Topological superconductivity in monolayer transition metal dichalcogenides

Theoretically, it has been known that breaking spin degeneracy and effectively realizing spinless fermions is a promising path to topological superconductors. Yet, topological superconductors are rare to date. Here we propose to realize spinless fermions by splitting the spin degeneracy in momentum space. Specifically, we identify monolayer hole-doped transition metal dichalcogenide (TMD)s as candidates for topological superconductors out of such momentum-space-split spinless fermions. Although electron-doped TMDs have recently been found superconducting, the observed superconductivity is unlikely topological because of the near spin degeneracy. Meanwhile, hole-doped TMDs with momentum-space-split spinless fermions remain unexplored. Employing a renormalization group analysis, we propose that the unusual spin-valley locking in hole-doped TMDs together with repulsive interactions selectively favours two topological superconducting states: interpocket paired state with Chern number 2 and intrapocket paired state with finite pair momentum. A confirmation of our predictions will open up possibilities for manipulating topological superconductors on the device-friendly platform of monolayer TMDs.

Π ph (p) and particle-particle channel Π pp (p) diverges faster as approaching the low energy limit. These susceptibilities of electrons with spin s and low-energy dispersion s (k) have the form and where spin s, s =↑ / ↓, ω n is the fermionic Matsubara frequency, k and p are momenta, G s (iω n , k) = 1 iωn− s (k) is the non-interacting Green's function, and f ( s k ) is the Fermi function at temerature T .
In general, Π ss pp (p) always diverges logarithmically at total-momentum p = 0 despite the low-energy dispersion s k , which indicates pair-momentum 0 superconductivity if dominates. On the other hand, Π ss ph (p) typically diverges at momentum-transfer p = 0 when the density of states diverges, i.e. near the van Hover singularity, or at some finite momentum-transfer p = Q when the Fermi surface is nested in the particle-hole channel at Q. The former and latter indicate instabilities such as ferromagnetism and density-waves respectively when they each dominates. In a two-pocket system, this requires a hole and an electron pocket to have the same low-energy dispersion (but opposite in energy). In the case where susceptibilities in the two channels diverge equally fast, one needs to further compare their corresponding driving interactions to determine the dominant instability.
In the current lightly p-doped monolayer TMD case, note that both pockets are hole pockets though they have the same low-energy dispersion ↑ (k) = − (k−K) 2 2m and ↓ (k) = − (k+K) 2 2m with respect to their own valley centers K and −K upon low-doping. Thus, the Fermi surface is in fact poorly nested at 2K in the particle-hole channel. To be precise, since ↓ (p) = ↑ (p + 2K), the particle-hole susceptibility has the relation for s =↑ ands = −s. Thus, is not diverging in the low energy limit as long as the density of states on the Fermi surface ν 0 is finite. Therefore, we do not consider particle-hole susceptibilities in this work.
On the other hand, since the Fermi surface is perfectly nested at 2K in the particleparticle channel, the particle-particle susceptibility diverges logarithmically as approaching the low-energy limit E → 0 with Λ being the UV cutoff scale. Note that the Cooper logarithmic divergence does not occur only at the usual p = 0, but also at p = 2K. This indicates that the superconductivity with pair momentum 0 (spatially uniform) and 2K (spatially modulated at 2K) could be equally dominant in the low energy. To determine which is truly more dominant, we need to study their pairing interactions using the RG analysis in the following section.

Supplementary Note 2: Inter-and intra-pocket effecitve interactions
We will calculate the inter-and intra-valley effective interactions g τ,τ (q, q ) and g τ,τ (q, q ) at energy Λ 0 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 channell. Before we start, notice that by omitting the valley index, which is inter-locked with the spin index τ for the low-energy fermions, the inter-and intra-valley interactions in the spin-valley locked two-pocket picture are just the opposite-and equal-spin interactions in a spin-degenerate single-pocket picture. Fortunately, Ref. 1 has already studied the pairing problem in a spin-degenerate single-pocket system under repulsive Hubbard interaction described by Eq. (1) and Eq. (2). Thus, we expect the same result as Ref.

+ (3j)
FIG. 1: Feymann diagrams for the two loop contributions to the inter-and intra-pocket interactions. 1, i.e. the largest attraction occuring in the angular-momentum-one channel, but with a different physical meaning when mapping back to the spin-valley locked two-pocket picture.
To make the mapping between the two pictures explicit, we will follow Ref. 1 to calculate inter (q, q ) and g (0) intra (q, q ) in the two-pocket picture and denote the spin s and valley ±K separately.

Tree level
At the tree level, the on-site repulsion U > 0 only contribute to the inter-pocket interaction because U acts between only electrons with opposite spins due to Pauli exclusive principle. Thus, with the superscript 1 denoting the tree-level contribution [see Supplementary Fig. 1(1a)].
This bare repulsion contributes to only thel = 0 component of g inter because U is independent of the incoming and outgoing momenta q and q . Since this is a perturbative analysis, the inter-pocketl = 0 pairing is suppressed regardless what the higher order contributions tol = 0 channel are. To have any finite contribution to aisotropic channels (l = 0) requires momentum-dependence from loop corrections.

Second order
The U 2 (one-loop) order contributions to inter-and intra-pocket interactions are and . Thus, the one-loop corrections are still momentumindependent and contribute to only thel = 0 channel. This is a consequence of isotropic parabolic dispersion in 2D. 1,2 Note that though g (0),2 intra seems to implyl = 0 intra-pocket pairing, pairings in evenl channels are not allowed since these are equal-spin pairs. Thus, if either the inter-or intra-pocket pairing were to occur at all, the effective attraction has to come from at least two-loop order.

Third order
The U 3 (two-loop) contribution of short-range repulsion for a spin-degenerate rotationalinvariant 2D system with a single pocket and parabolic dispersion has been proven to facilitate p-wave (angular momentum 1) pairing. 1 Since both pockets in p-doped TMDs have the same low-energy effective dispersion which is parabolic upon light doping, we can map this spin-valley locked two-pocket system to the spin-degenerate single-pocket system studied in Ref.1 by bringing the pocket centers K and K' both to k = 0. Thus, we expect to obtain the largest attractions in the partial-wave channell = 1 as well, but the partial-wave channels here are with respect to K and K' instead of Γ. This indicates degenerate inter-and intrapocket pairings withl = 1 after we map back to the two-pocket system. In the following, we will show the calculations of g  Supplementary Fig. 1(3a)-(3j), we can see that the two-loop contributions can be divided into two groups: the ones with one particle-particle and one particle-hole bubble (diagram 3a, 3b, 3g and 3h), and the ones with two particle-hole bubbles (diagram 3c, 3d, 3e, 3f, 3i, and 3j). We first calculate the former contributions to intra-pocket interaction, i.e. diagram (3g) and (3h). In the static limit, where ω n(ñ) is the fermionic Matsubara frequency, G s (iω n , l) = 1 iωn− s (l) is the non-interacting Green's function, t ± ≡ l ± q+q 2 , p ≡ q − q, and q and q are the external incoming and outgoing momenta relative to the valley center K. As the electrons have energy E = Λ 0 0, |q| = |q | ∼ q F with q F being the Fermi momentum of a single pocket. The over-all minus sign results from the closed fermion loop. The particle-particle loop integral can be calculated assuming t ± 2q F , which is the regime where the main momentum-dependence comes from. 1 Here,φ is the angle between the loop momentuml and t + , r −1 0 the UV cutoff for − sin 2φ , and c 0 contains terms independent of t ± . We will drop c 0 in the following since our purpose is to obtain the momentum-dependent part. Plugging Supplementary Eq. (11) back to Supplementary Eq. (10), we obtain the second loop integral where φ is the angle between p and l, 2 ≡ Here, 1 is a small parameter as we assumed t + /q F 1 in the first loop, which corresponds to the regime where the external momenta statisfy p ∼ 2q F and the loop momentum l/q F ∼l 1. Notice that the integral is dominated by the regime of φ where cos φ = O( ) is another small parameter besides . Since we are interested in the portion of scattering amplitude which depends on the external momenta, we will calculate g pp intra (q, q )− g pp intra (p = 2q F ) up to the leading order in the small parameters and cos φ. By keeping the small parameters in the upper and lower limitsl 1/2 while dropping those in the slowly varying logarithmic integrand, we obtain for the regime of external momenta satisfying 1.
The two-loop contributions to intra-pocket interaction involving only particle-hole bubbles, i.e. diagram (3i) and (3j), can be calculated in a similar way. In the same regime where the external momenta satisfy 1, we obtain where the minus sign is due to the closed fermion loop.
We then turn to the inter-pocket interaction. Among all the third-order contributions to g inter , diagram (3e) and (3f) in Supplementary Fig. 1 are both just the product of two second-order corrections and do not contribute to momentum-dependence. Thus, we will focus only on diagram (3a)∼ (3d). Note that similar to the case of intra-pocket interaction, diagram (3a) and (3b) involve vertex corrections from one particle-particle and one particlehole bubble just like (3g) and (3h), while diagram (3c) and (3d) involve corrections from two particle-hole bubbles just like (3i) and (3j). Thus, diagram (3a) and (3b) have similar amplitudes as diagram (3c) and (3d) except the momentum-transfer in the particle-hole bubble and the absence of closed fermion loop: where p ≡ q + q. On the other hand, diagram (3c) and (3d) have the same amplitudes as diagram (3i) and (3j) such that in the regime where is small. After collecting all the contributions, the U 3 corrections to the effective inter-and intra-pocket interactions at E = Λ 0 read and In summary, we have derived the effective inter-and intra-pocket interactions at E = Λ 0 from the bare repulsion U > 0 up to two-loop order: and Inter-pocket pairing where C > 0 and C < 0 are momentum-independent constants.
Supplementary Note 3: The real-space profile of the phase of the intra-pocket pairing wavefunction Since the intra-pocket pairs are spinless and the intra-pocket interaction is attractive in thẽ l = 1 channel, the intra-pocket pairing wavefunction on the spin-up pocket is expected to be ∆ ↑↑ q = ψ K+q,↑ ψ K−q,↑ ∝ q x ± iq y in terms of separate spin and valley indices. The p-wave pairing is expected to be chiral to avoid nodes due to energetics. The pairing wavefunction in real space can then be obtained by doing the following Fourier transform: where r and d = d(cos θ d , sin θ d ) are the center-of-mass and relative positions of the pair repectively, the relative momentum q = q F (cos θ q , sin θ q ) is confined on the circular pocket centered at K with q F being the Fermi momentum, and θ d (θ q ) is the angle between d(q) and K. While the phase winding from e iθ d [see Supplementary Fig. 2(a)] accounts for the q x + iq y pairing symmetry on a pocket, the spatial modulation in phase from e i2K·r [see Supplementary Fig. 2(b)] is a consequence of the finite pair-momentum 2K.