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Topological Superfluids with Finite Momentum Pairing and Majorana Fermions

Majorana fermions, quantum particles that are their own anti-particles, are not only of fundamental importance in elementary particle physics and dark matter, but also building blocks for fault-tolerant quantum computation. Recently Majorana fermions have been intensively studied in solid state and cold atomic systems. These studies are generally based on superconducting pairing between two Fermions with opposite momenta (\textit{% i.e.}, zero total momentum). On the other hand, finite total momentum Cooper pairings, known as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, were predicted 50 years ago and then widely studied in many branches of physics. However, whether FFLO superconductors can also support Majorana fermions has not been explored. Here we show that Majorana fermions can exist in certain types of gapped FFLO states, yielding a new topological quantum matter: topological FFLO superfluids/superconductors. We demonstrate the existence of such topological FFLO superfluids and the associated Majorana fermions using spin-orbit coupled degenerate Fermi gases and derive their physical parameter regions. The potential implementation of topological FFLO superconductors in semiconductor/superconductor heterostructures are also discussed.

These theoretical and experimental studies are based on the superconducting Cooper pairing (s-wave or chiral p-wave) with zero total momentum, that is, the pairing is between two fermions with opposite momenta k and −k (denoted as BCS pairing hereafter). On the other hand, the superconducting pairing can also occur between fermions with finite total momenta (pairing between k and −k + Q) in the presence of a Zeeman field, leading to spatially modulated superconducting order parameters in real space, known as FFLO states. The FFLO states were first predicted in 1960s [16,17], and now are a central concept for understanding exotic phenomena in many different systems [18][19][20][21][22][23]. A natural question to ask is whether MFs can also exist in a FFLO superconductor or superfluid?
In this Letter, we propose that FFLO superconductors/superfluids may support MFs if they possess two crucial elements: gapped bulk quasi-particle excitations and nontrivial Fermi surface topology. These new quantum states are topological FFLO superconductors/superfluids. In this context, traditional gapless FFLO states induced by a large Zeeman field do not fall into this category. Here we propose a possible platform for the realization of topological FFLO superfluids using two-dimensional (2D) or one-dimensional (1D) spinorbit (SO) coupled degenerate Fermi gases subject to in-plane and out-of-plane Zeeman fields. Recently, the SO coupling and Zeeman fields for cold atoms have already been realized in experiments [24][25][26][27][28], which provide a completely new avenue for studying topological superfluid physics. It is known that SO coupled degenerate Fermi gases with an out-of-plane Zeeman field support MFs with zero total momentum pairing [29][30][31][32]. We find in suitable parameter regions the in-plane Zeeman field can induce the finite total momentum pairing [33][34][35][36], while still keeps the superfluid gapped and preserves its Fermi surface topology. The region for topological FFLO superfluids depends not only on the chemical potential, pairing strength, but also on the SO coupling strength, total momentum and effective mass of the Cooper pair, as well as the orientation and magnitude of the Zeeman field, thus greatly increases the tunability in experiments. Finally, the potential implementation of the proposal in semiconductor/superconductor heterostructures is also discussed.

Results
System and Hamiltonian: Consider a SO coupled Fermi gas in the xy plane with the effective Hamiltonian where FIG. 1: Single particle band structure and Berry curvature. (a) Energy dispersion of the lower band. The green arrows represent the momenta of a Cooper pair of two atoms on the asymmetric Fermi surface. The red arrow represents the total finite momentum of the paring, which is along the deformation direction of the Fermi surface. (b) Berry curvature of the lower band, whose peak is shifted from the origin by hx/α along the ky direction.
α is the Rashba SO coupling strength, h = (h x , 0, h z ) is the Zeeman field and σ is the Pauli matrices.
is the binding energy, and k 1 + k 2 = k 3 + k 4 due to the momentum conservation. Without in-plane Zeeman field h x , the Fermi surface is symmetric around k = 0, and the superfluid pairing is between atoms with opposite momenta k and −k. While with both h x and SO coupling, the Fermi surface becomes asymmetric along the y direction (see Fig. 1a), and the pairing can occur between atoms with momenta k and −k + Q. In real space, such a finite total momentum pairing leads to a FF-type order parameter ∆(x) = ∆e iQ·x , where Q = (0, Q y ) is parallel to the deformation direction of the Fermi surface [33][34][35]. Notice that the energies of the superfluids with total momentum Q and −Q are nondegenerate, therefore FF phase with a single Q, instead of LO phase (∆(x) = ∆ cos(Q · x)) where pairing occurs at both ±Q, is considered here. Hereafter, if not specified, FFLO superfluids refer to FF superfluids. The dynamics of the system can be described by the following Bogliubov-de Gennes (BdG) Hamiltonian in the mean-field level, where the Nambu basis is chosen as (c k+Q/2,↑ , c k+Q/2,↓ , c † −k+Q/2,↓ , −c † −k+Q/2,↑ ) T . The gap, number and momentum equations are solved self-consistently to obtain ∆, µ and Q, see Methods, through which we determine different phases.
Physical mechanism for topological FFLO phase: Without SO coupling, the orientation of the Zeeman field does not induce any different physics due to SU(2) symmetry. The presence of both h x and SO coupling breaks this SU(2) symmetry, leading to a Fermi surface without inversion symmetry, see Fig. 1a. Here, h x deforms the Fermi surface, leading to FFLO Cooper pairings; while h z opens a gap between the two SO bands, making it possible for the chemical potential to cut a single Fermi surface for the topological FFLO phase. The Berry curvature of the lower band reads as Note that h x shifts the peak of Berry curvature from k = 0 to (0, −h x /α) (denoted by arrow in Fig. 1b). When atoms scatter from k to k ′ on the Fermi surface, they pick up a Berry phase, whose accumulation around the Fermi surface θ = d 2 kΩ k ≈ π. Such Berry phase modifies the effective interaction from s-wave (V kk ′ ∼ g is a constant) to s-wave plus asymmetric p-wave on the Fermi surface. Here we recover the well-known chiral p x + ip y pairing [29] in the limit h x = 0. The in-plane Zeeman field here creates an effective s-wave pairing component (although still hosts MFs), and the effective pairing is reminiscent to the (s+p)-wave pairing in some solid materials [37].
Parameter region for MFs: The BdG Hamiltonian (2) satisfies the particle-hole symmetry Ξ = ΛK, where Λ = iσ y τ y , K is the complex conjugate operator, and Ξ 2 = 1. The parameter region for the MFs is determined by the topological index M = sign(Pf{Γ}), where Pf is the Pfaffian of the skew matrix Γ = H BdG (0)Λ. M = −1(+1) corresponds to the topologically nontrivial (trivial) phase [38]. The topological phase exists when whereh x = h x + αQ y /2 andμ = µ − Q 2 y /8m. E g = min(E k,s ) defines the bulk quasi-particle excitation gap of the system with E k,s as the particle branches of the BdG Hamiltonian (2). The first condition reduces to the well-known h 2 z > ∆ 2 + µ 2 in BCS topological superfluids [6,7,[11][12][13][14]31]. The last condition ensures the bulk quasi-particle excitations are gapped to protect the zero energy MFs in the topological regime. The SO coupling and the FFLO vector shift the effective in-plane Zeeman field and the chemical potential. In contrast, in the BCS topological superfluids, the SO coupling strength, although required, does not determine the topological boundaries. Our system therefore provides more knobs for tuning the topological phase transition. To further verify condition (5), we calculate the Chern number in the hole branches C = n C n in the gapped superfluids [38], and confirm C = +1 when Eq. (5) is satisfied and C = 0 otherwise. Here C n = 1 2π d 2 kΓ n is the Chern number, Γ n = −2Im ∂Ψn ∂kx | ∂Ψn ∂ky is the Berry curvature [39], and |Ψ n is the eigenstate of two hole bands of the BdG Hamiltonian (2).  The transition from non-topological to topological phases defined by Eq. (5) can be better understood by observing the close and reopen of the excitation gap E g , which is necessary to change the topology of Fermi surface. In Fig. 2, we plot the change of E g along with the order parameter |∆|, the chemical potential µ, and the FF vector Q as a function of Zeeman fields. For a fixed h x but increasing h z , E g may first close and then reopen (Fig. 2a), signalling the transition from non-topological to topological gapped FFLO superfluids (Q y is finite for all h z , see Fig. 2b). For a fixed h z , the superfluid is gapped and Q y ∝ h x for a small h x (see Fig. 2d), thus any small h x can transfer the gapped BCS superfluids at h x = 0 to FFLO superfluids. However, such a small h x does not destroy the bulk gap of BCS superfluids (topological or non-topological), making gapped topological FFLO superfluids possible when the system is initially in topological BCS superfluids without h x . With increasing h x (Fig. 2c), E g may first close but does not reopen immediately, signalling the transition from gapped FFLO superfluids to gapless FFLO superfluids. For a small h z = 0.2E F , further increasing h x to ∼ 0.78E F , E g reopens again (Fig. 2c), signalling the transition from gapless FFLO to gapped topological FFLO superfluids. In this regime, Q y ∼ 0.6K F , which is not small. For a strong enough Zeeman field, the pairing may be destroyed and the system becomes a normal gas.
The complete phase diagrams are presented in Fig.  3. Since Q y and h x have the same sign, the phase diagram show perfect symmetry in the h x − h z plane. The BCS superfluids can only be observed at h x = 0, hence are not depicted. With increasing SO coupling strength, the topological FFLO phase is greatly enlarged through the expansion to the normal gas phase. For a small SO coupling (Fig. 3a), a finite h z is always required to create the topological FFLO phase; In the intermediate regime (Fig. 3b) we find an interesting parameter regime where the topological FFLO phase can be reached with an extremely small h z around h x ∼ 0.8E F . However, the topological FFLO phase can never be observed at h z = 0, as analyzed before from the Berry curvature and Chern number. From Fig. 3a-b we see that the topological gapped FFLO phase can be mathematically regarded as an adiabatic deformation of the topological BCS superfluids by an in-plane Zeeman field, although their physical meaning are totally different. In Fig. 3cd, we see that the gapless FFLO phase can be observed at small binding energy and small h z , while for large enough binding energy, the system can be either topological or non-topological gapped phases. In this regime, is required to close and reopen E g (see Fig. 3c-d).
The tricritical points marked by symbols in Fig. 3 are essential for understanding the basic structure of the phase diagram. Along the h z axis, the system only supports gapped BCS superfluids (topological or nontopological) and normal gas [31], while along the h x axis the system only supports trivial FFLO superfluids and normal gas [33][34][35]. So the adiabatic connection between the topological BCS superfluids and trivial FFLO phases is impossible, and there should be some points to separate different phases, which are exactly the tricritical points. In our model the transition between different phases is of first-order process. The existence of tricritical point here should be in stark contrast to the tricritical point at finite temperature in the same system without SO coupling, which arises from the accidental intersection of first and second order transition lines [23]. Therefore the tricritical points in Fig. 3 cannot be removed, although their specific positions vary with the system parameters.
Chiral edge modes: The topological FFLO superfluids support exotic chiral edge modes. To see the basic features more clear, we consider the same model in a square lattice with the following tight-binding Hamiltonian, where Here c iσ denotes the annihilation operator of a fermionic atom with spin σ at site i = (i x , i y ). Hereafter, we use t = 1 as the basic energy unit. For more details, see Methods.
In the following, we only present the chiral edge states in the topological gapped FFLO superfluid regime, and assume ∆ i = ∆e iQy iy . We consider a 2D strip with width W = 200, and the results for the strip along x and y directions in the topological FFLO phase are presented in Fig. 4. The linear dispersion of the edge states reads as where L and R define the left and right edges of the strip, and v L and v R are the corresponding velocities. We have also confirmed that the wavefunctions of the edge states  are well localized at two edges. For a strip along the x direction, the particle-hole symmetry as well as the discrete Z 2 symmetry for k x → −k x ensure the eigenenergies of Eq. 6 always come in pairs (E k ,−E k ), thus v R = v L . However, when the strip is along the y direction (parallel to the FFLO momentum Q), the eigenenergies no longer come in pairs, therefore v R = v L . The two chiral edge states with totally different velocities and density of states represent the most remarkable feature of our model. The Chern number C = 1 in our lattice model, thus only one pair of chiral edge states can be observed.
MFs in 1D Chain: Topological FFLO superfluid and associated MFs can also be observed in 1D SO coupled Fermi gas when the Hamiltonian (6) is restricted to 1D chain. In this case, the system is characterized by a Z 2 invariant, which can be determined using the similar procedure as discussed above. The only difference is that now not only k = 0, but also k = π needs be taken into account (see Methods). In Fig. 5a, we see Majorana zeroenergy state protected by a large gap (∼ 0.3t) emerges in a suitable parameter region. The superfluid order parameter (Fig. 5b) has the FF form. The local Bogoliubov quasi-particle operator γ( iσ at the left edge and u 0 iσ = −v 0 * iσ at the right edge (see Fig. 5c). This state supports two local MFs at two edges, respectively [6].

Discussions
Our proposed topological FFLO phase may also be realized using semiconductor/superconductor heterostruc-tures. Recently, topological BCS superconductors and the associated MFs have been proposed in such heterostructures [5][6][7][8] and some preliminary experimental signatures have been observed [11][12][13][14]. To realize a topological FFLO superconductor, the semiconductor should be in proximity contact with a FFLO superconductor, which introduces finite momentum Cooper pairs. The topological parameter region defined in Eq. (5) still applies except that the order parameter, chemical potential and FFLO vector are external independent parameters. The flexibility of Eq. (5) makes it easier for tuning to the topological region with MFs. Because the FFLO state can sustain in the presence of a large magnetic field, it opens the possibility for the use of many semiconductor nanowires with large spin-orbit coupling but small g-factors (e.g, GaSb, hole-doped InSb, etc.).
In summary, we propose that topological FFLO superfluids or superconductors with finite momentum pairings can be realized using SO coupled s-wave superfluids subject to Zeeman fields and they support exotic quasiparticle excitations such as chiral edge modes and MFs. The phase transition to the topological phases depends strongly on all physical quantities, including SO coupling, chemical potential, Zeeman field and its orientations, paring strength, FFLO vector Q and the effective mass of Cooper pairs explicitly, which are very different from topological BCS superfluids/superconductors that are intensively studied recently. These new features not only provide more knobs for tuning topological phase transitions, but also greatly enrich our understanding of topological quantum matters. The topological FFLO phases have not been discussed before, and the phases unveiled in this Letter represent a totally new quantum matter.

Momentum space BdG equations:
The partition function at finite temperature T is where Q is the total momentum of the Cooper pairs and ∆ is a spatially independent constant. Here the position dependent phase of ∆(x) can be gauged out by the transformation ψ σ → ψ σ e iQ·x/2 . Integrating out the fermion field ψ and ψ † , we obtain Z = D∆e −S eff , with effective action S eff = dτ dr |∆| 2 g − 1 2β ln DetβG −1 + Tr(H), where β = 1/T , and G −1 = ∂ τ + H BdG . The order parameter, chemical potential and FFLO vector Q are determined self-consistently by solving the following equation set In our model the deformation of Fermi surface is along the y direction, thus we have Q = (0, Q y ), and only three parameters need be determined self-consistently. We determine the different quantum phases using the following criterion. When E g > 0, ∆ = 0, we have gapped FFLO phases (M = −1 (C = +1) for topological, and M = +1 (C = 0) for non-topological). When there is a nodal line with E g = 0 and ∆ = 0, we have gapless FFLO phases. When ∆ = 0 (then Q = 0 is enforced), we get normal gas phases. It is still possible to observe gapless excitations in the gapless FFLO phase regime, however, we do not distinguish this special condition because gapless excitations are not protected by gaps. In our numerics, the energy and momentum are scaled by Fermi energy E F and its corresponding momentum K F in the case without SO coupling and Zeeman fields. The results in Fig.  2 and Fig. 3 are determined at n = K 2 F /2π and T = 0. Real space BdG equations: In the tight-binding model of (6), the many-body interaction is decoupled in the mean-field approximation. The particle number n iσ = c † iσ c iσ and superfluid pairing ∆ i = −U c i↓ c i↑ are determined self-consistently for a fixed chemical potential. Using the Bogoliubov transformation, we obtain the BdG equation i↑ v n * i↓ f (E n )−u n i↓ v n * i↑ f (−E n )], with f (E) = 1/ 1 + e E/T . In the tight-binding model, FF phase and LO phase can be determined naturally, which depend crucially on the parameters of the system as well as the position of the chemical potential. The results in Fig. 4 and Fig. 5 are obtained at T = 0.
Topological boundaries in lattice models: To determine the topological phase transition conditions, we transform the tight-binding Hamiltonian to the momentum space in Eq. 2. Here ξ k is replaced by −2t cos(k x ) − 2t cos(k y ) − µ for the kinetic energy, and k α by sin(k α ) for the SO coupling, where α = x, y. The topological boundary conditions can still be determined by the Pfaffian of Γ(K) = H BdG (K)Λ at four nonequivalent points, K 1 = (0, 0), K 2 = (0, π), K 3 = (π, 0), K 4 = (π, π) when the system is gapped. At these special points, Γ(k) is a skew matrix. The topological phase is determined by M = 4 i=1 sign(Pf(Γ(K i ))) = −1. For uniform BCS superfluids, the Pfaffian at K 2 and K 3 are identical, thus only K 1 and K 4 are essential to determine the topological boundaries. However, in our system, all four points affect the topological boundaries, and the exact expression of M is too complex to present here. In 1D chain, there are only two nonequivalent points at K 1 = 0 and K 2 = π. We find Pf(Γ(K 1 )) = ∆ 2 − (h z − µ −