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arXiv:2209.14811 · Physical review. B./Physical review. B

Tunable symmetry-protected higher-order topological states with fermionic atoms in bilayer optical lattices

Zhoutao Lei, Linhu Li, Yuangang Deng

Published Mar 29, 2023• 3 citations
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Abstract

Higher-order topological states that possess gapped bulk energy bands and exotic topologically protected boundary states with at least two dimensions lower than the bulk have opened a significant and new perspective for understanding topological quantum matter. Here, we propose to generate two-dimensional (2D) topological boundary states for implementing a synthetic magnetic flux of ultracold atoms trapped in bilayer optical lattices. It is shown that a Chern insulator, Dirac semimetals, and a second-order topological phase (SOTP) are generated by the interplay of the two-photon detuning and effective Zeeman shift. These observed topological phases can be well characterized by the energy gap of the bulk, the Wilson loop spectra, and the spin textures at the high-symmetry points of the system. We show that the SOTP exhibits a pair of $0\mathrm{D}$ boundary states. Meanwhile, the phases of the Dirac semimetals and Chern insulator support the conventional $1\mathrm{D}$ boundary states due to the principle of bulk-boundary correspondence. Strikingly, the boundary states that emerge for Dirac semimetals and the SOTP are topologically protected by $\mathcal{P}T$ symmetry and chiral-mirror symmetry (${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{M}}}_{\ensuremath{\alpha}}$), respectively. In particular, the location of $0\mathrm{D}$ corner states for the SOTP which are associated with ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{M}}}_{\ensuremath{\alpha}}$ symmetry are highly manipulable by tuning the magnetic flux. Our scheme herein provides a platform for emerging exotic topological boundary states, which may facilitate the study of higher-order topological phases in ultracold atomic gases.

DOI
10.1103/physrevb.107.115166

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We propose a realizable experiment scheme to construct a one-dimensional synthetic magnetic flux lattice with spin-tensor-momentum coupled spin-1 atoms and explore its exotic topological states. Different from the Altland-Zirnbauer classification, we show that our system hosts a symmetry-protected phase protected by a magnetic group symmetry (M\mathcal{M}M) and characterized by a Z2{Z}_{2}Z2​ topological invariant. In single-particle spectra, we show that the topological nontrivial phase supports two kinds of edge states, which include two (four) zero-energy edge modes in the absence (presence) of two-photon detuning. We further study the bulk-edge correspondence in a non-Hermitian model by taking into account the particle dissipation. It is shown that the non-Hermitian system preserves the bulk-edge correspondence under the M\mathcal{M}M symmetry but exhibits the non-Hermitian skin effect with breaking the M\mathcal{M}M symmetry at nonzero magnetic flux. This work provides insights in understanding the exotic topological quantum states of high-spin systems and facilitating their experimental explorations.
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.
Higher-order topological superfluids have gapped bulk and symmetry-protected Majorana zero modes with various localizations. Motivated by recent advances, we present a proposal for synthesizing multiorder topological superfluids that support various Majorana zero modes in ultracold atomic gases. For this purpose, we use the two-dimensional optical superlattice that introduces a spatial modulation to the spin-orbit coupling in one direction, providing an extra degree of freedom for the emergent higher-order topological state. We find the topologically trivial superfluids, first-order and second-order topological superfluids, as well as different topological phase transitions among them with respect to the experimentally tunable parameters. Besides the conventional transition characterized by the Chern number associated with the bulk gap closing and reopening, we find the system can support the topological superfluids with Majorana corner modes, but the topological phase transition undergoes no gap-closing of bulk bands. Instead, the transition is refined by the quadrupole moment and signaled out by the gap-closing of edge states. The proposal is based on the sss-wave interaction and is valid using existing experimental techniques, which unifies multiorder topological phase transitions in a simple but realistic system.
We propose a simple experimental scheme to realize simultaneously the one-dimensional spin-orbit coupling and the staggered spin flip in ultracold pseudospin-1/21/21/2 atomic Fermi gases trapped in optical lattices. In the absence of interspecies interactions, the system supports type-I and II Weyl semimetals in three-dimensional and gapped Chern insulators and gapless topological semimetal states in two-dimensional lattices. By turning on the sss-wave interactions, a rich variety of gapped and gapless inhomogeneous topological superfluids can emerge. In particular, a gapped topological Fulde-Ferrell superfluid, which supports the chiral edge states at opposite boundaries with the same chirality, is predicted.
The experimental realizations of spin-orbit coupling (SOC) in cold atom gases put topological superfluids (TSs) under the spotlight. The topological protection of suggested superfluid models comes from the local antiunitary symmetries of ten Altland-Zirnbauer symmetry classes. In this paper we theoretically study a two-dimensional BDI model describing spin-orbit-coupled bilayer fermionic gases with sss-wave interaction to realize mirror-symmetry- (nonlocal and unitary) protected gapped TS, beyond the paradigms suggested before. This gapped phase not only shows linearly dispersive Majorana zero modes on mirror invariant one-dimensional boundaries, but also shows Majorana corner states to realize the second-order TS (TS2) with sss-wave pairing, which is explicitly different from the nonconventional pairings in the suggested systems to realize TS2. Generally, engineering the single-particle band structures should be easier than directly engineering the complicated pairings. Considering the tremendous progress in manipulating cold atom systems, our work provides a possible route (by engineering the single-particle physics, such as SOC) to realize mirror-symmetry-protected gapped TS and TS2 in cold fermionic systems in the future.
Twisted bilayer materials provide a versatile platform for realizing correlated topological states. Motivated by the recent experimental realization of atomic Bose-Einstein condensates in twisted bilayer optical lattices, we investigate the emergence of fractional topological phases in such systems. For experimentally realistic parameters, the single-particle spectrum hosts a pair of quasi-degenerate nearly flat moiré bands. We show using self-consistent Hartree-Fock calculations that atomic interactions spontaneously break time-reversal symmetry, lift the quasi-degeneracy, and generate an isolated nearly flat band with a nonzero Chern number. At fractional filling, exact diagonalization of the Hamiltonian projected to this band reveals a threefold degenerate ground-state manifold and a characteristic particle entanglement spectrum, providing strong evidence for a fractional Chern insulator. The resulting many-body gap is of order 10−3ER10^{-3}E_R10−3ER​, corresponding to a nanokelvin energy scale accessible to state-of-the-art cold-atom experiments. Our scheme requires neither spin-orbit coupling, as in transition metal dichalcogenides, nor a finite magnetic field, as in existing twisted-bilayer-graphene experiments, providing a highly tunable route to strongly correlated topological phases in twisted bilayer optical lattices.