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Weyl Points and Topological Nodal Superfluids in a Face‑Centered‑Cubic Optical Lattice

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Weyl Points and Topological Nodal Superfluids in a Face‑Centered‑Cubic Optical Lattice This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg) Nanyang Technological University, Singapore. Weyl points and topological nodal superfluids in a face‑centered‑cubic optical lattice Lang, Li‑Jun; Zhang, Shao‑Liang; Law, K. T.; Zhou, Qi 2017 Lang, L.‑J., Zhang, S.‑L., Law, K. T., & Zhou, Q. (2017). Weyl points and topological nodal superfluids in a face‑centered‑cubic optical lattice. Physical Review B, 96(3), 035145‑. doi:10.1103/PhysRevB.96.035145 https://hdl.handle.net/10356/81397 https://doi.org/10.1103/PhysRevB.96.035145 © 2017 American Physical Society (APS). This paper was published in Physical Review B and is made available as an electronic reprint (preprint) with permission of American Physical Society (APS). The published version is available at: [http://dx.doi.org/10.1103/PhysRevB.96.035145]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law. Downloaded on 26 Sep 2021 15:43:45 SGT PHYSICAL REVIEW B 96, 035145 (2017) Weyl points and topological nodal superfluids in a face-centered-cubic optical lattice Li-Jun Lang,1,2 Shao-Liang Zhang,1,3 K. T. Law,4 and Qi Zhou1,5,* 1Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China 2Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore 3School of Physics, Huazhong University of Science and Technology, Wuhan, China 4Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China 5Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47906, USA (Received 29 December 2016; published 25 July 2017) We point out that a face-centered-cubic (fcc) optical lattice, which can be realized by a simple scheme using three lasers, provides one a highly controllable platform for creating Weyl points and topological nodal superfluids in ultracold atoms. In noninteracting systems, Weyl points automatically arise in the Floquet band structure when shaking such fcc lattices, and sophisticated design of the tunneling is not required. More interestingly, in the presence of attractive interaction between two hyperfine spin states, which experience the same shaken fcc lattice, a three-dimensional topological nodal superfluid emerges, and Weyl points show up as the gapless points in the quasiparticle spectrum. One could either create a double Weyl point of charge 2, or split it into two Weyl points of charge 1, which can be moved in the momentum space by tuning the interactions. Correspondingly, the Fermi arcs at the surface may be linked with each other or separated as individual ones. DOI: 10.1103/PhysRevB.96.035145 Fascinating progresses have been made in studying topolog- manipulate Weyl points in both noninteracting and interacting ical matters of ultracold atoms in the past few years. A number systems. An intrinsic property of a fcc lattice is that, the of topological models and topological phenomena difficult to lowest two bands, which are labeled as A and B, respectively, access in solid materials have been realized [1–5]. For instance, have “inverted” band structures, i.e., tunnelings with opposite the Harper-Hofstadter model [6] and the topological Haldane signs. This comes from a simple fact that the Brillouin zone model [7] have been delivered in optical lattices [1–4]. In (BZ) of a fcc lattice is the one folded from a simple cubic the continuum, a two-dimensional (2D) synthetic spin-orbit (sc) lattice, as shown in Figs. 1(a) and 1(b). As a result, coupling has created a single stable Dirac point that can be Weyl points naturally arise in the Floquet band structure, if moved anywhere in the momentum space [5]. So far, most of one simply uses a periodic shaking to overcome the band these studies have been focusing on one or two dimensions. A gap and couple the A and B bands. This is distinct from large class of three-dimensional (3D) topological phenomena the majority of previous proposals [21–25], which require remain unexplored at the moment. sophisticated designs to engineer tunnelings along all three A Weyl point is a characteristic 3D topological band directions. Moreover, uploading two hyperfine spin states onto structure [8], which provides an analog of Weyl fermions, such optical lattice, the highly tunable attractive interaction a building block in quantum field theory. It also serves as between fermionic atoms allows one to create a 3D nodal an ideal platform to explore a wide range of topological superfluid [26,27], which is composed of layered 2D chiral phenomena in gapless quantum systems, such as Fermi superfluids in the momentum space. Strikingly, Weyl points arc [9] and chiral anomalies [10]. Weyl points and Weyl show up in quasiparticle spectrum of such superfluid. One semimetals have recently been discovered in certain solid could either glue two Weyl points with the same chirality to materials [11–13]. Whereas this development represents a a monopole of charge 2, or further split such multiple-charge major advancement in the current frontier of condensed monopoles into multiple charge-1 ones, and move them around matter physics, challenges remain on manipulating Weyl in the BZ by tuning interactions. Correspondingly, two Fermi points since microscopic parameters are essentially fixed in a arcs emerge at the surfaces, and can be either linked with or given solid material. Furthermore, a fundamentally important separated from each other, depending on locations of the Weyl question regarding the interplay between Weyl semimetal and points in the bulk spectrum. interaction remains unsolved. Although theoretical studies have predicted a variety of interesting results, including novel I. HAMILTONIAN superconductivity in doped Weyl semimetals [14–20] and the emergent supersymmetry [19], there has been no experimental We consider a circularly shaken 3D fcc optical lattice, observation of such phenomena. It is therefore desirable to whose Hamiltonian in the Floquet framework is written as − have a highly controllable platform to investigate Weyl points H0 ih∂¯ τ , where τ is time, and the resultant quantum phenomena in interacting systems. P2 In this paper, we show that a face-centered-cubic (fcc) H0 = + V (x + f cos ωτ,y + f sin ωτ,z), (1) optical lattice provides physicists a unique means to create and 2M and f and ω are the shaken amplitude and frequency, respectively. M is the atom mass. Formally, the above equation *[email protected] is a direct generalization of the scheme of shaking a 2D 2469-9950/2017/96(3)/035145(5) 035145-1 ©2017 American Physical Society LI-JUN LANG, SHAO-LIANG ZHANG, K. T. LAW, AND QI ZHOU PHYSICAL REVIEW B 96, 035145 (2017) √ lattice, and G = (1,1,1)π/d0. d˜0 = 2d0 and d0 are the lattice spacings of the fcc and the sc lattices, respectively. Applying the shaking, the A band by absorbing a photonhω ¯ couples with the B band in the Floquet Hamiltonian, which can be written as H0 = (Ak + Bk)I/2 + K with ( − )/2 eiϕk = Ak Bk k K ∗ − , (3) iϕk − ke (Bk Ak)/2 = 0 + = 0 where Ak Ak hω¯ and Bk Bk. One-photon detun- ing, δ ≡ − hω¯ , can be defined to characterize the sep- aration of Ak and Bk. The interband coupling is writ- iϕk −ikx d0 ten as ke = (i sin kx d0 − sin ky d0)e with ϕk = −ikx d0 arg[(i sin kx d0 − sin ky d0)e ]. This coupling is the same as that in a shaken checkerboard lattice [28] since V (x,y,z) in Eq. (2) reduces to a 2D checkerboard lattice for each plane with a given value of z. FIG. 1. (a) Schematic of the fcc lattice with A and B sublattices. (b) The first BZ (red truncated octahedron) of the fcc lattice folded II. WEYL POINTS FROM SHAKING from that (black cubic) of the sc lattice. (c) The lowest two bands 0 0 A Weyl point requires that all matrix elements in Eq. (3) (blue solid), Ak and Bk, of the fcc lattice. Red dashed line represents 0 become zero at some k in BZ. In previous proposals [21–23], the subband of Ak after absorbing one photon via shaking. The 0 parameters are V = 8ER, hω¯ = 0.7ER,α= 0.06,f = 0.16d0.(d) this is realized by engineering the tunneling along all three One laser setup to realize the fcc lattice. Red (purple) arrows represent dimensions, which often requires sophisticated designs of the lasers with wavelength λ (λ) and frequency ω (ω). Details can microscopic models. Here, a shaken fcc lattice automatically be referred to in the Supplemental Material [29]. provides Weyl points. The off-diagonal term could vanish = 0 = at (kx,ky ) (0,0), (0,π), or (π,0). Meanwhile, since Bk checkerboard lattice [28] to a 3D fcc lattice, whose potential − 0 + − = Ak is readily satisfied in the static lattice, Ak Bk is written as 0 can be easily satisfied if δ is small enough, i.e., in the strong π π π interband hybridization regime with the shaken frequency V (x,y,z) =−V cos2 x + cos2 y + cos2 z a b c tuned near resonance. Whereas the total number of pairs of Weyl points depends on the microscopic parameters, here π π π + α cos x cos y cos z , (2) we focus on the simplest case with only one pair of Weyl a b c points, i.e., k0 = (0,0,k0z), so that |K|=0, where k0z satisfies + + + = where a, b, and c are constants.
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