state induced by antiferromagnetic order in 𝜅 -type organic conductors

,


I. INTRODUCTION
The relationship between magnetism and superconductivity has been of major interest in the field of strongly correlated electron systems.In particular, a coexistent state of the two orders can symptomize unconventional superconductivity since conventional BCS-type superconductors do not favor magnetism, resulting in an exotic state.For example, UGe 2 [1], UCoGe [2], and URhGe [3] are known as representative spintriplet superconductors coexisting with a ferromagnetic (FM) order [4].Another example is symmetry-protected nodal superconductivity in antiferromagnetic (AFM) systems [5], e.g., UPd 2 Al 3 [6] and Sr 2 IrO 4 [7].
Interestingly, theoretical studies showed that Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductivity [8,9] with finite center-of-mass (COM) momentum of the Cooper pairs can be realized under an odd-parity magnetic multipole order [7,[10][11][12], such as in the AFM state in locally noncentrosymmetric systems [13,14].In this case, the odd-parity magnetic quadrupole order breaks both spatial inversion () and time-reversal () symmetries, but preserves the combination of them.As a result, the energy band structure is asymmetric for the momentum flip  → −, while up and down spins or pseudospins are degenerate at any  point [Fig.1(a)].Note that the symmetry property is different from that in typical AFM systems, where magnetic translation symmetry (the combination of time-reversal and half-transition) is conserved.
The paper is organized as follows.In Sec.II we explain the two-dimensional (2D) tight-binding Hamiltonian of -(ET) 2  with an AFM molecular field, and show the symmetry and band structure of the model.In Sec.III we analyze an effective model with intraband attractive interactions as a first step, which helps us intuitively understand the AFM-induced FFLO superconductivity.Next, we discuss a more realistic situation, namely, a repulsive Hubbard model, and analyze it based on a fluctuation-exchange (FLEX) approximation and Eliashberg theory (Sec.IV).Finally, a discussion and a summary are given in Secs.V and VI, respectively.

II. MODEL AND ELECTRONIC BAND STRUCTURE
First, we introduce a 2D tight-binding model under the AFM ordering of  = 0 [17,[77][78][79].The conducting layer of -(ET) 2  has four independent ET molecules - in the unit cell, where - and - form dimers with different orientations [Fig.2(a)].By considering the frontier orbitals of the four molecules, the noninteracting Hamiltonian with the four sublattices in the unit cell is written as where  , is the annihilation operator of electrons carrying momentum  and spin  on the sublattice .The operators are defined by the Fourier transformation of the real-space operators: for example, the annihilation operators are given as  , =  e −i •  , , where  represents the position of the unit cell.In this convention, internal coordinates of the four ET molecules are neglected; we will address this point at the end of this section.The momentum-dependent Hamiltonian matrix is composed of two terms, Ĥ0 ( ) = Ĥkin ( ) + ĤAFM . ( The first term is the kinetic energy with the intradimer hopping (  ) and interdimer hoppings (  ,   ,   ) [see Fig. 2(a)], which is the Kronecker product of the 4 × 4 matrix in the sublattice space and the 2 × 2 identity matrix σ0 in the spin space.The lattice constants are set to unity.The second term in Eq. (3) represents a molecular field of the AFM order that breaks  symmetry, where MAF := diag[1, 1, −1, −1] and σ ( = , , ) are the matrix representing the antiferroic order between the - and - sublattices and the Pauli matrix in the spin space, respectively.Throughout this paper, we use the hopping parameters (  ,   ,   ,   ) = (−0.207,−0.067, −0.102, 0.043) eV obtained from a first-principles calculation for -Cl at 15 K [80], which are adopted in previous theoretical studies [15,16,81].However, the AFM-induced FFLO superconductivity presented in this paper should be feasible regardless of the details of the parameters.
The model belongs to a wallpaper group 2 (point group  2 ) [17] that is composed of translations, twofold rotation  2 , and glide symmetries   and   with respect to the  and  axes, respectively [82]: Here the unitary matrices representing the symmetry operations are defined by Note that all the symmetry properties in Eqs.(6) are conserved even for a nonzero AFM molecular field ℎ, since the unitary matrices in Eqs.(7) include the rotation in the spin space.If we do not consider the spin rotations (i.e., the spin part is equal to σ0 ), the glide symmetries   and   are broken in the AFM state (ℎ ≠ 0) [15,17].On the other hand, the time-reversal symmetry , which is preserved in the paramagnetic phase (ℎ = 0), is broken in the AFM phase (ℎ ≠ 0): where 14 represents the 4 × 4 identity matrix.
The noninteracting Hamiltonian Ĥ0 ( ) is diagonalized using a unitary matrix Vband ( ) as where ( )  is the th eigenenergy for spin  [83].for ℎ = 0 and ℎ = 0.2 eV, respectively.When the AFM molecular field ℎ is finite, the spin splitting appears at general  points, except along the  , axes and the Brillouin zone boundary [15], as shown in Fig. 2(c).Most of the -type salts possess three electrons per dimer on average, and then the electron density  =  ↑ +  ↓ per ET molecule is 1.5.In this case, by increasing ℎ the system turns at ℎ ≈ 0.14 eV to an insulating state with finite energy gap [Fig.2(c)].In the following discussions, we focus on the metallic region for ℎ ≤ 0.14 eV in the undoped case.Furthermore, we investigate the effect of carrier doping by changing the electron density  from a hole-doped regime ( = 1.3) to an electron-doped regime ( = 1.7) [see the green dashed lines in Figs.2(b) and 2(c)].In the doped regimes, we consider the wider range ℎ ≤ 0.2 eV of the AFM molecular field in the metallic regime.
As mentioned above, here we choose the expressions of the Hamiltonian [Eq.( 4)] and the symmetry operations [Eqs.(7)] without the phase factors concerning internal coordinates of the molecules, which are irrelevant to the one-particle properties.In the case of susceptibilities, however, a careful treatment on the phase factors must be conducted, as we will discuss later (see Sec. IV A and Appendix A).
In the following, to discuss SC states, we consider interaction effects in addition to the noninteracting tight-binding Hamiltonian introduced above.In Secs.III and IV, an effective theory with intraband attractive interactions and the Eliashberg theory with the repulsive Hubbard interactions, respectively, are discussed.

A. Attractive intraband interactions
In this section, we introduce effective attractive interactions to search for possible AFM-induced FFLO states.As a preparation, we transform the noninteracting Hamiltonian Eq. ( 1) to the band-based representation.By using the unitary matrix Vband ( ) in Eq. ( 10), we define the annihilation and creation operators of electrons on the band  as Then the Hamiltonian is rewritten in the band-based form As for the interaction term, we assume the intraband attractive pairing as an ideal situation, which is given by where is the creation operator of spin-singlet Cooper pairs with a COM momentum  on the band , and   ( ) indicates the momentum dependence of the order parameter with different anisotropy in  space indexed by .Although the pairing symmetry in -(ET) 2  is experimentally not fully determined, spin-singlet nodal superconductivity is supported by many measurements [84].Therefore, we here consider the extended  +   2 − 2 -wave superconductivity in the  1 irreducible representation (irrep) of point group  2 , and the    -wave one belonging to the  2 irrep (see Table I) [85].Following the previous studies [87][88][89][90][91], the basis function   ( ) is chosen as for the extended  +   2 − 2 -wave ( 1 ) order, and for the    -wave ( 2 ) order.In the following calculations, we assume two kinds of interaction parameters, Note that the results in Sec.III B are not qualitatively altered by the choice of the interaction parameters, as long as the transition temperature   is much smaller than the Fermi energy.

B. Linearized gap equation
Considering the Hamiltonian  =  0 +  (band) int , we now study whether the FFLO state is realized through the analysis of a linearized gap equation, and clarify the SC instability just above   .The linearized gap equation is formulated by calculating the SC susceptibility matrix whose matrix elements are defined by where   = 2 is the bosonic Matsubara frequency,  = 1/ is the inverse temperature, and    (, ) = e     ()e −  .The SC susceptibility is obtained by using the -matrix approximation as where  stands for (, i  ), and 1 represents the identity matrix.The irreducible susceptibility is given by where is the noninteracting band-based Green's function [92] and   = (2 + 1) is the fermionic Matsubara frequency. 0   is the chemical potential for the noninteracting Hamiltonian.Since the SC transition occurs when χSC () diverges, the criterion of the SC instability is obtained when the largest eigenvalue of χ(0) SC () Û(band) becomes unity.We assume that the bosonic Matsubara frequency   is always zero for the criterion.In this case, the irreducible susceptibility is simplified by performing the summation of the frequency in Eq. (20), where  () = (e  ( − 0 ) +1) −1 is the Fermi-Dirac distribution function.
Let  (0) SC () be the largest eigenvalue of χ(0) SC (, 0) Û(band) for a fixed .We then find the optimal COM momentum(s)  opt of the Cooper pairs near   such that the eigenvalue has a maximum value as For example, we show the structure of  (0) SC () for the   wave ( 2 ) order parameter in the electron-doped ( = 1.7) and hole-doped ( = 1.3) cases in Figs.3(a In the presence of the AFM field ℎ, on the other hand, the optimal COM momentum  opt becomes nonzero because of the spin splitting [Figs.3(b) and 3(d)].This can be understood as follows: when a spin-up state is located at a general  point on the Fermi surfaces, there is only a spin-up state at − for the -breaking and  2 -preserving AFM order, whereas the equal-spin pairing is prohibited in the intraband -wave superconductivity.Instead, to stabilize the spin-singlet superconductivity, a finite momentum shift  is necessary for the pairing between the spin-up  and spin-down − +  states [see Fig. 3(d)].This mechanism is similar to the FFLO state under Zeeman field.
One can see in Fig. 3 that the magnitudes of the splitting are quite different between the electron-and hole-doped cases; the Fermi surfaces and their splitting are quasi-one-dimensional and small (two-dimensional and large) in the electron-doped (hole-doped) regime.As a result,  opt in the hole-doped regime tends to be larger than that in the electron-doped regime.Also, reflecting the difference in the structures of the Fermi surfaces, the direction of  opt is parallel to  and  axes in the electron-and hole-doped cases, respectively.Furthermore,  (0) SC () =  (0) SC (−) is always satisfied because of the presence of the twofold rotation symmetry (or inversion symmetry), which results in the double-peak structure at ± opt of  (0) SC ().We will revisit this point in Sec.V.In this way, we analyze  (0) SC () by varying the electron density  and the AFM molecular field ℎ.The results for the extended  +   2 − 2 -wave ( 1 ) and    -wave ( 2 ) superconductivity are shown in Figs.4(a) and 4(b), respectively.Both phase diagrams contain a parameter region with ℎ > 0, where SC () has peaks at finite momenta ± opt (the colored circles), indicating the appearance of the AFM-induced FFLO state.The finite COM state is obtained in a broad parameter region for the  2 -type interaction, whereas it is restricted to a narrower regime in the  1 -type case.The reason is intuitively understood as follows, considering the gap structure for the two cases [87].When the interaction is  2 -type, the    -wave SC order parameter is zero on the  , axes and the Brillouin zone boundary, where the spin degeneracy is protected even in the AFM state [15].Therefore, the  2 superconductivity is more susceptible to the presence of the spin splitting; the finite COM momentum is necessary to form the spin-singlet Cooper pairs on the Fermi surfaces even when the splitting is small.On the other hand, the  1 case is not so sensitive to the spin splitting, since zeros of the extended  +   2 − 2 -wave order parameter are in general  points and therefore do not necessarily correspond to the spin-degenerate regions.

IV. ELIASHBERG THEORY BASED ON HUBBARD MODEL
In the previous section, we have demonstrated that the FFLO phase can be stabilized due to the spin splitting in the AFM state by using the effective model with attractive pairing interactions.In this section, we consider the Eliashberg theory based on the repulsive Hubbard model [77] as a more realistic description.The Hubbard Hamiltonian is written as  =  0 +  (Hub)  int , where the interaction term is given by being the on-site Coulomb repulsion on the ET molecule. , =  † ,  , is the electron-number operator; we remind that , , and  represent the unit cell position, the sublattice index and spin, respectively.Equation ( 24) is then rewritten as where   stands for (  ,   ), and For the later formulations, we define a matrix form of the Hubbard interaction: Û(Hub) := [ (Hub)  1  2 , 3  4 ].In the following Secs.IV A and IV B, a generalized susceptibility based on the FLEX approximation and a linearized Eliashberg equation are discussed, respectively.The source code used for the numerical calculations is available in Ref. [93], some parts of which are implemented based on the algorithm of the FLEX+IR package [94].
Before going into detail, we make several comments on the microscopic analyses in the -type salts.First, the correlation effects included in the FLEX approximation are important for the emergence of superconductivity.Indeed, our test calculations within the random phase approximation (RPA) indicate that the divergence of the magnetic susceptibility occurs far before the SC transition.Second, we emphasize the differences from previous related studies [87,88].They have discussed extended  +   2 − 2 -wave ( 1 ) versus    -wave ( 2 ) pairing instability in -type superconductors by using the FLEX approximation and the Eliashberg theory, but considered only the undoped ( = 1.5), paramagnetic (ℎ = 0), and zero-COMmomentum ( = 0) cases.In the present paper, we consider finite doping , extend to the presence of AFM field ℎ, and seek the possibility of a finite COM momentum .

A. Generalized susceptibility
Now let us formulate the generalized susceptibility within the FLEX approximation.The noninteracting Green's function for  = 0 is represented by the eight-dimensional (four sublattices × two directions of spin) matrix: In the interacting case ( ≠ 0), the dressed Green's function is given by Ĝ where  and Σ( , i  ) are the chemical potential selfconsistently determined in the interacting system and the (normal) self-energy, respectively; the matrix elements of the selfenergy are given by with  and  representing ( , i  ) and (, i  ), respectively.Within the FLEX approximation, the effective interaction vertex for the normal part is calculated as where are the bare and generalized susceptibilities, respectively.In the numerical study, we take 32 × 32 -point meshes and about 80 Matsubara frequencies generated by the SparseIR.jlpackage [95,96] based on the intermediate representation [97] and the sparse sampling [98].The temperature  and the energy cutoff  max are set to 1 meV and 2 eV, respectively.We show the results for  = 1 eV in the following.In general, the FLEX approximation is justified within the intermediate-coupling region in which  is smaller than half of the bandwidth , namely, / ≲ 0.5.Although our theory adopts the interaction comparable to the bandwidth ( ∼ ), in the three-quarter-filled dimerized system that we treat, the effective intradimer Coulomb interaction, which acts on the two electrons in the antibonding orbital in the dimer, is estimated as [77,99,100] Using the generalized susceptibility [Eq.( 33)], the dynamical susceptibility for any multipole operator Ô is generally given by where   is the internal coordinate of the sublattice , namely, the relative position from the origin of the unit cell.The phase factor e −i• (  1 −  3 ) in the summation is necessary to recover the glide symmetries (  and   ) of the susceptibility (see Appendix A for details).The multipole Ô can be represented by the Kronecker product of a 4 × 4 matrix Msl in the sublattice space and a 2 × 2 matrix Msp in the spin space.Although the former sublattice part is classified into 16 types of matrices using the cluster multipole description [17], we here focus on two representatives: the ferroic matrix 14 and the antiferroic one MAF .Table II shows the classification of the operator Ô in the two cases.We calculate Eq. ( 35) for all operators in the table, and confirm that the susceptibility for the longitudinal AFM (LAFM) or transverse AFM (TAFM) spin operator has the leading contribution, which are degenerate for ℎ = 0. Whether the LAFM or TAFM susceptibility is dominant for ℎ > 0 depends on the parameter choice (see Appendix C for details).
Here we discuss the relation between the Fermi surfaces and the AFM spin susceptibility by changing the AFM molecular field ℎ and electron density , which is directly related to the SC instability presented in the next subsection.Let us introduce the Hamiltonian taking the correlation effect into account, where the retarded self-energy in the static limit is evaluated by an approximation justified at low temperatures as, We calculate the real part of the right eigenvalues of Eq. ( 36) that is non-Hermitian in general; the Fermi surfaces renormalized by the correlations are determined by its zeros.

B. Linearized Eliashberg equation
We then investigate the SC instability within the framework of Eliashberg theory.Since the possibility of the FFLO state with the finite COM momentum  is of major interest in the present paper, we define the anomalous Green's function with finite  as   ()  , ′ := − ∫  0 d e i      , () −+, ′ , (39) where  , () = e   , e −  .Note that the function satisfies the following relation due to the Fermi-Dirac statistics: Using the normal and anomalous Green's functions in Eqs. ( 29) and ( 39), we can construct the Dyson-Gor'kov equation with finite .By linearizing the equation with respect to the anomalous term, the linearized Eliashberg equation is obtained as (41) The anomalous Green's function and the interaction vertex are calculated as where Δ () is the order parameter matrix and  := (, 0).We here assume that the interaction vertex V(a) is independent of the COM momentum .A similar formalism for the finite- Eliashberg theory has been discussed in a previous study [101].Within the Eliashberg theory, the phase transition into the SC state with the COM momentum  takes place when the eigenvalue   in Eq. ( 41) reaches unity; the  value whose transition temperature is highest is realized.Here, for numerical convenience, we evaluate the amplitude of the Eliashberg eigenvalue at a fixed temperature ( = 1 meV), which is far below the Neel temperature in -Cl, and judge which  is favored for each (, ℎ).By using the power method, we solve the linearized Eliashberg Eq. ( 41) and find the eigenvalue   and the corresponding order parameter Δ ().As is the analysis in Sec.III B, we consider the SC order parameter belonging to the  1 and  2 irreps of  2 .The initial functional form is chosen as for the  1 case, and for the  2 case, where the dominant -wave and subdominant -wave order parameter with  = 0.1 are taken into account.Note that, due to the sublattice degree of freedom, the subdominant spin-triplet component can be admixed in the even-parity order parameter; in particular, we confirmed that the -wave component has a comparatively large contribution when the spin splitting is present.We here take k :=  − /2, since the order parameter obeys the same relation as Eq. ( 40) for the anomalous Green's function, according to Eq. ( 42).In addition, we assume the even-frequency order parameter in all calculations.
By finding the optimal COM momentum(s)  opt such that the Eliashberg eigenvalue   reaches a maximum in the momentum space, we can draw the ℎ- phase diagram as in Fig. 6(a).First, the  2 order (circles) is more stable than the  1 order (squares) in the wide range of parameters.Second, the eigenvalue takes the peak at finite s in the hole-doped AFM regime, which indicates that the FFLO superconductivity is stabilized when the spin splitting is large.In Figs.6(b) and 6(c), for example, the momentum dependencies of   for (, ℎ) = (1.3,0) and (1.3, 0.2) are exhibited, respectively.The eigenvalue has a single peak at  = 0 in the paramagnetic state [Fig.6(b)], while the maximum is located at finite  opt ∥ x in the AFM state [Fig.6(c)].We also confirm that   =  − due to the existence of the  2 symmetry.
Although the parameter region that we find the FFLO state is relatively limited compared to the model in Sec.III, we believe that the results showing its stability using a realistic model and reliable many-body methodologies is encouraging.In recent numerical studies, the inclusion of off-site Coulomb repulsion terms, on top of the Hubbard model we considered, alters the competition of different symmetries and plays important roles in the SC instability, near  = 1.5 at ℎ = 0 [90,91].We expect modifications by the off-site terms for the finite AFM field ℎ as well, whose investigation is left for future studies.

V. DISCUSSION
We briefly discuss promising experimental setups to observe the AFM-induced FFLO superconductivity.First, the coexistence of the AFM and SC orders was reported in previous experiments, where the phase separation appears in bulk - [20,102,103].In particular, the deuterated sample is located in the vicinity of the Mott transition and is a candidate system for the realization of the FFLO state in the anisotropic AFM spin splitting.Another proposal is making use of recently developed experimental techniques to control phases by carrier doping and strain on a transistor device using a thin single crystal of -(ET) 2  [104,105].If the AFM phase and the SC phase appear in adjacent regions of the sample, the coexistence can be realized via the proximity effect.
Furthermore, our results show the double- structure because of the twofold rotation (or inversion) symmetry.Therefore, two possible forms of the order parameter are considered for the SC state below   : the single- (FF) state with Δ() ∼ exp(i opt • ) and the multi- (LO) state with Δ() ∼ cos( opt • ) (: real-space coordinate).In the case of the Zeeman-field-induced FFLO state, clean isotropic wave superconductors prefer the multi- phase rather than the single- phase [72,106], whereas both FF and LO states appear in anisotropic superconductors with nonmagnetic impurities [107].Since the impurity effect is generally considered to be small in organic conductors, we speculate that the multi- LO state is more stable in the coexisting state of the AFM and SC orders.If the LO state is realized, the modulation of the SC order may be observable by scanning tunneling spectroscopy.To actually determine which state is stabilized in the anisotropic splitting, solving the Bogoliubov-de Gennes equation or constructing the Ginzburg-Landau theory is needed, which is beyond the scope of the paper.
We here remark on the critical molecular field for the transition to the FFLO state.In the case of the FFLO superconductivity stabilized by an external magnetic field, the transition occurs when the Zeeman field becomes comparable to the magnitude of the SC order parameter at zero temperature Δ( = 0), which was shown by a mean-field analysis [108].A previous study on a single-band model introducing the anisotropic spin splitting demonstrated that the transition from a uniform wave SC state to a pair-density-wave (i.e., FFLO) state occurs also when the energy scale of the spin splitting approximately corresponds to Δ(0) [31].Therefore, one may expect that the critical AFM molecular field for the emergence of the FFLO state is given by |ℎ| ∼ Δ(0) in our case as well.However, such a clear criterion is not easily seen here, because of the following three reasons.First, the magnitude of the anisotropic spin splitting is not a simple function of ℎ since it sensitively depends on the carrier density, the real-space anisotropy of hopping integrals, and the resulting shape of Fermi surfaces.Second, interband interactions as well as intraband ones alter the critical field, as indicated by the comparison between the effective model (Fig. 4) and the Hubbard model (Fig. 6).Third, the stability of the FFLO state is strongly related to the symmetry of the SC order parameter, as shown by the difference between Figs. 4(a) and 4(b) in the effective model analysis.These reasons are attributed to the multiband and anisotropic properties of our model, both of which are inherent in realistic altermagnetic materials.
Our theoretical proposal of the AFM-induced FFLO superconductivity is applicable not only to -type organic conductors but also to many other materials hosting the unusual  = 0 AFM (or magnetic octupole/altermagnetic) order with anisotropic spin splitting, which have recently attracted much interest [21][22][23][24][25][26].Indeed, a recent theoretical study [35] has proposed that Cooper pairs in altermagnets acquire finite COM momentum through the analysis of a Cooper-pair propagator [109] in a continuum model.We should note that, unlike the case of the Zeeman field where highly two-dimensional compounds are required for the realization of the FFLO state to avoid orbital breaking, the mechanism here is free from such an effect; therefore, three-dimensional compounds can also be candidates, expanding the scope of target materials.

VI. SUMMARY
In this paper, we microscopically investigated the property of superconductivity coexisting with the AFM order in (carrierdoped) -(ET) 2 .By considering the effective theory with intraband attractive interactions, we found that FFLO superconductivity with finite COM momentum can be stabilized by the anisotropic spin splitting in the AFM phase.Then we considered the on-site repulsive Hubbard model as a more realistic situation, and analyzed the linearized Eliashberg equation based on the FLEX approximation.As a result, the FFLO state is realized in the hole-doped AFM state with comparatively large spin splitting.In the whole calculations, we adopt the anisotropic order parameter with extended  +   2 − 2 -wave ( 1 ) or    -wave ( 2 ) symmetry, the latter of which has nodes coinciding with the zeros of spin splitting and is more likely to induce the FFLO state.This indicates that the anisotropic superconductivity is important for the appearance of the finitemomentum state in the anisotropic spin splitting [31,110].
This finding paves a way to search for the FFLO state for the following reasons.As we mentioned in Introduction, the occurrence of the FFLO superconductivity was originally suggested in high magnetic fields with the strong Zeeman effect, whereas a concomitant orbital pair breaking effect with vortex states obstructs the observation of the FFLO state.On the other hand, the AFM-induced FFLO state is realized without an external magnetic field, and thus free from vortices.Therefore, the AFM state with spin splitting can be a good platform to realize the exotic FFLO superconductivity.Another distinction of the AFM state from the Zeeman field is that the magnitude of the spin splitting strongly depends on the carrier density, even when the energy scale of the AFM molecular is unchanged.Therefore, the spatial modulation of the FFLO order may be controllable by carrier doping.Finally, we believe that our theory provides a foundation for further investigations of exotic superconductivity in the spin-splitting AFM state.susceptibility for the model of -(ET) 2  that we treated in the main text with  = 2, as an example; the generalization to other susceptibilities and/or symmetries straightforward.For this purpose, the spin operator on the sublattice  in the unit cell  is defined by In the momentum space, the spin operator is transformed as in Convention 1, and in Convention 2. Using Eqs.(A9) and (A10), we next derive the transformation property of the longitudinal ( direction) spin under  ∈ .After some algebra, it is given by in Convention 1 and  Note that the spin operators in the integrand have no tilde symbol.Therefore, the broken (nonsymmorphic) symmetry can be recovered by multiplying the phase factor e −i• (  −  ′ ) to the integral calculated in Convention 2. That is the reason we include the phase factor in Eq. (35).In all calculations of the susceptibility except Fig. 7(b), we use the internal coordinates   = (0, 0),   = (1/2, 1/2),   = (1/2, 0), and   = (0, 1/2), where the four sublattices (molecules) are arranged like the Shastry-Sutherland lattice [113].

T
FIG. 1. Symmetry properties and schematic energy band structures in (a) magnetic quadrupole, (b) FM (or Zeeman field), and (c) magnetic octupole states.The red and blue dispersions represent (pseudo)spin up and down bands, respectively.The green dash-dotted lines are the Fermi surface in the paramagnetic state.
(a)], the AFM order breaks  symmetry but preserves  symmetry, which induces spin splitting in the band structure.The momentum-flip symmetry properties in the AFM state are the same as those in a FM state (or in a Zeeman field), whereas the band structure has anisotropic spin splitting in contrast to isotropic one in arXiv:2308.14227v3[cond-mat.supr-con]27 Nov 2023 the FM state [see Figs.1(b) and 1(c)].The AFM structure in -(ET) 2

FIG. 2 .
FIG. 2. (a) Schematic view of the molecular arrangement in the conducting layer of -(ET) 2 .The unit cell (green rectangle) contains four ET molecules labeled by -.The bonds with large hoppings are shown:  (orange bold),  (dotted),  (solid), and  (dashed).The red and blue arrows represent up and down spin moments, respectively, in the AFM phase.The noninteracting band structures for (b) ℎ = 0 and (c) ℎ = 0.2 eV.Energy bands of up-and down-spin electrons are represented by the red and blue lines, respectively.The green dashed lines show the Fermi levels for the electron density  = 1.3, 1.5, and 1.7 per ET molecule.The inset in (b) represents the path in the 2D Brillouin zone.Note that, in the orthorhombic wallpaper group 2,   =   =  is labeled by S, which corresponds to the M point in previous studies [15-17, 77].

FIG. 4 .
FIG. 4. Phase diagrams obtained from the analysis of the SC susceptibility for (a)  1 and (b)  2 interaction parameters.The temperature  is set to 1 meV.The closed (open) circles indicate that the largest eigenvalue of  (0) SC () is larger (smaller) than unity.The color of each circle represents the norm of the optimal COM momentum  opt = | opt |.The direction of  opt is represented by gray arrows in the circles.
Figure 5 displays the modified Fermi surfaces and LAFM or TAFM

FIG. 6 .
FIG. 6.(a) Phase diagram on the (ℎ, ) plane obtained from the analysis of the Hubbard model on the -type structure.The temperature  is set to 1 meV.The squares (circles) indicate that the stable order parameter belongs to the  1 ( 2 ) irrep.The color of each point represents the norm of the optimal COM momentum  opt = | opt |. (b), (c) The momentum dependence of the eigenvalue   of the Eliashberg equation in the hole-doped regime ( = 1.3) for ℎ = 0 and ℎ = 0.2, respectively.The eigenvalue has peaks at finite COM momenta in (c).

TABLE I .
Character table for the wallpaper group 2 (point group  2 ). represents the identity operation.Irrep   2     Basis functions