold Fermi superfluids induced by interactions and artificial spin-orbit fields. We construct the phase diagram for population-imbalanced systems at zero and finite temperatures, and analyze spectroscopic and thermodynamic properties to characterize various phase transitions. For balanced systems, the evolution from Bardeen-Cooper-Schrieffer to Bose-Einstein-condensate superfluids in the presence of spin-orbit effects is only a crossover as the system remains fully gapped, even though a triplet component of the order parameter emerges in the helicity basis. However, for imbalanced populations, spin-orbit fields induce a triplet component for the order parameter in the helicity basis, and may produce nodes in the quasiparticle excitation spectrum leading to bulk topological phase transitions of the Lifshitz type. Additionally a fully gapped phase exists, where a crossover from indirect to direct gap occurs, but a topological transition to a gapped phase possessing Majorana fermion edge states does not occur.