Superconductivity in Systems With Local Attractive Interactions

We discuss approaches based on the concepts of local electron pairing and the superconducting properties which they imply. The nature of the intermediate coupling regime is addressed and a recent progress in the BCS–Bose superconductivity crossover problem is outlined. We also survey the properties of systems with local attractive interactions consisting of a mixture of local electron pairs and itinerant fermions coupled via a charge exchange mechanism which mutually induces superconductivity in both subsystems. Finally, we briefly discuss the question of a pseudogap and a possible scenario of crossovers in high temperature superconductors.


Introduction
Detailed muon spin rotation ( SR) studies by Uemura and his co-workers indicate that high temperature superconductors (HTS) (cuprates, doped barium bismuthates and fullerenes) and also other nonconventional superconductors, such as Chevrel phases and organic superconductors, form a unique group of superconductors characterized by high transition temperatures relative to the values of n s =m (n s is the super uid carrier density and m the e ective mass) 1,2].These materials have their T c proportional to T F (the Fermi temperature) or T B (the Bose-Einstein condensation temperature), with T B (3 ?30)T c and T F (10 ?100)T c .
They generally exhibit a low carrier density, a small value of the Fermi energy (/ 0:1 0:3 eV), a short coherence length 0 and are extreme type II superconductors with the Ginzburg-Landau parameter 1 1{3].The HTS and other nonconventional superconductors are placed in the intermediate crossover region between the BCS physics and the preformed electron pair scenario (table 1).For most HTS; 0 k F 5 ?10, whereas 0 k F 1 is needed for the BCS microscopic c R.Micnas, S.Robaszkiewicz theory to hold.The short coherence length is the average diameter of the pair in the condensate and limits the range of interaction, k ?1  F is the average distance between the carriers.Another quantity of interest is the average distance between the pairs d p , which can be estimated as 1=d 3 p N(0) , where N(0) is the density of states at E F per spin and is the superconducting energy gap.In HTS 0 =d p 1, in contrast to conventional superconductors, where 0 =d p 1 4].The above facts point out that the interactions responsible for pairing in HTS are short-ranged.The relation T c ns m ?(for small doping), discovered by Uemura et al . 1] is universal for cuprates and this suggests that pairing is essentially nonretarded.
Moreover, for many HTS, regardless of a speci c microscopic mechanism leading to pairs, there are several universal trends in the scaled T c versus the hole density, T c versus the condensate density dependence, and T c dependences of the pressure and the isotope e ect coe cient 5].These universal trends are consistent with short-range, almost unretarded e ective interactions responsible for pairing, and, moreover, they suggest that there could be a common condensation mechanism and thermodynamic description of short-coherence length superconductors 7,4,6].
The above points also give some important hints for a phenomenological approach.Firstly, these systems have a small super uid density in the underdoped regime as superconductivity comes through the doping of the insulating parent compound.Secondly, since 0 =d p 1 for HTS, the phase uctuations are important and determine T c and can have a profound in uence on the normal state properties 5,4,8].Thirdly, the condition 0 =d p 6 1 gives the region of applicability of the description in terms of real space XY type models as a lattice of Cooper pairs on e ective sites R i of extension 0 .
All the empirical constraints and the universal trends and observations dis-cussed above put strong limitations on the microscopic theory of HTS and support the models with short-range, almost unretarded pairing interactions which will be discussed next.

Models of local electron pairing
The theoretical models of local pairing either start with a microscopic derivation of a local attractive interaction or simply postulate some e ective Hamiltonian 6,7,4].The simplest generic model, which can be thought of as a useful parametrization of the problem, is the extended Hubbard model (EHM) with an intra-site or inter-site attractive interaction: ( ?E i )n i ; (1) n = N e =N = 1 N X i hn i i; (2) n i = n i" + n i# ; n i = c y i c i , t ij denotes a transfer integral, U is an on-site and W ij is an inter-site interaction between tight-binding electrons.is a chemical potential and E i is a (random) site energy.Model (1) can be considered as rather general, resulting from a system of narrow band electrons strongly coupled to a bosonic eld.The bosonic modes can be phonons, excitons, acoustic plasmons etc.The parameters of (1) t ij ; U; W ij are e ective ones (renormalized from their bare values).The typical microscopic mechanism leading to an e ective short{ range attraction is a strong electron{lattice coupling which can give rise to the formation of polarons.For this polaronic mechanism the local anharmonic modes can play an important role 9,10].Also, the models introducing purely electronic (\chemical") mechanisms of a local attraction can be appropriately described by such an e ective Hamiltonian.
Two cases have been extensively studied: (i) U e < 0; W e > 0; when the induced local attraction outweights the on-site repulsion.This is the case of an intra-site attraction (or a negative U extended Hubbard model) and the problem of the formation of on-site electronic pairs in the strong U < 0 limit.The negative U Hubbard model is the simplest lattice model of a superconductor with a short coherence length displaying a crossover between the BCS-like superconductivity and the pair Bose condensation.It has been considered as an e ective model of superconductivity and charge orderings in the family of barium bismuthates (Ba 1?x K x BiO 3 , BaPb x Bi 1?x O 3 ), fullerides, the Chevrel phases, as well as cuprates.
(ii) W e < 0 but U e > 0; i.e. the case of an inter-site attraction when the induced attraction is strong enough to dominate over the inter-site Coulomb repulsion.This is a model for systems with inter-site pairing of various pairing symmetries having the most direct relevance to the family of cuprate HTS, and to heavy fermion superconductors, where it can describe e ective pairing of fermionic quasiparticles.
We should point out that, in certain cases, it is possible to introduce e ective sites and the inter-site attraction can still be mapped onto a negative U extended Hubbard model.This is, for example, the case of Alexandrov and Ranninger model of bipolaronic superconductivity 11] which begins with an inter-site attraction but is mapped onto a U < 0 problem in the strong attraction limit.
Besides U and W, there are other inter-site interaction terms which are not included in (1) and which can be of importance in real narrow band systems.They are given by where s + i = c y i" c i# ; s ?i = c y i# c i" ; s z i = 1 2 ((n i" ?n i# ); + i = c y i" c y i# ; ?i = c i# c i" : They describe the correlated hopping (K ij )interaction, the electron spin (J ij ) and charge exchange (I ij {pair hopping), respectively.Formally, J ij , I ij and K ij are the odiagonal terms of the Coulomb interaction V (r ?r 0 ): J ij = (iijV (r)jjj), Kij = (iijV (r)jij).These terms, involving a bond charge density, result from the fact that due to the translational invariance the electron density operator is not diagonal in the Wannier representation.The typical range of electron-electron interaction parameters arising from the Coulomb potential is U > W > K > jJj; jIj or, for a strongly screened potential, U > K > W > jJj.
A comprehensive study of the e ects of these terms on the superconductivity of systems with inter-site pairing has been carried out by us 15,6,7] and independently by Hirsch and Marsiglio 13], and the correlated hopping term interaction has been even supposed to be a universal mechanism of s-wave superconductivity in the system with a low concentration of holes.
In what follows we will focus on three issues within the theory of superconductivity with local pairing: Superconductivity of preformed pairs (local pairs, bipolarons); Superconductivity with inter-site pairing (extended s-wave(s ) vs d{wave symmetry); Crossover between the BCS and Bose condensation In section 5 we will also comment on the realistic generalization of the theory for the case of coexisting local pairs and itinerant carriers: a model of charged bosons (2e) and fermions (1e).

Tightly bound local pairs
In the model of local electron pairs (LPs), superconductivity resembles the super uid state of 4 HeII.The thermodynamic and electromagnetic properties of such a system of charged (hard-core) bosons have been extensively studied (for a review see 6,8], 12].In contrast to BCS superconductors, in the LP system the electron pairs (either on-site or inter-site) can exist above the transition temperature.T c is determined by the center-of-mass motion of pairs; it increases with decreasing the local attraction and increases with increasing the bandwidth (T c t 2 = j U j in the case of on-site pairing).In such a way an enhancement of T c with applying pressure is quite natural for the LP pair system.At some temperature T p > T c the local pairs nally break up into electrons.Hence, there will be, in general, three temperature regions.
(i) A low temperature region where the pairs are in the superconducting state with the properties analogous to the super uidity of charged bosons on a lattice.In the high density limit this phase can compete with the charge density wave (CDW) ordering (short-or long-range order).
(ii) An intermediate temperature regime with the state of dynamically disordered local pairs.
(iii) A high temperature regime above T p around which a dissociation of pairs takes place.
The regions (i) and (ii) are separated either via a single -type transition (SS !NO) or via a sequence of two ( SS-CDW !CDW !NO) or three ( SS !SS-CDW !CDW !NO) phase transitions in the case of on-site pairs.
It should be pointed out that since the gap in the single electron excitation spectrum persists across the SS transition, the single-electron conductivity of the normal phase (but below T p ) will be non-metallic and have an activated character.In such a case the transport properties can be dominated by charged LP.
In the regime well below T p , the system with preformed local pairs can be described by the Hamiltonian of hard-core charged bosons on a lattice 6,8] where n i = b y i b i ; b y i ; b j ] = (1 ?2n i ) ij : The rst term in equation ( 8) describes the transfer of electron pairs (hard-core bosons with charge 2e), whereas the second term stands for an e ective Coulomb interaction between the pairs.The number of bosons is xed by the condition ñ = 1 N P i hn i i.The operators b y i ; b j are commuting (Bose-like) for di erent sites, i.e. b y i ; b j ] = 0 for i 6 = j, but (b i ) 2 = (b y i ) 2 = 0, b y i b i + b i b y i = 1, for the same site, which re ects their fermionic nature.In the case of on-site pairing the Hamiltonian (4) can be derived by an exact mapping of the EHM in the strong attraction limit 6] and b + i = c y i" c y i# ; J ij = 2t 2 ij =jUj, V ij = J ij + 2W ij , B = 2 + jUj + J 0 , J 0 = P j(6 =i) J ij , ñ = n=2.It should be noted, however, that model ( 4) is common for the study of the superconductivity and CDW formation in systems with bound electron pairs, of either the on-site or inter-site nature.Also, model (4) is particularly useful for the extreme type II superconductors with a short coherence length, as far as the e ects of phase uctuations are concerned.
In the LP systems, which are equivalent to a hard core Bose gas on a lattice, T c strongly depends on n for any density.In particular, for the low density limit T c ñ2=3 (ñ) for d = 3(d = 2 + ), and such a dependence of T c on n can be displayed over a wide range of electron densities.
For the case of the anisotropic layered lattice structure this subject has been studied within the self-consistent RPA 6,8].It has been found that in the limit of small density (n 1), the concentration dependence of T c changes with the ratio = J ? =J k , where J ? and J k are inter-and intra-layer values of J ij , respectively.For < 1, T c is well described by k B T c = 4:17J k 1=3 (2ñ) 2=3 = 3:31(n ) 2=3 m ; (5) which is just the formula for a 3D (anisotropic) free Bose gas with the e ective mass m = (m ?m 2 k ) 1=3 ; m k = (2J k a 2 ) ?1 ; m ?= (2J ?d 2 ) ?1 , and with a density equal to the density of electron pairs in the system n = ñ(a 2 d) ?1 .(The quantities a and d denote the intraand inter-layer lattice spacing, respectively).However, for 1, T c is governed by the following expression: At higher densities the LP superconductors display a non-monotonic behaviour of T c versus n due to interpair repulsive interaction and the systems with on-site local pairs invariably show a maximum of T c near the phase boundary between the SS and CDW phases.In gure 1 we show an example of the nite temperature phase diagram of model ( 4) with near-neighbour (n.n) interactions for a simple cubic lattice calculated in MFA, RPA and BPW (Bethe-Peierls-Weiss) approximations.Indeed, MFA, being an exact theory for d ! 1, substantially overestimates T 0 c s for d 6 3.The collective excitation and quantum uctuation e ects extend the range of existence of a homogeneous SS phase and provide a power law ñ dependence of T c .Notice, that this phase diagram displays the main phases found in the doped barium bismuthates 6].For these materials the source of local attractive interaction is a chemical tendency toward disproportionation (Bi 4+ !Bi 3+ +Bi 5+ ) (valence skipping) and the electron-phonon coupling.
We remark that there are other factors not considered above which can reduce T c of charged bosons and also modify the concentration dependence of T c 6,8].It concerns: (i) the e ects of the diagonal and o -diagonal disorder (ii) reduction of T c by inter-subsystem density-density interaction if the LP coexist with itinerant carriers (section 5) and (iii) suppression of T c due to the Wigner crystallization in the low density regime (e ects of a long-range Coulomb interaction).
The LP superconductors are expected to have a short coherence length due to the short-range coupling between the pairs and the small radius of a pair.This leads to a relatively weak sensitivity of the SS phase in the low concentration limit to the presence of non-magnetic impurities (except for SS coupled to CDW), by analogy with the behaviour of the super uid phase of 4 HeII in the presence of a disorder.It also provides an enlarged width of the critical regime which should be experimentally accessible and a true critical behaviour of the quantum X-Y, s=1/2 model should be observable.For d=2 a Kosterlitz-Thouless type phase transition can occur 6].
The electromagnetic properties of LP superconductors are essentially di erent from those of BCS 6,8,12].The major e ect of a magnetic eld in a LP system occurs via its coupling to the orbital motion of the charged local pair.This leads to very enhanced values of H c 2 , proportionally reduced values of H c 1 , no Clogston limit for H c 2 as T !0 (H c 2 (0) E binding k B T c ) and a strongly enhanced penetration depth.Moreover, one obtains an upward curvature of H c 2 near T c , with H c 2 (1 ?(T=T c ) 3=2 ) 3=2 for the 3D system in the dilute limit.
The temperature dependences of and H c 1 can also be non-standard.In particular, for the screened Coulomb interaction in the dilute limit, there are three di erent regimes (d > 2) 8]: (i) a low-T \phonon" region for = T=T c < 0 = T 0 =T c with 1 ?( 0 = (T)) 2 d+1 , (where k B T 0 ñ V 0 , V 0 = 2(J 0 + V 0 )), (ii) an intermediate \free particle" region for 0 < < x 1 ?0 with 1 ?( (0)= (T)) 2 d=2 and (iii) the critical region for x < < 2? x with the XY model type critical behaviour and ( 0 = (T)) 2 = (1 ? ) , 2=3 for d = 3 ( 0 being the penetration depth amplitude).With increasing ñ and/or V 0 regions (i) and (iii) will expand at the cost of suppression of region (ii).Thus, the universal features of (0)= (T)] 2 in the considered model for d=3 are the T 4 behaviour in the T !0 limit and the 3D XY critical point behaviour close to T c .The temperature in the intermediate region of T depends on ñ and V 0 and re ects the location of the crossover from the dilute to the dense limit in terms of the exponent x in the formal expression: (0)= (T)] 2 = 1 ?(T=T c ) x ; with 3=2 < x < 4 (x !3=2 for ñ V 0 !0 and x ! 4 in the dense case).It is important to note that in the d = 2 + case there will be a regime of a linear in T dependence of the super uid density far from the critical region.
The super uid properties of the interacting hard{core charged bosons on a lattice for short{range and long{range inter-site interactions are summarized in table 2 8].
We note that the exponential T dependences of the thermodynamic characteristics for the case of unscreened Coulomb interactions (cf.table 2) will occur in a restricted temperature range.With increasing T, for d > 2 a crossover to power{low characteristics, c v (T) T d=2 ; h x ?x (T)i s ?s (T) T d=2 , can take place at higher T, if k B T c > k B T > = min E k 6 ~ 0 , where E k is the collective excitation spectrum with a long-range Coulomb interaction.
As far as other universal trends are concerned, we should point out the plots of T c versus ?2 (0) rst reported by Uemura et al. 1,2].In 4,14], the plots of T c = T c =T m c versus ?2 (0) = m jj (0)= jj (0) 2 for the hard core Bose gas on a lattice have been obtained with the use of a selfconsistent RPA theory for T c 6,8] and ?2 (0), for several values of V=J.(T m c corresponds to the maximum critical temperature in the 0 < n < 1 interval and m jj (0) is in plane penetration depth at T = 0K attained for T m c ).These scaled curves compare well with the corresponding experimental plots for various families of superconducting cuprates and Chevrel phases, for both the underdoped and overdoped regimes.The theory predicts an almost universal T c versus ?2 (0) behaviour (being only very weakly dependent on V=J) in the underdoped regime (low concentration) 4,14], and possible deviations from the universality for the systems in the overdoped regime.The general trends including the fact that the overdoped systems have lower T c and suppressed superconductivity are qualitatively reproduced 14].
Other points which can distinguish the LP superconductivity from the BCS 6,8,12] are: (i) The collective excitation spectrum with a sound wave-like excitation branch in the case of a screened Coulomb interaction.A reduced plasma frequency in the case of an unscreened Coulomb interaction.The existence of two energy gaps in the latter case: the gap in the single electron excitation spectrum which remains almost a T independent well in the normal state region (2 =k B T c 1) and the gap in the two electron spectrum 1 (2 1 =k B T c 1).
(ii) The heat capacity jump at T c : c=nk B 6 1 ( 1 for BCS), and a possibility of -like anomaly in the heat capacity.
(iii) The relaxation rate of nuclear spins T ?1 1 exp(?=k B T) (only thermally activated electrons can interact with the nuclear spin) (iv) Temperature dependent sound velocity (s(T)) with a negative temperature gradient, a jump of the 1st derivative of s(T) at T c and the sound wave damping ?T(? exp(?=k B T) in BCS).(v) Essentially the same e ect of nonmagnetic and magnetic impurities for s-type LP on T c .(vi) A possibility of the disorder induced LP superconductivity and superconducting glass behaviour of an LP superconductor.
The normal state properties for the on-site and inter-site local pairs can be di erent, due to the existence of triplet states for the latter.The main features common for both types of pair carriers are: (i) diamagnetic (or Van Vleck type) susceptibility (ii) eld independent resistivity and thermoelectric power S up to a very high magnetic eld( ) (iii) the possibility of a linear in T dependence of and a small value of thermopower in a wide range of temperature, in the case of a nondegenerate gas of charged bosons 12].

The extended Hubbard model with inter-site pairing
The superconductivity and magnetism in the extended Hubbard model (1) with on-site repulsion and inter-site attractive interaction, i.e.U e > 0; W e < 0 (and E i =const) have been studied in both the weak (U < 2zt) and the strong (U t) correlation limits, within various approximation schemes 15,6,4].This is a generic model incorporating magnetic correlations and inter-site pairing, having relevance for high?T c materials, as well as heavy fermion superconductors and organic superconductors.The model can be considered as the simplest model of oxygen holes pairing in high?T c cuprates due to the polaronic mechanism 15,12] or due to purely electronic mechanisms ( 15], 4]) or as an e ective model of quasiparticle (AF spin polarons) pairing ( 16,17]).These studies, including the analysis of various types of anisotropic superconducting pairings, the spin-density waves (SDW) and the phase separated state (electronic droplets formation) have been essentially carried out for a 2D square lattice with near neighbour (n.n) hopping and for arbitrary electron density.The e ects of next-nearest neighbour (n.n.n) hopping, antiferromagnetic exchange, the correlated hopping and longer range Coulomb repulsion have also been analyzed.In many of these studies the intersite interaction terms have been decoupled in the broken symmetry HFA, which, due to the extended nature of the pairing potential, is here better justi ed than in the case of on-site interaction.In contrast to the original BCS treatment for the phonon-mediated attraction, we did not impose any cuto in either momentum or frequency.The e ective short-range attraction in the considered model is essentially instantaneous on the time scale of the inverse bandwidth.
Let us brie y summarize the essential properties of the system with an intersite attraction having in mind that due to the complexity of the problem only partial and mainly qualitative conclusions can be drawn.The nature of the ordered state depends on the band lling and the values of the parameters involved.In gure 2 we show the ground state phase diagram at half-lling.For n 6 = 1 and for the nearest neighbour hopping only, the sequence of transitions d , p , s is possible with lowering the electron density (from n = 1) for U > 0(or small negative U). Close to half-lling the d-wave pairing dominates, which competes with the SDW state.The extended s-wave pairing is stable for lower electron densities and T s c versus n, with increasing n, and increases rst, then goes through the maximum and drops to zero above some critical density (cf.gures in 15]).Increasing jWj expands the range of stability of a dwave pairing towards higher values of jn ?1j.The stable superconducting solutions in the present model can be obtained upon an appropriate tuning of the parameters.Above some critical value of jWj, dependent on n, a condensed state of electronic droplets-phase separation becomes stable.The phase separation line rapidly shifts towards higher values of jWj=D upon increasing the long-range repulsive Coulomb interaction.
The mutual stability phase diagrams of anisotropic superconducting solutions and the condensed state for a 2D square lattice were given in 15,16].The increasing on-site repulsion suppresses the s-wave state and restricts it to low densities only.
The e ects of next-near neighbour hopping(t 2 ) are appreciable and can radically modify the mutual stability of superconducting and SDW states, as well as the variation of superconducting T c with the electron density 15].In the presence of t 2 the electron dispersion is of the form: " k = ?2t 1 (cos(k x a) + cos(k y a)) ?4t 2 cos(k x a) cos(k y a), where t 1 and t 2 are the near neighbour and next-near neighbour hopping parameters, respectively.For t 2 < 0 (which is the case suggested to reproduce the Fermi surface for hole doped cuprates), with increasing the ratio jt 2 =t 1 j the sequence of transitions with n can be changed from s !p ! d !s to s !d !p ! s and then to d !p ! s (see gure 14 in 4]).The p-wave pairing is strongly suppressed by an antiferromagnetic exchange interaction which, in turn, enhances the extended s and dwave singlet pairings.An example of the mutual stability diagram of s-d pairings (with excluded triplet p-wave pairing) for the square lattice with n.n and n.n.n hoppings in a t 2 =t 1 ?n plane is shown in gure 3.As we see, for t 2 < 0, with increasing jt 2 =t 1 j the range of stability of the d-wave pairing can extend up to n = 0, while the d ?s phase boundary for small (2 ?n) is only weakly dependent on jt 2 =t 1 j.
One should bear in mind recent calculations of van der Marel 18] which show that for singlet pairing potential g 1 , larger than the critical value (dependent on n), the ground state can be of a mixed s and d wave symmetry and the region of s ?d mixing almost coincides with the region of the p-wave symmetry if we use a spin independent interaction (jWj instead of g 1 ) (cf. gure 14 in 4]).
In the strong correlation limit (U t) model ( 1) can be mapped onto the generalized t ?J model with the additional inter-site interaction term (t ?J ?W model) and as such can generally result from the mapping of the multiple band extended Hubbard model to the single band problem 6].Determination of possible superconducting, magnetic and CDW solutions of the EHM on a 2D lattice is still a challenging problem.For a square lattice an electron pairing in the antiferromagnetic background via the spin-bag mechanism should also be considered.
In the dilute limit, formation of real bound inter-site pairs is possible.The properties of the extended bound states and resonances (formed by two lattice fermions interacting via a nonretarded potential) were studied in 6,19].This twobody problem is exactly soluble on any periodic lattice.The pair binding energies for di erent lattices and the pairing symmetry were determined 6,19].The symmetry of the stable bound state depends on the form of the pairing potential and the band structure.For model (1) with the interactions and hopping restricted to n.n., the two-body ground state is an extended s-wave for U > 0. The inclusion of n.n.n.hopping of the sign reversed to t 1 and the n.n.n.repulsion (W 2 > 0) favour the pairing of the d x 2 ?y 2 symmetry (compare gure 3).Moreover, both these factors reduce the minimum value of jW 1 j necessary for the bound state formation.Figure 4 shows the ranges of parameters for which the bound state of a given symmetry rst appears below the continuum band in the case of n.n attraction (W 1 < 0) and n.n.n.hopping(t 2 ) and n.n.n.repulsion (W 2 > 0).19].It is interesting to note that taking t 2 = ?0:45t 1 (which is close to the tight binding parameters inferred from the band structure calculations for 123 and 214 cuprates), one nds that the strength of the next neighbour repulsion must be at least 0:35t 1 in order to have for the ground state a dwave symmetry.
We stress that the e ective mass of a strongly bound inter-site pair can be small and even comparable with the mass of its constituent fermions, in contrast to the case of an on-site pairing.This is due to the fact that unlike the case of a strong on{site attraction, where the pair moves via virtual ionization only, the inter-site pair can easily move without breaking its bond, if the n.n.n.hopping is included.For example, in a square lattice m f = ~2=2(t 1 +2t 2 )a 2 , whereas the mass of a strongly bound inter-site pair The superconducting state requires a Bose condensation of such real bound pairs.For a low electron density the system behaves as a dilute gas of two-electron molecules and the problem can be mapped onto that of a hard-core charged Bose gas on a dual lattice, whose properties were discussed in section 3.1.
In the weak local attraction case, i.e. beyond the limit of real space pair formation, the physics becomes much more similar to BCS than in the strong coupling regime.However, even then, there remain some essential di erences in comparison with BCS superconductors.The fact that the attraction is static (without a cuto in the frequency dependence of the interaction), implies that many electrons inside the Fermi surface can contribute to the pairing and that the e ective halfbandwidth D (instead of !D ) will determine the energy scale.One also expects enhanced pair uctuation e ects which are beyond the standard mean-eld BCS approach.The consequences of this are the following.Firstly, T c and the energy gap in the BCS expressions are enhanced, since !D is replaced by D. Secondly, both these quantities are explicitly dependent on the electron density (see below).Thirdly, the ratio 2 =k B T c can deviate from the BCS value, being a function of the lattice structure (DOS), electron concentration, and the strength of attractive interaction.
At very small densities, analytic expressions for T s c can be obtained if the pairing potential is restricted either to on-site or nearest neighbour attraction: T s c n 1=3 (n 1=2 ) for d = 3(d = 2) lattice.However, if the attractive couplings fall o gradually with a distance: T s c n 2=3 (n) for d = 3(d = 2) lattice, i.e. one gets the same n dependence of T c as for LP superconductors.
In the systems with a local pairing interaction several types of phase separated states can develop besides homogeneous phases, particularly, if the long{ range Coulomb interactions are strongly screened.In these states, which can be favourable close to the half{ lling of the band, the system breaks into coexisting domains of two di erent charge densities and di erent types of electron ordering.For the on{site pairing these are the CDW{SS or CDW{NO states, whereas for the inter-site pairing the SDW{SC (s,d or p type), the CDW-SC or the state of electron droplets 15,16,21].In real materials the size of the domains will be nite and determined by the long{range Coulomb repulsion and disorder e ects (structural imperfections, disorder of doped ions, etc).

Crossover from BCS to the local electron pair limit
In the previous section, we presented properties of systems with a local electron pairing derived in the LP and weak coupling regimes.
Let us now brie y summarize the main physical ideas regarding the crossover from the cooperative Cooper pairing(BCS) to Bose condensation (BC) (see gure 5, cf. also gures 3 and 13 in 6]) 6,7,22,23]. 1.In both regimes there is only one phase transition at T c , as long as no other broken symmetry phases intervene or the system does not undergo a phase separation.The transition is from the normal state to the super uid one with an o -diagonal long range order (ODLRO)(or algebraic order in 2D). 2. The nature of the phase transition in both limits is quite di erent.In the BCS limit a formation of Cooper pairs and condensation at T c takes place simultaneously and T c exp(?1=N(0)V ) (N(0)-the density of states per spin, V -the parameter of attractive coupling).The rst deviation from this scenario can be described in terms of superconducting uctuations.In the preformed pair regime, however, the pair formation (T p ) and their condensation (T c ) are independent pro-cesses.T p and T c are widely separated and T p is a characteristic energy scale, not a phase transition temperature (at least for non-frustrated lattices).T c will decrease with the increase of coupling constant V .For T > T p local pairs are thermally dissociated.3.In the weak coupling limit, below T c , we have a BCS condensate of a large number of overlapping Cooper pairs ( a).Thermodynamics and T c are determined by single particle excitations (broken Cooper pairs) with an exponentially small gap.In the opposite, strong coupling regime one has the Bose condensate of tightly bound local pairs ( a), and the thermodynamics and T c are governed by the collective modes.With increasing coupling there is a smooth reduction of k B T c =E g (0)(T c over the energy gap) ratio from the BCS value.In the intermediate and strong coupling regime T c does not scale with the energy gap and k B T c =E g (0) decreases with the reduced concentration.
The collective modes evolve smoothly between the two regimes.In the BCS regime we have Anderson{ Bogolubov modes for a neutral case and plasmons for a charged case, respectively.In the BC limit there are either sound wave Bogolubov modes for the screened Coulomb repulsion or plasma modes for the charged boson super uid.As far as electromagnetic properties are concerned, there is a smooth evolution of the Meissner kernel from a Pippard type to a London type behaviour at T = 0K 22].Also, L ; H c and the coherence length evolve smoothly from the BCS to BC regime 24].In the BC case: = L = 1, superconductivity is clearly of extreme type II, H c2 is very large, H c2 (T) can exhibit a positive curvature vs T and H c1 =H c2 1. 4. For the BCS superconductor the normal state is a Fermi liquid.In the BC regime one has for T c < T < T p a normal Bose liquid (of bound pairs).The evolution of the normal state from the Fermi to Bose liquid is quite unusual, especially in 2D systems.From the recent studies it appears that even for a moderate attraction the (degenerate) Fermi liquid regime shows anomalous features due to pairing correlations such as a pseudogap, anomalous behaviour of spin susceptibility and a spin gap in the normal state.6,22,23,25{30] (cf.section 6).It is of interest that these anomalies (super uid precursor e ects) are similar to those experimentally observed in NMR and optical conductivity on underdoped high T c cuprates.From these studies it follows that deviations from the conventional Fermi liquid behaviour are generic to the normal state of short{coherence length superconductors.
The above results were mostly derived from the studies of one-component models.Both, for the case of on-site pairing and that of inter-site pairing the crossover between the \BCS" and the local pair limits can occur either by increasing the coupling strength, or by decreasing the carrier concentration (in the intermediate coupling regime).We should stress the importance of a mixed model of coexisting bosons and fermions 6,31,33].This model can be naturally considered as an extension of the extreme bosonic limit and a candidate to describe the intermediate coupling crossover regime.
Regarding open problems we point out the question of a crossover for anisotropic pairings of s or d{wave symmetry in systems of reduced dimensionality.Enhanced thermodynamic uctuations and short coherence length e ects are clearly challenging issues for the successful theory of cuprates and organic superconductors.

Coexisting local pairs and itinerant carriers
The coexistence of bound pairs, itinerant electrons and the e ects resulting from interactions between these two species constitute an important problem for understanding the intermediate crossover regime and application to real materials.Such a model of the mixture of local pairs and electrons (the mixture of charged Bosons and Fermions) interacting via a charge exchange was introduced by us a few years ago 31] and its extended version has been extensively analyzed in a number of more recent papers 32{46].It has been shown that in this type of systems a new mechanism of superconductivity can develop.It results from the intersubsystem charge exchange coupling, both hybridization induced and direct, and leads to the superconducting state involving both types of particles.The physical properties of such a mixture of interacting charged bosons (bound electron pairs) and electrons can show features which are intermediate between the features of pure local pair superconductors and those of classical BCS systems.The model may also have relevance to the problem of a single band system with a short-range attraction in the intermediate coupling regime where the bound and ionized pairs coexist 6,7,22].An e ective Hamiltonian of coexisting local d-electron pairs and itinerant c-electrons can be written as 31,32,6,7]: k refers to the energy band of c-electrons, 0 measures the relative position of the local pair level with respect to the bottom of the c-electron band. is the chemical potential which ensures that the total number of particles in the system is constant, i.e., J ij is the pair hopping integral, n d is the number of pairs per (e ective) site.I k;k 0 , represents the transverse component of the charge-charge coupling between the two subsystems.(For the sake of simplicity we take intersubsystem pair hopping of swave type).V k;k 0 is the density-density intersubsystem interaction.Depending on the relative concentration of \c" and \d" electrons we distinguish three essentially di erent physical situations.
(i) 0 < 0, so that all the available electrons form local pairs of \d" electrons (the d-regime or the \local pair" n d n c ) (ii) 0 > 0, so that the \c" electron band is lled up to the Fermi level = 0 and the remaining electrons are in the form of local pairs of \d" electrons (the c+d -regime or \intermediate", 0 < n d ; n c < 2). (iii) 0 > 0, so that the Fermi level < 0 and, consequently, at T = 0K all the available electrons occupy \c" electron states (the c-regime or \BCS", n c n d ).
For J ij = 0, in case (ii), superconductivity is caused by a perpetual interchange between local pairs of \d" electrons and pairs of \c" electrons.In this process \c" electrons become \polarized" into Cooper pairs and \d" electron pairs increase their mobility by decaying into \c" electron pairs.In this intermediate case neither the standard BCS picture nor the picture of local pairs ts, and superconductivity has a \mixed" character with a correlation length of the order of several interatomic spacings.The system shows features which are intermediate between the BCS and preformed local pair regimes.This concerns the energy gap in the single electron excitation spectrum, k B T c =E g (0) ratio, the thermodynamic critical eld, the Ginzburg ratio , the width of the critical regime and the normal state properties.
In case (i) the local pairs of \d" electrons move via virtual excitations into empty c-electrons states.Such a mechanism gives rise to the long-range hopping of pairs of \d" electrons (analogy to the Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction for s-d mechanism in the magnetic equivalent).The superconducting properties are analogous to those of a pure local pair superconductor, eventually with a reduced critical region due to the extended range of pair hopping.
In case (iii), on the contrary, we nd a situation which is similar to the BCS case: pairs of \c" electrons with opposite momenta and spins are exchanged via virtual transitions into local pair states.
The indirect long-range character of the charge exchange between the local pairs is an essential feature of the mixed model.This should be contrasted with the previously considered models of a local pair superconductivity in which the pair hopping term resulting from the kinetic exchange mechanism (J ij = 2t 2 ij =jUj) is obviously short-ranged.Thus, an indirect charge exchange can be e ective even if the local pair centers are well separated in space.The case of a small number of local pairs coupled by a long-range interaction resembles an RKKY \spin glass" and it might equally well exhibit a \superconducting glass" state or a \charge density wave glass" state.
The main features of the mixture of wide band electrons and local pairs are summarized below 31{33, 6,7].(1).The model avoids problems with small pair mobility (of on-site pairs) and can provide a screening mechanism of a long-range Coulomb interaction between charged bosons.(2).The origin of the energy gap can be distinct from BCS.An energy gap in a wide band can open due to the pair Bose condensate (hbi 6 = 0).
(3).As we proceed from the case of predominantly local pairs to that of predominantly wide band-electrons, we observe a non-monotonic behaviour of T c , which passes through a maximum of order I 2 0 =D when the two constituents have roughly equal concentrations and drops to zero when we approach regions (i) and (iii).( 4).The ratio E g (0)=k B T c (the energy gap over T c ) -is not universal.It varies around the BCS value 3.52 as the relative proportion of local pairs to wide band electrons is changed.Where T c is maximum, E g (0)=k B T c has a shallow minimum; it approaches the BCS value for predominantly wide band electrons and surpasses it as the concentration of local pairs increases above that of wide band electrons.(5).; H c ; ; GL evolve with a change of position of the LP level 0 from \LP" to \Crossover" and nally to a \BCS"-like regime.
As for the evolution of superconducting properties with increasing the total number of carriers, there are two possible types of change-overs 33,34]: (i) for 0 > 0, \BCS" !\Crossover"!"BCS", and (ii) for 0 6 0, \LP" !\BCS".The latter case is relevant to the doping dependence of superconducting characteristics observed in high T c cuprates.Only, in the case when the local pair level is deeply located below the bottom of the fermionic band, the system remains in the dregime for any n 6 2.
(6) The local pairs exist above T c together with itinerant fermions.As far as the state above T c is concerned, it has been shown that a system of local pairs and wide band electrons can exhibit a linear in T resistivity in the normal state where the Fermi level decreases linearly with T 35,36].The normal state properties deviate from the Fermi-liquid.In particular, numerical studies of the boson -fermion model (T > T c ) show: the existence of a pseudogap in single particle DOS, anomalies in one{electron self{energy (k; !), anomalies in charge and magnetic responses, which are similar to those observed in cuprates 37].Assuming a uniform distribution of LP states near E F it was demonstrated 38] that: Im (k; !) ?j!j (V{shaped form) (as in the marginal Fermi-liquid scenario).It can explain many normal state anomalies in HTS linear-like in T resistivity, tunneling conductance (g(V ) = g 0 + g 1 V ), photoemission data and optical conductivity.(7).The model also involves a Kondo lattice problem, but for charged pairs (double valence uctuations) (U e < 0) rather than for spins (U e > 0) 31,39].The Kondo type coupling for charge operators X i I( + di ?ci + H:c:) + V z di z ci instead of P i J i S i ( + di = b y i ; 2 z di + 1 = b y i b i ; + ci = c y i" c y i# ; 2 z ci + 1 = P c y i c i ).
Increasing I reduces the charge moment of local pairs.For I comparable or greater than the c-electron bandwidth, the charge Kondo lattice state with a local charge moment compensation (isospin singlet) can develop, suppressing superconductivity and CDW.At low T the narrow quasiparticle band appearing near E F is split by the coherence gap E c .E c disappears when T increases and at higher temperatures the system enters the incoherent charge Kondo regime, and then into the regime with properties similar to those of a single charge Kondo impurity.This new charge Kondo uid may have potential applications for the normal state of systems with enhanced double-valence charge uctuations, like doped BaBiO 3 oxides.
Various aspects of superconductivity in the boson-fermion model have been recently studied by many authors 40{46].A generalization to anisotropic pairing of extended s{wave or d-wave is also possible 41,44,46].

Scenario of crossovers in HTS
There is a growing consensus that the normal state of cuprate HTS is characterized by a pseudogap or a quasiparticle gap.Several experiments pointed out that above T c there is an anomalous reduction of spin response at temperatures much higher than the critical one (spin gap).This feature was initially associated with the bilayer nature of some cuprate families, however, it is now documented to occur in most of cuprate HTS.
The existence of a pseudogap (as probed by the spin and charge responses) in the excitation spectrum of the cuprates, opening at the characteristic temperature T ?, which can be much above T c in the underdoped regime, is well con rmed by a variety of measurements, including resistivity and Hall e ect 47], speci c heat 48], infrared studies 49,50], NMR 51], as well as by ARPES spectroscopy 52{54].In particular, the latter, for Bi2212, shows a gap in the normal state, which is almost nearly doping independent ( 33 meV), with a k dependence consistent with the d x 2 ?y 2 symmetry.T ? and T c are widely separated in the underdoped regime and eventually merge near the optimum doping 52,53] The recent tunneling spectroscopy data give evidence that a superconducting gap is temperature independent up to T c where it merges into a pseudogap.In the tunneling spectroscopy data a pseudogap is found to be present both in underdoped and overdoped samples and it scales with a superconducting gap 55].
These results and the evidence of a pseudogap behaviour in the normal state of doped barium bismuthates: BaPb 1?x Bi x O 3 56] and Ba 1?x K x BiO 3 57], together with universal features and trends of HTS discussed in the previous sections, give a strong support for the theories of superconductivity with a local pairing.In our opinion, these results are central for the microscopic theory of HTS and show that the conventional BCS theory, regardless of the symmetry of the order parameter, cannot describe superconductivity in HTS materials in the whole range of doping concentration.They suggest that the physics of these materials should be considered in terms of the crossover between \Bose(BE)(LP)" and \BCS" limits (as it was pointed out in 6] and 2]).Schematic plot in gure 6 illustrates possible crossovers in cuprate HTS.In this scenario we identify the pseudogap phenomena observed below T as a result of singlet pairing in the normal state.In that region a low temperature gap weakly depends on doping, while T c is proportional to the carrier density.With increasing the doping the state with preformed pairs without a longrange phase coherence crosses over to a \metallic" state and eventually to a Fermiliquid.In the underdoped regime the phase uctuations drive transition at T c .Beyond the optimum doping the amplitude uctuations control T c and a \BCS"-like behaviour is expected.As suggested by the Ginzburg-Landau phenomenology and the scaling theory of critical phenomena, with increasing the doping there is also possible a crossover from the essentially 2D behaviour of weakly coupled (Josephson type) CuO 2 planes to the (anisotropic) three-dimensional behaviour with the 3DXY critical point 58].At point x 1 in gure 6 there can be a quantum phase transition from insulator to superconductor.It has been recently observed in Znsubstituted high-T c cuprates in the underdoped region 59], that this transition is characterized by the universal 2D resistance 0 !h=4e 2 = 6:45k , as predicted by the scaling theory for insulator{superconductor transition in a bosonic system.
It is of interest to note that the character of several normal state anomalies related to tunneling, Raman scattering, optical conductivity and pseudogap, in cuprate HTS and doped bismuthates is quite similar.As we have discussed, a model of coexisting local pairs and itinerant fermions (which is essentially a twoband model) can provide a basis for the explanation of normal state anomalies and superconductivity in HTS.Moreover, there is a possibility that the mixed model describes generic features of the intermediate crossover regime of one-band models with a local attraction in the dilute limit.

Figure 1 .
Figure 1.Phase diagram of the LP model (4) with n.n pair hopping (J) and n.n inter-pair repulsion (V ), for sc lattice, and V=J = 2. Solid lines are from MFA, dotted ones are from RPA for the SS phase and a dashed line for CDW is from the BPW approximation.In the CDW+SS state two types of a longrange-order (LRO) may coexist forming a phase separated state or a supersolid depending on the range of inter-pair interactions.NO is a phase without LRO.
(k B T c m ?d 2 ) ; which shows a linear in n behaviour and reduces to the formula for the noninteracting Bose gas with a quasi-twodimensional spectrum E k = 1 2m k (p 2 x + p 2 y ) + 1 m ?d 2 (1 ?cos k z d) where the bandwidth in k z direction m ?d ?2 k B T c , and the density is n .The crossover of T c versus ñ behaviour from the Bose gas with anisotropic mass (T c ñ2=3 ) to quasi 2d(or d = 2 + ) Bose gas (T c ñ) takes place for 10 ?2 .

Figure 2 .
Figure 2. Ground state phase diagram of the t?U ?W model at a half-lled band for the two dimensional square lattice for near neighbour hopping, from the meaneld analysis.SDW { a spin density wave state, CDW { a charge density wave state, PS { phase separation, SS and DS are superconducting states with s and d x 2 ?y 2 symmetry, respectively 15,16].

Figure 3 .
Figure 3. Mutual stability diagram of extended s and d-wave pairings for the case of the n.n inter-site attractive interaction ((jW j + 3J=2)=4t 1 = 0:5, J is the antiferromagnetic exchange)and n.n and n.n.n hoppings on the square lattice.

Figure 4 .
Figure 4. Mutual stability diagram of the extended s and d-wave bound states in the plane of W 2 =4t 1 and t 2 =t 1 obtained for n.n.attraction W 1 < 0 and U = 1.(2D square lattice.After 19]).

Figure 5 .
Figure 5. Schematic diagram of the evolution from weak coupling BCS to preformed local pair regimes.Solid lines are for a lattice fermion model like the attractive Hubbard model (where T c exp(?t=jUj) in a weak coupling, whereas T c

Figure 6 .
Figure 6.Scenario for crossovers in cuprate HTS with possible BE-BCS and 2D-3D dimensional crossover.The dashed line marks a region of the onset of local pairing with a characteristic T , the heavy solid line is for the superconducting transition (T c ). Regime with local pairs is characterized by a pseudogap in oneelectron spectrum.AFM -antiferromagnet, SC -superconductor, x 0 corresponds to optimum doping, x 1 denotes a quantum critical point.

Table 1 .
Schematic evolution from BCS to Pair Bose condensation.d c 1=k F { the average distance between carriers, d p { the average distance between pairs.

Table 2 .
Super uid characteristics of the hard{core boson model.
H C includes the remaining c-c and d-d Coulomb interactions between charge carriers.The charge operators for local pairs b y i ; b i obey the Pauli spin 1/2 commutation rules which exclude multiple occupancy of a given pairing center.For on-site pairs b y