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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
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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. Interestingly, such a 3D 4t(1 2 cos k0zd0) 2t (2 cos 2k0zd0) δ/2. Near k0,the optical lattice of fundamental importance in solids has never matrix can be linearized been produced in ultracold atoms. We point out that it can vz(kz − k0z) ivx kx − vy ky be produced by three lasers with directions and polarizations K = , (4) −iv k − v k −v (k − k ) arranged in a few ways, one of which is shown in Fig. 1(d). x x y y z z 0z Three pairs of lasers are used, where lasers 1 and 2 interfere where vx = vy = d0 ≡ v,vz =−8d0 sin(k0zd0)(t + with each other and form a z-dependent checkerboard lattice t cos k0zd0). Equation (4) indeed describes a Weyl point in the in the x-y plane, while laser 3 with a different frequency BZ. When k = k0, |K|=0, and the Weyl point represents a forms a standing wave along the z direction (see Supplemental monopole in the momentum space with a topological charge Material [29]). This setup has already been realized by ±1. The sign of the charge is determined by the sign of Esslinger’s group [3]. vx vy vz. Since for each k0z, there is always another solution To concretize the discussion, we focus on a symmetric −k0z to satisfy |K|=0 with an opposite chirality, one sees case a = b = c. All results here can be easily generalized that the total chirality in BZ is zero. The positions of the Weyl to an arbitrary choice of a,b,c. The exact band structure points are determined by the microscopic parameters in the of the static lattice V (x,y,z), where f = 0, can be solved system, such as the detuning δ. Changing the value of |δ|, exactly using plane-wave expansions. The results for the Weyl points move in BZ, and once a pair of Weyl points with 0 0 lowest two bands, denoted as Ak and Bk,areshown opposite chiralities meet in BZ, they annihilate each other. For in Fig. 1(c). Both of them can be well approximated large enough |δ|, the disappearance of Weyl points indicates 0 = + by the tight-binding results k 4t(cos kx d0 cos ky d0 that the 3D band structure becomes topologically trivial in A cos ky d0 cos kzd0 + cos kzd0 cos kx d0) + 2t (cos 2kx d0 + such cases with weak interband hybridization. + 0 =− 0 + cos 2ky d0 cos 2kzd0), and Bk Ak , where t and t characterize the nearest- and next-nearest-neighbor tunnelings III. 3D NODAL SUPERFLUID in A sublattice, and is the band separation of them. The inverted structure between A and B bands could be understood We now turn to the interaction effects. Whereas the 0 ≈ 0 ≈ qualitatively from the folding of BZ, i.e., Ak ck, Bk interplay between Weyl fermions and interaction has been ck+G + , where ck is the ground band dispersion of the SC studied in the literature [14–20], our shaken lattices provide
035145-2 WEYL POINTS AND TOPOLOGICAL NODAL SUPERFLUIDS . . . PHYSICAL REVIEW B 96, 035145 (2017)
two Weyl semimetals or two normal metals with small Fermi surfaces. Define the pairing order parameters A(B) = −UA(B) ˆ A(B),−k↓ˆ A(B)k↑ /N, where N is the number of unit cells, the BCS Hamiltonian can be written as ⎛ ⎞ iϕk Ak − μ ke A 0 ⎜ − ⎟ ⎜ e iϕk − μ 0 ⎟ ˆ = k Bk B HBCS ⎜ ∗ − ⎟, ⎝ − − iϕ−k ⎠ A 0 μ Ak ke ∗ − iϕ−k − 0 B ke μ Bk (5)
where A and B are solved self-consistently for a fixed density n = n↑ + n↓. Since UA and UB could be controlled independently, very rich physics emerges. 1 ↑ = ↓ We first consider n n > 2 , i.e., a finite Fermi surface FIG. 2. (a)–(c) Fermi arcs with (100) surface of (a) noninteracting surrounding each Weyl point in the BZ. Turning on one of single particles for either spin up or spin down, and (b), (c) Bogoli- the interactions, say U or U , it turns out that there exists ubov quasiparticles for half-filling with (U ,U ) = (0.15,0)E and A B A √B R one point on each Fermi surface remaining gapless, whereas (0.15,0.08)ER, respectively. k˜y,z = (kz ± ky )/ 2. (d)–(f) Schematics of nodal points in the bulk with no interaction, only one interaction, the superfluid gap opens anywhere else, as shown in Fig. 2(e). We note that the BCS Hamiltonian (5) is block-diagonalized and both interactions, respectively. Pink (green) and red (blue) = = = dots represent the nodal points with +(−)1 and +(−)2 charges, if one sets kx ky 0. When UA 0, which naturally = respectively. Spheres in (e) and (f) enclosing the nodal points have the leads to A 0, all matrix elements of one block vanish ∗ = ± ± same total Chern number, −2, of the lowest two bands. Parameters at the momenta k (0,0, k0z kF ), i.e., the Bogoliubov for single particles are the same as those in Fig. 1(c). quasiparticle spectrum remains gapless. The other branch, which comes from the other block, opens a gap due to a finite B . Alternatively, when UB = 0 the spectrum is gapless at ∗ = ± ∓ one a unique system to explore new physics that has not been k (0,0, k0z kF ). explored before. We introduce two hyperfine spin states into We point out that the emergent 3D nodal superfluid is the fcc optical lattice, each of which has the same single- a topological one, which can be seen from the topological particle Hamiltonian specified by Eq. (3). This corresponds charges carried by the nodal points in the quasiparticle to a spin-independent shaken lattice, where we have two spectrum. Expanding the quasiparticle spectrum near such Weyl points with the same chirality at the same positions gapless point, an effective Hamiltonian is obtained, ∗ in the momentum space. Without interaction, the many-body ±v qz 0 H ± = z ground state is simply composed of two identical copies of eff ∓ ∗ 0 vz qz Weyl semimetals, if the chemical potential is tuned right at 2 2 2 −2iθ the Weyl point. Unlike the electronic spins, the hyperfine v q + q −2ημ e q + x y ξ spin is conserved in ultracold atoms. This gives rise to two , (6) | |2 + 4μ2 e2iθq 2ημ copies of topological charge of 1 in the single-particle level, as ξ ξ shown in Fig. 2(d). However, introducing attractive interaction where q is the momentum measured from the gapless point ∗ ∗ = − ∗ = + + inevitably leads to particle-hole mixing between the two k or k , θq arg(iqx qy ), and vz 8sin(k0z ηkF )[t hyperfine spin states. A natural question is then as follows: t cos(k0z + ηkF )]. η =+1 and ξ = AforUB = 0 while η = What is the fate of such Weyl semimetals? Alternatively, one −1 and ξ = BforUA = 0. The superscript ± is the valley could consider tuning the chemical potential away from Weyl index, representing the two Fermi surfaces surrounding the ± points so that the Fermi surface becomes finite. Such finite two Weyl points in noninteracting systems. Heff describes a Fermi surface indicates that the low-lying excitations in the Bogoliubov quasiparticle spectrum, which is linear along the interacting system are actually located at momenta away from qz direction and quadratic along the qx and qy directions, 2 the Weyl points of the noninteracting system. It is thus desired since the off-diagonal term ∼(iqx ± qy ) . It thus describes a to explore whether the Weyl point, which is now embedded monopole of charge 2 in the momentum space. We thus see that inside the Fermi sea, is relevant to the emergent superfluidity the topological charge when introducing attraction interaction when the attractive interaction is turned on. shows up in Bogoliubov quasiparticle spectrum. Using σ =↑,↓ to denote these two hyperfine spin Turning on the other interaction, the monopole of charge states, the onsite interaction can be written as Vˆ = 2 splits to two charge-1 ones, i.e., two Weyl points, in the − − ∗ ∗ UA i∈A nˆi↑nˆi↓ UB i∈B nˆi↑nˆi↓, where nˆiσ is the density kx -ky plane, the positions of which are denoted as k1 and k2, operator for spin-σ particles at site i, and UA(B) > 0isthe respectively. This can be understood from the conservation of onsite interaction strength for A(B) sites. Our Hamiltonian the total charge of the monopoles enclosed in the spheres, as is different from the ones previously studied in the litera- shown in Figs. 2(e) and 2(f). Near these two Weyl points, the ture, where the degree of freedom that participates in the dispersion becomes linear along all three directions. Since the interaction is the same as that provides the band crossing noninteracting system has a fourfold rotation symmetry about [14–20]. Here, we consider the attractive interaction between the kz axis, and such a splitting reduces the symmetry to a
035145-3 LI-JUN LANG, SHAO-LIANG ZHANG, K. T. LAW, AND QI ZHOU PHYSICAL REVIEW B 96, 035145 (2017) twofold one, the choice of the direction for the splitting is a of the spectrum is no longer linear, and is also important to consequence of spontaneous symmetry breaking, along either the critical interaction. Thus, there is no simple expression c c the kx or ky direction. By changing the ratio UA/UB , these two of UA. Nevertheless, if UA >UA, the results become similar Weyl points move in BZ before meeting their counterparts to those with a finite Fermi surface. A double Weyl point of with opposite chiralities emerged from the other valley. For charge 2 emerges in the quasiparticle spectrum. Turning on noninteracting systems, it is known that tuning the parameters UB , such multiple-charge monopole splits to two Weyl points leads to the movement of Weyl points without opening the gap. of 1. Using typical values in Fig. 1(c) for single particles and in In our system, tuning interaction also offers such opportunity Fig. 2(c) for interactions, we can estimate the order parameters to control the positions of Weyl points in the Bogoliubov as A ∼ 12.02 nK and B ∼ 1.71 nK. quasiparticle spectrum in BZ. There is an alternative way to understand why the 3D IV. FERMI ARCS superfluid remains nodal with turning on a small UA.Asshown A characteristic feature of Weyl points in noninteracting in Eq. (5) and Figs. 2(d)–2(f), in 3D BZ each 2D plane with systems is the existence of Fermi arc [9] at the surface, an + + fixed kz defines a 2D s (d id) superfluid, similar to the one unclosed line as the zero energy state. Whereas in the absence emerged from a 2D shaken lattice [28]. Whereas tuning kz ef- of interaction, such Fermi arc is indeed observed in our system fectively changes the chemical potential of such 2D superfluid, as shown in Fig. 2(a), it is more interesting to explore the its Chern number can be computed straightforwardly. When interacting case when the nodal superfluids have emerged. | | ∗ ∗ ∗ ∗ kz
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