Fermi Liquid instabilities in two-dimensional lattice models
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Fermi Liquid instabilities in two-dimensional lattice models
C.A. Lamas, D.C. Cabra,
2, 1, 3 and N. Grandi Departamento de F´ısica, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina Laboratoire de Physique Th´eorique, Universit´e Louis Pasteur,3 Rue de l’Universit´e, 67084 Strasbourg, C´edex, France. Facultad de Ingenier´ıa, Universidad Nacional de Lomas de Zamora,Cno. de Cintura y Juan XXIII, (1832) Lomas de Zamora, Argentina.
We develop a procedure for detecting Fermi liquid instabilities by extending the analysis of Pomer-anchuk to two-dimensional lattice systems. The method is very general and straightforward to apply,thus providing a powerful tool for the search of exotic phases. We test it by applying it to a latticeelectron model with interactions leading to s and d -wave instabilities. PACS numbers:
1. INTRODUCTION
The Landau theory of the Fermi Liquid (FL) is one ofthe most important frameworks to understand conven-tional weakly interacting metallic systems . The low en-ergy physics of interacting fermions in three dimensions isusually described by Landau’s FL theory whose centralassumption is the existence of single particle fermionicexcitations, or “quasiparticles”, with a long lifetime atvery low energies. In lower dimensions, however, thesituation is much more interesting: in one dimensionalsystems Landau’s quasiparticles are typically unstable,giving rise to the so called Luttinger Liquid. On theother hand, two dimensional lattice models are far morecomplicated to treat since conventional perturbation the-ory breaks down for densities close to half-filling, wherecompeting infrared divergences appear as a consequenceof Fermi Surface (FS) nesting and van Hove singularities.In Ref. Pomeranchuk developed a method to diag-nose instabilities of the FL, by “deforming” the FS andstudying the resulting energy gain. In its original form, itapplies to systems with a three dimensional spherical FS,but it can be easily generalized to the two-dimensionalcontinuum case.The experimental observation of exotic phases instrongly correlated systems has triggered an enormouseffort from the theoretical community to try to under-stand their microscopical origin. One possible route todetect instabilities of a FL is precisely the analysis doneby Pomeranchuk. Due to that, the Pomeranchuk insta-bility has been studied by several authors with differenttechniques in the last few years and in particu-lar, the instability of the FL towards the nematic phasewas investigated for several models .In this paper we develop a general method to tracesuch instabilities in lattice models, in a simple and rig-orous way. It allows for the study of systems which havean arbitrary shape of the FS in the absence of interac-tions, thus being applicable to models relevant to hightemperature superconductors, manganites, ruthenates,etc. . as long as one can rely on a perturba-tive analysis. It can be applied in principle to any lattice problem in a systematic way.We test our method within two examples, the attrac-tive Hubbard model and a model with forward scatteringinteractions that give rise to d -wave FS deformation (theso-called “ d -wave Pomeranchuk instability”).The paper is organized as follows: Section 2 contains adetailed derivation of the method, with the proof of ourformulas in subsection 2.1, and a shorthand recipe for theapplication of the results in subsection 2.2. Then in Sec-tion 3 we apply the method to a two dimensional squarelattice with the various interactions studied in , the s -wave interaction being studied in subsection 3.2, whilethe d -wave instability in subsection 3.3. Finally, section4 contain the conclusions, and some specific calculationsare presented in the Appendix.
2. TWO DIMENSIONAL POMERANCHUKINSTABILITY2.1. Derivation of the method
In the theory of Landau’s FL, the free dynamics atzero temperature is determined by the dispersion relation ε ( k ). In terms of it, the FS is defined as the set of pointsin momentum space satisfying the equation µ − ε ( k ) = 0 . (2.1)In the ground state of the system, all single-particle statesinside the FS µ − ε ( k ) > µ − ε ( k ) < E = Z d k ( ε ( k ) − µ ) δn ( k ) + 12 Z d k Z d k ′ f ( k, k ′ ) δn ( k ) δn ( k ′ ) , (2.2) where δn ( k ) is the change in the distribution function n ( k ), and we have assumed that only two-particle inter-actions are present. The interaction function f ( k, k ′ ) canbe related to the low energy limit of the two particle ver-tex.Pomeranchuk criterion allows to identify low energyexcited states of the system that make (2.2) negative.This signals an instability, and the breakdown of the FLdescription. In what follows, we will carefully go throughall the steps needed to perform such analysis.First let us define, associated to any given state of thesystem, a smooth function g ( k ) such that it takes posi-tive values at occupied single-particle states and negativevalues at unoccupied ones. Then at the frontier betweenthese two regions we have the equation g ( k ) = 0 . (2.3)For the ground state, such frontier coincides with the FSallowing us to choose g ( k ) = µ − ε ( k ) . (2.4) Under a variation δn ( k ) of the distribution function n ( k ), we get an excited state that can be described interms of a new function g ′ ( k ) = g ( k )+ δg ( k ). The frontierbetween occupied and unoccupied single-particle states isnow located at points satisfying g ′ ( k ) = g ( k ) + δg ( k ) = 0 . (2.5)By an abuse of language we will call the solution of thisequation the deformed FS .Since at T = 0 we have δn ( k ) = ± δn ( k ) = H [ g ′ ( k )] − H [ g ( k )] , (2.6)where H ( x ) is the unit step function, defined by H ( x ) = 1if x > H ( x ) = 0 if x <
0. This can be replacedin (2.2) to write the energy of the quasiparticles as afunctional of g ( k ) and g ′ ( k ), namely E = Z d k ( ε ( k ) − µ ) ( H [ g ′ ( k )] − H [ g ( k )]) + 12 Z d k Z d k ′ f ( k, k ′ )( H [ g ′ ( k )] − H [ g ( k )])( H [ g ′ ( k ′ )] − H [ g ( k ′ )]) . (2.7)To go further, we have to take into account the constraintimposed by the Luttinger theorem , or in other wordsthe preservation of the area of the FS under the defor-mation Z d k δn ( k ) ≡ . (2.8)By using (2.6) this can be rewritten as a functional con-straint on the functions g ( k ) and g ′ ( k ) Z d k H [ g ′ ( k )] = Z d k H [ g ( k )] . (2.9)In two-dimensions the constraint (2.9) can be easilysolved as follows. We first rename the integration vari-ables on the right hand side to k ′ . Next we assume thata change of variables k ′ = k + δk ( k ) can take the righthand side into the form of the left hand side. Writing g ′ ( k ) = g ( k ) + δg ( k ) we get two unknown functions tobe solved for, namely δg ( k ) and δk ( k ), together with theequation Z d k H [ g ( k ) + δg ( k )] = Z d k (cid:12)(cid:12) ∂ j δk i (cid:12)(cid:12) H [ g ( k + δk ( k ))] , (2.10) where i, j ∈ { , } label the orthogonal directions in mo-mentum space.A particular class of solutions can then be obtained bysolving the following equations (cid:12)(cid:12) ∂ j δk i (cid:12)(cid:12) = 1 ,g ( k ) + δg ( k ) = g ( k + δk ( k )) . (2.11) The first line (2.11) implies that the change of variablesgoing from k ′ to k is an area preserving diffeomorphism.The second line on the other hand, can be interpreted assaying that the variation δg ( k ) is a translation of g ( k ) byan amount δk . We can solve (2.11) as δk i = ( e ǫ jk ∂ j λ∂ k − k i ,δg = ( e ǫ ij ∂ i λ∂ j − g . (2.12)Where λ is a free function parameterizing the deforma-tion. If we assume that the deformation of the FS issmall, then δg ( k ) is also small and we can parameterizeit in terms of a slowly varying λδk i ≃ ǫ ij ∂ j λ ,δg ≃ ǫ ij ∂ j λ∂ i g . (2.13)In what follows, each specific form of λ will characterizean excited state, the sign of the resulting energy will giveus information about the instabilities.Now that we have solved the constraint, we go backto the energy of the quasiparticles (2.7) and write it interms of the free unconstrained variable λ . To simplifythe resulting expression, we need to change variables to amore convenient coordinate system in momentum space.We choose a special set of variables g = g ( k x , k y ) ,s = s ( k x , k y ) , (2.14)where the new variable g varies in the direction trans-verse to the unperturbed FS. The variable s varies inthe longitudinal direction tangent to the FS, namely itsatisfies ∂ i s∂ i g = 0.Separating the energy (2.7) into a linear and an inter-action term E = L + I , we get for the linear part L = Z d k ( ε ( k ) − µ ) ( H [ g + δg ] − H [ g ]) == Z ds dg J ( s, g )( ε ( s, g ) − µ ) ( H [ g + δg ] − H [ g ]) == Z ds Z − δg dg J ( s, g )( ε ( s, g ) − µ ) , (2.15)where J = | ∂ ( k x , k y ) /∂ ( g, s ) | is the Jacobian of the trans-formation (2.14). Expanding in powers of the integrationvariable g around the unperturbed FS g = 0 we get L = Z ds Z − δg dg ∂ ¯ g (cid:2) J ( s, ¯ g )( ε ( s, ¯ g ) − µ (cid:1) ] ¯ g =0 g + O ( δg ) == 12 Z ds [ J ( s, ¯ g ) δg ] g = g ′ =0 + O ( δg ) , (2.16)where in the second line we have integrated out the vari-able g and made use of the fact that ( ε ( s, g = 0) − µ (cid:1) = 0.In order to replace the explicit form of δg eq.(2.13) in theintegrand of (2.16) we make use of the identity ǫ ij ∂ i g∂ j λ = ǫ ij ( ∂ i g∂ g g + ∂ i s∂ s g )( ∂ j g∂ g λ + ∂ j s∂ s λ ) = = ǫ ij ∂ i g∂ j s∂ s λ ≡ J − ∂ s λ , (2.17)where we have used the fact that, according to our def-initions, ∂ g g = 1 and ∂ s g = 0. Now replacing in (2.16)we get L = 12 Z ds (cid:2) J − ( g, s ) ( ∂ s λ ) (cid:3) g =0 (2.18)The calculus of I is analogous and gives I = 12 Z ds Z ds ′ [ f ( g, s ; g ′ , s ′ ) ( ∂ s λ )( ∂ s ′ λ )] (cid:12)(cid:12) g = g ′ =0 . (2.19) Adding the two contributions we finally have E = 12 Z ds Z ds ′ (cid:16) f (0 , s ; 0 , s ′ ) + J − (0 , s ) δ ( s − s ′ ) (cid:17) ×× ∂ s λ (0 , s ) ∂ s ′ λ (0 , s ′ ) . (2.20) As the functions λ (0 , s ) characterizing the excited statesare arbitrary, we can equally work with functions ψ ( s ) = ∂ s λ ( g, s ) | g =0 . In what follows we will be interested inexcited states such that ψ ( s ) ∈ L [0 , S ]. Assuming that s makes a complete turn around the FS when it runsfrom 0 to S , we also need to impose periodicity in thatinterval.Since the sign of E in eq. (2.20) determines the sta-bility of the FL, from all the above we conclude that thestability condition reads E = Z ds ′ Z ds ψ ( s ′ ) 12 (cid:18) J − ( s ) δ ( s − s ′ ) + f ( s, s ′ ) (cid:19) ψ ( s ) > , (2.21)where have we defined f ( s, s ′ ) = f ( g, s ; g ′ , s ′ ) (cid:12)(cid:12) g = g ′ =0 ,J − ( s ) = J − ( g, s ) (cid:12)(cid:12) g =0 . (2.22)Note that the stability condition has two terms, thefirst of which contains the information about the formof the FS via J − ( s ), while the second encodes the spe-cific form of the interaction in f ( s, s ′ ). There is a clearcompetition between the interaction function in the sec-ond term of the integrand and the first term that onlydepends of the geometry of the unperturbed FS.We see that E is a bilinear form, acting on the realfunctions ψ ( s ) parameterizing the deformations of theFS E = h ψ, ψ i , (2.23)where h u, v i = Z ds ′ Z ds u ( s ) 12 (cid:16) f ( s, s ′ ) + J − ( s ) δ ( s − s ′ ) (cid:17) v ( s ′ ) . (2.24) The stability condition is then equivalent to asking thisform to be positive definite for any possible deformation, i.e. ∀ ψ : h ψ, ψ i > . (2.25)In consequence, the natural way to diagnose an instabil-ity is to diagonalize this bilinear form and to look fornegative eigenvalues.We can expand the functions ψ ( s ) in some basis of L [0 , S ] that we will denote { ξ i ( s ) } ψ ( s ) = X i a i ξ i ( s ) , (2.26)and then write E = X i ,i a i a i h ξ i , ξ i i . (2.27)The bilinear form h , i can be taken as a pseudo-scalarproduct, which is linear and symmetric but, in general,not positive-definite. Only in the free case f ( s, s ′ ) ≡ { ξ i ( s ) } are taken to beorthogonal with respect to this pseudo-scalar product,then the functional (2.21) is given by E = X i a i χ µi , (2.28)where χ µi = h ξ i , ξ i i is the square pseudo-norm of the or-thogonal functions. If χ µi has a negative value for some i , then by choosing the corresponding a i = 1 and a j = 0for j = i , we see that the energy is negative denoting aninstability. In this case we say that we have an insta-bility in the i -th channel. In other words, the stabilitycondition has been mapped into ∀ i : χ µi > , (2.29)the χ µi being taken as the stability parameters. If any ofthese quantities is negative, then the FS is unstable.We perform such diagonalization by choosing a basison L [0 , S ] as a given set of functions { ψ i } and then mak-ing use of the Gram-Schmidt orthogonalization proce-dure to transform it into an orthogonal basis { ξ i } . Notethat, being the bilinear form not necessarily positive def-inite, the new basis cannot be normalized to 1 but to ±
1. Get the dispersion relation ε ( k ) and the interactionfunction f ( k, k ′ ) for the model under study.2. Change variables according to (2.14). The variable g is completely fixed by the dispersion relation ac-cording to (2.4). The choice of s is arbitrary ex-cept for the constraint of being tangential to theFS, ∂ i s∂ i g = 0.3. Write the bilinear form E as in (2.21).4. Choose an arbitrary basis of functions { ψ i } of L [0 , S ].5. Apply the Gram-Schmidt orthogonalization proce-dure, verifying at each step whether h ξ i , ξ i i >
06. If for a given channel i one finds that h ξ i , ξ i i < L [0 , S ] is infinite dimensional, thepresent method is not efficient to verify stability: at anystep i it may always be the case that for some j , χ µi + j <
3. INSTABILITIES IN THE SQUARE LATTICE3.1. Contribution of the free Hamiltonian
We start considering free fermions in the square lattice,with a Hamiltonian given by H = X ( ε ( k ) − µ ) c † k c k , (3.1)where ε ( k ) = − t (cos k x + cos k y ) , (3.2)where only hopping to nearest neighbors has been takeninto account. The FS is defined by g ( k ) = µ − ε ( k ) = µ + 2 t (cos k x + cos k y ) = 0 , (3.3)where µ is the chemical potential. Notice that g > g ( k x , k y ) = µ + 2 t (cos k x + cos k y ) , (3.4) s ( k x , k y ) = arctan (cid:18) tan( k y / k x / (cid:19) . (3.5)It is straightforward to see that g and s are mutuallyorthogonal variables. Using the following shorthand no-tation α = cos k x , (3.6) β = cos k y , (3.7)we can write g = µ + 2 t ( α + β ) , tan ( s ) = (cid:18) − β β (cid:19) (cid:18) α − α (cid:19) , (3.8)and the Jacobian takes the form J = t (cid:18) αβ − α + β − (cid:19) . (3.9)Notice that J ≥
0. Writing α and β as functions g and s we have for the Jacobian evaluated at g = 0. J [ g = 0 , s ] = 12 t p − β ( µ ) cos (2 s ) , (3.10)with β ( µ ) = 1 − ( µ t ) . The limits for the variable s canbe taken as − π < s ≤ π .The inverse of the jacobian J − ( s ) can be expandedin series of sin( ns ) and cos( ns ) and it is straightforwardto show that only the coefficients of cos(4 ns ) are non-vanishing. This results in the following expansion J − ( s ) = X n j µn cos(4 ns ) , (3.11) U = 0 U = − . U = − . U = − . U = − . U = − . µ µ -4 -2 0 2 402.557.51012.515 -4 -2 0 2 402.557.51012.515-4 -2 0 2 402.557.51012.515 -4 -2 0 2 402.557.51012.515-4 -2 0 2 402.557.51012.515 -4 -2 0 2 402.557.51012.515 FIG. 1: The instability parameters. For U = 0 we show thefirst 10 parameters χ µi as a function of µ . For other valuesof the interaction we show only the parameters correspondingto the three lower channels that show instabilities, namelychannels χ , χ and χ (colors are identified in Fig 2). Noticethat when we increase U , the FL breakdown occurs first forthe higher channels and closer to half filling. where the coefficients j µn are fixed by the expansion.Some of them are given in the Appendix. The simplicityof this expansion suggest to take as our starting base inthe Gram-Schmidt orthogonalization procedure the set { sin( ns ) , cos( ns ) } .In the next subsections we will analyze as an exam-ple of application the possible instabilities in this two-dimensional fermion model when subjected to various in-teractions. In particular we are interested in interactionsof the form . f ( k, k ′ ) = Constant × d ( k ) d ( k ′ ) , (3.12)with d ( k ) = 1 , ( s -wave) , (3.13) d ( k ) = (cos k x + cos k y ) , (extended s -wave) ,d ( k ) = (cos k x − cos k y ) , ( d -wave d x − y ) . s -wave instability First we consider a constant interaction correspondingto take d ( k ) = 1 so that f ( k, k ′ ) = U , (3.14) U being a constant measuring the strength of the inter-action. This form of the interaction can be obtained by µ/t U t FL FLNFL χ χ χ -4 -2 0 2 4-0.6-0.5-0.4-0.3-0.2-0.10 FIG. 2: Phase diagram for f ( s, s ′ ) = U , are displayed theregions of instability for the first three channels. For smallerinteractions, higher channels are unstable closer to half filling. a Mean Field approximation or a first order perturbativeexpansion for the interaction function on the Hubbardmodel .Using the product defined in eq. (2.24) we can calcu-late the first instability parameters. χ µ = 2 π ( U π + j µ ) , (3.15) χ µ , = πj µ ,χ µ = π ( j µ − j µ ) ,χ µ = π ( j µ + 12 j µ ) ,χ µ , = − π j µ j µ + j µ π + π (cid:18) j µ j µ (cid:19) ! ,χ µ = π ( j µ − j µ ) ,χ µ = π U j µ π U + j µ ) − π j µ π U + j µ ++ j µ π + π j µ π U + j µ ) ! + π j µ , etc. . . (3.16)The stability parameters for the first channels areshown in Fig 1. For U = 0 all the χ µn are positive asexpected. When we increase the interaction, among thefirst 20 parameters, only χ µ , χ µ and χ µ change. Forsimplicity only these parameters are plotted for U = 0.With these first instability channels we can draw aqualitative phase diagram in the ( µ, U ) space as in Fig 2where the first instability zones are shown and a tentativeglobal phase diagram is drawn.Note that, when the interaction is increased, the firstinstability channel corresponds to the highest of the threeshown in the figure ( e.g. χ µ ). This behavior is main-tained for channels χ µi with higher index i , and we as-sume that generically these higher channels will show theinstability closer to half filling and for interactions arbi-trarily small. On the other hand, the higher the channel,the closer the instability region is to half-filling. Extrap-olating this behavior we see that the instabilities on thelarge- i channels take place only very close to µ = 0.The behavior for the extended s -wave with a form fac-tor d ( k ) = (cos k x +cos k y ) = α + β can be studied writing d ( k ) in terms of the variables g and s using the solutionsof (3.8) and evaluating at g = 0. We have d ( s ) = − µ t andthe interaction function reads f ( s, s ′ ) = U (cid:16) µ t (cid:17) . (3.17)Again f ( s, s ′ ) is independent of the variables s and s ′ butnow dependent of the chemical potential µ .The corresponding instability parameters are obtainedby changing U → U (cid:0) µ t (cid:1) in (3.15), and the phase dia-gram can then be inferred to be analogous to that of Fig.2 but with the vertical axes replaced by U (cid:0) µ t (cid:1) . d -wave Pomeranchuk instability Now we investigate d -wave Fermi Surface Deformation(dFSD) instability in the charge channel on a squarelattice. The forward scattering interaction driving thedFSD has the form f ( k, k ′ ) = − G d ( k ) d ( k ′ ) (3.18)with G > d -wave form factors d ( k ) = (cos k x − cos k y ). The above expression for this effective interac-tion was obtained by Metzner et al using functionalrenormalization group methods.Using the shorthand notation (3.6) the interactionreads f ( s, s ′ ) = − G ( α − β )( α ′ − β ′ ) (3.19)and using the solution of eq. (3.8) we have d ( s ) = 2cos(2 s ) (cid:18) J − ( s )2 − (cid:19) (3.20)Notice that this interaction contains the Jacobian but itsorigin is totally independent of the treatment developedin the last sections.The form factor d ( s ) can be expanded in a series of theform d ( s ) = ∞ X k =0 d k cos((4 k + 2) s ) , (3.21)where the first coefficients of the expansion are presentedin the Appendix.Performing the Gram-Schmidt orthogonalization as inprevious case, we find the χ µ -parameters correspondingto this interaction. The results are very similar, and we µ/t G t FL FLNFL χ χ χ -4 -2 0 2 40.20.40.60.811.2 FIG. 3: Phase diagram for the d -wave interaction. The firstthree unstable channels are shown. The dashed line corre-sponds to the the critical value for the interaction parameterstudied in reference 11 within a Mean Field treatment. will only display here the first two parameters that showan instability of the system, namely χ µ = − G π d + π j + π j , (3.22) χ µ = π gπ d − j + j ) (cid:0) j − j +2 j j ++ j − j j − j ( j + j ) + gπ (cid:0) d (2 j + j ) − d d ( j + j ) + 4 d (2 j + j ) (cid:1)(cid:1) (3.23)Again, by making use of this first instabilities we cansketch the phase diagram corresponding to this interac-tion, as shown in Fig. 3. Similarly to the previous case,as the interaction grows instabilities appear first in thehigher channels. The dashed line corresponds to the crit-ical value of the interaction found in ref 11 by means ofa Mean Field procedure. Notice that the lowest channelscover most of the instability zone. The phase diagramshown in Fig. 3 is consistent with the results presentedin [11,14].Unlike treatments using Mean Field, with the presentformalism it is possible to identify the region in the pa-rameter space where each channel presents a breakdownof the Fermi liquid behavior.
4. SUMMARY AND CONCLUSIONS
In this paper we have developed a general procedurefor detecting instabilities in two dimensional lattice mod-els. It is an extension of the formalism of Landau-Pomeranchuk, in particular for lattice systems with anarbitrary shaped FS and allows to describe the phasediagram of the system as an alternative to the usual pro-cedures. The steps are simple and applicable to a widevariety of systems.Complementarily to other descriptions, our procedurepermits to identify the breakdown of the Fermi liquidbehavior on each instability channel independently.As a form of testing our procedure, we have analyzedthe stability of the Fermi Liquid in a square lattice, forvarious interactions already studied in the literature. The s -wave instability in the electronic channel and the insta-bility produced by d -wave forward scattering interactionswere studied at T = 0. The instabilities corresponding tolow channels produce a breakdown of the FL behavior fora wide range of fillings, while those occurring for higherchannels are closer to half filling. Our result are in goodagreement with those obtained by different methods thatwere previously published by other authors.Generalization to higher dimensions, spin-dependentinteractions or finite temperature can be achieved fol-lowing the same lines and the results will be presentedelsewhere .
5. APPENDIX: ORTHOGONAL BASIS ANDSERIES EXPANSION.
In this Section we present the coefficients in the expan-sion of the functions used in this paper. For the Jacobian,the series takes the form J − ( s ) = X n j µn cos(4 ns ) , (5.1)where the first coefficients in the expansion are given by j µ = 4 π E (cid:20) − µ (cid:21) j µ = 1 π (cid:18) | µ | E (cid:20) − µ (cid:21) − E (cid:20) − µ (cid:21) −− | µ | π F (cid:20) − , , , − µ (cid:21) ++ 2 π F (cid:20) − , , , − µ (cid:21)(cid:19) ,j µ = 4 π (cid:18) E (cid:20) − µ (cid:21) + − π F (cid:20) − , , , − µ (cid:21) ++ 3 π F (cid:20) − , , , − µ (cid:21)(cid:19) ,j µ = − π (cid:18) E (cid:20) − µ (cid:21) ++4 π (cid:18) F (cid:20) − , , , − µ (cid:21) −− F (cid:20) − , , , − µ (cid:21) ++ 10 F (cid:20) − , , , − µ (cid:21)(cid:19)(cid:19) , (5.2) where E [ m ] is the complete elliptic integral E [ m ] = Z π q − m sin ( t ) dt , (5.3)and F ( a, b ; c ; z ) is the hypergeometric function F ( a, b ; c ; z ) = Γ( c )Γ( b ) Γ( − b + c ) × (5.4) × Z t b − (1 − t ) c − b − (1 − tz ) − a dt . The form factor for the d-wave forward scattering in-teraction can be expanded as follows d ( s ) = ∞ X k =0 d k cos((4 k + 2) s ) , (5.5)with d µ = − (cid:16) π − E h − µ i(cid:17) πd µ = 4 π (cid:18) π − E (cid:20) − µ (cid:21) ++ 2 π F (cid:20) − , , , − µ (cid:21)(cid:19) ,d µ = − π (cid:18) π − E (cid:20) − µ (cid:21) ++ 10 π F (cid:20) − , , , − µ (cid:21) −− π F (cid:20) − , , , − µ (cid:21)(cid:19) ,d µ = 4 π (cid:18) − E (cid:20) − µ (cid:21) ++ π (cid:18) F (cid:20) − , , , − µ (cid:21) −− F (cid:20) − , , , − µ (cid:21) ++ 20 F (cid:20) − , , , − µ (cid:21)(cid:19)(cid:19) . The orthogonal basis { ξ i } depends of course on the spe-cific form of the interaction, but in all the cases studiedhere it satisfy the following properties:1 - The functions are either linear combinations ofsin( s ) or of cos( s ) separately. There is not mixtures ofsin and cos.2 - All the functions reduce to the expressions corre-sponding to the free case in the limit when the interactionparameter is sent to zero. ACKNOWLEDGMENTS:
We would like to thank E. Fradkin for helpful discus-sions. This work was partially supported by the ESFgrant INSTANS, ECOS-Sud Argentina-France collabo-ration (Grant No A04E03), PICS CNRS-Conicet (Grant No. 18294), PICT ANCYPT (Grant No 20350), and PIPCONICET (Grant No 5037). A Clear and interesting introduction to Fermi Liquid the-ory can be founded in: A. J. Leggett, Rev. of Mod. Phys. , 331 (1975). I. J. Pomeranchuk, Sov. Phys. JETP , 361 (1958). H. Yamase, W. Metzner, cond-mat/0701660 (2007). A. Numayr, W. Metzner, Physical Review B , 035112(2003). W. Metzner, J. Reiss, D. Rohe, cond-mat/0701660 (2007). J. Quintanilla, A. J. Shofield, cond-mat/0601103 (2006). J. Quintanilla, C. Hooley, B. J. Powell, A. J. Shofield, M.Haque, cond-mat/07042231 (2007). C. Wu K. Sun, E. Fradkin, S. Zhang, cond-mat/06010326(2006). J. Nilson, A. H. Castro Neto, Phys. Rev. B , 195104(2005) C. J. Halboth, W. Metzner, Phys. Rev. Lett. , 5162(2000). I. Khavkine, Chung-Hou Chung, Vadim Oganesyan, Hae-Young Kee, Phys. Rev. B , 155110 (2004). V. Hankevich, F. wegner, Physical Review B , 333(2003). A. P. Kampf, A. A. Katanin, Physical Review B , 125104(2003). Ying-Jer Kao, Hae-Young Kee, Phys. Rev. B 76, 045106(2007). C. Puetter, H. Doh, Hae-Young Kee, cond-mat/07061069(2007). S. A. Grigera, P. Gegenwart, R. A. Borzi, F. Weickert, A.J. Schofield, R. S. Perry, T. Tayama, T. Sakakibara, Y.Maeno, A. G. Green, et al , Science. , 1154 (2004). S. A. Grigera, R. A. Borzi, A. P. Mackenzie, S. R. Julian,R. S. Perry, Y. Maeno, A. G. Green, et al , Phys. Rev. B , 214427 (2003). H. -Y. Kee, A. A. Katanin, J. Phys. Soc. Jpn. , 073706(2007). H. Yamase, Y. B. Kim, Phys. Rev. B , 184402 (2005). J. M. Luttinger, Phys Rev P. A. Frigeri, C. Honerkamp, T. M. Rice,cond-mat/0204380 (2002). Y. Fuseya, H. Maebashi, S. Yotsuhashi, J. Phys. Soc. Jpn. , 2158 (2000). For an exelent report about d x − y paring in cuprate su-perconductors see: D. J. Scalpino, Phys. Rep. , 329(1995).24