Topological Defects in Systems with Two Competing Order Parameters: Application to Superconductors with Charge- and Spin-Density Waves
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec Topological Defects in Systems with Two Competing Order Parameters:Application to Superconductors with Charge- and Spin-Density Waves
Andreas Moor, Anatoly F. Volkov, and Konstantin B. Efetov
1, 21
Theoretische Physik III, Ruhr-Universität Bochum, D-44780 Bochum, Germany National University of Science and Technology “MISiS”, Moscow, 119049, Russia (Dated: May 8, 2018)On the basis of coupled Ginzburg–Landau equations we study nonhomogeneous states in systems withtwo order parameters (OP). Superconductors with superconducting OP ∆ , and charge- or spin-density wave(CDW or SDW) with amplitude W are examples of such systems. When one of OP, say ∆ , has a form ofa topological defect, like, e.g., vortex or domain wall between the domains with the phases 0 and π , theother OP W is determined by the Gross–Pitaevskii equation and is localized at the center of the defect. Weconsider in detail the domain wall defect for ∆ and show that the shape of the associated solution for W depends on temperature and doping (or on the curvature of the Fermi surface) µ . It turns out that, providedtemperature or doping level are close to some discrete values T n and µ n , the spacial dependence of the func-tion W ( x ) is determined by the form of the eigenfunctions of the linearized Gross–Pitaevskii equation. Thespacial dependence of W corresponding to the ground state has the form of a soliton, while other possi-ble solutions W n ( x ) have nodes. Inverse situation when W ( x ) has the form of a topological defect and ∆ ( x )is localized at the center of this defect is also possible. In particular, we predict a surface or interfacial su-perconductivity in a system where a superconductor is in contact with a material that suppresses W . Thissuperconductivity should have rather unusual temperature dependence existing only in certain intervals oftemperature. Possible experimental realizations of such non-homogeneous states of OPs are discussed. PACS numbers: 71.45.Lr, 71.55.-i, 74.81.-g, 74.72.-h, 75.30.Fv
I. INTRODUCTION
Materials with two order parameters (OP) have long beenknown. For example, superconductivity in stoichiometricternary compounds ErRh B and M x Mo S (with M mean-ing Ho, Dy, or Er and x = In the last decades interest in the systemswith two OPs has increased drastically in connection withthe discovery of high- T c superconductors (see, e.g., 2–8).Very recently, it has been experimentally established that,in cuprates, the superconducting OP ∆ coexists with a statewith a charge modulation (see recent papers Refs. 9–18 andreferences therein).In another class of recently discoveredsuperconductors—the so-called Fe-based pnictides—superconductivity may coexist with a spin densitywave (SDW) (for a review see Refs. 19 and 20).Coexistence of OPs of different types results in several in-teresting phenomena. One can mention the enhancementof the London penetration depth or a peak in the spe-cific heat jump at the doping level at which the SDW isformed, the peculiar dynamics of the OPs (see recent papersRefs. 26 and 27 and references therein).Nonhomogeneous states in systems with two OPs arealso very interesting and unusual. For example, a CDWarises in the center of vortices in cuprates. Nonhomoge-neous states in superconductors may arise even in the ab-sence of magnetic field. For example, Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) states may appear in superconductorsin the presence of an exchange field and the so called am-plitude solitons can be energetically favorable in conduc-tors with CDW or SDW. The latter have been predicted inRefs. 30, 31 and observed in Ref. 32 in systems with a single OP—in quasi-one-dimensional conductors with a CDW.Amplitude solitons in systems with a CDW mean that theamplitude W ( x ) of the CDW drops to zero at some point,and a local energy level ǫ arises in the system. For ex-ample, if W = W ∞ tanh( x / p ξ w ), the phase χ of W changesfrom χ = x = ∞ to χ = π at x = −∞ . The energy level ǫ may vary in time when a sufficiently high current passesthrough the system. In this case, the stationary state isunstable and one deals with a dynamical amplitude soli-ton with ǫ ( t ). Another case of a non-stationary (mov-ing) soliton was studied in a recent work.
In both cases,the structure of the amplitude soliton can be found analyti-cally from a solution of microscopic equations.
It is rele-vant to mention the stripes in high- T c superconductors that are higher dimensional relatives of the solutions ofRefs. 30, 31.Fulde-Ferrel-Larkin-Ovchinnikov states in superconduc-tors coexisting with other OPs, such as CDW or SDW,were studied recently in several works. In particular, ithas been pointed out in Ref. 43 that the states similar tothe FFLO state are possible in superconductors competingwith CDW or SDW. Note that another type of nonhomoge-neous states in systems with two OPs has been studied inRefs. 44–46 using the so-called Brazoskii-type model. Inparticular, it has been shown that a “glassy” phase may arisein these systems.However, the FFLO-like state may arise, e.g., in Fe-basedpnictides, only at low enough temperatures T when thedependence of W on the curvature µ is a multivaluedfunction which is in a full analogy with superconduc-tors with an exchange field h . In the latter case, the de-pendence ∆ ( h ) at low T is a multivalued function in a cer-tain interval of h . Spatial dependence of ∆ and W can bedescribed by a generalized Eilenberger equation comple-mented by self-consistency relations but their analyticalstudy is not simple.In this Paper, we analyze nonuniform states of the OPson the basis of the Ginzburg–Landau equations that areconsiderably more transparent than the Eilenberger equa-tion. We are interested in nonuniform states correspond-ing to topological defects. We concentrate here on one-dimensional structures and consider the dependence onlyon one coordinate x . This means that we consider a sit-uation when the superconducting OP ∆ changes its phasefrom 0 to π across this defect, while the amplitude of theCDW (or SDW) W ( x ) is localized at the center of the defectdecaying to zero away from this point. Of course, the oppo-site situation is also possible, i.e., when the function W ( x )changes sign having opposite values at −∞ and ∞ and ∆ ( x )is a localized function.Using a system of coupled Ginzburg–Landau equationswe show that, while the superconducting OP may vary inspace as ∆ ( x ) = ∆ ∞ tanh( x / p ξ s ), the form of the ampli-tude W ( x ) depends on temperature or doping. It is de-scribed by the Gross–Pitaevskii equation resulting in a pe-culiar quantization of the solutions for W ( x ). The func-tion W ( x ) corresponding to the ground state has the formof a soliton whereas the functions W n ( x ) corresponding toexcited states have nodes.In the opposite case, when a solution W ( x ) = W ∞ tanh( x / p ξ w ) is brought about, the super-conducting OP ∆ is localized near the point x = and the appearance of boundedge states with possible formation of Majorana fermions atthe surface of superconductors are remarkable examplesof these phenomena. II. FREE ENERGY AND GINZBURG–LANDAU EQUATIONS
We consider a model which is described by the G–L equa-tions. As has been shown in Ref. 27, it is applicable to quasi-one-dimensional superconductors with a CDW and to two-band superconductors with an SDW. The latter model hasbeen developed in detail in Refs. 48, and 49 for Fe-basedpnictides. After certain modification, this model can beapplied also to cuprates. On its basis, one can deriveG–L equations for the OPs ∆ and W . We neglect space vari-ations of the phases of ∆ and W and consider these OPs asreal quantities.To make the physical meaning of the coefficients in the G–L expansion more transparent, we write the G–L equa-tions first for the case of superconductors with a CDW(or SDW). In the notation of Refs. 48, and 49, these equa-tions have the form − ξ s ∇ ∆ + ∆ £ W s m + ∆ s − ln( T s / T ) ¤ = − ξ w ∇ W + W £ 〈 µ s m 〉 + W s m + ∆ s m − ln( T w / T ) ¤ = ξ s , w are the coherence lengths (at low temperatures)for ∆ and W , respectively, and T s , w are, respectively, thecritical temperatures for the transition into the pure super-conducting state or into a state with a CDW or an SDWonly. In other words, T w is the critical temperature forthe transition into the charge-ordered state in absence of ∆ and µ , while T s is the superconducting transition temper-ature in absence of W . The angle brackets mean the angleaveraging (in Fe-based pnictides) or integration along thesheets of the Fermi surfaces in quasi-one-dimensional su-perconductors. The functions s m , s m , etc., are functions ofthe normalized curvature (see Appendix) m = µ /( π T s ) and µ = µ + µ ϕ cos £ ( p y + p z ) a ¤ is a curvature in quasi-one-dimensional superconductors with a doping-dependentvalue of µ . It is assumed that the Fermi surface of these su-perconductors consists of two slightly curved sheets whichare perpendicular to the x axis. In the case of Fe-basedpnictides, µ = µ + µ ϕ cos(2 ϕ ) is a quantity that describes anelliptic ( µ ϕ
0) and circular ( µ ϕ =
0) Fermi surfaces of elec-tron and hole bands.
All quantities— ∆ , W and µ —aremeasured in units of π T s . The expressions for the coeffi-cients in the G–L expansion with account for impurity scat-tering have been calculated in Ref. 53 (see also Ref. 53).Replacing the derivative ∇ → ∇ − i2 π A / Φ , one can useEqs. (1) and (2) to describe vortices in superconductors witha CDW , where Φ is the magnetic flux quantum and A –themodulus of the vector potential of a magnetic field.As it is seen from Eq. (2), the critical temperature T w depends on doping, i.e., on the parameter µ . We choosethis parameter µ = µ c in such a way that T w ( µ c ) = T s . Thismeans that at T = T s , the quantities ∆ = W =
0, and, thus, µ c obeys the equation 〈 µ c s m ( µ c ) 〉 = ln( T w / T s ) ≡ ln r , (3)where r = T w / T s and µ c is a function of two parameters,i.e., µ c = µ c ( µ , µ ϕ ).Then, we expand the function s m ( µ , T ) in the de-viations δ [ µ ] = µ − µ c and δ T = T s − T , thus obtaining s m ( µ , T ) = s m ( µ c , T s ) + β δ T + 〈 β δ [ µ ] 〉 , and use Eq. (3)to obtain equations in a general standard form (assumingthat all the functions depend only on one coordinate x ), − ξ s ∆ ′′ + ∆ £ − a s + b s ∆ + γ W ¤ =
0, (4) − ξ w W ′′ + W £ − a w + b w W + γ ∆ ¤ =
0, (5)with ∆ ′ and W ′ as well as ∆ ′′ and W ′′ denoting the first andsecond derivatives with respect to x , respectively. Theseequations determine extrema of the free energy functional F = Z d x © ξ s ∆ ′ − a s ∆ + b s ∆ + γ ∆ W + ξ w W ′ − a w W + b w W ª (6)with respect to ∆ and W , and the corresponding coefficientsof the G–L expansion are related to variables in Eqs. (1)and (2) via a s = η , b s = s ≃ a w = η (1 − β ) − 〈 β δ [ µ ] 〉 , b w = s m , γ = s m , where η = − T / T s . The expressionsfor β are given in the Appendix.The coupled G–L Eqs. (4) and (5) are, of course, not newand have been used long time ago in, e.g., Ref. for studyingcompetition between superconductivity and CDW in thepresence of disorder or commensurability. III. SOLITON-LIKE SOLUTIONS AT QUANTIZEDTEMPERATURES AND DOPING
Our aim now is to find new non-trivial inhomogeneoussolutions of Eqs. (4) and (5). For simplicity, we consider thecase when the last term in Eq. (4) can be neglected, whichis legitimate when the coupling constant γ or a small ampli-tude W is small (we will see that at temperatures T or dop-ing level µ near some critical values T N and µ N the ampli-tude W is indeed small). In the zero-order approximationwe obtain for ∆ ( x ) − ξ s ∆ ′′ + ∆ £ ∆ b s − a s ¤ =
0. (7)Equation (7) has the well-known nonuniform solution (seefor example Ref. 56), ∆ ( x ) = ∆ ∞ tanh( κ s x ), (8)where ∆ ∞ = p a s / b s and κ s = a s /2 ξ s . This equation de-scribes, for instance, the behavior of ∆ ( x ) in a vicinity ofS/N interface at the superconductor side, where N is a nor-mal metal with a strong depairing. We consider this solutionin an infinite superconductor.Substituting this expression into Eq. (5), we obtain anequation for the amplitude of the CDW or SDW˜ ξ w W ′′ + W £ E + U w cosh − ( κ s x ) ¤ = gW , (9)where E = a w b s − a s γ , U w = a s γ , g = b s b w , and ˜ ξ w = ξ w s .These quantities may be written in notations used for quasi-one-dimensional supercondcutors and Fe-based pnictidesas E = η £ s (1 − β ) − s m ¤ , U w = η s m , g = s s m . Equa-tion (9) for spatial variation of the CDW amplitude W has aform of the well known Gross–Pitaevskii equation. Solu-tions of this equation can be written rather easily in limitingcases. We consider the simplest situation when the RHS ofEq. (9) is small, i.e., gW ≪ U w .We are interested in solutions with ∆ ( x ) given by Eq. (8)and W ( x ) decaying to zero at x → ±∞ . In particular, the so-lution for W ( x ) may have the form of a soliton. Such a state with a finite ∆ and zero W at infinity is stable if the con-ditions ∂ F / ∂ ∆ > ∂ F / ∂ W > ∆ = ∆ ∞ W = ∂ F / ∂ ∆ | W = ∼ b s isalways positive and ∂ F / ∂ W | W = ∼ a w − a s γ / b s ∼ − E / b s is positive if the quantity E is negative. We will see that justat negative E , Eq. (9) has a solution in the form of a soliton.In zero-order approximation we obtain for W ˜ ξ w W ′′ + W £ E + U w cosh − ( κ s x ) ¤ =
0. (10)This equation is integrable and its solutions ψ n correspond-ing to a discrete spectrum of E n are expressed in terms ofhypergeometric functions. In our notations, the “energy”levels of discrete spectrum are given by E n = − a s b s ξ w ξ s · − (1 + n ) + s + γξ s ξ w b s ¸ , (11)and their maximal number n max is determined by2 n max ≤ q + γξ s / ξ w b s −
1. Note that in Fe-based pnic-tides, ξ s / ξ w ≃ T w /( T s s m ) = r / s m in the ballistic case.We expand the correction δ W to the zero-order solu-tion W in terms of the normalized eigenfunctions ψ n of theoperator ˆ L = − ˜ ξ w ∂ xx − U w cosh − ( κ s x ). These functionsobey the equation ˆ L ψ n = E n ψ n . (12)Solutions of Eq. (9) can be written explicitly if the quan-tity E = E ( η , δ [ µ ]) is close to a certain “energy” level E n , sayto E N , such that E ≃ E N = E ( η N , δ [ µ N ]) (in the language ofthe original electronic model, the “temperature” η or dop-ing δ [ µ ] should be chosen properly). We write Eq. (9) in theform ˆ L W = E N W + R ( W ), (13)with R = gW + ( E − E N ) W and represent W as W ( x ) = c N ψ N ( x ) + δ W N ( x ), where δ W N ( x ) = P ′ n c N , n ψ n ( x ),and the summation runs over all n except the term n = N .We substitute this W ( x ) into Eq. (13) and multiply thisequation first by ψ N and then by ψ n with n N , then inte-grating the obtained result each time over x . Thus, takinginto account the orthogonality of different eigenfunctions,we find the coefficients c n c N = E − E N g 〈〈 ψ N 〉〉 , (14) c N , n = g c N 〈〈 ψ N ψ n 〉〉 E n − E N with n N , (15)where 〈〈 f ( x ) 〉〉 = R ∞−∞ d x f ( x ), where the double angle brack-ets are used to distinguish this operation from the averagingover the angles introduced in the Appendix. Obviously, inEq. (15), ψ n and ψ N have to have same parity (both even orboth odd).The obtained expressions are valid provided the condi-tion | E − E N | ≪ | E n − E N | is satisfied. This condition meansthat if the “temperature” η or doping δ [ µ ] is chosen in such FIG. 1. (Color online.) The coefficient C in Eq. (17) on µ c for r = µ c is calculated from Eq. (3) and represents aline in the ( µ ϕ , µ ) plane, which is shown in the upper part of thefigure. In the lower part, the value of C along the obtained µ c -lineis presented as function of µ ϕ (inserting the corresponding valueof µ ). The coefficient C is negative, thus, as η > δ [ µ ] < a way that the quantity E ( η , δ [ µ ]) is close to E N , i.e., thedifference on the LHS of this condition is smaller than thedifference between any energy level E n and E N , the spa-tial dependence of W ( x ) is given by the leading order whilethe second term, δ W N ( x ), gives a small correction. Sincewe assumed that the RHS of Eq. (9) is small compared tothe term a s ∆ , the condition | E − E N | ≪ a s b s b w / γ should bealso satisfied.The ground state is realized if at some “temperature” η the quantity E ( η ) is close to E ( η ). In this case, W ( x ) hasthe form of a soliton. If E ( η ) is close to E , the amplitudeof the CDW is an odd function of x . For the ground state,Eq. (11) yields µ A + δ [ µ ] B ® η ¶ = s s m r ·s + r s m s m s − ¸ , (16)where A = s m − s (1 − β ) and B = s β . At a given δ [ µ ] = µ − µ c , that can be both positive and negative, thisequation determines the “temperature” η at which the so-lution of the Gross–Pitaevskii equation has a soliton-like so-lution W . Similarly, setting n = η corresponding to the first excited statewith an odd function of the OP W ( x ) etc. In Fig. 2 weplot the spatial dependence of the CDW amplitude W ( x ) for n =
0, 1, 2 and 3. Note that if r s m /( s m s ) <
1, only a singlesoliton-like solution exists.
FIG. 2. (Color online.) Coordinate dependence of W (red), W (black), W (green) and W (blue) near the corresponding “en-ergy” levels E N for the case when the superconducting state is fa-vored far from the defect at x =
0. Exactly at E N , as follows fromEqs. (14) and (15), W =
0. Note that in the opposite case when theCDW or the SDW state is more favorable at x → ∞ , one needs tomake the exchange ∆ ↔ W and, correspondingly, U w ↔ U s and κ s ↔ κ w , and the shown curves will describe the dependence ∆ ( x )while W = W ∞ tanh( κ w x ). As an example, we calculate for the ground state the de-pendence of η on δ [ µ ]. It follows from Eq. (16) that, as-suming δ [ µ ] independent on ϕ , η = C · δ [ µ ], (17)where the coefficient C is given by C = 〈 B 〉 · s s m r µs + r s m s s m − ¶ − A ¸ − . (18)The coefficient C depends on µ c defined by Eq. (3). It isnegative and, thus, since η > δ [ µ ] should also be nega-tive. We plot the dependence of C on µ c in Fig. 1. Moreprecisely, the critical doping µ c is calculated from Eq. (3)and represents a line in the ( µ ϕ , µ ) plane (the upper partof Fig. 1). Projecting this line onto the µ ϕ axis and in-serting the corresponding values of µ we obtain the plotof C = C ( µ c ) ≡ C ( µ , µ ϕ ) presented in the lower part of Fig. 1.Consider the temperature interval where the soliton-likesolution for W ( x ) exists. As follows from Eq. (14), the dif-ference E − E must be positive if the constant g = b s b w ispositive. This implies that the difference E − E = A ( T − T )has to be positive as well (at A > T < T ,no W appears at the topological defect, but at T > T , asoliton-like solution for W ( x ) arises with the amplitude W (0) ∼ p ( T − T ). On the other hand, as follows fromEq. (16), the temperature T is less than the tempera-ture T ≡ c µ B / A , where c µ = − δ [ µ ]. This means that thesoliton-like solution for W as well as solutions correspond-ing to excited states exist in the interval T < T < T . (19)The solutions found above are valid if the free en-ergy F s of the superconducting state at x → ±∞ is lowerthan F w for a state with W
0. This is possibleif F s − F w ∼ ( ∆ − W ) ∼ £ a s ( η )/ b s − a w ( η )/ b w ¤ <
0. Thiscondition determines a temperature interval in which ourconsiderations are valid.If the difference F s − F w is positive, then the same proce-dure of finding solutions of G–L equations can be repeatedwith an exchange ∆ ↔ W adapting correspondingly E n andother quantities. In particular, W = W ∞ tanh( κ w x ), with W ∞ = a w / b w and κ w = p a w / ξ w , and the superconduct-ing OP ∆ ( x ) is expressed in terms of hypergeometric func-tions, i.e., it is localized at x =
0. Consider, for example, anN/S system where in the superconductor S there exists notonly the superconducting OP ∆ , but also a density wave W ,and N is a normal metal with a strong suppression of W (for example, with a strong interband impurity scatteringwhich suppresses the OP W ). Then, at the S-side, thedependence W ( x ) is determined by the above written ex-pression, and at a certain temperature which may be evenhigher than T s , at the N/S interface superconductivity mayarise spreading over a distance ∼ ξ s from the interface.Note that the found nonhomogeneous solutionsfor ∆ ( x ) and W ( x ) are energetically favorable in com-parison with uniform solutions, ∆ ∞ and W ∞ , pro-vided the energy loss [due to the gradient of ∆ ( x )] δ F s ∼ R d x ∆ ∞ [1 − tanh ( κ s x )] ∼ ∆ ∞ ξ s / p a s is less thanthe energy gain δ F w (due to the appearance of W ) δ F w ∼ R d x W ( x ) ∼ W ∞ ξ w . This cannot occur in theconsidered case of small W . However, in heterostructures,like an N/S system, the solution Eq. (8) (at x >
0) is dictatedby a boundary condition in case of strong depairing inthe N metal and, therefore, there is no energy loss in thesuperconducting part of the free energy. Thus, the consid-ered states may be realized in heterostructures. The case ofuniform superconductors with a not small OP W requires aseparate consideration. IV. CONCLUSION
On the basis of Ginzburg–Landau equations we studied apossibility of nonhomogeneous states in systems with twoOPs. Materials, where the superconducting OP ∆ and theOP W related to a CDW (or an SDW) may exist, belong tothis class of systems. In the situation when the supercon-ducting state is more favorable, the Ginzburg–Landau equa-tions have nonhomogeneous solutions which describe ∆ ( x )in the form of a topological defect, Eq. (8), and W ( x )—inthe form of a function localized near the center of the de-fect, x =
0. The form of W ( x ) is described by the Gross–Pitaevskii equation and depends essentially on the proxim-ity of the function E ( η , δ [ µ ]) to the eigenvalues E N of thelinearized Gross–Pitaevskii equation. If E ( η , δ [ µ ]) = E N atsome temperature T N = (1 − η N ) T s and doping δ [ µ N ], thenthe amplitude of the function W ( x ) turns to zero and in-creases as W ∼ p | E N − E ( η , δ [ µ ]) | when η or δ [ µ ] deviatefrom η N and δ [ µ N ]. At a given temperature T in the inter-val Eq. (19), there are, generally speaking, several solutionsfor W ( x ). The most stable one is the solution which cor- responds to the ground state (soliton-like solution). There-fore, in the equilibrium case one can observe only this solu-tion for W ( x ). Other solutions may affect the response of thesystem to the influence of fluctuations or of external pertur-bations.On the other hand, if the state with W ∆ = W ( x ) determinedby Eq. (8) (correspondingly adapted as ∆ → W , ξ s → ξ w )and ∆ ( x ) is localized near the point x =
0. In principle, suchsolutions may arise in the bulk (especially near some de-fects) and in heterostructures of type N/N s , w , where N s , w is a material under consideration in which ∆ and/or W mayexist, and N is a material with a strong depairing towards theOPs ∆ and W . For example, in an N/N w heterostructure, theOP in the vicinity of the interface has inevitably the form ofEq. (8) and a localized ∆ ( x ) ACKNOWLEDGMENTS
We appreciate the financial support from the DFG via theProjekt EF 11/8-1; K. B. E. gratefully acknowledges the finan-cial support of the Ministry of Education and Science of theRussian Federation in the framework of Increase Competi-tiveness Program of NUST “MISiS” (Nr. K2-2014-015).
Appendix A: Coefficients in the Ginzburg–Landau equations
The free energy has the form (see also Refs. 48, and 49) Φ ( ∆ , W , µ ) = − (2 π T ) E m X ω = ℜ ( P ) + ∆ λ sc + W λ dw , (A1)where P = p ( ς sc ω + i µ ) + W , ς sc ω = p ω + ∆ and ω is theMatsubara frequency with a cut-off E m ; λ sc and λ dw are theinteraction constants of the superconductivity and spin- orcharge-density wave, respectively. Expanding this expres-sion in ∆ and W and performing variation with respect tothese variables, we come to Eqs. (1) and (2) with the coeffi- cients defined as s = ∞ X n = (2 n + − , (A2) s m = ∞ X n = (2 n + − £ (2 n + t + m ¤ − , (A3) s m = ∞ X n = £ (2 n + − m ¤ (2 n + − £ (2 n + + m ¤ − ® ,(A4) s m = ∞ X n = (2 n + £ (2 n + − m ¤£ (2 n + + m ¤ − ® ,(A5) β = ∞ X n = m (2 n + £ (2 n + + m ¤ − ® , (A6) β = ∞ X n = n + − £ (2 n + + m ¤ − , (A7)where t = T / T s and the angle brackets denote the angleaveraging (in Fe-based pnictides) or integration along thesheets of the Fermi surfaces in quasi-one-dimensional su-perconductors. L. Bulaevskii, A. 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