## Introduction

The quest for material realizations of topological chiral superconductors with nontrivial Chern numbers1,2,3,4 is fuelled by predictions of exotic signatures, such as Majorana zero modes and quantized Hall effects. Unfortunately, natural occurrence of bulk topological superconductors are rare with the best candidates being superfluid 3He (ref. 5) and Sr2RuO4 (ref. 6). Instead, much recent experimental progress relied on proximity-inducing pairing to a spin–orbit-coupled band structure building on the proposal of Fu and Kane7. Their key insight was that a paired state of spinless fermions is bound to be topological and that the surface states of topological insulators are spinless in that the spin degeneracy is split in position space (r-space): the two degenerate Dirac surface states with opposite spin textures are spatially separated. Nevertheless, despite much experimental progress along this direction8,9,10,11,12,13,14, the confinement of the helical paired state to the interface of the topological insulator and a superconductor limits experimental access to its potentially exotic properties.

Another type of exotic paired states that desires material realization is the finite-momentum-paired states, which has long been pursued since the first proposals by Fulde and Ferrell15 and by Larkin and Ovchinnikov16. Most efforts towards realization of such modulated superconductors17,18, however, relied on generating finite-momentum pairing using spin-imbalance under a (effective) magnetic field in close keeping with the original proposals. Exceptions to such a spin-imbalance-based approach are refs 19, 20 that made use of spinless Fermi surfaces with shifted centres. More recently, there have been proposals suggesting modulated paired states in cuprate high-Tc superconductors21,22,23. However, unambiguous experimental detection of a purely modulated paired state in a solid-state system is lacking.

We note an alternative strategy that could lead to pairing possibilities for both topological and modulated superconductivity: to split the spin degeneracy of fermions in momentum space (k-space). This approach is essentially dual to the proposal of Fu and Kane, and it can be realized in a time-reversal-invariant non-centrosymmetric system when a pair of Fermi surfaces centred at opposite momenta ±k0 consist of oppositely spin-polarized electrons (see Fig. 1a). When such a spin-valley-locked band structure is endowed with repulsive interactions, conventional pairing will be suppressed. Instead, there will be two distinct pairing possibilities: interpocket and intrapocket pairings, where the latter will be spatially modulated with pairs carrying finite centre-of-mass momentum ±2k0.

What is critical to the success of this strategy is the materialization of such k-space-split spinless fermions. A new opportunity has arisen with the discovery of a family of superconducting two-dimensional (2D) materials, monolayer group-VI transition metal dichalcogenides (TMDs) MX2 (M=Mo, W, X=S, Se)24,25,26,27. Although the transition metal atom M and the chalcogen atom X form a 2D hexagonal lattice within a layer as in graphene, monolayer TMDs differ from graphene in two important ways. First, TMD monolayers are non-centrosymmetric, that is, inversion symmetry is broken (see Fig. 1b,c). As a result, monolayer TMDs are direct-gap semiconductors28 with a type of Dresselhaus spin–orbit coupling29,30 referred to as Ising spin–orbit coupling31. This spin–orbit-coupled band structure leads to the valley Hall effect30,32, which has established TMDs as experimental platforms for pursuing valleytronics applications30,32,33,34,35,36. Our focus, however, is the fact that there is a sizable range of chemical potential in the valence band that could materialize the k-space spin-split band structure we desire (see Fig. 1d). Second, the carriers in TMDs have strong d-orbital character and, hence, correlation effects are expected to be important. Interestingly, both intrinsic and pressure-induced superconductivity have been reported in electron-doped (n-doped) TMDs24,25,26,27 with the debate regarding the nature of the observed superconducting states still on-going37,38,39,40,41.

Here we propose to obtain k-space-split spinless fermions by lightly hole-doping (p-doping) monolayer TMDs such that the chemical potential lies between the two spin-split valence bands. We investigate the possible paired states that can be driven by repulsive interactions42 in such lightly p-doped TMDs using a perturbative renormalization group (RG) analysis going beyond mean-field theory38,43. We find two distinct topological paired states to be the dominant pairing channels: an interpocket chiral (p/d)-wave paired state with Chern number |C|=2 and an intrapocket chiral p-wave paired state with a spatial modulation in phase. The degeneracy can be split by the trigonal warping or Zeeman effect.

## Results

### Spin-valley-locked fermions in lightly p-doped monolayer TMDs

The generic electronic structure of group IV monolayer TMDs is shown in Fig. 1d. The system lacks inversion symmetry (see Fig. 1b,c), which leads to a gapped spectrum and a Sz-preserving spin–orbit coupling. Such Ising spin–orbit coupling31 acts as opposite Zeeman fields on the two valleys that preserve time-reversal symmetry. Furthermore the spin–orbit coupling is orbital-selective44 and selectively affects the valence band with a large spin-split29.

By lightly p-doping the TMDs with the chemical potential μ between the spin-split valence bands, spin-valley-locked fermions can be achieved near the two valleys (see Fig. 2a,b). Assuming negligible trigonal warping at low doping, we can use a single label τ=↑,↓ to denote the valley and the spin. Denoting the momentum measured from appropriate valley centres ±K by q, the kinetic part of the Hamiltonian density is

where μ is the chemical potential, m is the effective mass of the valence band and cq,↑ψK+q,↑ and cq,↓ψK+q,↓ each annihilates a spin-up electron with momentum q relative to the valley centre K or a spin-down electron with momentum q relative to the valley centre−K (see Fig. 2a). Hence, the spin-valley-locked two-valley problem is now mapped to a problem with a single spin-degenerate Fermi pocket. Nonetheless, the possible paired states with total spin τz=±1 and τz=0 in fact represent the novel possibilities of intrapocket modulated pairings with total τz=±1 and interpocket pairing with total τz=0, respectively (see Fig. 2a,b).

### Pairing possibilities

To discuss the pairing symmetries of the two pairing possibilities, it is convenient to define the partial-wave channels with respect to the two-valley centres ±K. Since a total spin τz=±1 intrapocket pair consists of two electrons with equal spin, Pauli principle dictates such pairing to be in a state with odd partial-wave . Stepping back to microscopics, such pairs carry finite centre-of-mass momentum ±2K and form two copies of phase-modulated superconductor15. This case may or may not break time-reversal symmetry due to the absence of locking between the s of the two pockets τ=↑,↓. For the total τz=0 interpocket pairing, the allowed symmetries of a superconducting state is further restricted by the underlying C3v symmetry of the lattice. In particular, the absence of an inversion centre allows the pairing wavefunction in each irreducible representation to be a mixture between parity-even and -odd functions with respect to the Γ point45. Specifically, s-wave mixes with f-wave and d-wave mixes with p-wave (see Fig. 2c,d). Among the irreducible representations of C3v, two fully gapped possibilities are the trivial A1 representation, which amounts to (s/f)-wave pairing () and a chiral superposition of the 2D E representation, which amounts to a mixture of p±ip and did pairing (). The mixing implies that the non-topological f-wave channel that is typically dominant in trigonal systems as a way of avoiding repulsive interaction will be blocked together with s-wave by the repulsive interaction in the p-doped TMDs. Hence, it is clear that the pairing instability in channel is all one needs for topological pairing in the p-doped TMDs.

### Two distinct topological paired states

To investigate the effects of the repulsive interactions between transition metal d-orbitals, we take the microscopic interaction to be the Hubbard interaction, which is the most widely studied pardignamtic model of strongly correlated electronic systems

where W is the ultraviolet energy scale, U>0 and ni,s is the density of electrons with spin s on site i. By now, it is well established that the interaction that is purely repulsive at the microscopic level can be attractive in anisotropic channels for low-energy degrees of freedom, that is, fermions near Fermi surface. The perturbative RG approach has been widely used to demonstrate this principle on various correlated superconductors. For the model of p-doped TMDs defined by Equations (1) and (2), the symmetry-allowed effective interactions at an intermediate energy scale close to the Fermi level in the Cooper channel (see Supplementary Note 1) would be:

where q and q′ are the incoming and outgoing momenta. Now, the remaining task is to derive the effective inter- and intrapocket interactions g↑,↓(q,q′) and g↑,↑(q,q′) perturbatively in the microscopic repulsion U and check to see whether attraction occur in the channel (see Methods and Supplementary Note 2).

Before going into the details of calculation, it is important to note that isotropic pairing with is forbidden by Pauli principle in the total τz=±1 channel and blocked by the bare repulsive interaction in the total τz=0 channel. Hence, we need to look for attraction in the anisotropic channel, which is given by the momentum-dependent part of gττ(0). With our assumption of isotropic dispersion at low-doping, one needs to go to the two-loop order to find momentum dependence in the effective interaction. Fortunately, it has been known for the model of Equations (1) and (2) that effective attraction is indeed found in anisotropic channels at the two-loop order46. Here we carry out the calculation explicitly (see Methods and Supplementary Note 2) and find the effective interactions in the channel to be attractive, that is,

for τ,τ′=↑,↓, where is the angle associated with the momentum transfer, and is the normalized angular-momentum-one eigenstate in 2D.

In the low-energy limit, the effective attractions in the channel at the intermediate energy scale Λ0 in Equation (4) will lead to the following two degenerate topological paired states (see Methods): the interpocket (p/d)-wave pairing, which is expected to be chiral (see Fig. 3a) and the modulated intrapocket pairing (see Fig. 3b). The degeneracy is expected for the model of Equations (1) and (2) with its rotational symmetry in the pseudo spin τ. There are two ways this degeneracy can be lifted. First, the trigonal warping will suppress intrapocket pairing as the two points on the same pocket with opposing momenta will not be both on the Fermi surface any more (see Fig. 3c). On the other hand, a ferromagnetic substrate will introduce an imbalance between the two pockets, which promotes intrapocket pairing47 (see Fig. 3d).

## Discussion

The distinct topological properties of the two predicted exotic superconducting states lead to unusual signatures. The interpocket paired state (see Fig. 3a) is topological with Chern number |C|=2 because of the two pockets (see Methods). The Chern number dictates for two chiral edge modes, which in this case are Majorana chiral edge modes each carrying central charge (1)/(2) (refs 1, 48). This is in contrast to d+id paired state on a single spin-degenerate pocket, which is another chiral superconducting state49,50,51,52,53,54 with four chiral Majorana edge modes. An unambiguous signature of two Majorana edge modes in the interpocket chiral paired state will be a quantized thermal Hall conductivity1 of

at temperature T, where c=1 is the total central charge. In addition, signatures of the chiral nature of such state could be revealed by a detection of time-reversal symmetry breaking in polar Kerr effect and muon spin relaxation measurements. Finally, a sharp signature of anisotropy of the pairing will be the maximization of the critical current in a direct current superconducting quantum interference device (dc SQUID) interferometry set-up of Fig. 4a at some finite flux Φmax≠0.

The intrapocket paired state (see Fig. 3b) is not only topological, but also its phase of the gap is spatially modulated with ei2K˙r and ei2K˙r for spin-up and -down pairs, respectively, where r is the spatial coordinate of the centre-of-mass of the pair (see Supplementary Note 3). Since the gaps on the two pockets are not tied to each other in principle, the system may be either helical respecting time-reversal symmetry (C=0) or chiral (C=2). Either way, there will be a Majorana zero mode of each spin species at a vortex core so long as τz is preserved. What makes the intrapocket paired state distinct from existing candidate materials for topological superconductivity, however, is its spatial modulation. Smoking gun signature of the modulation in phase would be the halved period (hc)/(4e) of the oscillating voltage across the dc SQUID set-up in Fig. 4b in flux Φ due to the difference between the pair momenta on the two sides of the junction. Another signature of the intrapocket paired state will be the spatial profile of the modulated phase directly detected with an atomic resolution scanning Josephson tunnelling microscopy23,55.

In summary, we propose the k-space spin splitting as a new strategy for topological superconductivity. Specifically, we predict lightly p-doped monolayer TMDs with their spin-valley-locked band structure and correlations to exhibit topological superconductivity. Of the monolayer TMDs, WSe2 may be the most promising as its large spin-splitting energy scale56 allows for substantial carrier density within the spin-valley-locked range of doping57. The rationale for the proposed route is to use a lower symmetry to restrict the pairing channel. The merit of this approach is clear when we contrast the proposed setting to the situation of typical spin-degenerate trigonal systems. With a higher symmetry, trigonal systems typically deals with the need for anisotropic pairing due to the repulsive interaction by turning to the topologically trivial f-wave channel2,49. The n-doped TMDs whose low-energy band structure is approximately spin-degenerate fall into this category. Hence, experimentally realized superconductivity in n-doped systems would likely be topologically trivial even if the superconductivity is driven by the same repulsive interaction we consider here. The predicted topological paired states in p-doped TMDs are a direct consequence of the spin-valley locking, which breaks the spin degeneracy in k-space and creates two species of spinless fermions. Experimental confirmation of the predicted topological superconductivity in p-doped TMDs will open unprecedented opportunities in these highly tunable systems.

## Methods

### Perturbative RG calculation

For the RG calculation, we follow the perturbative two-step RG procedure in ref. 49, which has been used to study superconductivity in systems such as Sr2RuO4 (ref. 58) and generic hexagonal lattices with spin degeneracy49. Taking the Hubbard on-site repulsion in Equation (2) as the microscopic interaction, the first step is to integrate out higher-energy modes and obtain gτ,τ(0) in Equation (3), the low-energy effective interactions in the Cooper channel at an intermediate energy close to the Fermi level. The second step is to study the evolution of these effective interactions as the energy flows from Λ0 to 0, which is governed by the RG equations.

In the first step, we calculate the inter- and intrapocket effective interactions ginter(0)(q,q′)≡(q,q′) and gintra(0)(q,q′)≡gτ,τ(0)(q,q′) in terms of the incoming and outgoing momenta q and q′ order by order in U until we obtain attraction in one of them in certain partial-wave channel . Following ref. 46, we find the effective interactions to be (see Supplementary Note 2)

and

where p=q±q′ is the momentum transfer, C>0 and C′<0 are momentum-independent constants coming from tree level and one-loop order, and the momentum-dependent terms come solely from two-loop order.

Each partial-wave component is given by the projection of ginter/intra(0)(q,q′) on to the normalized angular momentum eigenstate in 2D, , where is the angle associated with the momentum transfer p. We find

where Hn is the nth harmonic number and α≡(U3m2)/(2π3) and β≡(U3m2)/(64π3) are postive constants related to density of states and interaction strength. Here terms with α and β come from contributions with one particle–particle and one particle–hole bubble, and two particle–hole bubbles, respectively (see Supplementary Note 2). The α term in acquires an extra minus sign on top of from the closed fermion loops in Supplementary Fig.1 (3g) and (3h). Meanwhile, the α term in contains an implicit factor because of the fact that the outgoing external momenta in Supplementary Fig. 1 (3a) and (3b) are exchanged, which is equivalent to setting .

Note that with even s is forbidden since intrapocket pairs have equal spin, and that for odd s since they correspond to the spin-triplet states with τz=0 and ±1, respectively. While λinter(0),0>0 as expected from the bare repulsion, the most negative values are .

In the second step, we derive and solve the RG equations to study the evolutions of the effective interactions as the energy E lowers from Λ0 to 0. Using in Equation (8) as the initial values for the RG flows, the channel with the most relevant attraction in the low-energy limit E→0 is the dominant pairing channel. Under the assumption that the energy contours for 0<E0 are isotropic, different partial-wave components do not mix while the inter- and intrapocket interactions with the same can in principle mix. By a procedure similar to that in refs 50, 59, we find the RG equations up to one-loop order to be

and

where the inverse energy scale is the RG running parameter, d1(y)≡(δΠppss(±2K))/(δ y), d2(y)≡, and d3(y)≡(δΠphss(0))/(δ y). Here Πpp/phss(k) is the non-interacting static susceptibility at momentum k in the particle–particle or particle–hole channel defined in Supplementary Note 1. Since the low-energy band structure is well nested at ±2K in the particle–particle channel, the Cooper logarithmic divergence appears not only at k=0 but also ±2K (see Supplementary Note 1). Thus, d1(y)=1. On the other hand, since the low-energy band structure is poorly nested at any k in the particle–hole channel and is far from van Hove singularity, the particle–hole susceptibilities do not diverge in the low-energy limit (see Supplementary Note 1). Thus, in the low-energy limit y→∞. Therefore, with logarithmic accuracy, the inter- and intrapocket interactions renormalize independently with the well-known RG equation in the Cooper channel

with i=inter, intra. The RG flow , which solves the RG equation, shows that the pairing interaction in channel becomes a marginally relevant attraction only if the initial value . Since we concluded that the most negative initial values occur in the channels for both inter- and intrapocket interactions in the first step of the RG procedure, we expect degenerate inter- and intrapocket pairings in the low-energy limit.

### The Chern number of interpocket paired state

The interpocket chiral paired state becomes just a spinful p+ip paired state with total spin τz=0 when we map the spin-valley-locked two-pocket problem to a spin-degenerate single-pocket problem. The spinful p+ip pairing comprises two copies of ‘spinless’ p+ip pairings as the Bogoliubov-de Gennes Hamiltonian of the former can be written as

where the low-energy dispersion ɛq=−(q2)/(2m)−μ, the gap function and cq≡(cq,↑ ±cq,↓ )/. Since a spinless p+ip paired state has Chern number C=1, where with , the τz=0 spinful p+ip paired state in the single-pocket system has C=2. Hence, the interpocket chiral pairing in the two-pocket system has C=2 as well.

### Data availability

The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Information file.