Topological Fulde-Ferrell Superfluids in Triangular Lattices
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Topological Fulde-Ferrell Superfluids in Triangular Lattices
Long-Fei Guo,
1, 2
Peng Li,
1, 2, ∗ and Su Yi
3, 4, † College of Physical Science and Technology, Sichuan University, 610064, Chengdu, China Key Laboratory of High Energy Density Physics and Technology ofMinistry of Education, Sichuan University, Chengdu, 610064, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China School of Physics, University of Chinese Academy of Sciences, P. O. Box 4588, Beijing 100049, China (Dated: August 12, 2018)Fulde-Ferrell (FF) Larkin-Ovchinnikov (LO) phases were proposed for superconductors or su-perfluids in strong magnetic field. With the experimental progresses in ultracold atomic systems,topological FFLO phases has also been put forward, since it is a natural consequence of realizablespin-orbital coupling (SOC). In this work, we theoretically investigate a triangular lattice modelwith SOC and in-plane field. By constructing the phase diagram, we show that it can producetopological FF states with Chern numbers, C = ± C = −
2. We get the phase boundaries bythe change of the sign of Pfaffian. The chiral edge states for different topological FF phases are alsoelucidated.
PACS numbers: 67.85.-d, 03.65.Vf, 03.75.Lm, 05.30.Fk
I. INTRODUCTION
Cooper pair was first proposed in 1956 to explain su-perconductors [1]. It describes a pair of fermions boundtogether due to attractive interaction. The fermions haveopposite momentum so that the pair has zero momen-tum totally. However, Cooper pair with finite center-of-mass momentum may also exist in the presence ofstrong magnetic field. This consideration led to an ex-otic superconductor with inhomogeneous order parame-ter in real space, known as Fulde-Ferrell [2] and Larkin-Ovchinnikov (FFLO) phases [3]. There are two typesof FFLO phases: phase modulated FF state and spatialmodulated LO state. In the past two decades, FFLOphases attract tremendous interests in both experimentand theory [4–9]. But only ambiguous experiment evi-dences from heavy-fermion superconductors and organicsuperconductors are available [10–12].On the other hand, topology is also a hot topic in con-densed matter field for several decades [13–15]. Recently,the spin-orbit coupling (SOC) has been realized by ul-tracold atoms as condensates or on an optical lattice[16–19], which paves the way to the topological FFLOstates [20–25]. Theoretical researches show that topo-logical FFLO states can be induced in one and two di-mensional Fermi gas. And according to the bulk-edgecorrespondence, edge modes are supported when thereare boundaries [26, 27].In two dimensions, topological FF state with Chernnumber C = 1 can be produced for cold atoms in a squarelattice [21, 23]. To realize topological state with higherChern number, one can resort to complicated hoppings ∗ Electronic address: [email protected] † Electronic address: [email protected] [28] or lattices [29]. Nonetheless, the simple triangularlattice favors some topologically nontrivial states, it canproduce topological state with higher Chern number bymerely the nearest-neighbor hoppings [30, 31]. In thiswork, we investigate a system with SOC and in-planeZeeman field on the triangular lattice to achieve topo-logical FF states. We found, in the noninteracting case,our system with nearest-neighbor hoppings supports thegapped Chern insulators with Chern number C = 1 aswell as C = −
2. The boundaries of both of the topolog-ical phases are ellipses in a two-parameter plane consti-tuted by the in-plane and out-plane fields. In the pres-ence of on-site attractive s-wave interactions, we solve thesystem by self-consistent equations. The non-uniform FFsuperfluid states are obtained in a large area. And weconfirm that all the FF states are topologically nontriv-ial. Their Chern numbers are C = 1 , − −
2. We usea set of signs of Pfaffians at high symmetry points in thefirst Brillouin zone (1st BZ) to characterize the differenttopological phases. Each one of these signs of Pfaffianschanges with the energy gap closing and reopening at thecorresponding point. We calculate the chiral edge states,whose wave functions are spatially localized at the edgesin open boundary situation, which can help us to con-firm that the bulk is in a topologically nontrivial state.Different FF phases exhibit different pairs of chiral edgestates. The edge current is directly determined by Chernnumber, i.e. the summation of chiralities of the edgemodes.This paper is organized as follows. In Sec. II, we in-troduce a system in the triangular lattice with SOC andin-plane field and solve it by self-consistent method. InSec. III, we construct the phase diagram in the noninter-acting and interacting case. The topological FF phaseswith C = − , − (b) k y k x a a a (a) FIG. 1: (Color line) (a) Triangular lattice with direction vec-tors a , a , and a = − ( a + a ). (b) Brillouin zone of trian-gular lattice. summary. II. MODEL HAMILTONIAN
We consider a two-dimensional (2D) spin-orbit coupledfermionic gas trapped in a triangular lattice subjected toan external magnetic field. In the position space, themodel Hamiltonian reads H = X h i,j i φ † i H s φ j + H int , (1)where φ † i = ( c i, ↑ , c i, ↓ ) with c † i,σ ( c i,σ ) being the creation(annihilation) operator for the spin σ particle at site i , H s and H int represent the single-particle and interactionHamiltonian, respectively, and the summation is over thenearest-neighbor pairs. More specifically, we assume thatthe single-particle Hamiltonian takes the form H s = t i,j σ z + it i,j so ( v i,j × σ ) z + ( h · σ − µ ) δ i,j , (2)where σ α ( α = x, y, z ) are a Pauli matrices, t i,j is thehopping matrix elements which, when combined with σ ,gives rise to opposite signs for the spin-up and -down par-ticles [32], t i,j so are the Rashba SOC coefficients, v i,j arethe vectors connecting lattice sites i and j , h = ( h x , , h z )is the magnetic field which includes an in-plane compo-nent h x along the x direction and an out-of-plane one h z , and µ is chemical potential. The s -wave interactionHamiltonian can be expressed as H int = − U X i c † i, ↑ c i, ↑ c † i, ↓ c i, ↓ , where U > a = (1 , a = ( − / , √ / a = (1 / , √ / G = (0 , π/ √
3) and G = (2 π, π √ x direc-tion, the Fermi surface becomes asymmetric along the y axis. Consequently, the BCS pairs may carry a finitecenter-of-mass momentum Q y along the y direction [33].Such state is described by the FF order parameter, which,in the position space, is defined as ∆ i = U h c i, ↓ c i, ↑ i = ∆ e i Q · R i with Q = (0 , Q y ) [34]. After transformed into the mo-mentum space, the mean-field Hamiltonian in the Namburepresentation reads H = 12 X k Ψ † k H BdG Ψ k + N | ∆ i | U − µ ! , where Ψ k = (cid:16) c k + Q / , ↑ , c k + Q / , ↓ , − c †− k + Q / , ↓ , c †− k + Q / , ↑ (cid:17) T is the Nambu spinor with k = ( k x , k y ), N is the totalnumber of lattice sites, and the Bogoliubov-de Gennes(BdG) Hamiltonian is H BdG = ( a k + h z ) τ ⊗ σ z + b k τ z ⊗ σ z + c k τ z ⊗ σ x + ( d k + h x ) τ ⊗ σ x + e k τ z ⊗ σ y + f k τ ⊗ σ y + ∆ τ x ⊗ σ − µτ z ⊗ σ (3)with ⊗ being the Kronecker product and τ α ( α = x, y, z )being the Pauli matrices acting on the particle-hole space.The explicit expressions for the elements of H BdG matrix, a k , b k , c k , e k , f k , and g k , can be found in the AppendixA. We note that the BdG Hamiltonian only possesses aparticle-hole symmetry, Ξ H BdG ( k )Ξ − = Λ H ∗ BdG ( k )Λ = − H BdG ( − k ), where Ξ = Λ K with Λ = iσ y ⊗ τ y and K be-ing the complex conjugation operator. It can be verifiedthat Ξ = 1. Apparently, the system belongs to the classD according to Wigner-Dyson symmetry classification ofrandom matrix [35] which has a topological invariant Zin two dimension [36–38].Following the standard treatment, we diagonalize theBdG Hamiltonian, H BdG | ψ να ( k ) i = E να, k | ψ να ( k ) i , whichleads to the quasiparticle eigenenergy E να, k and the quasi-particle wave function | ψ να ( k ) i , here ν = ± represent theparticle (+) and hole ( − ) bands, α = 1 and 2 denote theupper (1) and lower (2) helicity branches. Now, the ther-modynamic potential at temperature T can be calculatedthrough Ω = NU | ∆ | − N µ + 12 X α, k E − α, k − k B T X α, k ln (cid:16) e E − α, k / ( k B T ) (cid:17) , (4)where k B the Boltzmann constant and the summation isrestricted to the hole bands ( ν = − ) due to the inherentparticle-hole symmetry in the Nambu spinor represen-tation. The order parameter can be numerically deter-mined by the mean-field saddle equations ∂ Ω /∂ ∆ = 0, h x /t h z /t C 0
C 2
C 1
FIG. 2: (Color line) Quantum phases of the single-particleHamiltonian Eq. (6) with t so /t = 1. Chern insulators arelocated within two ellipses as expressed by Eq. (8) and (9). ∂ Ω /∂Q y = 0, as well as the equation for the conservationof the total particle number ∂ Ω /∂µ = − N . Finally, thetopological property of the system is characterized by theChern number [15, 39] C = X α π Z dk x dk y z · ∇ k × A − α ( k ), (5)where A − α ( k ) = i h ψ − α ( k ) | ∇ k | ψ − α ( k ) i is the vector po-tential. III. RESULTS
In this section, we present the results about the zero-temperature quantum phases of our model. For conve-nience, we assume that the hopping matrix element is siteindependent, i.e., t = t i,j and t so = t i,j so . Moreover, weselect t as the energy unit such that the model Hamilto-nian is completely specified by the parameters t so /t , h x /t , h z /t , U/t , and µ/t . In below, the value of the chemicalpotential is fixed at µ/t = 1 for simplicity.To start, let us briefly discuss the quantum phases ofour model in the absence of interaction. In the momen-tum space, the single-particle Hamiltonian reduces to H s = X k ( d x σ x + d y σ y + d z σ z ) , (6)where d x = √ t so (sin k − sin k )+ h x , d y = t so (2 sin k − sin k − sin k ), and d z = t (cos k + cos k + cos k ) + h z with k α = k · a α . Due to the breaking of the time-reversaland chirality symmetries, the single-particle Hamiltonianbelongs to the symmetry class C. Therefore, the possi-ble topologically nontrivial ground state in 2D systemis characterized by Z invariant [36–38]. To identify thetopological state, we focus on the gapless points defined tFF[ 1] tFF[ 2] tFF[1]tFF[ 1] CI NGNGIN h x /t h z /t FIG. 3: (Color line) Zero-temperature phase diagram with t so /t = 1, U/t = 5, and µ/t = 1. The dashed curve repre-sents the boundary between the gapped (left side) and gapless(right side) topological superfluid phases. The integers in thesquare brackets specify Chern numbers.TABLE I: Calculation of Chern number in different parameterregion for the single particle system H s .parameter region o ( k i ) p ( k i ) C k k k ± k ± upper ellipse 1 -1 -1 -1 -2lower ellipse 1 1 -1 1 1 by | d ( k ) | = 0 , (7)where d ≡ ( d x , d y , d z ). It can be shown that Eq. (7) givesrise to two ellipses, h x t + (cid:18) h z t − (cid:19) = 14 (8)and 4 h x t + (cid:18) h z t + 2 (cid:19) = 16 , (9)which divide the h x h z parameter plane into three regions.The topological property of each region is determined bythe Chern number C = π R dk x dk y n · ∂ k x n × ∂ k y n ,where n = d / | d | is a unit vector.In Fig. 2, we plot the phase diagram of the Hamilto-nian (6) for t so /t = 1. Without loss of generality, onlythe result for h x > (a) E g /t /t (b)(c) Q y ( / a ) h z /t FIG. 4: (Color line) h z dependence of the quasiparticle bandgap E g (a), superfluid order parameter ∆ (b), and pairingmomentum Q y (c) for h x /t = 0 .
3. Other parameters are thesame as those in Fig. 3. upper ellipse is −
2. These topologically nontrivial statescan be understood by noting that the Chern number canbe reexpressed as [40, 41] C = 12 X k i o ( k i ) p ( k i ) , (10)where k i are the roots of the equations d x ( k ) = d y ( k ) = 0(see Appendix B), o ( k i ) is the chirality of the vector field( d x ( k ) , d y ( k )) around k i , and p ( k i ) = sgn( d z ( k i )) is thepolarity at k i . In Tab. I, we summarize the contributionof each root to the Chern number.We now turn to study the superfluid phases of the sys-tem by taking into account the attractive s -wave interac-tion. Fig. 3 summarizes the quantum phases in the h x h z parameter plane with U/t = 5 and µ/t = 1. Here thequantum phases are characterized by the superfluid or-der parameter ∆, the center-of-mass momentum Q y , theChern number C , and the chemical potential µ . Specif-ically, when ∆ = 0, we may either have an insulating(IN) phase or a normal gas (NG) phase. For the formerstate, the chemical potential locates in the band gap;while, for the latter state, it lies in the band such thatthe excitation is gapless [42]. Moreover, an IN phaseis Chern insulator (CI) if it is topologically nontrivial( C = 1 here). Next, a superfluid state (∆ = 0) withnonzero Q y is denoted by ‘FF’. In our model, the ap-pearance of FF states is due to the in-plane magneticfield h x that deforms the Fermi surface [33]. In fact, thestandard Bardeen-Cooper-Schrieffer superfluid ( Q y = 0)only exists at h x = 0. Additionally, all superfluid phasesare found to be topologically nontrivial and they are fur-ther specified by the letter ‘t’ and by Chern numbers inthe square bracket. Finally, the dashed line in Fig. 3marks the boundary between gapped and gapless super-fluid phases. More specifically, for small h x , the energy of the lower helicity particle branch E +2 , k is always positivefor the gapped superfluid phases. However, E +2 , k may be-come less than zero as h x is increased, which leads to thegapless superfluid phases [43, 44].In Fig. 4, we plot the h z dependence of the quasipar-ticle band gap E g ≡ max { , min( E +2 , k ) } , the superfluidorder parameter ∆, and the pairing momentum Q y fora fixed h x /t = 0 .
3. As can be seen, for small h z , theFermi surface locates in the band gap such that the sys-tem remains in the insulating phase. Then as h z is in-creased, E +2 , k moves downward, while E − , k moves oppo-sitely, which leads to a vanishing E g , signaling the en-tering of the NG phase. As one further increases h z , thetFF phases emerge. Within the superfluid phases, the en-ergy gap closes and reopens whenever a topological phasetransition is encountered. Eventually, at very large h z ,the large population difference between the spin- ↑ and - ↓ particles prohibits the formation of the superfluid pairs.As a result, the system falls into insulting phases again.The phase boundaries between topological superfluidphases in Fig. 3 can be alternatively determined byconsidering the high symmetry points k ′ that satisfyΞ H BdG ( k ′ )Ξ − = − H BdG ( k ′ ). To this end, we introducean auxiliary matrix W ( k ′ ) ≡ H BdG ( k ′ )Λ [45, 46] that isantisymmetric, i.e., W T ( k ′ ) = − W ( k ′ ). A topologicalindex can then be defined as P ( k ′ ) = sgn[Pf[ W ( k ′ )]] , (11)where Pf[ W ( k ′ )] denotes the Pfaffian of W ( k ′ ). SincePf[ W ( k ′ )] = ± p det[ H BdG ( k ′ )], P ( k ′ ) will never changeits sign unless the energy gap at k ′ is closed, indicatingthat P ( k ′ ) is indeed topologically protected. This alsosuggests that the phase boundaries are determined bythe condition P ( k ′ ) = 0 or, formally,[ d ( k ′ ) + h x ] + [ a ( k ′ ) + h z ] − ∆( h x , h z ) = µ − f ( k ′ ) . (12)In our model, the high symmetry points include Γ =(0 , M = (0 , π/ √ M = ( ± π, π/ √ Σ =( ± − [ − cos( √ Q y / / , P ( k ′ = Γ) = − M , M ,and Σ leads to the three phase boundaries between thetopological superfluid phases through Eq. (12).To gain more insight into these topological phases, wefurther consider the edge modes of the system by impos-ing a hard-wall boundary condition along the y direc-tion. Fig. 5 shows the quasiparticle spectra of distincttFF phases and the probability distribution of the corre-sponding edge states. Consider, for example, the tFF[-1]phase in Fig. 5(a), the lines of the edge modes cross theFermi level three times. To distinguish each edge stateclearly, we plot a horizontal dashed line slightly abovethe Fermi level to reveal the Hall current. It shows thatthe dashed line intersects with the lines of edge modes atA, B, C, D, E, and F successively. Correspondingly, theprobability distributions over the lattice sites for these -0.50.00.5 -0.50.00.5 -0.30.00.3 E /t FEDC A B k x (a) C lattice site DE p r ob a b ilit y * A B C D E FA x y FB k x E /t (b) k x A B yx p r ob a b ilit y * lattice site A BB A (c) E /t A B C D p r ob a b ilit y * lattice site BC D
A B C D yx A FIG. 5: (Color line) Quasiparticle spectra (left panels) andprobability distributions of the edge states (right panels) fortFF[ −
1] (a), tFF[1] (b), and tFF[ −
2] (c) phases. Insets in theright panels show the directions of the currents correspondingto the edge states. From top to bottom row, the out-f-planemagnetic fields are h z /t = 1 .
6, 2, and 3 .
1, respectively. Otherparameters are t so /t = 1, U/t = 5, µ/t = 1, and h x /t = 0 . states are plotted in the right panel of Fig. 5(a), whichshows that states A, D, and E locate at the lower edgeand states B, C, and F at the upper edge. These edgestates also give rise to edge currents, I ∝ − ∂E/∂k x , atthe upper or lower edges. To proceed further, let us fo-cus on the edge states A and F, which locate at upperand lower edges, respectively. Due to the translationalsymmetry along the x direction, the band is symmet-rical about the k x axis, which suggests that the groupvelocities of the states A and F have the same magni-tude but with opposite signs. As a result, states A andF form a clockwise current loop at the edges and con-tribute a Chern number, C AF = −
1, to the system. Sim-ilar analysis can be carried out for other pairs of statesand leads to the Chern numbers C CD = 1 and C BE = − C = C AF + C CD + C BE = −
1. Likewise, from Fig. 5(b)and (c), it can be verified that the the Chern numbers ofthe tFF[1] and tFF[-2] phases are indeed C = 1 and − IV. SUMMARY
In summary, we have investigated a triangular latticewith SOC and in-plane magnetic field. We construct thephase diagram in the noninteracting andinteracting casesrespectively. In the noninteracting case, we distinguishdifferent phases by Chern numbers. The Chern insulatorswith Chern number C = 1 and C = − C = − , , − ACKNOWLEDGMENTS
We acknowledge useful discussions with Yan He andYuangang Deng. This work was supported by theNSFC (Grants No. 11074177, No. 11421063, and No.11674334) and SRF for ROCS SEM (20111139-10-2).
Appendix A: Components of the BdG Hamiltonian
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12 cos ( 12 sin − η )] , √ − η (cid:19) , k ± = (cid:18) ± − [ −
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