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Physical Review B

Topological superfluid state of fermions on a p -band optical square lattice

Ya-Jie Wu, Jing He, Chun-Li Zang, Su-Peng Kou

Published Aug 20, 2012• 11 citations
Publisher

Abstract

In this paper we study an interacting mixture of ultracold spinless fermions on the $s$ band and bosons on the $p$ band in a 2D square optical lattice, of which the effective model is reduced to a $p$-band fermionic system with nearest-neighbor attractive interaction. From this effective $p$-band model, we find a translation symmetry protected ${\mathbb{Z}}_{2}$ topological superfluid that is characterized by a special fermion parity pattern at high-symmetry points in momentum space $\mathbf{k}=(0,0)$, $(0,\ensuremath{\pi})$, $(\ensuremath{\pi},0)$, $(\ensuremath{\pi},\ensuremath{\pi})$. Such ${\mathbb{Z}}_{2}$ topological superfluid supports the robust Majorana edge modes and a new type of low-energy excitation---(supersymmetric) ${\mathbb{Z}}_{2}$ link excitation.

DOI
10.1103/physrevb.86.085128

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In this paper we study an interacting mixture of ultracold spinless fermions on the sss band and bosons on the ppp band in a 2D square optical lattice, of which the effective model is reduced to a ppp-band fermionic system with nearest-neighbor attractive interaction. From this effective ppp-band model, we find a translation symmetry protected Z2{\mathbb{Z}}_{2}Z2​ topological superfluid that is characterized by a special fermion parity pattern at high-symmetry points in momentum space k=(0,0)\mathbf{k}=(0,0)k=(0,0), (0,\ensuremathπ)(0,\ensuremath{\pi})(0,\ensuremathπ), (\ensuremathπ,0)(\ensuremath{\pi},0)(\ensuremathπ,0), (\ensuremathπ,\ensuremathπ)(\ensuremath{\pi},\ensuremath{\pi})(\ensuremathπ,\ensuremathπ). Such Z2{\mathbb{Z}}_{2}Z2​ topological superfluid supports the robust Majorana edge modes and a new type of low-energy excitation---(supersymmetric) Z2{\mathbb{Z}}_{2}Z2​ link excitation.
We propose a physical scheme for the realization of two-dimensional topological odd-parity superfluidity in a spin-independent bond-centered square optical lattice based upon interband fermion pairing. The D4{D}_{4}D4​ point-group symmetry of the lattice protects a quadratic band crossing, which allows one to prepare a Fermi surface of spin-up fermions with odd parity close to the degeneracy point. In the presence of spin-down fermions with even parity populating a different energetically well-separated band, odd-parity pairing is favored. Strikingly, as a necessary prerequisite for pairing, both Fermi surfaces can be tuned to match well. As a result, topological superfluid phases emerge in the presence of merely sss-wave interaction. Due to the Z2{Z}_{2}Z2​ symmetry of these odd-parity superfluids, we infer their topological features simply from the symmetry and the Fermi-surface topology as confirmed numerically.
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.
Weyl points exist in a fascinating topological state of matter with linear band crossings analogous to magnetic monopoles. Tremendous efforts have been devoted to investigating fermionic topological materials with Weyl points in the single-particle band dispersion. It remains elusive for realizing interaction-induced Weyl points, especially for bosons. Motivated by recent experimental progress in ultracold atoms, we propose a scheme to create Weyl points for Bogoliubov excitations of a bosonic superfluid in a three-dimensional cubic optical lattice. The unique design of the lattice leads to interaction-induced time-reversal symmetry breaking for a ppp-orbital superfluid, which in turn induces Weyl Bogoliubov excitations. Analogous to Weyl semimetals of electronic systems, the superfluid also supports topologically protected edge modes due to the bulk-boundary correspondence.
ppp-wave interaction in cold atoms may give rise to exotic topological superfluids. However, the realization of ppp-wave interaction in a cold atom system is experimentally challenging. Here we propose a simple scheme to synthesize effective ppp-wave interaction in conventional sss-wave interacting quantum gases. The key idea is to load atoms into a spin-dependent optical lattice potential. Using two concrete examples involving spin-1/2 fermions, we show how the original system can be mapped into a model describing spinless fermions with nearest-neighbor ppp-wave interaction, whose ground state can be a topological superfluid that supports Majorana fermions under proper conditions. Our proposal has the advantage that it does not require spin-orbit coupling or loading atoms onto higher orbitals, which is the key in earlier proposals to synthesize effective ppp-wave interaction in sss-wave quantum gases, and may provide a completely new route for realizing ppp-wave topological superfluids.
We propose to realize a ppp-wave superfluid using bosons mixed with a single species of fermions in a deep optical lattice. We analyze with a self-consistent method its excitation spectrum in presence of a vortex, and we point out the range of interaction strengths in which the zero-energy mode with topological character exists on a finite optical lattice. Lattice effects are strongest close to fermionic half filling: here the linearity of the low-lying spectrum is lost, and a new class of extended zero-energy modes with checkerboard structure and ddd-wave symmetry appears.
The recent experimental realization of spin-orbit (SO) coupling for ultracold bosons and fermions opens an exciting avenue for engineering quantum matter that may be challenging to realize in solid-state materials such as SO-coupled pseudospin-1 fermions. While one-dimensional SO coupling for spin-1 bosons has been experimentally realized, the generation of two-dimensional (2D) SO coupling and its topological properties are largely unexplored. Here we propose an experimental scheme for realizing a 2D Rashba-type SO coupling in a square lattice for pseudospin-1 Fermi gases. Because of the extended spin degree of freedom, many interesting topological phases could exist without relying on lattice point group symmetries that are crucial in solid-state materials. These exotic phases include triply degenerate points, quadratic band touching, a large Chern number (C=5C=5C=5) superfluid with five Majorana modes, triple-Weyl fermions, and so on. Our scheme can be generalized to larger spins and provides a distinctive route for engineering topological quantum matter by utilizing large spin degrees of freedom, instead of specific lattice symmetries.