EExtremely Correlated Superconductors
B Sriram Shastry Physics Department, University of California, Santa Cruz, CA, 95064
February 15, 2021
Abstract
Superconductivity in the t - J model is studied by extending therecently introduced extremely correlated fermi liquid theory. Exactequations for the Greens functions are obtained by generalizing Gor’kov’sequations to include extremely strong local repulsion between elec-trons of opposite spin. These equation are expanded in a parameter λ representing the fraction of double occupancy, and the lowest orderequations are further simplified near T c , resulting in an approximateintegral equation for the superconducting gap. The condition for T c isstudied using a model spectral function embodying a reduced quasi-particle weight Z near half-filling, yielding an approximate analyticalformula for T c . This formula is evaluated using parameters represen-tative of single layer High- T c systems. In a narrow range of electrondensities that is necessarily separated from the Mott-Hubbard insula-tor at half filling, we find superconductivity with a typical T c ∼ K. The single band t - J model Eq. (1), and the closely related strong cou-pling Hubbard model have attracted much attention in recent years. * [email protected] a r X i v : . [ c ond - m a t . s up r- c on ] F e b INTRODUCTION In large part the interest is due to the potential relevance of these mod-els in describing the phenomenon of High T c superconductivity, dis-covered in cuprate materials in 1987 [1] and later, in other materials.These models lead to a single sheet of the fermi surface, and are spec-ified by fixing the band hopping t and the exchange energy J for the t - J model, or equivalently 4 t / U for the strong coupling ( U (cid:29) t )Hubbard model. The exotic possibility of superconductivity arisingfrom such inherently repulsive systems, is surprising from a theoret-ical perspective, and also challenging. Significant theoretical workusing a variety of tools on the t - J and the strong coupling Hubbardmodels [2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13] has given useful insights intothe role of strong correlations in cuprate superconductivity. Howevergiven the non-triviality of the theoretical task of solving these mod-els, and the consequent sweeping approximations frequently made, ithas been difficult to reconcile the different methods, or to agree on theresults.In this work we extend the extremely correlated fermi liquid the-ory [14] recently formulated to overcome the technical difficulties ofthe strong coupling models, to include superconducting type brokensymmetry. We obtain exact equations for the electron Greens functionthat generalize Gor’kov’s equations for BCS type superconductivity[15].Our equations include the effect of extremely strong local repulsionbetween electrons. They are further expanded in terms of a param-eter λ , representing the fraction of doubly occupied states [16]. Thiscontrolled expansion leads to a set of successive equations that areamenable to numerical study. Notable results within the normal statefrom O ( λ ) version of the equations include calculations of the asym-metric photoemission lines[17], and most recently the calculation ofthe almost T-linear resistivity in single layer cuprates[18].In order to obtain explicit, if approximate results, the O ( λ ) termsfor the superconductor are further simplified near T c , and the low-est order condition for T c is formulated in Eq. (70). This condition isevaluated using a simple phenomenological electronic spectral func- INTRODUCTION tion, modeling strong correlations near half filling in terms of a den-sity dependent quasiparticle weight Z and a wide background. Thismodel has the advantage of leading to an explicit analytical formulafor T c , in terms of the various parameters of the t - J model, thusallowing for a thorough understanding of the role of different pa-rameters on the result. Evaluating this expression we find that themodel supports a d-wave superconducting phase consistent with data[19, 20], located away from half filling. The T c is found to be typically ∼ K, i.e. an order of magnitude smaller than that of the uncorre-lated model, in a range of densities determined by the band param-eters. The temperature-density phase diagram has the form of a ta-pered tower Fig. (1). A smooth dome structure reported in cuprates,is replaced here by a somewhat narrow density range and an exagger-ated height near the peak. The location of the peak can be varied bychoosing the hopping parameters, but always remains well-separatedfrom the insulating limit as seen in Fig. (3).The paper is organized as follows. In Section (2) we define the t - J Hamiltonian, express it in terms of the correlated fermionic operators,and outline the method of external potentials employed to generatethe exact dynamical equations for the electron Greens function G andthe Gor’kov anomalous Greens function F . In Section (3) the equa-tion is expanded in λ and further simplified near T c . In Section (4)the condition for T c is evaluated using a model spectral function. Thissection contains expressions that involve only the electronic spectralfunction, and might be directly accessible to readers who are primar-ily interested in the concrete results. In Section (5) we conclude witha discussion of the results. THEORETICAL PRELIMINARIES The t - J Hamiltonian is H tJ = H t + H J (1) H t = − ∑ ij σ t ij (cid:101) c † i σ (cid:101) c j σ − µ ∑ i n i H J = ∑ ij J ij ( (cid:126) S i . (cid:126) S j − n i n j ) with the density operator n i = ∑ σ (cid:101) c † i σ (cid:101) c i σ , and spin density operator S α i = ∑ σσ (cid:48) (cid:101) c † i σ τ ασσ (cid:48) (cid:101) c i σ (cid:48) , τ α is a Pauli matrix and the correlated fermidestruction operator (cid:101) c i is found from the plain (i.e. canonical or un-projected) operators c i , by sandwiching it between two Gutzwillerprojection operators (cid:101) c i σ = P G c i σ P G . The creation operators follow bytaking their hermitean conjugate. The physical meaning of this sand-wiching process is that the fermi operators act within the subspacewhere projector P G enforces single occupancy at each site. In the fol-lowing work we will also find it useful to study the uncorrelated t-JmodelH unc-tJ = − ∑ ij t ij c † i σ c j σ − µ ∑ i n i + ∑ ij J ij ( (cid:126) S i . (cid:126) S j − n i n j ) (2)where all operators, including the density and spin, are defined by thesame expression as Eq. (1) but with the unprojected fermion operators c i σ , c † i σ ’s. This model serves as a useful reference point in the study ofthe correlated problem, since it has a superconducting ground statethat is expected to be well described by the BCS-Gor’kov mean-fieldtheory for small enough J . Our aim is to study the effect of strongcorrelations on this mean-field theory solution of the model Eq. (2)It is convenient for our calculations to use the operators inventedby Hubbard to represent this projection process, where (cid:101) c † i σ ↔ X σ i , (cid:101) c i σ ↔ X σ i , (cid:101) c † i σ (cid:101) c i σ (cid:48) ↔ X σσ (cid:48) i . (3)These operators satisfy the following fundamental anti-commutation THEORETICAL PRELIMINARIES relations: { X σ i i , X σ j j } = = { X σ i i , X σ j j } (4) { X σ i i , X σ j j } = δ ij (cid:16) δ σ i σ j − σ i σ j X ¯ σ i ¯ σ j i (cid:17) , ¯ σ = − σ . (5)In physical terms, for a given site index i , and with indices { a , b } ∈{ ↑ , ↓} limited to the three states of the projected Hilbert space, theoperation of X abi to its right causes transitions between the initial state b and the final state a . To yield the correct fermion antisymmetry,the creation operator X σ i i anti-commutes with creation or destructionoperators at different sites with any spin. In terms of these operatorswe can rewrite H t = − ∑ ij t ij X σ i X σ j − µ ∑ i X σσ i (6) H J = = − ∑ ij J ij σ i σ j X σ i σ j i X ¯ σ i ¯ σ j j (7)In order to calculate the Greens functions for this model, we addan imaginary time τ dependent external potential (or source term) A to the definition of thermal averages. The expectation of an arbitraryobservable Q ( τ , . . . ) , composed e.g. of a product of several (imag-inary) time ordered Heisenberg picture operators, is written in thenotation (cid:104)(cid:104) Q ( τ , . . . ) (cid:105)(cid:105) = Tr P β T τ { e −A Q ( τ , . . . ) } . (8)Here T τ is the time-ordering operator, an external potential term A = (cid:82) β d τ A ( τ ) , and P β = e − β H / Tr (cid:0) e − β H T τ e −A (cid:1) is the Boltzmann weightfactor including A . Here A ( τ ) is a sum of two terms, A ρ ( τ ) involv-ing a density-spin dependent external potential V , and A C ( τ ) involv-ing J ( J ∗ ) Cooper pair generating (destroying) external potentials.These are given by A ρ ( τ ) = ∑ i V σ i σ j i ( τ ) X σ i σ j i ( τ ) A C ( τ ) = ∑ ij (cid:16) J ∗ j σ j i σ i ( τ ) X σ i i ( τ ) X σ j j ( τ ) + J i σ i j σ j ( τ ) X σ i i ( τ ) X σ j j ( τ ) (cid:17) (9) THEORETICAL PRELIMINARIES where we require the antisymmetry J i σ i ; j σ j = −J j σ j ; i σ i and likewisefor J ∗ . The external potentials J , J ∗ in Eq. (9) couple to operatorsthat add and remove Cooper pairs of correlated electrons, and are es-sential to describe the superconducting phase. At the end of the cal-culations, the external potentials are switched off, so that the averagein Eq. (8) reduces to the standard thermal average. Tomonaga[21] in1946 and Schwinger[22] in 1948 (TS) pioneered the use of such exter-nal potentials [23, 24]. We next illustrate this technique for the presentproblem. The advantage of introducing these external potential ( or “sources”)is that we can take the (functional) derivatives of Greens function withrespect to the added external potentials in order to generate higherorder Greens functions. If we abbreviate the external term as A = ∑ i U j ( τ ) V j ( τ ) , where U j ( τ ) is one of the above c-number potential,and V j ( τ ) is the corresponding operator in the imaginary-time Heisen-berg picture, and Q i ( τ ) an arbitrary observable, straightforward dif-ferentiation leads to the TS identity TrP β T τ { e −A Q i ( τ (cid:48) ) V j ( τ ) } = (cid:104)(cid:104) Q i ( τ (cid:48) ) (cid:105)(cid:105) (cid:104)(cid:104) V j ( τ ) (cid:105)(cid:105) − δδ U i ( τ ) (cid:104)(cid:104) Q i ( τ (cid:48) ) (cid:105)(cid:105) (10)This important identity can be found by taking the functional deriva-tive of Eq. (8) with respect to U j ( τ ) , and is now illustrated with vari-ous choices of the external potential.From Eq. (10) we note the frequently used result (cid:104)(cid:104) σ i σ j X ¯ σ i ¯ σ j i ( τ ) Q ( τ (cid:48) ) (cid:105)(cid:105) = (cid:16) γ σ i σ j ( i τ ) − D σ i σ j ( i τ ) (cid:17) (cid:104)(cid:104) Q ( τ (cid:48) ) (cid:105)(cid:105) (11)where γ σ i σ j ( i τ ) = σ i σ j (cid:104)(cid:104) X ¯ σ i ¯ σ j i ( τ ) (cid:105)(cid:105)D σ i σ j ( i τ )) = σ i σ j δδ V ¯ σ i ¯ σ j i ( τ ) , (12) THEORETICAL PRELIMINARIES The singlet Cooper pair operator is (cid:16) X ↑ i X ↓ j − X ↓ i X ↑ j (cid:17) = σ X σ i X σ j , (13)where summation over repeated spin indices is implied, and its Her-mitean conjugate − (cid:16) X ↑ i X ↓ j − X ↓ i X ↑ j (cid:17) = ¯ σ X σ i X ¯ σ j . (14)We denote the Cooper pair correlation functions at time τ as C ij ( τ ) = (cid:104)(cid:104) σ X σ i ( τ ) X σ j ( τ ) (cid:105)(cid:105) (15) C ∗ ij ( τ ) = (cid:104)(cid:104) ¯ σ X σ i ( τ ) X ¯ σ j ( τ ) (cid:105)(cid:105) (16)We note that C ∗ ij equals the complex conjugate of C ij only after theexternal potentials are finally turned off, but not so in the intermediatesteps.The basic equation Eq. (10) for the Cooper pair operators for anarbitrary operator Q are δδ J ∗ i σ i j σ j ( τ ) (cid:104)(cid:104) Q (cid:105)(cid:105) = (cid:104)(cid:104) X σ j j ( τ ) X σ i i ( τ ) (cid:105)(cid:105)(cid:104)(cid:104) Q (cid:105)(cid:105) − (cid:104)(cid:104) X σ j j ( τ ) X σ i i ( τ ) Q (cid:105)(cid:105) (17) δδ J i σ i j σ j ( τ ) (cid:104)(cid:104) Q (cid:105)(cid:105) = (cid:104)(cid:104) X σ i i ( τ ) X σ j j ( τ ) (cid:105)(cid:105)(cid:104)(cid:104) Q (cid:105)(cid:105) − (cid:104)(cid:104) X σ i i ( τ ) X σ j j ( τ ) Q (cid:105)(cid:105) (18)From these relations the Cooper-pair correlations can be found bysumming over the spins (cid:104)(cid:104) σ X σ i ( τ ) X σ j ( τ ) Q (cid:105)(cid:105) = (cid:2) C ij ( τ ) − K ij ( τ ) (cid:3) (cid:104)(cid:104) Q (cid:105)(cid:105) (19) (cid:104)(cid:104) ¯ σ X σ i ( τ ) X ¯ σ j ( τ ) Q (cid:105)(cid:105) = (cid:104) C ∗ ij ( τ ) − K ∗ ij ( τ ) (cid:105) (cid:104)(cid:104) Q (cid:105)(cid:105) (20)where K ij ( τ ) = ∑ σ ¯ σ δδ J ∗ i σ ; j ¯ σ ( τ ) (21) K ∗ ij = ∑ σ ¯ σ δδ J i σ ; j ¯ σ ( τ ) . (22) THEORETICAL PRELIMINARIES We are interested in the electron Greens function G i σ i j σ j ( τ , τ (cid:48) ) = −(cid:104)(cid:104) X σ i i ( τ ) X σ j j ( τ (cid:48) ) (cid:105)(cid:105) , (23)where the usual time ordering is included in the definition of thebrackets Eq. (8). To describe the superconductor, following Gor’kov[15] we define the anomalous Greens function : F i σ i j σ j ( τ , τ (cid:48) ) = ¯ σ i (cid:104)(cid:104) X ¯ σ i i ( τ ) X σ j j ( τ (cid:48) ) (cid:105)(cid:105) (24)where ¯ σ ≡ − σ .We note that the Cooper pair correlation functions Eq. (16), whichplays a crucial role in defining the order parameter of the supercon-ductor, can be expressed in terms of the anomalous Greens functionusing C ∗ ij ( τ ) = ∑ σ F i σ j σ ( τ , τ ) . (25)We will also need the equal time correlation of creation operators C ij ( τ ) Eq. (15). It is straightforward to show that when the externalpotentials A are switched off, this object is independent of τ and canbe obtained by complex conjugation of C ∗ ij . It is possible to add an-other anomalous Greens function with two destruction operators as inEq. (24), corresponding to Nambu’s generalization of Gor’kov’s work.In the present context it adds little to the answers and is avoided bytaking the complex conjugate of C ∗ ij to evaluate C ij . G The equations for the Greens functions follow quite easily from theHeisenberg equations, followed by the use of the identity Eq. (10),and has been discussed extensively by us earlier. There is one newfeature, concerning an alternate treatment of the H J (exchange) term,necessary for describing superconductivity described below. THEORETICAL PRELIMINARIES Taking the τ derivative of G we obtain ∂ τ (cid:104)(cid:104) X σ i i ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105) = δ ( τ − τ (cid:48) ) δ i f ( δ σ i σ f − γ σ i σ f ( i τ ))+ (cid:104)(cid:104) [ H t + H J + A ( τ ) , X σ i i ( τ )] X σ f f ( τ (cid:48) ) (cid:105)(cid:105) (26)We work on the terms on the right hand side. At time τ we note [ H t + A ρ , X σ i i ] = µ X σ i i − V σ i σ j i X σ j i + t ij ( δ σ i σ j − σ i σ j X ¯ σ i ¯ σ j i ) X σ j j . (27)From this basic commutator, using Eq. (10), Eq. (11) and the defini-tions Eq. (12) we obtain (cid:104)(cid:104) [ H t + A ρ ( τ ) , X σ i i ( τ )] X σ f f ( τ (cid:48) ) (cid:105)(cid:105) = (cid:16) µ δ σ i σ j − V σ i σ j i (cid:17) (cid:104)(cid:104) X σ j i ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105) + t ij (cid:104)(cid:104) X σ i i ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105) − t ij ( γ σ i σ j ( i τ ) − D σ i σ j ( i τ )) (cid:104)(cid:104) X σ j i ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105) (28)For the exchange term [ H J , X σ i i ] = J ij ∑ σ j σ i σ j X σ j i X ¯ σ i ¯ σ jj (29) = − J ij σ i X ¯ σ i j (cid:16) X ↑ i X ↓ j − X ↓ i X ↑ j (cid:17) . (30)In order to obtain Eq. (30) from Eq. (29), we used X ¯ σ i ¯ σ j j = X ¯ σ i j X σ j j andanticommuted the equal time operators X σ j i X ¯ σ i j into − X ¯ σ i j X σ j i , fol-lowed by summing over σ j . This subtle step is essential for obtainingthe superconducting phase, since the role of exchange in promotingCooper pairs manifests itself here. Using Eq. (19) we find (cid:104)(cid:104) [ H J , X σ i i ( τ )] X σ f f ( τ (cid:48) ) (cid:105)(cid:105) = − J ij σ i (cid:0) C ij ( τ + ) − K ij ( τ + ) (cid:1) (cid:104)(cid:104) X ¯ σ i j ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105) . (31)In treating this term we could have proceeded differently by stick-ing to Eq. (29), using Eq. (10) with a different external potential termas in Eq. (11) to write (cid:104)(cid:104) [ H J , X σ i i ( τ )] X σ f f ( τ (cid:48) ) (cid:105)(cid:105) = J ij ∑ σ j σ i σ j (cid:104)(cid:104) X σ j i ( τ ) X ¯ σ i ¯ σ jj ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105) = − J ij (cid:0) γ ¯ σ i ¯ σ j ( j τ ) − D ¯ σ i ¯ σ j ( j τ ) (cid:1) (cid:104)(cid:104) X σ j i ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105) . (32) THEORETICAL PRELIMINARIES These two expressions Eq. (31) and Eq. (32) are alternate ways of writ-ing the higher order Greens functions [25]. In order to describe a bro-ken symmetry solution with superconductivity, we are required to useEq. (31), since using the other alternative disconnects the normal andanomalous Greens functions altogether, thereby precluding a super-conducting solution.The term (cid:104)(cid:104) [ A C ( τ ) , X σ i i ( τ )] X σ f f ( τ (cid:48) ) (cid:105)(cid:105) generates a term that is lin-ear in J which is treated similarly and the final result quoted in Eq. (34).We summarize these equations compactly by defining G − i σ i j σ j = δ ij δ σ i σ j ( µ − ∂ τ ) + t ij δ σ i σ j − δ ij V σ i σ j i Y i σ i j σ j = t ij γ σ i σ j ( i ) X i σ i j σ j = − t ij D σ i σ j ( i ) , (33)and using a repeated spin index summation notation we write theexact equation ( G − i σ i j σ j − Y i σ i j σ j − X i σ i j σ j ) G j σ j f σ f ( τ , τ (cid:48) ) = δ ( τ − τ (cid:48) ) δ i f ( δ σ i σ f − γ σ i σ f ( i ))+ J ij (cid:0) C ij ( τ ) − K ij ( τ ) (cid:1) F j σ i f σ f ( τ , τ (cid:48) )+ J j σ j ; i σ k (cid:0) δ σ i , σ k − γ σ i σ k ( i ) + D σ i σ k ( i ) (cid:1) σ j F j ¯ σ j f σ f ( ττ (cid:48) ) . (34)The final term drops off when we switch off the external potential J . F The Gor’kov Greens function F in Eq. (24) satisfies an exact equationthat can be found as follows. First we note ∂ τ (cid:104)(cid:104) X ¯ σ i i ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105) = (cid:104)(cid:104) [ H t + H J + A ( τ ) , X ¯ σ i i ( τ )] X σ f f ( τ (cid:48) ) (cid:105)(cid:105) (35)A part of the right hand side satisfies (cid:104)(cid:104) [ H t + A ρ ( τ ) , X ¯ σ i i ( τ )] X σ f f ( τ (cid:48) ) (cid:105)(cid:105) = − (cid:16) µ δ σ i σ j − V ¯ σ i ¯ σ j i (cid:17) (cid:104)(cid:104) X ¯ σ j i ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105)− t ij (cid:104)(cid:104) X ¯ σ i i ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105) + t ij ( γ ¯ σ j ¯ σ i ( i τ ) − D ¯ σ j ¯ σ i ( i τ )) (cid:104)(cid:104) X σ j i ( τ ) ; X σ f f ( τ (cid:48) ) (cid:105)(cid:105) (36)The exchange term is treated similarly to Eq. (29) [ H J , X ¯ σ i i ] = J ij (cid:16) X ↑ i X ↓ j − X ↓ i X ↑ j (cid:17) σ i X σ i j (37) THEORETICAL PRELIMINARIES so that using Eq. (20) we get (cid:104)(cid:104) [ H J , X ¯ σ i i ] X σ f f ( τ (cid:48) ) (cid:105)(cid:105) = − J ij σ i (cid:16) C ∗ ij ( τ − ) − K ∗ ij ( τ − ) (cid:17) (cid:104)(cid:104) X σ i j ( τ ) X σ f f ( τ (cid:48) ) (cid:105)(cid:105) (38)We gather and summarize these equation in terms of the variablesthat are “time-reversed” partners of Eq. (34) and hence denoted withhats: (cid:98) G − i σ i j σ j = δ ij δ σ i σ j ( µ + ∂ τ ) + t ij δ σ i σ j − δ ij V ¯ σ i ¯ σ j i (cid:98) Y i σ i j σ j = t ij γ ¯ σ j ¯ σ i ( i ) (cid:98) X i σ i j σ j = − t ij D ¯ σ j ¯ σ i ( i ) (39)So that (cid:16) (cid:98) G − i σ i j σ j − (cid:98) Y i σ i j σ j − (cid:98) X i σ i j σ j (cid:17) F j σ i f σ f = − J ij (cid:16) C ∗ ij − K ∗ ij (cid:17) G j σ i f σ f + σ i ∑ m J ∗ i ¯ σ n m σ m ( δ σ i , σ n − γ ¯ σ n ¯ σ i ( i ) + D ¯ σ n ¯ σ i ( i )) G m σ m f σ f (40)The final term arising from (cid:104)(cid:104) [ A C , X ¯ σ i i ] X ¯ σ f f ( τ (cid:48) ) (cid:105)(cid:105) drops off when weswitch off the external potential J ∗ . The equations Eq. (34) and Eq. (40) are exact in the strong correla-tion limit. Noting that all terms containing γ and D in Eq. (34) andEq. (40) arise from Gutzwiller projection, we obtain the correspond-ing equations for the uncorrelated t - J model in Eq. (2) by droppingthese terms. Recall also that the external potentials J , J ∗ representthe imposed symmetry-breaking terms that force superconductivity,and are meant to be dropped at the end. In this uncorrelated case, letus understand the role of the terms with the Cooper pair derivatives K , K ∗ . If we ignore these terms and also set J , J ∗ → THEORETICAL PRELIMINARIES providing a self consistent determination of C ∗ ij in terms of F . Thusby neglecting the terms with K , K ∗ , the role of the exchange J is con-fined to providing the lowest order electron-electron attraction in theCooper channel, and amounts to neglecting the higher order dress-ings of the electron self energies and irreducible interaction by termsinvolving J .In the correlated problem strong interactions already strongly mod-ify the Greens function G from G , and the self energy terms due to J are minor[14]. The role of J is of course significant in providinga mechanism for superconducting pairing, so we shall retain that.Keeping this in mind, we drop the terms involving K , K ∗ , J , J ∗ inEq. (34) and Eq. (40). This suffices for our initial goal, of generaliz-ing a Gor’kov type[15] mean-field treatment of Eq. (2) to the stronglycorrelated problem Eq. (1).Multiplying the γ and D terms, or equivalently the X and Y termswith λ and expanding the resulting equations systematically in thisparameter constitutes the λ -expansion that we discuss below.With these remarks in mind we make the following changes to theequations Eq. (34) and Eq. (40):(i) We drop the terms proportional to J , J ∗ and the correspondingderivative terms K , K ∗ .(ii) Defining the gap functions: ∆ ij = J ij C ij and ∆ ∗ ij = J ij C ∗ ij (41)(iii) We scale the each occurrence of γ , X , Y , (cid:98) X , (cid:98) Y by λ .With these changes we write the modified Eq. (34) and Eq. (40): ( G − i σ i j σ j − λ Y i σ i j σ j − λ X i σ i j σ j ) G j σ j f σ f = δ ( τ − τ (cid:48) ) δ i f ( δ σ i σ f − λγ σ i σ f ( i )) + ∆ ij F j σ i f σ f (42) (cid:16) (cid:98) G − i σ i j σ j − λ (cid:98) Y i σ i j σ j − λ (cid:98) X i σ i j σ j (cid:17) F j σ i f σ f = − ∆ ∗ ij G j σ i f σ f (43)and the self consistency condition Eq. (16) and Eq. (25) fix the correla-tion functions C ’s in terms of F . As λ → EXPANSION OF THE EQUATIONS IN λ equations of Gor’kov for the uncorrelated-J model. The λ parametergoverns the density of doubly occupied states, and hence a series ex-pansion in this parameter builds in Gutzwiller type correlations sys-tematically. We expand the Greens functions to required order in λ and finally set λ = ( g − − λ Y − λ X ) . G = ( − λγ ) + ∆ . F (44) ( (cid:98) g − − λ (cid:98) Y − λ (cid:98) X ) . F = − ∆ ∗ . G (45)where the symbols G , F etc are regarded as matrices in the space, spinand time variables, with the dot indicating matrix multiplication ortime convolution. In the case of X , (cid:98) X it also indicates taking the nec-essary functional derivatives. λ We decompose of both Greens functions in Eq. (44) and Eq. (45) as G = g . (cid:101) µ , F = f . (cid:101) µ (46)where (cid:101) µ is a function of spin, space and time that is common to bothGreens function. As an example of the notation, the equation G = g . (cid:101) µ stands for G i σ i j σ j ( τ i , τ j ) = ∑ k σ k (cid:82) β d τ k g i σ i k σ k ( τ i , τ k ) (cid:101) µ k σ k j σ j ( τ k , τ j ) .Here (cid:101) µ is called the caparison (i.e. a further dressing) function, ina similar treatment of the normal state Greens function. The terms g and f are called the auxiliary Greens function. The basic idea isthat this type of factorization can reduce Eq. (44), to a canonical typeequation fo g , where the terms − λγ is replaced by . We remark thatthis is a technically important step since the term − λγ modifies thecoefficient of the delta function in time, and encodes the distinctionbetween canonical and non-canonical fermions.To simplify further, we note that X contains a functional derivativewith respect to V , acting on objects to its right. When acting on a pairof objects, e.g. X . G = X . g . (cid:101) µ , we generate two terms. One term is EXPANSION OF THE EQUATIONS IN λ ( X . g ) . (cid:101) µ , where the bracket, temporarily provided here, indicates thatthe operation of X is confined to it. The second term has the derivativeacting on (cid:101) µ only, but the matrix product sequence is unchanged fromthe first term. We write the two terms together as X . g . (cid:101) µ = X . g . (cid:101) µ + X . g . (cid:101) µ , (47)so that the ‘contraction’ symbol refers to the differentiation by X , andthe ‘.’ symbol refers to the matrix structure. We may view this as theLeibnitz product rule.Let us now operate with X on the identity g . g − = , where g − isthe matrix inverse of g . Using the Leibnitz product rule, we find X . g = − (cid:18) X . g . g − (cid:19) . g (48)and hence we can rewrite Eq. (47) in the useful form X . g . (cid:101) µ = − (cid:18) X . g . g − (cid:19) . g + X . g . (cid:101) µ . (49)With this preparation we rewrite Eq. (44) the equation for G as ( g − − λ Y + λ (cid:18) X . g . g − (cid:19) ) . g . (cid:101) µ = ( − λγ ) + ∆ . f . (cid:101) µ + λ X . g . (cid:101) µ (50)We now choose g , f such that ( g − − λ Y + λ (cid:18) X . g . g − (cid:19) ) . g = + ∆ . f . (51)Substituting Eq. (51) into Eq. (50), we find that (cid:101) µ satisfies the equation (cid:101) µ = ( − λγ ) + λ X . g . (cid:101) µ . (52)Note that Eq. (51) has the structure of a canonical equation since wereplaced the − λγ term by in Eq. (50). Thus the non-canonicalEq. (44) for G , F is replaced by a pair of canonical equations for g , (cid:101) µ .In Eq. (51) we note that the action of X is confined to the bracket λ (cid:18) X . g . g − (cid:19) , unlike the term λ X . G in the initial Eq. (44) . We maythus view the term in bracket in Eq. (51) as a proper self energy for g . EXPANSION OF THE EQUATIONS IN λ For treating the equation for F Eq. (45) we use the same schemeEq. (46) and find (cid:98) X . F = (cid:98) X . f . (cid:101) µ (53)With this we rewrite Eq. (45) after cancelling an overall right multi-plying factor (cid:101) µ ( (cid:98) g − − λ (cid:98) Y + λ (cid:98) X . f . f − ) . f = − ∆ ∗ . g + λ (cid:98) X . f . (cid:101) µ . (cid:101) µ − (54)Summarizing we need to solve for f , g , (cid:101) µ , ∆ ∗ from Eqs. (51,52,54) byiteration in powers of λ . T c For the present work, we note that the equation Eq. (54) simplifiesconsiderably, if we work close to T c . This truncated scheme is suffi-cient to determine T c for low orders in λ . In the regime T ∼ T c , f maybe assumed to be very small, enabling us to throw away all terms of O ( f ) and also to discard terms of O ( λ f ) giving f = − (cid:98) g . ∆ ∗ . g + o ( λ f ) , (55)so that Eq. (51) can be written as g − = g − − λ Y + λ (cid:18) X . g . g − (cid:19) + ∆ . (cid:98) g . ∆ ∗ (56)In this limit the above two are the equations O ( λ ) required to besolved, together with Eq. (52) and the self consistency condition Eq. (41),Eq. (25). The latter can be combined with Eq. (24) as ∆ ∗ ij = J ij C ∗ ij = − J ij ∑ σ F i σ , j σ ( τ + , τ ) (57)and further reduced using Eq. (46). On turning off the external poten-tials we recover time translation invariance. We next perform a fouriertransform to fermionic Matsubara frequencies ω n = πβ ( n + ) usingthe definition F ( τ ) = β ∑ n e − i ω n τ F ( i ω n ) , and write Eq. (46) in thefrequency domain as F p σ ( i ω n ) = f p σ ( i ω n ) (cid:101) µ p σ ( i ω n ) . (58) EXPANSION OF THE EQUATIONS IN λ Thus taking spatial fourier transforms with the definition J ( q ) = J (cid:0) cos q x + cos q y (cid:1) , (59)so that the self consistency condition Eq. (57) finally reduces to ∆ ∗ ( k ) = − β ∑ p σω n J ( k − p ) f p σ ( i ω n ) (cid:101) µ p σ ( i ω n ) (60)We may write Eq. (55) as f p σ ( i ω n ) = − (cid:98) g σ ( p , i ω n ) ∆ ∗ ( p ) g σ ( p , i ω n ) (61)where the time reversed free Greens function (cid:98) g ( p , i ω n ) = − i ω n + µ − ε − p = − i ω n − ξ p (62)with ξ = ε p − µ and by using ε p = ε − p , and µ is taken as the non-interacting system chemical potential, discarding the corrections of µ due to λ . Therefore Eq. (60) becomes ∆ ∗ ( k ) = β ∑ p σω n J ( k − p ) (cid:98) g σ ( p , i ω n ) ∆ ∗ ( p ) g σ ( p , i ω n ) (cid:101) µ p σ ( i ω n ) (63)Here g is taken from Eq. (56), i.e. the O ( λ ) Greens function with asmall correction (for T ∼ T c ) from the gap ∆ . Performing the spinsummation and recombining g . (cid:101) µ = G , we get the equation in terms ofthe physical electron Greens function ∆ ∗ ( k ) = β ∑ p ω n J ( k − p ) ∆ ∗ ( p ) (cid:98) g ( p , i ω n ) G ( p , i ω n ) . (64)Here again the physical electron Greens function G is taken from the O ( λ ) theory if we neglect the corrections from the gap, which van-ishes above T c anyway. We express the physical Greens function interms of its spectral function A ( p , ν ) G ( p , i ω n ) = (cid:90) d ν A ( p , ν ) i ω n − ν (65) ESTIMATE OF T C The frequency integral in Eq. (57) can be performed as1 β ∑ ω n (cid:98) g ( p , i ω n ) G ( p , i ω n ) = (cid:90) d ν A ( p , ν ) − f ( ν ) − f ( ξ p ) ν + ξ p . (66)where f is the fermi distribution f ( ν ) = ( + exp βν ) . Hence ∆ ∗ ( k ) = ∑ p J ( k − p ) ∆ ∗ ( p ) (cid:90) d ν A ( p , ν ) − f ( ν ) − f ( ξ p ) ν + ξ p . (67)In summary this eigenvalue type equation for ∆ ∗ ( k ) , together with thespectral function A ( p , ν ) determined from the O ( λ ) Greens functionin Eq. (56), gives the self-consistent gap near T c . At sufficiently hightemperatures, i.e. in the normal state T > T c , the gap ∆ ∗ vanishes, sothat A is independent of ∆ ∗ . In this case Eq. (67) reduces to a linear in-tegral equation for ∆ ∗ . We may then determine T c from the conditionthat the largest eigenvalue crosses 1. For this purpose we only needthe normal state electron spectral function of the strongly correlatedmetal. T c T c The condition for obtaining a d-wave superconducting state is givenby setting T = T + c in Eq. (67) writing ∆ ∗ ( k ) = ∆ ( cos k x − cos k y ) ,using the normal state spectral function for A and canceling an overallfactor ∆ ( cos k x − cos k y ) . Following these steps we get1 = J ∑ p (cid:8) cos ( p x ) − cos ( p y ) (cid:9) (cid:90) d ν − f ( ν ) − f ( ε p − µ ) ν + ε p − µ A ( p , ν ) (cid:12)(cid:12)(cid:12)(cid:12) T c . (68)Instead of working with Eq. (68), it is convenient to make a usefulsimplification for the average over angles. Since Eq. (68) is largestwhen (cid:126) p is on the fermi surface, we factorize the two terms and write1 = J Ψ ( µ ) Γ (69) Γ = ∑ p (cid:90) d ν − f ( ν ) − f ( ε p − µ ) ν + ε p − µ A ( p , ν ) (cid:12)(cid:12)(cid:12)(cid:12) T c (70) ESTIMATE OF T C where Γ is a particle-particle type susceptibility. Here Ψ ( µ ) is morecorrectly the weighted average of (cid:8) cos ( p x ) − cos ( p y ) (cid:9) with a weightfunction that is the integrand in Eq. (70). We simplify it to the fermisurface averaged momentum space d-wavefunction Ψ ( µ ) = n ( µ ) ∑ p (cid:8) cos ( p x ) − cos ( p y ) (cid:9) ) δ ( ε p − µ ) (71)where n ( (cid:101) ) is the band density of states (DOS) per spin and per site,at energy (cid:101) , n ( (cid:101) ) = N s ∑ p δ ( ε p − (cid:101) ) . (72)Using this simplification and performing the angular averaging overthe energy surface ε (cid:126) p = (cid:101) , we write the (particle-particle) susceptibil-ity Γ (Eq. (70)) as Γ = (cid:90) d (cid:101) (cid:90) d ν n ( (cid:101) ) A ( (cid:101) , ν ) − f ( ν ) − f ( (cid:101) − µ ) ν + (cid:101) − µ (cid:12)(cid:12)(cid:12)(cid:12) T c , (73)where A ( (cid:101) , ν ) is the angle-averaged version of the spectral function A ( p , ν ) . We estimate this expression below for the extremely corre-lated fermi liquid, by using a simple model for the spectral function A . In Eq. (73) if we replace the spectral function A by the (fermi gas)non-interacting result A ( (cid:101) , µ ) = δ ( ν − (cid:101) + µ ) , we obtain the Gor’kov-BCS mean-field theory, where the susceptibility Γ reduces to (cid:90) d (cid:101) n ( (cid:101) ) tanh β c ( (cid:101) − µ ) ( (cid:101) − µ ) .This expression is evaluated by expanding around the fermi energy,and utilizing the low T formula (cid:82) W d (cid:101)(cid:101) tanh β c (cid:101) ∼ log (cid:104) ζ W k B T c (cid:105) , where W is the half-bandwidth and ζ = Γ to 1/ J Ψ ( µ ) gives the d-wave superconducting transition temperature for the un-correlated t - J model k B T ( un ) c ∼ W e − g , (74) ESTIMATE OF T C with the superconducting coupling constant g = J Ψ ( µ ) n ( µ ) , (75)is positive by our definition. We next use a simple model spectral function to estimate these in-tegrals. It has the great advantage that we can carry out most inte-grations analytically and get approximate but closed form analyticalexpressions for T c , which provide useful insights. The model spectralfunction contains the following essential features of strong correla-tions namely:• A quasiparticle part with fermi liquid type parameters, wherethe quasiparticle weight Z goes to 0 at half filling n =
1, and• A wide background.The model spectral function used is in the spirit of Landau’s fermiliquid theory[26, 27, 28] with suitable modifications due to strong cor-relation effects[14]. We take the spectral function as A ( (cid:101) , ν ) = Z δ ( ν − mm ∗ (cid:101) ) + ( − Z ) W Θ ( W − | ν | ) . (76)Here Θ ( x ) = ( + x | x | ) , W the half-bandwidth mm ∗ is the renormal-ized effective mass of the fermions, and Z is the fermi liquid renor-malization factor. The first term is the quasiparticle part with weight Z , and second part represents the background modeled as an invertedsquare-well. Integration over ν gives unity at each energy (cid:101) . Z is cho-sen to reflect the fact that we are dealing with a doped Mott-Hubbardinsulator so it must vanish at n =
1. For providing a simple estimatewe use Gutzwiller’s result [29, 30] Z = − n . (77) ESTIMATE OF T C The effective mass is related to Z and the k-dependent Dyson self en-ergy Σ through the standard fermi liquid theory[26, 27, 28] formula mm ∗ = Z × ( + ∂ Σ ( (cid:126) k , µ ) ∂ε k (cid:12)(cid:12)(cid:12)(cid:12) k F ) . (78)The Landau fermi liquid renormalization factor mm ∗ can be inferredfrom heat capacity experiments provided the bare density of states isassumed known.Using Eq. (76) in Eq. (73) and decomposing the susceptibility Γ into a quasiparticle and background part, the equation determining T c is: (cid:0) Γ QP + Γ B (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) T → T c = J Ψ ( µ ) (79) Γ QP = Z (cid:90) d (cid:101) n ( (cid:101) ) − f ( (cid:101) − µ ) − f ( mm ∗ ( (cid:101) − µ ))( (cid:101) − µ )( + mm ∗ ) (80) Γ B = ( − Z ) W (cid:90) d (cid:101) n ( (cid:101) ) (cid:90) W − W d ν − f ( (cid:101) − µ ) − f ( ν )( (cid:101) − µ ) + ν .(81)Using the same approximations that lead to Eq. (74) the Γ QP can beevaluated as Γ QP = Zn ( µ ) + mm ∗ (cid:90) W d (cid:101)(cid:101) (cid:18) tanh (cid:101) k B T − tanh (cid:101) m / m ∗ k B T (cid:19) , (82)and hence at low enough T the estimate Γ QP ∼ n ( µ ) Z + mm ∗ log ζ W (cid:113) mm ∗ k B T . (83)Unlike the quasiparticle part with this log T behavior at low T, thebackground part is nonsingular as T →
0, since a double integralover the region of small (cid:101) − µ and ν is involved. It can be estimatedby setting T = (cid:101) − µ ∼ (cid:101) and replacing n ( (cid:101) ) ∼ n ( µ ) . With Γ B ≡ n ( µ ) γ B , (84) γ B = ( − Z ) W (cid:90) W − W (cid:90) W − W
12 sign ( (cid:101) ) + sign ( ν ) (cid:101) + ν d (cid:101) d ν . (85) ESTIMATE OF T C Integrating this expression we obtain γ B = ( − Z ) log 4. (86)Combining Eqs. (79,83,86) we find k B T c ∼ W × (cid:114) mm ∗ × e − gef f (87)where the effective superconducting coupling: g e f f = Z (cid:0) + mm ∗ (cid:1) (cid:110) J e f f Ψ ( µ ) n ( µ ) (cid:111) (88)and an effective exchange J e f f = J − γ B J Ψ ( µ ) n ( µ ) , (89)where the denominator represents an enhancement due to the back-ground spectral weight. In comparing Eq. (87) with the uncorrelatedresult Eq. (74) several changes are visible. The bandwidth prefactoris reduced by correlations due to the factor of (cid:113) mm ∗ (cid:28)
1. This fac-tor vanishes as n → T c in theclose proximity of the insulator. A similar but even more drastic ef-fect arises from multiplying factor Z ( + mm ∗ ) in the coupling g e f f Eq. (88).This term reflects the quasiparticle weight in the pairing process, andvanishes near the insulating state. Being situated in the exponential,it kills superconductivity even more effectively than the bandwidthprefactor. Away from the close proximity of the insulator other termsin g e f f become prominent, allowing for the possibility of supercon-ductivity. Amongst them is the replacement of the exchange energyby J e f f . In a density range where Ψ ( µ ) n ( µ ) is appreciable, this en-hances J e f f over J due to the feedback nature of Eq. (89), and has animportant impact on determining the phase region with superconduc-tivity. T c We turn to the task of estimating the order of magnitude of the Tc inthis model. When we take typical values for cuprate systems: W ∼ ESTIMATE OF T C K (i.e ∼ J ∼ K (i.e. ∼ T ( un ) c Eq. (74) is a few thousand K,at most densities. It remains robustly non-zero at half filling, sincein this formula correlation effects are yet to be built in and the Mott-Hubbard insulator is missing. For the correlated system, we estimate T c from Eq. (87) using similar values of model parameters. The termsarising from correlations in Eq. (87) are guaranteed to suppress super-conductivity near the insulating state, since Z → δ >
0) can support superconductivity. And if so, whether thetemperature scales are robust enough to be observable. The answer isyes to both questions, within the approximations made.
In order to estimate the order of magnitude of the Tc, its dependenceon J and band parameters, we choose parameters similar to thoseused in contemporary studies for the single layer High T c compound La − x Sr x CuO . The hopping Hamiltonian − ∑ ij t ij (cid:101) C † i σ (cid:101) C j σ , gives rise toband energy dispersion ε ( (cid:126) k ) = − t ( cos k x + cos k y ) − t (cid:48) cos k x cos k y − t (cid:48)(cid:48) ( cos 2 k x + cos 2 k y ) on a square lattice. Thus the hopping ampli-tudes t ij are equal to t when i , j are nearest neighbors, t (cid:48) when i , j are second-nearest neighbors, and t (cid:48)(cid:48) when i , j are third-nearest neigh-bors. For this system we will use the values [31, 13] t = eV , t (cid:48) / t = − ± t (cid:48)(cid:48) / t = .01. (90)This parameter set is roughly consistent with the experimentally de-termined fermi surface of La − x Sr x CuO [31], we comment below onconsiderations leading to a more precise choice. The tight bindingband extends from − W ≤ (cid:101) ≤ W , where W = t , neglecting asmall shift due to t (cid:48) . The exchange energy is chosen to be J / t = J / k B ∼ K , (91)as determined from two magnon Raman experiments[32] on the par-ent insulating La CuO . Note that the t - J model is obtainable from ESTIMATE OF T C the Hubbard model by performing a large U / t super-exchange ex-pansion, giving J = t U . Thus our choice of J corresponds to a strongcoupling type magnitude of U / t ∼ mm ∗ [33]. In theproximity of the Mott-Hubbard insulating state n →
1, an enhance-ment in m ∗ m is expected on general grounds, reflecting a diminishedthermal excitation energy scale due to band narrowing. For illustrat-ing the role of this parameter we use two complementary estimates mm ∗ ∼ ( − n ) , (a) (92) mm ∗ = Z , (b) (93)where estimate (a) gives a two-fold enhancement of m (cid:63) m at δ = .15,while estimate (b), obtained by neglecting the k dependence of theself energy in Eq. (78), gives a seven-fold enhancement. The formu-las used are simple enough so that the effect other estimates for mm ∗ should be easy enough for the reader to gauge. In Fig. (1) the superconducting transition temperature for d-wave sym-metry is shown as a function of the hole density δ = − n where theband parameters are indicated in the caption. It shows that T c is max-imum at δ ∼ T c is a few hundred K, whichis an order of magnitude lower than that of the uncorrelated system.The small kink-like features to the right of the peak reflect structurein the DOS shown as inset in Fig. (2). In Fig. (2) the effective super-conducting coupling g e f f is shown for three different symmetries ofthe Cooper pairs: d -wave, extended s -wave, and s + id -wave. It isclear that within this theory, only d-wave symmetry leads to robustsuperconductivity, the other two symmetries lead to effects too smallto be observable. From Fig. (3) we see that the peak density is shifted ESTIMATE OF T C δ T c K Figure 1:
The superconducting transition temperature for the correlated model T c (Eq. (87)) ( t (cid:48) / t = − t (cid:48)(cid:48) / t = mm ∗ = Z ). The scale of the maximum transitiontemperature is smaller by an order of magnitude from the uncorrelated model. As the in-sulator is approached δ →
0, and T c decreases drastically. This is easy to understand sincethe quasiparticle weight Z shrinks on approaching the insulating state, killing the coupling g e f f Eq. (88). When δ goes beyond the peak (optimum) value, the effective superconduct-ing coupling g e f f agains falls off as seen in Fig. (2) and in Fig. (5) due to the other factors inEq. (88). When g e f f drops below ∼ T c is negligible. ESTIMATE OF T C ���� ���� ������������� δ � ( μ � ) � ���� ���� ����������������������������� ���������������� δ � � �� �� � � - � � __ � �� � d s+id s Figure 2:
The figure and inset use t (cid:48) / t = − t (cid:48)(cid:48) / t = mm ∗ = Z . The effective superconducting coupling g e f f (Eq. (88)) for three Cooperpair symmetries: (i)(blue) d-wavefunction (cid:104) (cid:8) cos ( k x ) − cos ( k y ) (cid:9) (cid:105) FS , (ii)(brown) ex-tended s-wavefunction (cid:104) (cid:8) cos ( k x ) + cos ( k y ) (cid:9) (cid:105) FS , and (iii)(magenta) s + id wavefunction (cid:104) (cid:8) cos ( k x ) + cos ( k y ) (cid:9) (cid:105) FS . For the d-wavefunction, the drastic decrease of T c on both sidesof the peak values in Fig. (1) can be understood by referring to the the second y-axis, givingthe temperature scale T appxc = × e − gef f K. This scale provides an order of magnitude of T c at a given g e f f by assuming a prefactor 10 K. It illustrates the rapid reduction of T c when g e f f < ∼ Inset:
The band DOS at the fermi energy shows an enhancement around thehole density δ = ESTIMATE OF T C ���� ���� ������������ δΨ δ T c K Figure 3:
The superconducting transition temperature for the correlated model T c (Eq. (87)) for three parameter sets- (i) (red) t (cid:48) / t = − t (cid:48)(cid:48) / t = δ peak = t (cid:48) / t = − t (cid:48)(cid:48) / t = .01 with δ peak = t (cid:48) / t = − t (cid:48)(cid:48) / t = δ peak = mm ∗ = Z for and the dashed lines mm ∗ = δ . Inset: Ψ ( µ ) the fermi surface averaged d-wavefunction (cid:104) (cid:8) cos ( k x ) − cos ( k y ) (cid:9) (cid:105) FS is shown forthe three sets of band parameters. The peak values occur at the densities where T c is high-est. Their peak magnitude ∼ (cid:126) k ∼ {± π , 0 } , { ± π } , where | cos ( k x ) − cos ( k y ) | ∼ ESTIMATE OF T C by varying the band hopping parameters. As the peak density movestowards small δ , its height falls rapidly. This is understandable as theeffect of the quasiparticle weight Z in the formulas Eqs. (87,88). Wealso note that the use of different expressions for the effective mass inEqs. (92,93) change the width of the allowed regions somewhat, butare quite comparable.The inset in Fig. (3) displays the d-wavefunction averages corre-sponding to the same sets of parameters. It is interesting to notethat the height of the peaks Ψ max ∼ n ( µ ) = π (cid:82) k max dk x | v y ( (cid:126) k ) | , where the ve-locity v y ( (cid:126) k ) = ( k y )( t + t (cid:48) cos k x + t (cid:48)(cid:48) cos k y ) is evaluated with k y → k y ( k x , µ ) on the fermi surface. Thus the region of small | v y | dominates the integral. If v y vanishes on the fermi surface, we get a(logarithmic van Hove) peak in the DOS. Now the average of Ψ ( µ ) is largest, when (cid:126) k is close to {± π , 0 } and { ± π } . Therefore if thefermi surface passes through {± π , 0 } and { ± π } for an “ideal den-sity”, then we simultaneously maximize the average of Ψ , and obtaina large DOS . The condition for this is found by equating the bandenergy at {± π , 0 } to the chemical potential µ = t (cid:48) − t (cid:48)(cid:48) , therebyfixing the corresponding density δ . It follows that a given δ can befound from several different sets of the parameters t (cid:48) , t (cid:48)(cid:48) . The inset ofFig. (3) shows the average Ψ ( µ ) displays peaks, the middle one (red)coincides in location with the peak in the DOS in the inset of Fig. (2).In Fig. (4) we illustrate the role of the feedback enhancement ofthe exchange J due to the background spectral function discussed inEq. (89). For each set of parameters, there is a density region whereboth the DOS at the fermi energy and the averaged d-wavefunctionare enhanced, and the confluence directly enhances J e f f . In turn this isreflected in the superconducting coupling g e f f . In Fig. (5) we see howthe confluence of enhancements in the DOS and in the d-wavefunction Ψ ( µ ) , further boosts the superconducting coupling g e f f and offsets to ESTIMATE OF T C ���� ���� ���� ���� ���� δ ������������������ � ��� / � � ��� �� δ Figure 4:
The effective exchange J e f f from Eq. (89) for the three parameter sets- (i) (red) t (cid:48) / t = − t (cid:48)(cid:48) / t = δ peak = t (cid:48) / t = − t (cid:48)(cid:48) / t = .01 with δ peak = t (cid:48) / t = − t (cid:48)(cid:48) / t = δ peak = J / t ∼ J e f f / t is considerably enhanced in the range of densities exhibiting high T c . This enhancement in turn boosts up g e f f , via Eq. (88), and hence plays an important rolein giving an observable magnitude of T c in Fig. (1) and Fig. (3). ���� ���� ���� ���� ������������������������� ���������������� δ � � �� �� � e - � _ _ _ � �� � Figure 5:
The effective superconducting coupling g e f f (Eq. (88)) for the three curves inFig. (3), with parameter sets- (i) (red) t (cid:48) / t = − t (cid:48)(cid:48) / t = δ peak = t (cid:48) / t = − t (cid:48)(cid:48) / t = .01 with δ peak = t (cid:48) / t = − t (cid:48)(cid:48) / t = δ peak = T c on both sides of the peak values in Fig. (3) can beunderstood by referring to the the second y-axis, giving the approximate temperature scale T appxc = × e − gef f K. CONCLUSIONS some extent the suppression due to a small magnitude of Z , as seenin Eq. (88). As a result of this competition T c turns out to be in theobservable range. The additional y-axis in Fig. (5) translates the su-perconducting coupling g e f f to an order of magnitude type transitiontemperature T appxc = e − g ef f K . This scale helps us to understandwhy T c falls off so rapidly when δ increases beyond the peak valuewhere the coupling g e f f falls below ∼ T c . This work presents a new methodology for treating extremely corre-lated superconductors. The exact equations of the normal and anoma-lous Greens functions in the superconductor are derived. These arefurther expanded in powers of a control parameter λ related to thedensity of double occupancy, and the second order equations are givenin Eqs. (51,52,54), together with the self consistency conditions Eqs. (25,41).A further simplification is possible for T ∼ T c where the anomalousterms are small. This leads to a tractable condition for T c given inEq. (69), expressed in terms of the electron spectral function. Fur-ther analysis uses a model spectral function Eq. (76), which is simpleenough to yield an explicit expression for T c in Eq. (87). More elab-orate calculations should be feasible upon the availability of reliablespectral functions, when one may directly solve Eq. (68).The present approximate calculation gives an insight into the fac-tors determining T c in this model. We see that in the optimal T c casethat the average of the d-wavefunction Eq. (71) peaks simultaneouslywith the DOS. This is possible when the energy bands at the fermi en-ergy pass through { π , 0 } and symmetry related points in the Brillouinzone. This peaking of the d-wavefunction average has the potentialfor verification in photoemission studies.In the approximation used here, the maximum T c is nominally un-bounded in a narrow density range here due to the logarithmic singu- ACKNOWLEDGEMENTS: larity of the DOS. It is expected to be cutoff to a finite value of O ( K ) due to a more exact integration over energies, when using a reliablespectral function instead of Eq. (76). Such an integration would alsosupersede the Gor’kov-type approximation of expanding around thefermi surface ( (cid:82) d (cid:101) n ( (cid:101) ) ∼ n ( µ ) (cid:82) d (cid:101) ) employed here, thereby flat-tening out the sharp peak into a smoother shape. Finally this mean-field description of the superconductor is expected to be corrected byfluctuations of the phase, in a strictly two dimensional case, and byinterlayer coupling, in the physically realistic case of a three dimen-sional system of weakly coupled layers.In conclusion this work contains the outline of a new formalism totreat superconducting states of models with extremely strong correla-tions. A transparent calculation for the t - J model is carried out withina low order scheme, using typical model parameters. It leads to su-perconductivity with T c ’s of O ( K ) , in a finite range of densities lo-cated slightly away from the insulator, in parallel to the experimentalsituation in cuprate superconductors. These results demonstrate thatthe exchange energy J can indeed provide the fundamental bindingforce between electrons forming Cooper pairs in cuprates. The work at UCSC was supported by the US Department of Energy(DOE), Office of Science, Basic Energy Sciences (BES), under AwardNo. DE-FG02-06ER46319.
References [1] J. G. Bednorz and K. A. Muller, Z. Phys.
B64 , 188 (1986).[2] J. E. Hirsch, Phys. Rev. Letts. , 1317 ( 1985).[3] P. W. Anderson, Science , 1196 (1987).[4] G. Baskaran, Z. Zou and P. W. Anderson, Sol. St. Comm. , 973(1987). EFERENCES [5] G. Kotliar, Phys. Rev. B37 , 3664 (1988).[6] C. Gros, Ann. Phys. , 53 (1989).[7] H. Yokoyama and H. Shiba, J. Phys. Soc. Jpn. , 2482 (1988).[8] A. Paramekanti, M. Randeria and N. Trivedi, Phys. Rev. Lett. ,217002, (2001).[9] C.S. Hellberg and E.J. Mele, Phys. Rev. Lett. , 2080 (1991).[10] F.C. Zhang, T.M. Rice, Phys. Rev. B 37 , 3759 (1988).[11] P. A. Lee, N. Nagaosa and X-G. Wen, Rev. Mod. Phys. , 17(2006).[12] T. Giamarchi and C. Lhuillier, Phys. Rev. B 43 , 12943 (1991).[13] M. Ogata and H. Fukuyama, Rep. Prog. Phys. , 036501 (2008).[14] B. S. Shastry, Phys. Rev. Letts. , 056403 (2011); Ann. Phys. ,164-199 (2014). http://physics.ucsc.edu/~sriram/papers/ECFL-Reprint-Collection.pdf [15] L. P. Gor’kov, Sov. Phys. JETP , 505 (1958).[16] B. S. Shastry, Phys. Rev. B , 125124 (2013); Appendix-A.[17] G.-H. Gweon, B. S. Shastry and G. D. Gu, Phys. Rev. Letts. ,056404 (2011).[18] B. S. Shastry and P. Mai, Phys. Rev. B ,115121(2020).[19] C. C. Tsuei, J. R. Kirtley, M. Rupp, J. Z. Sun, A. Gupta, M. B.Ketchen, C. A. Wang, Z. F. Ren, J. H. Wang, and M. Bhushan,Science, , 329 (1996).[20] D. J. Scalapino, Phys. Repts. , 329 (1995).[21] S. Tomonaga, Prog. Theor. Phys. , 27 (1946).[22] J. Schwinger, Phys. Rev. , 1439 (1948).[23] F. Dyson, Phys. Rev. , 486 (1949).[24] P. C. Martin and J. Schwinger, Phys. Rev. , 1342 (1959).[25] S. Engelsberg, Phys. Rev. , 1251 (1962). EFERENCES [26] L. D. Landau, Sov. Phys. J.E.T.P. , 1058 (1956), ibid , 59[27] P. Nozi`eres, in Theory of Interacting Fermi Systems , (W. A. Ben-jamin, New York, 1964).[28] A. A. Abrikosov, L. Gor’kov and I. Dzyaloshinski,
Methods ofQuantum Field Theory in Statistical Physics , Prentice-Hall, Engle-wood Cliffs, NJ (1963).[29] M. Gutzwiller, Phys. Rev. Letts. , 159 (1963).[30] W.F. Brinkman and T.M. Rice, Phys. Rev. B 2 , 4302 (1970).[31] T. Yoshida, T. Yoshida, X. J. Zhou, D. H. Lu, S. Komiya, Y. Ando,H. Eisaki, T. Kakeshita, S. Uchida, Z. Hussain, Z.-X. Shen, andA. Fujimori, J. Phys.: Condens. Matter , 125209 (2007); M.Hashimoto, T. Yoshida, H. Yagi, M. Takizawa, A. Fujimori, M.Kubota, K. Ono, K. Tanaka, D. H. Lu, Z.-X. Shen, S. Ono, and Y.Ando, Phys. Rev. B , 094516 (2008).[32] R. R. P. Singh, P. A. Fleury and K. Lyons, Phys. Rev. Letts. ,2736 (1989).[33] For this purpose we could, in principle use the Sommerfeld typelow T formula for a fermi liquid. Estimating mm ∗ from data turnsout to be somewhat problematic. One must not only choose thebare band hopping parameters, but also neglect features of thedata, such as the strong T dependence of the Sommerfeld coeffi-cient γ in the data for La − x Sr x CuO , as reported in J. W. Loram,J. Luo, J. R. Cooper, W. Y. Liang and J.L. Tallon, J. Phys. C Solids62