Formation of d-wave superconductivity caused by proximity to the chiral spin liquid

# Formation of d-wave superconductivity caused by proximity to the chiral spin liquid

## Abstract

We investigate pairing instabilities of free electrons caused by the proximity-effect of the chiral spin liquid. We explore a theoretical model where a frustrated spin-half system on a moat-band lattice with degenerate dispersion is immersed in an environment of free fermion bath. Ordering of spins, or condensation of hardcore bosons, is prevented by spontaneous formation of the chiral spin liquid on the moat-band square lattice. We use the spin exchange interaction between spins in the CSL state and the fermionic environment as a starting point to demonstrate, that (i) the Chern-Simons gauge field of the CSL induces a nontrivial effective interaction between fermions; (ii) the fermion interaction further induces a superconducting instability. Modulation of the Chern-Simons flux within the unit cell of the lattice, which is an inherent characteristic of the CSL, gives rise to a d-wave pairing symmetry of the induced superconductor. This microscopic model provides a scenario describing how a quantum spin-liquid state may lead to d-wave superconductivity upon carrier doping. We argue that this effect can be attributed to the pairing interaction induced by the proximity to the quantum spin-liquid.

## I introduction

Since the discovery of superconductivity by Kamerlingh and Onnes in 1911 and later the Bardeen-Cooper-Schrieffer (BCS) theory in 1957 (1), most of the effort has been put to study the mechanism of pairing of electrons. For traditional BCS superconductors (SCs), it is well accepted that it is the electron-phonon interaction that induces an effective attractive force between electrons and leads to the formation of pairs. For high- superconductors, though still being debated, there have been different theories including the dopped Mott insulator (2); (12); (3); (4); (5); (6); (7); (8); (9); (10); (11) and the resonant valence bond (RVB) picture (13); (12); (21); (14); (15); (16); (17); (18); (19); (20), the interlayer-tunneling mechanism(22); (23), the antiferromagnetic spin-fluctuation (24); (25); (26); (27); (32); (31); (30); (29); (28), spin-gap proximity effect mechanism (33), the magnon pairing (34), the strong electron-phonon coupling in transition metal oxides (35); (36) and more.

In this work, we explore the possibility of a new route to generate high superconductivity. Recalling that the quantum spin liquid (QSL) state (13); (21); (37)–describing the disordered state of the frustrated antiferromagets–could give rise to superconductivity after doping (38); (39); (41); (42); (40); (43); (44), it is interesting to study the proximity effect of a QSL state to the doped carrier bath. Even though in realistic materials there is only one kind of electronic state, a model that separately deals with the itinerant carriers and the QSL can be used to capture the essential coupling between the disordered spins and the doped fermionic degrees of freedom. To this end, we put forward a theoretical model that is related to and mimics the doped QSL. Specifically, the model includes two components: one is a chiral spin liquid (CSL) state and the other is free fermionic environment. They interact with each other through spin exchange coupling.

In this work, we are particularly interested in the CSL state (45); (46); (47); (48); (49); (50); (51); (60); (53); (54); (55); (56); (57); (58); (59); (52); (61); (62); (63), suggested by V. Kalmeyer and R. B. Laughlin (64). CSL has been predicted to be likely groundstate in frustrated antiferromagnets, resembling quantum Hall states. Even though numerical evidences have been accumulating in Kagome and honeycomb lattice (65); (66); (67); (68); (69); (70); (71), the general condition under which the frustrated exchange interaction will spontaneously break the time-reversal symmetry (TRS) is still unclear. In Refs. (72); (73), it is argued that a 2D hardcore boson system with a moat-shape band can hardly exhibit condensation at a generic momentum, and a TRS breaking chiral state of fermions coupled to U(1) Chern-Simons gauge field will be more favorable as it has a lower free energy (72). The - XY model on the honeycomb lattice is a vivid example, where the CSL state is predicted due to the emergence of moat band (74), reproducing certain results of numerical simulation using density matrix renormalization group (DMRG) (71), exact diagonalization (ED) (68), and variational Monte Carlo (QMC) (69); (70) techniques.

An interesting topic is to extend the above prediction of CSL state to the square lattice, being compatible with the lattice structure of high- SCs, and to consider the effect of the CSL state on the nearby free fermionic environment. Remarkably, in a series of publications by X.G. Wen it has been conjectured that a possible CSL state (75) based on Heisenberg model in square lattice may emerge(76); (77); (78); (79); (80). In this work, we focus on the XY model on the square lattice and show the possible emergence of a CSL state due to strong frustration when up to third neighbor spin-spin interactions are taken into account resulting in the moat band degeneracy of hardcore bosons. Then, we find interesting results after considering its coupling to an environment system of free fermions. The results are summarized as follows: (1) Coupling of disordered half-spins in the CSL state to fermions leads to an interesting proximity effect. It induces an effective local fermion-fermion interaction mediated by the Chern-Simons (CS) gauge of the CSL. (2) This induced interaction favors superconductivity of the fermions in the intermediate regime of the induced CS magnetic fluxes. (3) The self-consistent mean-field solution of the CSL state determines the specific value of the CS flux, which lies in the parameter regime favoring superconductivity. (4). The CS gauge field further modulates the pairing potential and gives rise to a d-wave () pairing symmetry of the induced SC. The above results is quite general and applicable to any frustrated spin-1/2 system supporting a moat-band for the hardcore bosons. The interaction with a free fermion bath is assumed here through spin exchange. These results may potentially be applicable to cuprate SCs and also to heavy fermion SCs.

## Ii Moat band and CSL on the square lattice

The model we study contains two sub-systems. One is a antiferromagnetic (AFM) XY model in square lattice with a local spin for each site. The other is the environment of a fermionic bath, say electrons. The AFM model is immersed in the electrons gas which is also defined in square lattice. We can envision a two-layer total system, with the top and bottom layer being the spin model and the electron gas, respectively. To describe the coupling between them, we consider the s-d spin exchange interaction between electrons and the local spin degrees of freedom in spin model. The total Hamiltonian reads as , where

 Hafm=J1∑r,j(SxrSxr+ej+SyrSyr+ej)+J2∑r,j(SxrSxr+aj+SyrSyr+aj)+J3∑r,j(SxrSxr+bj+SyrSyr+bj), (1)

where we consider an isotropic model in XY plane without losing any generality. , and are the nearest neighbour (NN), next nearest neighbour (NNN) and the next next nearest neighbour (NNNN) bond vectors in square lattice. describes the electron bath with free nearest hopping in the square lattice, and

 Extra open brace or missing close brace (2)

where is the Pauli matrix describing the true spin degrees of freedom of electrons. Since we consider a XY model for the upper layer and no component is involved, we also focus on a XY exchange coupling between electrons and local spins in . One can make generalization of the above XY models to Heisenberg models, but it is not the main focus of the current work.

We first consider only. For spin-half case, Eq.(1) can be mapped directly to a hardcore boson model. For the XY model considered here, the hardcore bosons are noninteracting, and we can obtain their energy spectrum straightforwardly. We explicitly show in Fig.1 the hardcore boson dispersion for different parameters. As shown, with increasing while keeping fixed (), an energy minimum loop degeneracy, namely the moat band, will gradually occur which are centered around in lattice momentum space. For , the moat shows up and gradually grow larger for larger . Besides, the moat band does not change much for a large region of around . Therefore, even though we focus mainly on in this work, the same conclusions can be expected for a large parameter region of .

Along the moat, there is an infinite degeneracy at the energy (with being the energy of the moat) and the single particle density of states diverges near the moat bottom as , highlighting the similarity with a 1D hard-core boson system where the groundstate should be given by the the Tonks-Girardeau gas of free fermions (81); (82); (83); (84). The moat band in 2D was also found in honeycomb lattice, where both analytic (72); (74); (73) and numerical results (68); (69); (70); (71) show the absence of hardcore boson condensation and the emergence of a TRS-breaking spinless fermionic states, namely the CSL.

Rather than presenting a strict investigation of the groundstate of the -- XY model (85), in this work, we only focus on the problem of how a CSL affects the nearby fermionic bath, and therefore we construct one type of CSL mean-field state that is compatible with the symmetry requirement and is also energetically more favorable, with a finite self-consistent solution of order parameters for some parameter range. Then, we focus on the interaction between this CSL state and nearby electron bath.

To this end, we firstly proceed with the lattice version of CS transformation (86); (87); (88); (89); (90), where the spin raising/lowering operator is mapped to a spinless fermion attached to string operator ,

 S±r=f±r^U±r, (3)

where , where the summation runs over all sites of lattice. operators obey fermionic commutation rules and represent spinless fermions, whose number operators is given by . As will be clear later, since the fermions is intrinsically coupled to a CS gauge field, we term them the CS fermions.

Inserting the CS representation to Eq.(1), the Hamiltonian is cast into the form,

 H=J1∑rjf†rfr+ejei∑~rϕ~r,rr+ejn~r+J2∑rjf†rfr+ajei∑~rϕ~r,rr+ajn~r+J3∑rjf†rfr+bjei∑~rϕ~r,rr+bjn~r, (4)

where is defined as , i.e., the angle of the bond seen by the site at . The phase accumulation around a generic loop centered at site can be obtained . Accordingly, we define the flux . Then the total path integral action is captured by a spinless CS fermion gas coupled to fluctuating CS gauge field,

 S=∫dt[∑rf†r(i∂t−A0r)fr+12π∑rA0rBr−J1∑rjf†rfr+ejeiAr,r+ej−J2∑rjf†rfr+ajeiAr,r+aj−J3∑rjf†rfr+bjeiAr,r+bj], (5)

where we have defined the gauge field on bond as . is the Lagrangian multiplier field introduced to satisfy the “Stocks theorem” of CS gauge fields (74). acts the same as the time component of the four-vector CS field. In the following, we focus on half-filling case, and consider a mean-field ansatz that describes a CSL state, as is predicted by the above moat band dispersion.

For half-filling, we do not expect the uniform CSL state (73) stabilized for small filling factor but expect the non-uniform CSL state similar to that in honeycomb lattice at half-filling (74). For the simplest case with , the planar Néel order favors a -flux semimetal phase of CS fermions (91); (92), generating A and B two sublattices. Nonzero and will introduce further modulation of hopping terms between the two sublattices. To this end, we can separate the nearest bond gauge field into a uniform component and a fluctuating component, . The uniform gauge field gives rise to the -flux state in the square lattice. If one firstly do not consider the Brillouin zone (BZ) folding effect due to CS gauge fields on further neighbours, it is convenient to separate symmetric and antisymmetric combinations of gauge fields between A and B sites, belong to the same unit cell: and . In a closed trajectory that includes a unit cell (A and B site), the total flux threading through it is at half-filling. Therefore, similar to the honeycomb lattice (74), it is easy to see that the homogeneous symmetric components can be gauged out at half filling due to the periodicity. Due to above analysis, only the antisymmetric gauge fields on neighbour bonds, i.e., and (and thus ), remain. Fig.2 shows a mean-field ansatz of a possible CSL state with the NNN and NNNN CS flux attachment. As shown in Fig.2(a), for NNN hopping along the (reversed) direction of each arrows on NNN bonds, the CS fermions acquire a phase accumulation (). Similarly in Fig.2(b), the NNNN hopping generates the phase, .

The mean-field ansatz in Fig.2(a)(b) further enlarges the unit cell and introduces the folding of BZ. In the enlarged unit cell, the total flux is still gauge equivalent to zero due to the periodicity. As seen in Fig.2(b), the CS flux threading in the square (BCBC) that centered at H site is , and the flux in each triangle in the side (ACB) is . This gives rise to a staggered chiral state of CS fermions. For , the flux state displays two inequivalent Dirac cones. After the folding of BZ, one Dirac cone is shifted to point and the other is located at , as is shown in Fig.2(c). For generic values of and , the TRS and parity symmetry are broken. For example, in the simplest case with and , one can see directly that the accumulated CS phases in any triangular or square plaquette are all gauge nontrivial and are variant under timer reversal. This further gaps out the CS fermions spectrum. After turning on and , the Dirac cones are gapped out for a generic values of and , generating a fully gapped insulating phase of CS fermions as depicted in Fig.2(d). Even though the above theory is based on mean-field approximation of string operator where the fluctuation of particle number is neglected, the resulting full gap state with the breaking of parity and TRS symmetry give confidence that the mean-field theory is a stable description of the CSL up to NNNN CS gauge field in square lattice. In the following, using the constructed CSL as an example, we focus on the main problem in this work, i.e., how a CSL state in square lattice affects nearby electron baths (a more strict numerical study on the ground state of XY model will be presented in a later study).

## Iii Superconducting instability due to proximity to CSL

### iii.1 Induced effective interaction

Before employing a self-consistent mean-field approximation to determine values of phases and , we firstly shift our attention to the proximity effect to CSL. We start from a model of fermions coupled to the U(1) CS gauge field that can describe the CSL state,

 Scsl=∫dt[∑rf†r(i∂t−A0r)fr+12π∑rA0rBr−∑r,r′Jr,r′f†rfr′eiAr,r′+H.C.], (6)

here the second term in the bracket is equivalent to a gauge invariant full Chern-Simons term (the difference is a boundary term) which can be temporarily put aside when considering the proximity effect, as it does not contain any matter fields. Eq.(5) is a specific example of general Eq. (6) corresponding to the square lattice. Then we consider the coupling between the CSL and the bottom layer of the electron bath. In the CSL state, the original spin degrees of freedom is melted down to fractional excitations. After choosing an appropriate representation, we mathematically mapped to the CS fermions attached to fluctuating gauge field. Hence, the CS representation also has to be inserted into the spin operators in the coupling term in Eq.(2), which gives rise to,

 Hc=Jc∑rc†rαcrβ(σ+αβfrU−r+σ−αβf†rU+r). (7)

Since the CS fermion operator comes from CSL state where a full gap and a full Chern-Simons action stabilizes the mean-field state and suppresses the fluctuation of gauge field (arising from the string operator), the mean-field approximation in the string operators can be inherited in , i.e., we require in and define . Since the total action is bilinear in terms of , we can exactly integrate out the CS fermions in Eq.(6) and (7). Then apart from the free electron bath, the total action is cast into,

 S=−J2c∫dt∑rr′c†rασ+αβcrβ^Vr,r′c†r′ρσ−ρσcr′σ, (8)

where , with . Since is the exchange interaction of neighbouring spins which is orthogonal to the onsite term with , the above action can be decomposed into two terms. In the frequency space, the first term describes a onsite retarded interaction between electrons which reads as,

 SO=J2c∫dωdω′dν∑r×[1ν−A0rc†r↑(ω)cr↑(ω′−ν)c†r↓(ω′)cr↓(ω+ν)]. (9)

The second term includes all the neighbouring exchange interactions between electrons, which arises due to the self-energy correction from the nearby CSL. It is of the form:

 SN=−∫dωdω′dν∑r,r′×[J2cJr,r′e−2iAr,r′c†r↑(ω)cr′↑(ω′−ν)c†r′↓(ω′)cr↓(ω+ν)]. (10)

and are the two effects received by electrons from the nearby CSL. is a Hubbard-type onsite retarded interaction. It is due to the microscopic process where an electron perturbs a local CS fermion at site and at time , and then the perturbed CS fermion in turn interacts with the local electron at and at a later time . Since the two couplings takes place at the same site, no phase accumulation would occur in as in the CSL state. For , the two coupling processes take place at different spatial site and . A phase will be accumulated due to the propagation of the CS fermion from to . Moreover, as known from , when an electron interacts with CS fermions at site and followed by the reversed process at another site , the two string operators at and will combine and form another phase term . Therefore, the total CS phase accumulated is , which is associated to the induced neighbouring exchange interactions between electrons, Eq.(10). In the following, we focus on the weak coupling regime, . After making Fourier transformation of Eq.(9) to time space and then neglecting the retarding effect by letting , Eq.(9) then gives rise to a non-negative weak Hubbard interaction between electrons. In the weak-coupling regime and without considering any coincidences like Fermi surface nesting, this term may only trivially renormalize the bare Green’s function of electron bath and therefore is not our focus in this work. We focus on the latter term , which, after Fourier transformation, gives rise to a instantaneous spin exchange interaction between electron at and , namely,

 VN=∑r,r′J2cJr,r′e−2iAr,r′c†r↑cr′↑c†r′↓cr↓. (11)

Even though being weak-coupling, there is a nontrivial CS phase factor modulated the interaction, . These phase factors, which are determined intrinsically by the CSL mean-field state, suggest that the induced interaction could give rise to instability of the original free electron baths.

### iii.2 Superconductivity on a square lattice

A brief inspection of Eq.(11) results in the following observation: If the phase term between a link reaches , then a negative real component of interaction will emerge. The effective negative interaction between electrons may favor the formation of Cooper pairing between electrons at and at . Therefore, in the following, we consider the mean-field theory of the induced interaction in the particle-particle channel. In this section, we temporarily regard the CS phases and to be tuning parameters, as this will give rise to a more general phase diagram. In the later section, we will combine the self-consistently determined values of and to examine the realistic superconducting phases that could take place.

Application of the discussed method of induced interaction to initially noninteracting fermions on the square lattice yields the following effective Hamiltonian,

 H=−t∑⟨r,r′⟩c†rσcr′σ+H.C.+V1∑⟨r,r′⟩e−2iAr,r′c†r↑c†r′↓cr↓cr′↑+V2∑⟨⟨r,r′⟩⟩e−2iAr,r′c†r↑c†r′↓cr↓cr′↑+V3∑⟨⟨⟨r,r′⟩⟩⟩e−2iAr,r′c†r↑c†r′↓cr↓cr′↑, (12)

where we have considered the first, second, and third neighbor interaction terms of XY model on a square lattice, and with . are the CS gauge field inherited from the CSL state, depicted in Fig.3(a)(b). Then we solve the Hamiltonian using self-consistent mean-field theory. In the particle-particle channel, we introduce Hubbard-Stratonovich bosonic field, for pairing term on the nearest bond , and similarly for pairing on and , we define them as and , respectively. Further neglecting the fluctuation of the Hubbard stratonovich field, we consider constant mean-field orders with for the NN and NNN pairing, with . For the NNNN pairing, as the CS flux is not uniform and there are no NNNN gauge fields on and bonds, we introduce two order parameters defined respectively as following, and . Then, the mean-field Hamiltonian is obtained as,

 H=−t∑⟨r,r′⟩c†rσcr′σ+H.C.+V1∑⟨r,r′⟩Δ1e−2iAr,r′cr↓cr′↑+H.C.+V2∑⟨⟨r,r′⟩⟩Δ2e−2iAr,r′cr↓cr′↑+H.C.+V3∑⟨⟨⟨r,r′⟩⟩⟩Δ3e−2iAr,r′cr↓cr′↑+H.C.+V3∑⟨⟨⟨r,r′⟩⟩⟩′Δ′3cr↓cr′↑+H.C.−V1Δ21∑⟨r,r′⟩e2iAr,r′−V2Δ22∑⟨⟨r,r′⟩⟩e−2iAr,r′−V3Δ23∑⟨⟨⟨r,r′⟩⟩⟩e−2iAr,r′−8NV3Δ′23, (13)

where denotes the sum over the NNNN bonds and . Inserting the CS-flux in Fig.3(a)(b) and then transforming the Hamiltonian to lattice momentum space and then minimizing the free energy of the ground state, we can self-consistently determine the order parameters , , and . For , we do not find any finite self-consistent solution for the nearest pairing , i.e., . Interestingly enough, nonzero self-consistent solutions for () can be found for any finite value of and as long as the CS phases and are located in intermediate region. Eight different phases are found and we denote them by , with . and represents that whether and , respectively. For example the denotes the trivial order with . The calculated phase diagram is shown in Fig.4. It is found that, in order to lower the ground state free energy, for , becomes nonzero, for , becomes finite, and for , develops a finite value. The finite values of the pairing potentials spontaneously breaking the global symmetry.

The above results can be simply understood as following. For , the CS phase factor on the NNN bond , , has the phase angle between and . Moreover the total interaction term on the bond can be decomposed into the real and imaginary part, whose strength is respectively given by and . In mean-field level, the imaginary part has a canceling effect due to the Hermitian counterpart, while the real part plays important role in generating SC instability. For , the real component becomes negative and therefore an effective attractive interaction between electrons at and is generated, giving rise to superconductivity in the electron bath. The reason is the same for the NN and the NNNN pairing term. Whereas, since the total CS phase on two subsequent bonds and is equivalent to the CS phase on . The fluctuating phase on the NN bond is equal to half of that in the NNNN bond, i.e., (while the uniform component is gauge equivalent to zero). Therefore, the condition for effective attraction between nearest bond electrons becomes , leading to the region that supports finite solution of .

The above picture is based on microscopic second-quantized formalism, whereas, a straightforward classical interpretation for the attraction between electrons can be attributed to the Ampere force generated by the CS gauge field. As is shown in Fig.2(a)(b), the CSL state exhibits a staggered CS flux in adjacent plaquette. The opposite flux boosts electrons and generates their movement in opposite direction (clockwise and anticlockwise) in two adjacent plaquette. Therefore, on the same bond, electrons from adjacent plaquette acquire momentum and move in the same direction, which further generates an attractive Ampere force, responsible for the pairing between electrons.

## Iv Self-consistent solution for CS-fluxes and the nature of d-wave pairing

### iv.1 Solution for the CS flux

We have shown that the CSL state can generate a CS gauge field-modulated interaction in the nearby electron bath, and moreover, the interaction is able to induce a SC instability in a large parameter region in terms of the gauge flux and , leading to the phase diagram in Fig.3. Therefore, the remaining questions is whether the gauge flux in the CSL indeed lies in the SC region. This can be understood by further considering the action of CSL state, whose general form reads as,

 Scsl=∫dt[∑rf†r(i∂t−A0r)fr+12π∑rA0rBr−∑r,r′Jr,r′f†rfr′eiAr,r′+H.C.]. (14)

To determine the gauge flux and , which should be solely dependent on the upper layer CSL state, we can temporality turn off the coupling . Then the above remaining action is bilinear in terms of CS fermions . After integrating out the CS fermions exactly, the fermionic free energy functional can be obtained as , where is a field dependent on CS flux and . The mean-field equations are obtained by taking variation of and , i.e., and .

We still consider the mean-field ansatz in Fig.2(a)(b) as an example. Let us firstly examine the terms with CS Lagrangian multiplier . In the enlarged unit cell, after expanding every term, the Lagrangian multiplier terms gives a “staggered chemical potential” of CS fermions (74). This can be seen clearly as following. As has been discussed before, for a closed loop whose the CS flux is gauge equivalent to zero, we can gauge out the symmetric CS gauge field. The square plaquette (formed by NNN bonds) centered at A and at B contribute a staggered CS flux whose total contribution is gauge equivalent to zero. Therefore, we define the symmetric gauge field component and the antisymmetric component , so that we have . Similarly, we have other terms as , . Hence, we can define and . Then, the CS term can be expanded taking into account each site in the unit cell. The NNN bonds contribute the following CS term as

 Misplaced & (15)

where the terms with is canceled. Similarly, the NNNN bonds gives rise to the CS terms,

 12π∑riA0irBir=12π∑rA0(BAr−BHr)+A0(BCr−BBr)=8NA0′γπ. (16)

The above CS terms generate constant terms in the free energy functional of CS fermions and are important to determining the self-consistent solution of and . The functional can be obtained via the energy spectrum of the CS fermions, i.e., , where the sum of means the sum of all the filled bands. Finally, by minimizing , we can self-consistently determine the CS phases and , which are essential as their values are directly related to the resulting SC phase in the nearby electron baths (Fig.3). We calculated and for different parameters ( is fixed to ). The results are shown in Fig.4. No finite order parameter are found until reaches a value around . This is quantitatively in accordance with the moat band feature of the XY model, which emerges around . With increasing , increases and crosses at , while decreases and crosses the line around . For large (when it is comparable to ) the CS flux and changes quite slowly.

The inset in Fig.4 shows the trajectory of as a function of when is increased from 0 to 1.05. From this trajectory, one can directly know what phases the nearby electron bath will be formed by comparing it to the phase diagram in Fig.3. We plot the obtained trajectory in the phase diagram in Fig.3. We replot the trajectory in the inset of Fig.4 into the phase diagram in Fig.3, as depicted by the orange data points. We clearly observe that, with increasing , the trajectory of CS phases will go through three different states. The first is the trivial Fermi liquid state for small . The second is the SC state where the NNN electron pairing is induced by the proximity effect of the CSL. This phase is stabilized in , corresponding to the curve between A and B boundary point. The third phase is the SC state with both NNN and NNNN pairing amplitudes, which is stable for .

### iv.2 Pairing symmetry of the induced superconductor

Now we turn our attention to study the properties of the induced SC in the bottom layer. As it is clear from Fig.3, only two SC phases are supported by the realistic self-consistent solution of the CS phases, and . The SC state in is relatively simple since only is nonzero. We firstly focus on this region, and we denote and to be A and B point, corresponding to the two boundary points in the phase in Fig.3.

Before proceeding, we firstly give a coarse analysis of the pairing symmetry by examining Eq.(13). For the state, we have and and being zero. The Hamiltonian contains the term . This term shows that even though being constant, it is further modulated by the CS gauge field , after absorbing which, acquires spatial dependence. After transforming to the lattice momentum space, the -dependent pairing potential indicates a nonuniform pairing symmetry. As a simple analysis, to consider the state, we forget about the NNNN interaction. Moreover, we firstly neglect the folding effect of the BZ due to the third neighbour CS gauge field. In this case, one can readily extract the pairing potential term on NNN bonds by transforming to -space. After inserting the CS gauge field on NNN bond (Fig.2(a)), we can show that the dominant pairing term enjoys the analytic form in -space as . It is easily seen that when , , giving rise to a d-wave () pairing symmetry, while for , is reduced to , generating a s-wave pairing symmetry and a full gap at small . In between the two values of , a mixed s-wave wave and d-wave SC is obtained.

We plot the module and the phase angle of in Fig.5. As is clearly seen, at boundary A (Fig.3) with , the module of the pairing potential has gapless nodes along and (the red curve in (a)) while the phase angle has a difference between adjacent quadrants (the red curve in (b)). The sign change of the pairing potential clearly exhibits the d-wave feature. When gradually deviates from , a s-wave component will grow and mix with the d-wave component which gradually is decreased, as is depicted by the purple curve in Fig.5. Finally, at , i.e., the B boundary point in Fig.3, the s-wave pairing completely replace the d-wave symmetry.

The above analysis gives us the predicted pairing symmetry without considering the effect of BZ folding due to the NNNN CS gauge attachment. Now we take into account this effect and examine how the folding of BZ will modify the pairing symmetry. To gain realistic knowledge of the pairing symmetry of the induced SC, we have to firstly obtain the self-consistent determined amplitude , the CS phases and , and a more realistic value of the exchange coupling strength . The most straightforward way to uncover the pairing symmetry of is to extract the realistic energy spectrum of the SC state by numerically looking into Eq.(13). The energy spectrum contains both the contribution from the pairing term and the kinetic energy of the normal state. Therefore, we can extract the pairing term by artificially setting . Through this way, we monitor the evolution of the energy spectrum (at ) as one gradually varies and tunes the system from A to B boundary point along the trajectory in the state in Fig.3. For each point along the trajectory (corresponds to a value of from to ), we self-consistently obtain all the needed ingredients and calculated the energy spectrum, which is shown in Fig.6. As is shown, the order parameter decreases monotonously with . Interestingly, different from the earlier results without considering the BZ folding, for both A and B boundary of the state, we obtain gapless energy spectrum along and . For any generic parameter point along the trajectory, the spectrum is also qualitatively the same. All data point clearly show the symmetry of the induced SC state. Therefore, we can claim that the SC state is a d-wave SC generated by the proximity effect of the CSL. The d-wave pairing symmetry is not only an effect of the modulation of the CS gauge field, but also a result of the BZ folding due to NNNN CS flux attachment.

Last, as shown in Fig.3, the tail of the trajectory also enters into the phase where both NNN and NNNN pairing amplitude are finite. However, our self-consistent calculation show that is much less than up to realistic parameter value . Therefore, the NNNN pairing induced by the CSL state is negligible compared to the NNN component. Moreover, it can be straightforwardly verified that the NNNN pairing is also d-wave. But rather than the symmetry of , it displays the pairing symmetry. Therefore, we identify the induced state to be a SC with mixed pairing symmetry, including a dominant and a small portion of pairing amplitude, and we identify the state to be a pure SC.

## V discussion

We have investigated a theoretical model where a frustrated magnetic spin-half system, which supports a underlying moat band of hardcore bosons, is immersed in an environment of free fermion bath. The spin system and the environment can exchange their spin through the s-d coupling. We demonstrated that as long as there is a moat shape emerge in some parameter region, the hard core bosons condensation could be energetically ruled out by a TRS-breaking CSL state. We showed that the CSL state has a proximity effect on the fermionic environment in the sense that it is able to induce local CS gauge field-modulated interactions in the fermionic bath, which are further able to generate a SC instability. We present a classical interpretation where the Ampere force generated by the staggered CS flux in the adjacent square plaquette is responsible for the pairing of electrons.

The interesting features and discussion are in the following. First, we found that the CS phases and determined self-consistently indeed lie in the SC instability region. However, this conclusion is not lattice independent. We have also checked the self-consistent solution on honeycomb lattice (see also Ref.(74)), the CS flux is too small and cannot generate a SC phase. This is an interesting observation and may provide some understanding on high- SC which only are discovered in material with square lattice structure. Second, interestingly, the pairing symmetry of the induced SC is found to be d-wave. This is not only due to the modulation of the CS gauge field but also due to the folding effect of BZ. Third, it is interesting to note that one can also construct other mean-field ansatz that breaks the parity and TRS. Our theory on the SC instability is, in some sense, general, as the formalism of the proximity effect and the mean-field theory of the SC instability do not rely on the specific mean-field ansatz. However, the specific pairing symmetry and the pairing amplitude should be dependent on the mean-field ansatz. Nevertheless, we expect that for a large class of CSL mean-field ansatz, a d-wave SC instability could be the most favorable state as long as the CS gauge fields display a similar two-in two-out pattern at a certain lattice site (see Fig.2). Fourth, in general, our theory should be applicable to any frustrated system with a moat band of hardcore bosons which is coupled to additional carriers. This might be related to cuprate, where we believe a moat band of hardcore bosons might also emerge as a result of the frustrated further neighbour hoppings. Fifth, our model may also be relevant to the SC phase in the heavy fermions system (93); (94), where there indeed occurs a realistic s-d coupling between charge carriers and the local moment.

###### Acknowledgements.
Rui Wang acknowledges Lei Hao, Sudeshna Sen for useful discussion. This work was supported by the National Key R&D Program of China (Grant No. 2017YFA0303200), and by NSFC under Grants No. 11574217 and No. 60825402. T.A.S. acknowledges startup funds from UMass Amherst.

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