Theory of pair density wave on a quasi-one-dimensional lattice in the Hubbard model
TTheory of pair density wave on a quasi-one-dimensional lattice in the Hubbard model
Soma Yoshida , Keiji Yada and Yukio Tanaka Department of Applied Physics, Nagoya University, Nagoya 464–8603, Japan
In this study, we examine the superconducting instability of a quasi-one-dimensional lattice inthe Hubbard model based on the random-phase approximation (RPA) and the fluctuation exchange(FLEX) approximation. We find that a spin-singlet pair density wave (PDW-singlet) with a center-of-mass momentum of 2 k F can be stabilized when the one-dimensionality becomes prominent towardthe perfect nesting of the Fermi surface. The obtained pair is a mixture of even-frequency and odd-frequency singlet ones. The dominant even-frequency component does not have nodal lines on theFermi surface. This PDW-singlet state is more favorable as compared to RPA when self-energycorrection is introduced in the FLEX approximation. I. INTRODUCTION
Unconventional superconductors have been studied ex-tensively in condensed matter physics[1]. Theoretical re-search on unconventional pairing in strongly correlatedsuperconductors was initiated by the discovery of high-T c cuprate [2]. It is known that antiferromagnetic spin fluc-tuation favors spin-singlet d -wave pairing in the Hubbardmodel in a two-dimensional square lattice [3, 4]. Fur-thermore, several unconventional superconductors with d -wave spin-singlet pairing have been found. However, ithas been clarified that in a quasi-one-dimensional Hub-bard model, spin-singlet d -wave pairing becomes unsta-ble because of the overlap of the nodal lines of the gapfunction and the Fermi surface. In this case, the odd-frequency spin-singlet p -wave pairing becomes dominantbecause the nodal lines of the gap function can avoid theFermi surface [5–8].Although there have been several studies on odd-frequency gap functions in strongly correlated systemssince the proposal by Berezinskii, the physical propertiesof odd-frequency gap functions have not been examinedyet. [5, 6, 9–19] There is a fundamental difficulty in for-mulating a uniform odd-frequency gap function withoutthe center-of-mass momentum [20]. Thus, it is necessaryto consider a nonuniform Cooper pair for the realiza-tion of odd-frequency pairing. One such possible Cooperpair is the so-called pair density wave (PDW), where thecenter-of-mass momentum is of the order of the Fermimomentum. [13, 21–23]. It is interesting to study thestability of a PDW in a quasi-one-dimensional systemfrom this viewpoint.This state is distinct from the so-called Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, where the center-of-mass momentum is significantly less compared with k F .Research on PDWs started in the context of η pairing[24]. At present, the PDW has become a hot topic inthe research on stripe and pseudogap phases in high- T C cuprates [25 ? –34]. The PDW singlet pairing state isone of the candidates for the pairing in the quasi-one-dimensional Hubbard model because the standard spin-singlet d -wave pairing becomes unstable. Thus, it is es-sential to check whether a PDW is possible in the quasi-one-dimensional Hubbard model. The PDW has a remarkable property from the pointof view of symmetry of the Cooper pair. ConventionalCooper pairs are formed between electrons with momenta k and − k , which are related by parity. In contrast, themomenta of two electrons in the Cooper pair of a PDWstate are k and − k + Q , and these are not related byinversion symmetry. This means that the gap functionof the PDW do not have any relation about the exchangeof the momenta of two electrons k and − k + Q . Therefore,even- and odd-parity pairs can mix with each other. Inthis sense, the study of PDW pairing is essential fromthe viewpoint of the presence of odd frequency state inbulk.In this study, based on the standard random phaseapproximation (RPA), we solved the linearized Eliash-berg equation by comparing six possible pairings:(i) even-frequency spin-singlet even-parity (ESE), (ii)even-frequency spin-triplet odd-parity (ETO), (iii) odd-frequency spin-singlet odd-parity (OSO), (iv) odd-frequency spin-triplet even-parity (OTE), (v) 2 k F spin-singlet pair density wave (PDW-singlet), and (vi) 2 k F spin-triplet pair density wave (PDW-triplet). The center-of-mass momentum of the Cooper pair is zero for (i)to (iv). Among these, the competitive dominant pair-ings are the ESE d -wave, OSO p -wave, and 2 k F spin-singlet PDW. When one-dimensionality becomes promi-nent with the good nesting condition of the Fermi surface,the most dominant Cooper pair becomes a PDW-singletpair. The present PDW-singlet state is supported by thenesting of the Fermi surface because the regions where aCooper pair can be formed are enhanced. The resultinggap function is a mixture of the even- and odd-frequencycomponents. It is remarkable that the dominant even-frequency component displays a momentum dependencesimilar to that of the OSO p -wave component. Under thefluctuation exchange (FLEX) approximation, the OSOpairing becomes unstable owing to the self-energy effect,and the parameter regions where the PDW-singlet stateis stabilized become wider.The rest of this paper is organized as follows. InSection II, we describe the model and the formulationof the RPA and FLEX approximation. In Section III,we present the results of the calculations based on theEliashberg equation and illustrate the eigenvalue and mo- a r X i v : . [ c ond - m a t . s up r- c on ] F e b mentum dependencies of the gap function. In Section IV,we summarize the results obtained. II. MODEL AND FORMULATION
We analyze a two-dimensional single-band Hubbardmodel on an anisotropic triangular lattice, as shown inFig. 1, where t x , t y , and t d are the hopping integrals alongthe x − , y − , and diagonal direction, respectively. Here-after, we choose t x as the unit of energy.The Hamiltonian is expressed asˆ H = (cid:88) (cid:104) i,j (cid:105) ,σ (cid:16) t ij ˆ c † iσ ˆ c jσ + h.c. (cid:17) + (cid:88) i U ˆ n i ↑ ˆ n i ↓ , (1)where t ij is the hopping integral between the i and j sites, (cid:104) i, j (cid:105) represent the sets of nearest neigh-bors, ˆ c † iσ (ˆ c iσ ) and ˆ n iσ = ˆ c † iσ ˆ c iσ denote the creation(annihilation) and number operators, respectively,and U represents the on-site interaction. The disper-sion relation in the normal state is represented as follows: ε k = − t x cos k x − t y cos k y − t d cos ( k x + k y ) . (2)In this study, we consider the half-filling case, where theparticle number ˆ n i = ˆ n i ↑ + ˆ n i ↓ is equal to unity. In thepresent model, the Fermi surface has a nesting vector Q nest = ( π, π/
2) in a quasi-one-dimensional region with t y /t x = t d /t x (cid:28)
1, as shown in Fig. 2. ty txtd xy FIG. 1. Schematic of two-dimensional anisotropic Hubbardmodel
In this study, we evaluate the superconducting pairinginteractions based on the RPA [3, 4] and FLEX approx-imations [35, 36].Under these approximations, the irreducible suscepti-bility is expressed as follows: χ ( q ) = − TN (cid:88) k G ( k ) G ( k + q ) , (3) Q nest =( π , π /2) FIG. 2. Fermi surface in a quasi-one-dimensional region.Nesting vector Q nest = ( π, π/ t y /t x = t d /t x = 0 . where G ( k ) is the single-particle Green’s function and N is the number of k -meshes. Here, k ≡ ( k , ε n ) and q ≡ ( q , ω m ) are the short-hand notations of the momentaand Matsubara frequency for the fermion, ε n = (2 n − πT , and boson, ω m = 2 mπT , respectively. Using theirreducible susceptibility, the spin susceptibility χ sp ( q )and charge susceptibility χ ch ( q ) are expressed as follows: χ sp ( q ) = χ ( q )1 − U χ ( q ) , (4) χ ch ( q ) = χ ( q )1 + U χ ( q ) . (5)Here, the Stoner factor is defined as f s = U χ ( Q nest , χ ( q ) is maximum at q = ( Q nest , f s →
1, and the spin fluctu-ation becomes strong near the magnetic phase. In thisstudy, we choose the on-site interaction U as f s = 0 . G ( k ) = ( iε n − ε k + µ ) − , where µ is the chemical poten-tial and the self-energy is neglected.(b) Under the FLEX approximation, Green’s functionis determined self-consistently using the Dyson equa-tion as follows [35, 36]. First, we start from G ( k ) =( iε n − ε k + µ ) − as Green’s function. Next, we deter-mine the spin and charge susceptibilities by (3),(4), and(5). Then, the effective interaction is calculated as fol-lows: V n ( q ) = 32 U χ sp ( q ) + 12 U χ ch ( q ) − U χ ( q ) . (6)The self-energy is represented as follows:Σ( k ) = TN (cid:88) q V n ( q ) G ( k − q ) . (7)Finally, we obtain the new Green’s function as follows: G − ( k ) = G − ( k ) − Σ( k ) . (8)We iterate these sequential calculations until Green’sfunction converges sufficiently.From the calculated spin and charge susceptibilities,we determine the effective pairing interaction for a spin-singlet and spin-triplet pairing as follows: V s eff ( q ) = U + 32 U χ sp ( q ) − U χ ch ( q ) , (9) V t eff ( q ) = − U χ sp ( q ) − U χ ch ( q ) . (10)In the present study, we consider the possible realizationof PDWs with the center-of-mass momentum Q . There-fore, the anomalous Green’s function is defined as follows: F σσ (cid:48) ( k, Q ) = − (cid:90) β (cid:104) T [ˆ c k σ ( τ )ˆ c − k + Q σ (cid:48) (0)] (cid:105) e iε n τ dτ. (11)This function must satisfy the following relation owingto Fermi–Dirac statistics: F σσ (cid:48) ( k, Q ) = − F σ (cid:48) σ ( − k + Q , Q ) . (12)The Dyson-Gorkov equation of the PDW state is notedas G ( k ) = G ( k ) + G ( k )Σ( k ) G ( k )+ G ( k )∆ σσ (cid:48) ( k, Q ) F † σ (cid:48) σ ( k, Q ) (13) F σσ (cid:48) ( k, Q ) = G ( k )Σ( k ) F σσ (cid:48) ( k, Q )+ G ( k )∆ σσ (cid:48) ( k, Q ) G ( − k + Q ) , (14)where we abbreviate Q = ( Q ,
0) and omit the summan-tion of the spin subscripts that appear repeatedly. Welinearize (13) and (14) with respect to the anomalousterm. Thus, the linearized Eliashberg equation is repre-sented as λ ( Q )∆( k, Q ) = − TN (cid:88) k (cid:48) V s,t eff ( k − k (cid:48) ) F ( k (cid:48) , Q ) , (15) F ( k (cid:48) , Q ) = G ( k (cid:48) ) G ( − k (cid:48) + Q )∆( k (cid:48) , Q ) . (16)We calculate the maximum eigenvalue λ and the gapfunction ∆( k ) using the power method for possible pair-ings. It is known that the superconducting transitiontemperature is determined by the condition under which λ becomes unity. Because λ increases monotonically withdecrease in temperature, the pairing with a larger valueof λ is more favorable.In the PDW state, the inversion symmetry of the no-mal state require the degenerate of the gap function∆( k, Q ) and e iθ ∆( k, − Q ) as the solution of Eq. (15)and (16). θ is an arbitrary phase and is not determinedin the linearlized theory. However, the inversion symme-try of the nomal state does not give any relation between∆( k, Q ) and ∆( − k + Q, Q ). Therefore, the even- andodd-parity states mix with each other. However, theseanomalous terms do not break the spin-rotational sym-metry of the system. Thus, the pairing function is classi-fied into the spin-singlet or spin-triplet states. We refer to the singlet and triplet states that have a finite center-of-mass momentum as the PDW-singlet and PDW-tripletstates, respectively. From the Hermition of the Hamilto-nian, the gap function satisfies ∆( k , ε n ) = ∆ ∗ ( k , − ε n ).Therefore, we can choose the phase of the gap function,where the real (imaginary) part of the gap function haseven (odd) in Matsubara frequency. For example, thegap function of the PDW-singlet state satisfiesReal[∆( k , ε n , Q )] = Real[∆( − k + Q , ε n , Q )]= Real[∆( k , − ε n , Q )] , (17)Imag[∆( k , ε n , Q )] = − Imag[∆( − k + Q , ε n , Q )]= − Imag[∆( k , − ε n , Q )] . (18)We take sufficiently large N = N x × N y k -point mesheswith the cutoff Matsubara frequencies ε M/ and ω M/ . III. RESULTS
In the following, we study the PDW state in a quasi-one-dimensional parameter region, where the magnitudesof t y and t d are sufficiently lower than that of t x .We calculate the eigenvalues of the linearized Eliash-berg equation by changing the center-of-mass momentum Q for t y /t x = t d /t x = 0 .
1, as shown in Fig. 3, to ob-tain a value of Q that stabilizes the PDW state. In thisparameter region, the maximum value of the eigenvalueare obtained as the center-of-mass momentum Q = .This means that a spatially uniform pairing without thecenter-of-mass momentum is obtained. In this case, theOSO pairing is obtained to be consistent with previousresults [5, 6]. On the other hand, the eigenvalue λ ( Q )has a peak at Q = Q nest (= ( π, π/ Q = Q nest (= ( π, π/ Q = , it is expected that the solution with this nest-ing vector Q is realized by choosing smaller values of t y /t x and t d /t x . This is explained by the fact that twoelectrons on the Fermi surface form a Cooper pair. Wesuppose that two electrons have the wave numbers k and − k + Q . As shown in Fig. 4, if an electron with k is lo-cated on the Fermi surface, then an electron with − k isalso located on the Fermi surface owing to inversion sym-metry. However, it is not evident that an electron with awave number − k + Q is located on the Fermi surface. Toincrease the number of electrons with momentum − k + Q on the Fermi surface under the condition that an electron k is located on the Fermi surface, it is desirable that oneside of the Fermi surface overlaps with the other side bythe translation of momentum Q . Because this require-ment agrees with the property of the nesting vector, itis appropriate to choose the nesting vector Q nest as thecenter-of-mass momentum of a Cooper pair.After we obtain a preferable center-of-mass momen-tum, we study the stability of the PDW state in a quasi-one-dimensional system by changing t y /t x = t d /t x andcalculating the eigenvalues of each symmetry, as shownin Fig. 5, where we choose the nesting vector Q nest as the Q x /π Q y / π . . . . . . λ ( Q ) FIG. 3. Center-of-mass momentum dependencies of the eigen-value λ ( Q ) of the linearized Eliashberg equation in the PDW-singlet state. T /t x = 0 . N = 128 × M = 512.FIG. 4. Two electrons that form a Cooper pair on the Fermisurface. center-of-mass momentum Q of the PDW states. We ob-tain the nesting vector as the momentum that maximizesthe irreducible susceptibility χ ( q , k , πT ). Figs.6(c), 6(d), 7(b),and 8(b) show the Matsubara frequency dependencies of∆( k max , ε n ). k max is the momentum that gives the max-imum value of | ∆( k , πT ) | . The solid and green dashedlines in Figs. 6(a), 6(b), 7(a), and 8(a) represent theFermi surfaces of the normal state and nodes of the gapfunctions, respectively. The ESE and OSO states shownin Figs. 7 and 8 are consistent with the previous results . . . . . t y /t x = t d /t x . . . . . . . λ ESEOSOETOOTEPDW-singletPDW-triplet
FIG. 5. t y /t x = t d /t x dependencies of the eigenvalue λ of thelinearized Eliashberg equation. T /t x = 0 . N = 128 × M = 4096 [6, 8].As referred to in the preceding study [6, 8], the OSOstate is relatively stabilized because the ESE d -wave stateis unstable in a quasi-one-dimensional parameter regionowing to the overlapping of the nodes of the d -wave gapfunction and Fermi surface. On the other hand, accord-ing to the momentum dependence of the PDW-singletstate in Fig. 6(a), the line shape of the real part ofthe momentum dependence (even frequency part of thegap function) resembles that of the OSO state shown inFig. 8(a) and does not have a node on the Fermi surface.The corresponding line shape of the momentum depen-dence of the imaginary part (odd-frequency part of thegap function) of the PDW-singlet state in Fig. 6(b) hasfour nodes on the Fermi surface, and the momentum de-pendence near k x = π/ ↑↓ ( k + Q / , ε n , Q )]= Real[∆ ↑↓ ( − k + Q / , ε n , Q )] . (19)In contrast, the corresponding imaginary part satisfiesthe equationImag[∆ ↑↓ ( k + Q / , ε n , Q )]= − Imag[∆ ↑↓ ( − k + Q / , ε n , Q )] . (20)This means that the even-frequency (odd-frequency) partof the PDW-singlet state has an inversion symmetry witheven(odd) parity with respect to k = Q / π/ , π/ (a) − k x /π − k y / π − R e a l[ ∆ ↑↓ ( k , π T ) ] (b) − k x /π − k y / π − − I m ag [ ∆ ↑↓ ( k , π T ) ] (c) − ε n R e a l[ ∆ ↑↓ ( k m a x , ε n ) ] (d) − ε n − I m ag [ ∆ ↑↓ ( k m a x , ε n ) ] FIG. 6. Momentum and Matsubara frequency dependencies of the real (a1),(a2) and imaginary (b1),(b2) parts. The solidlines represent the Fermi surface. Green dashed lines represent the node of the gap function. Center-of-mass momentum Q = ( π, π/ t y /t z = t d /t x = 0 . T /t x = 0 . N = 128 × M = 4096. have a momentum dependence similar to that of the OSOstate when Q is a nesting vector. Noted that even (odd)frequency part of the gap function of the PDW-singletstate is not strictly odd (even) function with respect toan inversion of k to -k and it is accidental that evenfrequency part has the momentum dependence similarto OSO p-wave.The obtained ratio of the norms of the realand imaginary parts of the PDW-singlet state (cid:80) k ,ε n | Real[∆( k , ε n )] | : (cid:80) k ,ε n | Imag[∆( k , ε n )] | at t y /t x = t d /t x = 0 . Q nest = ( π, π/ t y /t x = t d /t x = 0 .
02 and t y /t x = t d /t x = 0 .
01, asshown in Fig. 9. The overlap of the red and blue linesnear k x = − π/
2, as displayed in Fig. 9, is not sufficient.Because two electrons can form a Cooper pair only in the overlapping region of the two lines shown in Fig. 9,as the nesting condition becomes worse, the more PDW-singlet state is destabilized as compared with the super-conducting states with zero center-of-mass momentum.To confirm this, we illustrate the momentum dependenceof the real part of the anomalous Green’s function of thePDW-singlet state Real[ F ( k , πT )] calculated under theRPA, as shown in Fig. 10(a), where the maximum valueof | Real[ F ( k , πT )] | is normalized to be unity. As shownin Fig. 10(a), Real[ F ( k , πT )] has a nonzero value only inthe overlap region of the red solid and blue dashed linesin Fig. 9(a), because G ( k (cid:48) ) G ( − k (cid:48) + Q ) in eq. (16) reachesa maximum in this region. The effective interaction inthe RPA has peaks at Q nest and − Q nest , and the gapfunction is given by eq. (15). Therefore, we define f ( k )as f ( k ) = F ( k + Q nest , πT ) + F ( k − Q nest , πT ) (21)and calculate it as shown in Fig. 10(b) to estimate theregions where the real parts of the gap functions of thePDW-singlet state can have large values. We confirmthat the regions where it has large values, as shown inFig. 6(a) and 10(b), are almost the same. Therefore, (a) − k x /π − k y / π − − ∆ ↑↓ ( k , π T ) (b) − ε n ∆ ↑↓ ( k m a x , ε n ) FIG. 7. (a) Momentum and (b) Matsubara frequency dependencies of gap function of ESE state. Solid lines represent the Fermisurface. Green dashed lines represent nodes of the gap function. t y /t z = t d /t x = 0 . T /t x = 0 . N = 128 × M = 4096(a) − k x /π − k y / π − − ∆ ↑↓ ( k , π T ) (b) − ε n − ∆ ↑↓ ( k m a x , ε n ) FIG. 8. () Momentum and (b) Matsubara frequency dependencies of gap function of the OSO state. Solid lines represent theFermi surface. Green dashed lines represent nodes of the gap function. t y /t z = t d /t x = 0 . T /t x = 0 . N = 128 × M = 4096 when the degree of nesting is lowered, the regions wherethe gap function of the PDW-singlet state can have largevalues are more reduced. The nesting of the Fermi sur-face displayed in Fig. 9(b) is more prominent than thatdisplayed in Fig. 9(a). This means that as the one-dimensionality becomes stronger, the nesting conditionbecomes better. Therefore, the PDW-singlet state be-comes stable at an extreme one-dimensional region inFig. 5.We fix t d = 0 to eliminate the instability from theincompleteness of the nesting. Therefore, the Fermi sur-face has the perfect nesting vector Q nest = ( π, π ). Weillustrate the t y /t x dependence of the eigenvalue calcu-lated using the RPA, as shown in Fig. 11. In Fig. 11,a stable region of the PDW-singlet state is expanded to t y /t x (cid:39) .
25 owing to the perfect nesting. However, theESE state is most stable in the region that is not quasi-one-dimensional. Therefore, similar to the OSO pairing,the PDW-singlet state is stabilized because the ESE pair-ing becomes unstable in a quasi-one-dimensional param- eter region [5, 6].We exhibit the phase diagram in a quasi-one-dimensional parameter region in Fig. 12. In this calcula-tion, we use the RPA and do not fix the on-site interac-tion U as before. As displayed in Fig. 12, the OSO stateis stabilized in the region with t d ∼ t y where the spinfrustration effect is prominent. In contrast, the PDW-singlet state becomes dominant in the region with strongone-dimensionality where the nesting of the Fermi surfacebecomes prominent.Finally, we illustrate the t y /t x = t d /t x dependence ofthe eigenvalue calculated by the FLEX approximation,as shown in Fig. 13. We choose the nesting vector as thecenter-of-mass momentum of a Cooper pair, as in thecase of the RPA. In Fig. 13, the PDW-singlet state isevidently stabilized in a wider parameter region as com-pared to the calculated results based on the RPA. Tounderstand the reason for this stabilization, we calculatethe Fermi surface considering the self-energy obtained bythe analytic continuation from the Matsubara frequency (a) − k x /π − k y / π (b) − k x /π − k y / π FIG. 9. Nesting of the Fermi surface. (a) t y /t x = t d /t x = 0 .
2, (b) t y /t x = t d /t x = 0 .
1. The red solid and blue dashed linesrepresent the Fermi surface and one that is parallelly moved by Q = ( π, π/ − k x /π − k y / π − . − . . . . R e a l[ F ( k , π T ) ] (b) − k x /π − k y / π − . − . . . . f ( k ) FIG. 10. Momentum dependence of real part of (a) the anomalous Green’s function and (b) f ( k ) of PDW-singlet state. Thecenter–of-mass momentum Q = ( π, π/ t y /t z = t d /t x = 0 . T /t x = 0 . N = 128 × M = 4096. . . . . . . t y /t x . . . . . . . λ ESEOSOETOOTEPDW-singletTRI
FIG. 11. t y /t x dependence of eigenvalue λ of the linearizedEliashberg equation calculated under RPA. The PDW-singletstate appears from t y /t x = 0 . Q = ( π, π ), t d = 0, T /t x = 0 . N = 128 × M = 4096. to the real one using the Pade approximation. The solidred and blue lines in Fig. 14 represent the Fermi surfacescalculated using the FLEX approximation and withoutthe self-energy correction, respectively. The line shapeof the resulting Fermi surface becomes increasingly one-dimensional because of the self-energy. Therefore, thePDW-singlet state becomes stable owing to the improve-ment of the nesting. In contrast, the OSO pairing is sup-pressed by the broadness of the peak width of the effec-tive interaction by self-energy, as mentioned in previousworks [8]. The peak width of the effective interaction cal-culated using the FLEX approximation is wider than thatcalculated using the RPA, and the PDW-singlet statedoes not suffer the instability, as described above, be-cause the principal component of the PDW-singlet stateis the even-frequency state. For these reasons, the PDW-singlet state becomes more stable using the FLEX ap-proximation as compared with the RPA. . . . . t y /t x . . . . t d / t x ESEOSOPDW-singlet
FIG. 12. Phase diagram in a quasi-one-dimensional system.It should be noted that on-site interaction U is not fixed;instead, the Stoner factor is fixed. T /t x = 0 . N = 128 × M = 4096, f s = 0 . . . . . . . t y /t x = t d /t x . . . . λ PDW-singletOSOESE
FIG. 13. t y /t x = t d /t x dependence of eigenvalue λ of the lin-earized Eliashberg equation calculated under FLEX approxi-mation. T /t x = 0 . N = 128 × M = 4096. IV. CONCLUSION
In this study, we examined the possible pairing sym-metry mediated by spin fluctuation in the quasi-one-dimensional Hubbard model considering the finite center-of-mass momentum of a Cooper pair, referred to as apair density wave (PDW). Based on the random phaseapproximation (RPA) calculation, we have compared thestability of the superconducting states such as the even-frequency spin-singlet even-parity (ESE), even-frequency − k x /π − k y / π FIG. 14. Fermi surface considering the self-energy. The redsolid line represents the Fermi surface calculated using theFLEX approximation, and the red dashed line represents theno-interaction Fermi surface. t y /t x = t d /t x = 0 . T /t x =0 . N = 128 × M = 4096. spin-triplet odd-parity (ETO), and odd-frequency spin-singlet odd-parity (OSO) pairings without the center-of-mass momentum and PDW with the center–of-massmomentum 2 k F by comparing the maximal eigenvaluesof the linearized Eliashberg equation. Among these,the ESE d -wave pairing, OSO p -wave pairing, and spin-singlet PDW are the dominant competing states in thepresent model. In the quasi-one-dimensional parameterregion t y , t d (cid:28) t x , the ESE d -wave pairing becomes un-stable owing to the overlapping of the nodes of the gapfunction and the Fermi surface. Therefore, the obtainedPDW-singlet state with the center-of-mass momentum Q ∼ k F becomes the most dominant state when one-dimensionality becomes prominent. The nodes of thepresent PDW state do not cross the Fermi surface, similarto odd-frequency spin-singlet p -wave pairing (OSO). Thepresent PDW-singlet state is supported by the nestingof the Fermi surface and becomes unstable in the caseof incomplete nesting of the Fermi surface because theregions where a Cooper pair can be formed are reduced.It should be noted that the obtained PDW-singletgap function has both even-frequency and odd-frequencycomponents. In general, the obtained gap function of thePDW-singlet state is neither an odd nor an even functionowing to the inversion of the momentum k to − k . Inthe present case, because the magnitude of the center-of-mass momentum is almost 2 k F , the even-frequencypart has a momentum dependence similar to that of theOSO component. The even-frequency component of thegap function is dominant, and this node does not overlapwith the Fermi surface.Under the FLEX approximation, the parameter regionwhere PDW is stabilized becomes wider as compared tothat of RPA because the OSO state becomes suppressedowing to the self-energy effect of the quasiparticle.In this study, we have only considered the on-siteCoulomb repulsion. It is known that charge fluctuation isenhanced in the presence of the off-site Coulomb repul-sion, which also enhances the spin-triplet f -wave pair-ing in the Q1d superconductor [37]. The charge fluctua-tion can also enhance odd-frequency spin-triplet pairingin Q1d superconductors [6]. It is interesting to see howthe PDW triplet state is stabilized in the presence of theoff-site Coulomb repulsion.The physical properties of the PDW-singlet stateare also interesting. Because this pairing has a spatialoscillation, the translational invariance is broken mi-croscopically, and we expect a sufficient odd-frequency pair amplitude even if the major component of the gapfunction is even-frequency pairing [38–42]. Furthermore,it is known that odd frequency pair density wave isinduced by the coexsistence of the charge density waveand d-wave superconductor in underdoped cuprates [ ? ].It is also interesting to study the tunneling, Josephson,and proximity effects in this pairing [43, 44]. ACKNOWLEDGMENTS
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