A Basis functions

# Cooperon condensation and intra-valley pairing states in honeycomb Dirac systems

## Abstract

Motivated by recent developments in the experimental study of superconducting graphene and transition metal dichalcogenides, we investigate superconductivity of the Kane-Mele (KM) model with short-range attractive interactions on the two-dimensional honeycomb lattice. We show that intra-valley spin-triplet pairing arises from nearest-neighbor (NN) attractive interaction and the intrinsic spin-orbit coupling. We demonstrate this in two independent approaches: We study superconducting instability driven by condensation of Cooperons, which are in-gap bound states of two conduction electrons, within the -matrix approximation and also study the superconducting ground state within the mean-field theory. We find that Cooperons with antiparallel spins condense at the and points. This leads to the emergence of an intra-valley spin-triplet pairing state belonging to the irreducible representation A of the point group . The fact that this pairing state has opposite chirality for and identifies this state as a “helical” valley-triplet state, the valley-analog to the He-B phase in two dimension. Because of the finite center of mass momentum of Cooper pairs, the pair amplitude in NN bonds exhibits spatial modulation on the length scale of lattice constant, such that this pairing state may be viewed as a pair-density wave state. We find that the pair amplitude spontaneously breaks the translational symmetry and exhibits a -Kekulé pattern. We also discuss the selection rule for pairing states focusing the characteristic band structure of the KM model and the Berry phase effects to the emergence of the intra-valley pairing state.

###### pacs:
74.78.-w,74.20.-z,74.70.Wz

## I Introduction

Since the discovery of graphene, electronic properties of atomically thin two-dimensional (2D) materials have attracted wide-spread interest. Indeed remarkable features arise through the interplay of spin and valley degrees of freedom in the unusual band topology. Among other properties also superconductivity has been studied, despite great experimental difficulties in sample preparation and doping, particularly in graphene as well as transition metal dichalcogenides (TMDs). Superconductivity has been observed in Li-decorated monolayer graphene (1), ion gated MoSe, MoTe, WS (2), ion gated MoS (3); (4), and monolayer NbSe (5). In addition to their potential impact on applications, the superconducting states in such 2D materials also stimulate theoretical studies. Although the superconducting state observed in Li-decorated monolayer graphene is most likely due to conventional BCS-pairing arising from enhanced electron-phonon coupling by the adatoms (6), various exotic superconducting states have been suggested for pure and doped graphene (7); (8); (9); (10); (11); (12); (13). Furthermore, unconventional Ising pairing protected by spin-valley locking is predicted for the superconducting state in NbSe atomic layers (5) and ion-gated MoS (3); (4).

Motivated by these experimental advances, we investigate superconductivity in the 2D honeycomb lattice structure that is common to graphene and TMDs. Our main purpose in this paper is to analyze the structure of the superconducting phase in the honeycomb lattice with special emphasis on topological aspects. For this purpose, we employ the Kane-Mele (KM) model (14) that was proposed as a minimal model of topological insulators (15); (16). We assume generic short-range attractive interactions and discuss the symmetry of superconducting ground states. In contrast to most studies on superconductivity our starting point will be the insulating state where we explore the pairing states that could arise through Cooperon condensation for sufficiently strong pairing interactions. As we will discuss below a particularly interesting case of unconventional Cooper pairing appears for nearest-neighbour (NN) attractive interaction.

The two possible pairing states on the honeycomb lattice considering the valley-structure of the electronic bands are illustrated in Fig. 1: Inter-valley pairing state and intra-valley pairing state. The former is the simple BCS pairing state involving electrons with opposite momenta in the different valleys near the and points. In contrast, electrons form pairs within the same valley in the latter case. Namely, they have opposite momenta with respect to or points, and, therefore, an electron pair has finite center of mass momentum equivalent to and , respectively. Because of the finite center of mass momentum of Cooper pairs, this pairing state may be viewed as a pair-density wave (PDW) state (17), in which the gap function spatially modulates on length scales of the lattice constant. The possibility of the intra-valley pairing has been pointed out in graphene (9); (12) as well as in doped Weyl semimetals (18).

In this paper, we show that the intra-valley spin-triplet pairing state can arise due to the interplay of the NN attractive interaction and the intrinsic spin-orbit (SO) coupling in the KM model. The interesting feature of the intra-valley pairing state is that it involves two gap functions associated with Cooper pairs condensed at each of the two valleys, and point (see Fig. 1). In the intra-valley spin-triplet pairing state, the gap functions have both the components of and -wave symmetry in the vicinity of and , and constitute a parity-mixed superconducting state, as we will show. We demonstrate the emergence of this exotic superconducting state by employing two independent microscopic approaches: We first study superconducting instability in the insulating state within the -matrix approximation, and then we examine the most stable superconducting state within the mean-field (MF) theory. In the former, we find that bound states of two conduction electrons called “Cooperons” (19); (20); (21); (22) are formed within the band gap and the intra-valley pairing state is preempted by condensation of Cooperons at the and points at the same interaction strength. We also discuss the origin and nature of the intra-valley pairing state. We find that it may arise due to the Berry phase effects associated with the Dirac points, i.e., and points.

The paper is organized as follows: In Sec. II, we describe the system and the model. In Sec. III, we discuss the selection rule for pairing states based on the characteristic feature of the energy band. In Sec. IV, we study formation of Cooperons and their condensation in the topological insulating state. In Sec. V, we study the superconducting ground state within the MF theory and discuss its various aspects. We conclude in Sec. VI.

## Ii Model

We study the KM model (14) with short-range attractive interaction on the honeycomb lattice depicted in Fig. 2. The Hamiltonian reads

 H = HKM+Hint, (1) HKM = −t∑⟨i,j⟩∑σ(c†iσcjσ+h.c.)−μ∑i,σniσ (2) −it′∑⟨⟨i,j⟩⟩∑σ,σ′νij(σz)σσ′c†iσcjσ′, Hint = −U∑ini↑ni↓−V∑⟨i,j⟩ninj, (3)

where annihilates an electron at site with spin , the chemical potential, and denotes the summation over all the NN/next-nearest-neighbor (NNN) sites. The first term in Eq. (2) describes the NN hopping and the third term the intrinsic SO coupling (14), where () is the Pauli matrix of electron spin and (-1) if electrons make a left (right) turn to get to the site from the site . We consider the on-site and NN attractive interactions in Eq. (3) and assume .

Turning to -space, we introduce

 ciσ=1√M∑kckσe−ik⋅ri , (4)

where is the total number of unit cells that is half of the total lattice sites . The KM Hamiltonian (2) in momentum space reads

 Unknown environment '% (5)

where, , , and . Here, () annihilates an electron on the A (B) sublattice with momentum and spin . and () are the bond vectors that connect the NN sites and NNN sites, respectively, as shown in Fig. 2. We set the lattice constant unity ().

The dispersion relations of the conduction and valence bands are obtained by diagonalizing Eq. (5) as

 E=±√|γk|2+ζ2k=±ϵk. (6)

is approximated in the vicinity of the point as

 γK+p≃vF(px−ipy), (7)

and the point as

 γK′+p=γ∗K−p≃−vF(px+ipy), (8)

where denotes momentum measured relative to the and points (, , ). Here, we introduced the Fermi velocity . Thus, at half-filling () without the SO coupling (), the conduction and valence bands have linear dispersions that describe massless Dirac fermions in the vicinity of the and points.

On the other hand, the diagonal elements in Eq. (5) are approximated as

 ζK+p≃ΔSO,ζK′+p≃−ΔSO, (9)

where (we assume throughout the paper). The dispersion in the vicinity of the and points at half-filling is given by

 E=±√v2Fp2+Δ2SO . (10)

Figure 3 schematically shows the dispersion (10) that has the energy gap at the and points. Thus, the low-energy physics is dominated by massive Dirac fermions.

The effective Hamiltonian at half-filling linearized in the vicinity of the and points reads

 HKM=∑pψ†Kp(vFτ⋅p+ΔSOσzτz)ψKp Missing or unrecognized delimiter for \left (11)

where and () is the Pauli matrix of sublattice-pseudospin. Precisely at the or point, since the off-diagonal terms vanish, Eq. (11) is diagonalized in the sublattice basis. This means that the wave functions at the bottom of the conduction band and the top of the valence band localize on either A or B sublattice. Figure 3 shows the sublattices assigned to them. It exhibits a peculiar character of the wave function in momentum space: The sublattices assigned to the and points are different within the same band. This implies that the insulating state due to the SO coupling described by the KM Hamiltonian (2) does not reduce to the trivial band insulator with decoupled A and B sublattices in the limit of large energy gap . Thus, it is topologically distinct from the trivial band insulator (14).

In this peculiar insulating state, the spin Hall conductivity is quantized, which is characterized by the topological number called spin Chern number. The nonzero spin Chern number guarantees the existence of the helical edge modes that are predicted by the bulk/boundary correspondence (14); (15); (16).

Note that the Berry phase of Bloch electrons associated with adiabatic evolution around the and points in momentum space has opposite signs. In particular, for a massless Dirac fermion (), the Berry phase of conduction band upon going around the and points are and , respectively. This feature plays a crucial role in the emergence of the intra-valley pairing state, as we will discuss in Sec. IV.

## Iii Selection rule for pairing states

The special character of the wave function of the KM model described in the last section enables us to identify possible pairing states induced by the local attractive interactions which we choose to be of density-density type to avoid any bias on the spin configuration. On the other hand, through the choice of sublattices we select at the outset different sublattice pseudo-spin configurations.

Figure 3 implies that in the inter-valley pairing state two conduction electrons in different valleys form a pair. With the on-site attractive interaction electrons pair on the same sublattice with opposite spins, while the NN interaction couples electrons on different sublattices and favors pairing with parallel spins. On the other hand, in the intra-valley pairing state the NN interaction prefers opposite spins. The same applies to two holes in the valence band.

We can extend the above observation further to more general attractive interactions to derive the following selection rule: If the attractive interaction dominantly works between electrons (holes) on the same sublattice, it induces inter-valley pairing of electrons (holes) with opposite spins or intra-valley pairing with parallel spins. If the attractive interaction dominantly works between electrons (holes) on different sublattices, it causes inter-valley pairing of electrons (holes) with parallel spins or intra-valley pairing with opposite spins.

Indeed, the on-site attractive interaction naturally induces the inter-valley pairing, i.e., the conventional spin-singlet -wave BCS pairing. In contrast, the NN attractive interaction induces the unconventional intra-valley pairing state with mixed parity, as we will see in the next section.

## Iv Cooperon condensation

In an insulator, superconducting fluctuation due to attractive interaction leads to formation of Cooperons within the band gap and a superconducting instability could be driven by condensation of Cooperons (19); (20); (21); (22). In this section, to verify the selection rule of the previous section from a microscopic approach, we study formation and condensation of Cooperons in the topological insulating state at half-filling based on the tight-binding Hamiltonian (1).

The Green’s function in a matrix form in the sublattice-pseudospin space is given by

 ^Gσ(k,~t−~t′)=−⟨T~tψkσ(~t)ψ†kσ(~t′)⟩, (12)

where denotes imaginary time. The Green’s function for spin-up electrons in momentum space reads

 ^G↑(k) = 1iωn−(ζkγkγ∗k−ζk)+μ (13) = ^Pk↑iωn−ϵk+μ+^Qk↑iωn+ϵk+μ, ^Pk↑ = (u2kukvkeiθkukvke−iθkv2k), (14) ^Qk↑ = (v2k−ukvkeiθk−ukvke−iθku2k), (15)

where is the fermionic Matsubara frequency and . and are defined as

 uk = √12(1+ζkϵk),vk=√12(1−ζkϵk). (16)

The Green’s function for spin-down electrons can be obtained by substituting in as

 ^G↓(k) = 1iωn−(−ζkγkγ∗kζk)+μ (17) = ^Pk↓iωn−ϵk+μ+^Qk↓iωn+ϵk+μ, ^Pk↓ = (v2kukvkeiθkukvke−iθku2k), (18) ^Qk↓ = (u2k−ukvkeiθk−ukvke−iθkv2k). (19)

Note that the phase factor in the off-diagonal elements is associated with the flip of the sublattice-pseudospin.

The interaction Hamiltonian (3) in momentum space reads

 Hint = 12M∑k,k′,q∑σ,σ′∑τ,τ′gττ′σσ′(k′−k) (20) ×c†kτσc†−k+qτ′σ′c−k′+qτ′σ′ck′τσ , gττ′σσ′(k) = −Uδσ′,¯σδτ,τ′ (21) −V[δτAδτ′Bf∗(k)+δτBδτ′Af(k)],

where annihilates an electron with momentum and spin at sublattice , denotes opposite spin of , and .

We employ the -matrix approximation that describes the superconducting instability due to pair formation. The Bethe-Salpeter (BS) equation for the -matrix approximation diagrammatically represented in Fig. 4 is given by

 Γτ1τ2,τ3τ4σσ′(k,k′;q)=Γ0τ1τ2,τ3τ4σσ′(k,k′) −1βM∑k′′,ω′′n∑ν5,ν6gτ1τ2σσ′(k′′−k′)Gτ1τ5σ(k′′) ×Gτ2τ6σ′(q−k′′)Γτ5τ6,τ3τ4σσ′(k′′,k′;q), (22)

where is the vertex part. In lowest-order, it reduces to the bare interaction:

 Γ0τ1τ2,τ3τ4σσ′(k,k′)=δτ1,τ3δτ2,τ4gτ1τ2σσ′(k′−k). (23)

We denote and , where is the bosonic Matsubara frequency. Hereafter in this section, we restrict ourselves within the insulating state at half-filling and set .

### iv.1 On-site attractive interaction

We first set to examine pairing due to the on-site attractive interaction. In this case, Eq. (22) greatly simplifies to

 ^Γ(q)=−U^I+U^Π(q)^Γ(q), (24) Πτ1τ2(q)=1βM∑k,ωnGτ1τ2σ(k)Gτ1τ2¯σ(q−k), (25)

where , , and . Eq. (24) is easily solved:

 ^Γ(q)=(−U)(^I−U^Π(q))−1. (26)

From the condition for to have poles,

 det[^I−U^Π(q,Ω)]=0, (27)

we obtain the energy spectrum of Cooperons.

Figures 5 (a) and (b) show the energy spectrum of Cooperons obtained by solving Eq. (27). They illustrate the formation of Cooperons below the edge of the two-particle continuum. Any small induces Cooperons below the continuum. The on-site attractive interaction boosts the formation of a Cooperon bound state, particularly, in the vicinity of the point at which the dispersion has its minimum. This implies that the inter-valley pairing of two electrons is energetically favorable.

The minimum energy gap at the point () is plotted as a function of in Fig. 6. progressively decreases as is increased and the Cooperon softens and eventually reaches zero energy at the point for the critical strength , as shown in Fig. 5 (b) indicating an instability. The condensation of Cooperons at the point leads to the proliferation of Cooper pairs with zero total momentum, i.e., the inter-valley pairing state. Thus, the conventional -wave spin-singlet superconducting state is realized due to the on-site attractive interaction.

### iv.2 NN attractive interaction

We next set and examine pairing due to the NN attractive interaction. Since vanishes if or , the nonzero matrix elements of are those with , , , and . Thus, Eq. (22) can be rewritten in a matrix form as

 ^Γσσ′(k,k′;q)=^Γ0(k,k′) −1M∑k′′^Γ0(k,k′′)^πσσ′(k′′;q)^Γσσ′(k′′,k′;q). (28)

Here, we define

 ^Γσσ′(k,k′;q)=(ΓAB,ABσσ′(k,k′;q)ΓAB,BAσσ′(k,k′;q)ΓBA,ABσσ′(k,k′;q)ΓBA,BAσσ′(k,k′;q)), (29) ^πσσ′(k;q) =1β∑ωn(GAAσ(p)GBBσ′(q−k)GABσ(p)GBAσ′(q−k)GBAσ(p)GABσ′(q−k)GBBσ(p)GAAσ′(q−k)), (30) ^Γ0(k,k′)=−V3∑i=1^mik^mi†k′, (31)

where

 ^mik=(e−ik⋅δi00eik⋅δi). (32)

We then obtain

 ^Xiσσ′(k;q)=^X0iσσ′(k;q)+V3∑j=1^Πijσσ′(q)^Xjσσ′(k;q), (33)

where

 ^Xiσσ′(k;q)=1M∑k′^mi†k′^πσσ′(k′;q)^Γσσ′(k′,k;q), (34) ^X0iσσ′(k;q)=1M∑k′^mi†k′^πσσ′(k′;q)^Γ0(k′,k), (35) ^Πijσσ′(q)=1M∑k^mi†k^πσσ′(k;q)^mjk. (36)

Eq. (33) can be further cast into the following form

 ~Xσσ′(k;q)=~X0σσ′(k;q)+V~Πσσ′(q)~Xσσ′(k;q), (37) ~Xσσ′(k;q)=⎛⎜ ⎜ ⎜⎝c^X1σσ′(k;q)^X2σσ′(k;q)^X3σσ′(k;q)⎞⎟ ⎟ ⎟⎠, (38) Unknown environment '% (39) ~Πσσ′(q)=⎛⎜ ⎜ ⎜⎝^Π11σσ′(q)^Π12σσ′(q)^Π13σσ′(q)^Π21σσ′(q)^Π22σσ′(q)^Π23σσ′(q)^Π31σσ′(q)^Π32σσ′(q)^Π33σσ′(q)⎞⎟ ⎟ ⎟⎠. (40)

Then, Eq. (37) can be solved by

 ~Xσσ′(k;q)=[~I−V~Πσσ′(q)]−1~X0σσ′(k;q). (41)

The condition for the matrix to have poles is given by

 det[~I−V~Πσσ′(q)]=0. (42)

Figures 5 (c) and (d) show the energy spectrum of Cooperons obtained by solving Eq. (42). Multiple branches of Cooperons appear below the edge of the continuum, because the spin-orbit coupling breaks the rotational symmetry in spin space and lifts the degeneracy between Cooperons with different spin configurations. Figure 5 (c) illustrates that a bound state of electrons with opposite spins appears in the vicinity of the point for any . The dispersion is symmetric under a rotation of 60 degrees, so the bound state forms also in the vicinity of the point. On the other hand, electrons with parallel spins form a bound state in the vicinity of the point. This difference between pairs of electrons with parallel and opposite spins can be qualitatively understood by the selection rule in the previous section. Namely, the formation of Cooperons at the point corresponds to the inter-valley pairing and at the and points to the intra-valley pairing.

As is increased, the minima of the dispersions of Cooperons decrease progressively and the condensation of Cooperons with opposite spins first takes place at the and points simultaneously, as shown in Fig. 5 (d). If is increased further, Cooperons with parallel spins condense at the point.

Figure 6 shows the gap of Cooperons with opposite spins at the point () as well as that of Cooperons with parallel spins at the point () as functions of . indeed indicates that the NN interaction favors formation of Cooperons in the vicinity of the point. The fact that for a fixed and the critical value at the onset of the Cooperon condensation is smaller than in Fig. 6 also shows that the NN attractive interaction is more effective than the on-site attractive interaction for pair formation. The condensation of Cooperons with opposite-spin configuration at the point leads to the spin-triplet intra-valley pairing state, as we will see in the next section.

The two branches within the same spin configuration in Figs. 5 (c) and (d) correspond to singlet and triplet states of sublattice-pseudospin, whose energy splitting increases as increases as shown in the figures. In the limit of , restoring the SU(2) symmetry in spin space, each of the upper and lower branches becomes doubly degenerate for different spin configurations and there remain two branches of Cooperon bound states.

### iv.3 Berry phase effects

In this subsection, we illustrate Berry phase effects on the Cooperon condensation at the and points and the intra-valley pairing. For simplicity, we set and .

The Green’s function, which is independent of electron spin without the SO coupling (), reads

 ^G(k) = 1iωn−(0γkγ∗k0) (43) = 12(1eiθke−iθk1)iωn−ϵk+12(1−eiθk−e−iθk1)iωn+ϵk.

The diagonal and off-diagonal elements of diagrammatically shown in Fig. 7 are given by

 πττ(k;q) = 142(ϵk+ϵq−k)(ϵk+ϵq−k)2−(iΩn)2, (44) π12(k;q) = ei(θk−θq−k)42(ϵk+ϵq−k)(ϵk+ϵq−k)2−(iΩn)2, (45)

where . The phase factor of the off-diagonal elements arises from the exchange of electrons in different sublattices as described in Fig. 7(b).

We consider intra-valley pairing and set . Assuming the momenta of paired electrons are in the vicinity of the point, i.e., , , and by linearizing in the momentum , the phase factor in the off-diagonal elements reduces to

 ei(θk−θq−k)=eiπ=−1. (46)

The phase factors compensate each other such that the off-diagonal elements of remain finite. This leads to the interference of the direct and exchange processes in Figs. 7 (a) and (b). As a result, the condition of poles (42) with reduces to

 1M∑pvFp4v2Fp2−(iΩn)2=16V. (47)

Evaluating the critical value of the interaction strength for Cooperon condensation with , we obtain

 Vc=8πvF3√3pc, (48)

where is a momentum cut-off.

For comparison, we consider now the inter-valley pairing and set . Assuming and linearizing by , the phase factor reduces to

 ei(θk−θq−k)=e2iϕp, (49)

where is the polar angle of in the plane. The cancelation of phase factors is absent in this case because of the opposite signs of the Berry phase around and . The integration over yields vanishing off-diagonal elements of , so the condition (42) with reduces to

 1M∑pvFp4v2Fp2−(iΩn)2=13V. (50)

Setting , we find the interaction strength for Cooperon condensation as

 V′c=16πvF3√3pc=2Vc. (51)

The critical interaction strength for the onset of the inter-valley pairing is twice as large as that of the intra-valley pairing.

In comparison with the above two cases, we conclude that the interference of the direct and the exchange processes for the intra-valley pairing lowers the energy of Cooperons and yields the Cooperon condensation at and . This is consistent with the observation in the previous subsection that the two branches of Cooperon correspond to sublattice-pseudospin singlet and triplet states for , which arise as an interference effect between the direct and exchange processes in Fig. 7. Note that the same mechanism indeed works for the Cooperon condensation at and in the case of due to the phase factors in the off-diagonal elements of the Green’s functions in Eqs. (13) and (17).

## V Mean-field theory

In the previous section, we demonstrated that Cooperons composed of electrons with opposite spins condense at and , if the NN attractive interaction dominates. This implies the emergence of the intra-valley pairing state in the superconducting phase. In this section, we examine this unconventional superconducting ground state of the KM model with the NN attractive interaction within a mean-field theory. We confirm that the Cooperon condensation at the and points indeed leads to the intra-valley pairing state. We use the mean field approach to elucidate some remarkable properties of this state. To simplify the discussion we set and assume only the NN attractive interaction throughout this section.

The NN interaction in momentum space can be written in a standard form (23) as

 Hint = 12∑k,k′,q∑τ1∼τ4∑σ1∼σ4Vτ1τ2τ3τ4σ1σ2σ3σ4(k,k′,q) (52) ×c†kτ1σ1c†−k+qτ2σ2c−k′+qτ3σ3ck′τ4σ4,

where denotes the center of mass momentum of electron pairs. The matrix element of the interaction reads

 Vτ1τ2τ3τ4σ1σ2σ3σ4(k,k′,q) = −V2M[(f(k−k′)δABBA+f(−k+k′)δBAAB)(δ↑↑↑↑+δ↑↓↓↑+δ↓↑↑↓+δ↓↓↓↓) (53) −(f(k+k′−q)δABAB+f(−k−k′+q)δBABA)(δ↑↑↑↑+δ↑↓↑↓+δ↓↑↓↑+δ↓↓↓↓)].

Here, we define and . The matrix element satisfies the following relations due to the fermionic anticommutation relations:

 Vτ1τ2τ3τ4σ1σ2σ3σ4(k,k′,q)=−Vτ2τ1τ3τ4σ2σ1σ3σ4(−k+q,k′,q) =−Vτ1τ2τ4τ3σ1σ2σ4σ3(k,−k′+q,q). (54)

The condensation of Cooperons at and with the same interaction strength implies the emergence of two distinct condensates of electron pairs with and . To describe these condensates, we introduce the two mean-field gap functions with total momenta () as

 Δτ1τ2σ1σ2(k;Ks) = ∑k′∑τ3,τ4∑σ3,σ4Vτ1τ2τ3τ4σ1σ2σ3σ4(k,k′,Ks) (55) × ⟨c−k′+Ksτ3σ3ck′τ4σ4⟩,

where we denote and . From Eq. (54), the gap functions are antisymmetric with respect to exchange of fermions

 Δτ1τ2σ1σ2(k;Ks)=−Δτ2τ1σ2σ1(−k+Ks;Ks). (56)

We also set the components of the gap functions for equal spins to be zero: , because only Cooperons with opposite spins condense in the presence of the SO coupling. Thus, the non-vanishing components of the gap functions are

 ΔAB↑↓(k;Ks) =