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On the Theory of the Pseudogap Formation in 2D Attracting Fermion Systems

Two-dimensional system of the fermions with the indirect Einstein phonon-exchange attraction and added local four-fermion interaction is considered. It is shown that in such a system at resulting attraction between particles a new nonsuperconducting phase arises along with the normal and superconducting phases. In this, called"abnormal normal", or pseudogap, phase the absolute value of the order parameter is finite but its phase is a random quantity. It is important that the new phase really exists at low carrier density only, i.e. it shrinks with doping increasing in the case of phonon attraction. The relevance of the results for high-temperature superconductors is speculated. Key words: 2D metal, arbitrary carrier density, normal phase, abnormal normal phase, pseudogap, suderconducting phase, Berezinskii-Kosterlitz-Thouless phase, electron-electron and electron-hole pairing


Introduction
The problem of an adequate description of the physical properties of high-temperature superconductors (HTSCs) still remains one of the actual problems of the modern solid state physics. It is connected with some peculiar properties of HTSCs. Among them there are such as quasi-2D character of electronic (and magnetic) properties, a relatively low and changeable carrier density n f and its influence on properties of HTSCs (see, for example, review [1]).
Nowadays, one of the widely discussed questions on HTSCs is the problem of the so called pseudogap (or spin gap if magnetic subsystem of HTSCs is taken into account) [2,3,4], which is usually experimentally observed as a loss in the spectral weight of quasiparticle (or spin) excitations in normal state samples with lowered carrier density n f [5,6,7]. Corresponding underdoped samples reveal some specific spectral, magnetic and thermodynamic pecularities which still continue to be not sufficiantly understood now. Moreover, the striking difference between the low (underdoped) and high (overdoped) density regions in HTSCs is increasingly debated and is considered as one of the very central and key questions in physics of the cuprates [8,9].
The possibility of experimental changing of n f value in HTSCs puts a rather general theoretical problem of the description of the crossover from composite boson superfluidity (low n f ) to Cooper pairing (large n f ) when n f increases (in other words, a description of the transition from the so called underdoped regime to the overdoped one). Such a crossover was already studied for 3D and quasi-2D systems (see reviews [10,11]). 2D case has been considered for the present at temperature T = 0 only [10,12] what is connected with the Hohenberg-Mermin-Wagner theorem which forbids any homogeneous (long-range) order in pure 2D systems at T = 0 due to the long-wave fluctuations of the charged order parameter (OP).
The problem of the inhomogeneous condensate (Berezinskii-Kosterlitz-Thouless, or BKT, phase) formation was also considered despite of some difficulties in 2+1 relativistic field models [13] where the fermion concentration effects are irrelevant. At the same time these effects were studied in nonrelativistic model in [14], for example, without taking into account the existence of the neutral OP ρ. Its consideration proves to be very important (see [15]) and results in the formation of a separate equilibrium phase with ρ = 0 which is located on the phase diagram of a system between normal and superconducting (here -BKT) ones. Due to fluctuations of the OP phase this new state of a system is of course also non-superconducting.
In this paper an attempt is made to study the crossover as well as the above mentioned new phase formation possibility in 2D fermion system with both a more realistic indirect (phonon) and also a direct (local) four-fermion (4F-) interactions. Thus, the work is to a certain extent a specific and non-trivial generalization of the preliminary short communication [15] where this non-superconducting phase appearance was studied for 4F-case only and of the paper [16] where Fröhlich model was used for the investigation of the crossover at T = 0.
As it will be seen in the boson exchange model (in contrast to the pure 4F-case), the new phase really exists when n f is rather small what allows to relate this result to underdoped HTSC compounds. But actually it is interesting to take into account a more real situation with an indirect attraction and some kind of local repulsion which may correspond to the short-range (screened) Coulomb interaction between carriers. In general case we, however, suppose that 4F-interaction can be repulsive and attractive as well. Besides, the case of total repulsion allows to explore the fermion-antifermion (electron-hole) pairing channel which in spite physical difference can be formally described by the same manner.

Model and main equations
Let us choose the simplest Hamiltonian density in the form: where ψ σ (x) is a fermionic field with an effective mass m and spin σ =↑, ↓; µ is the chemical potential of the fermions which fixes n f ; ϕ(x) is a phonon field operator, g ph and g 4F are the electron-phonon and the 4F-interaction coupling constants, respectively. As we said above, g 4F can be positive (fermion-fermion attraction) or negative (fermion-antifermion attraction); in (1) we seth = k B = 1.
In (1) H ph is the Hamiltonian of free phonons which can be described by the propagator where Ω n = 2nπT (n is an integer) is the Matsubara frequency [17]. As it follows from (2), the propagator D(iΩ n ) has been chosen in the simplest form with ω 0 being the Einstein (dispersionless) phonon frequency. It was done because of several reasons: first, this propagator gives a possibility to integrate the equations obtained; second, it is precisely the optic phonon and quadrupolar exciton modes with their relatively weak dispersion are widely considered as exchange bosons that can contribute into the hole-hole attraction in HTSCs [1,18,19], and third, the qualitative results concerning retardation effects do not strongly depend on the model studied. But on the other hand, the propagator (2) for the model under consideration can hardly be used for quantitative description of the cuprates and also their spin-wave branches which, as it is well-known, have linear dispersion law.
It is important that the Hamiltonian (1) is invariant under symmetry transformations of two types [20], namely: and which must be taken into account. The phase α(x) in (3) and (4) is real.
With the purpose to calculate the phase diagram of the system it is necessary to find its thermodynamic potential. It can be calculated by making use of the auxiliary bilocal field method (see, for example, [21]), which is a generalization of the standard Hubbard-Stratonovich one for the boson-exchange case. Then the grand partition function Z can be expressed through a path integral over the fermionic ψ σ (x) and the complex auxiliary fields In the case of the model (1) it is convenient following the Ref. [22] to introduce the bispinor and its hermitian conjugate one which here are the analogous of the Nambu spinors [23].
After the substitution of (5) in (1) the Hamiltonian takes the form: where I ⊗ τ z is the direct product of the unit I and Pauli τ z 2 ⊗ 2 matrices; D(x) is defined by (2). In such a representation of the Hamiltonian (6) and the field variables (5) the Feinman diagram technique becomes applicable in its ordinary form [22]. Thus, after standard excluding of the boson field ϕ(x), the Lagrangian of the system can be expressed by the formula: The kernel K is the effective non-local inter-particle interaction function and will be explicitly defined in the momentum space below.
In order to explore the pairing possibility in the system let us introduce the bilocal auxiliary field, or OP, (the integration over x 2 and y 2 is assumed). Here φ ch ∼< ψ † ↑ ψ † ↓ > and φ ins ∼< ψ † ↓ ψ ↑ > are electron-electron (charged) and electron-hole (insulating) OP, respectively (we neglect non-zero spin pairing). The auxiliary fields φ ch and φ ins are responsible for the dynamical symmetry breaking (in according with (3) and (4), correspondingly).
Adding to (7) a zero term to cancel the 4F-interaction, one could obtain the Lagrangian in the form: Let us transform the expression for the kernel K; then in the momentum space it is where p i = ( p i , ω i )(i = 1, 2) and P = ( P , ω) designate the relative and the centre of mass momenta, respectively. By the definition the kernel K P (p 1 ; p 2 ) is in fact independent of P (so we omit index P henceforth) and aquires the simple form which will be used in (9). The last expression evidently demonstrates that the total character of the effective inter-particle interaction as it always takes place in such a situation [24,23] defined by the possible competition between the first (retarded) and second (retardless) terms in (10), or their common action.
The partition function can be written as: where Ω[G] is the thermodynamic potential which in the "leading order" is where Tr includes 2D spatial r and "time" 0 ≤ τ ≤ β integrations as well as the standard trace operation. The full Green function of a system is From (11) and (12) we arrive to the φ-equation (the Schwinger-Dyson one): Substituting (13) into (11) one can obtain the expression for Ω(G): The last is the well-known Cornwell-Jackiw-Tomboulis formula for the effective action in the one-loop approximation [25]. Using (13) we can rewrite this expression in the form As it was shown by Thouless et al. [26] (see also [15]) in 2D case it is natural to pass to a new parametrization of the OP (8) -its absolute value and the phase, namely: where ρ ch and ρ ins are real.
As it will be shown below, with the given kernel (10) there can arise only one (φ ch or φ ins ) OP. Therefore, it is necessary to make, simultaneously with (15), the spinor transformation (in according with (3) and (4)) (the spinor χ(x) is real and formally corresponds to chargeless fermions). It is easy to see from (15), (16) and (17) that the phase dependences of the charged and insulating OPs are similar.
Below we shall obtain θ-corrections for the φ ch case, because the final equations for φ ins will be the same up to substitution ρ ch → ρ ins . The reason is that when K(p 1 , p 2 ) describes the attraction (charge pairing channel) the symmetry of the Lagrangian under operations (3) proves to be crucial for the representation (16); while when K(p 1 , p 2 ) corresponds to the repulsion (chargeless, or insulating, pairing channel) the symmetry (4) becomes already important and the representation (17) must be used as a "working" one. With this difference the rest of the calculations are almost identical and so we shall consider in detail the charge channel which is most interesting for metallic (superconducting) systems.
In variables (16) the Green function (12) transformes to Then using (18) supposing θ gradients are small (the hydrodynamic approximation) and taking them into account up to the second order the effective potential (14) can be naturally divided it two parts: Ω = Ω kin (ρ ch , ∇θ) + Ω pot (ρ ch ) where in (∇θ) 2 approximation Assuming now that ρ ch (x, y) is homogeneous 1 after somewhat tedious but otherwise straightforward calculation one can obtain from (19): where J(µ, T, ρ ch (µ, T )) = plays the role of neutral OP stiffness. Note that in comparison with the retardation free 4Fmodel [15] the last expression contains one more term, namely: the term with the derivative.
The equation for the temperature T BKT of the BKT transition can be written down after direct comparison of the kinetic term (20) in the effective action with the Hamiltonian of 1 Equations for ρ ch and ρ ins will be obtained below and, as was shown in [16], it is an admissible approximation to put in them the value ρ ch (and ρ ins ) independent of spatial and time variables.
the 2D XY-model which has the formally identical form [27]. Hence it easy to conclude that The essential difference of this equation obtained from that for the XY-model is its spontaneous dependence on µ (or n f ) and ρ ch .
To complete the set of self-consitent equations which allow to trace an explicit dependence of T BKT on n f , the equations for ρ ch and µ have also to be given. In particular, the equation for ρ ch (iω n ) is nothing else but (13) with ∇θ = 0, i.e. the Green function G of the neutral fermions substitutes G, so that (13) in frequency-momentum represantation takes the form where ω n = (2n + 1)πT is the Matsubara fermionic frequency [23] and kernel K(ω m , ω n ) is defined above. We cited the final equations for both OPs, ρ ch and ρ ins , in order to show only that they indeed are the same but alternative if the kernel K changes the sign.
An analitical solution of these equations, as well as obtaining both the equation (22)  Making use of this approximation the number equation which follows from the ordinary condition V −1 ∂Ω[G]/∂µ = −n f (V is a volume of a system) and is crucial for crossover description has to be added to (22) and (23) for self-consistency; so one comes to where ǫ F = πn f /m is the Fermi energy of free 2D fermions with the simplest quadratic dispersion law. Thus, in the case under consideration all unknown quantities ρ ch , µ and T BKT are the explicit functions of n f .

Analysis of the solutions
Unlike the usual (with T -independed unit vector) XY-model, in the superconducting one there exist two critical temperatures: T ρ where formally the complete OP given by (8) arises but its phase is a random quantity, i.e. < φ(x, y) >= 0 2 , and another one, where the phase of the OP becomes ordered, so that < φ(x, y) > = 0. In other words, the temperature T ρ is in fact the temperature of appearance of neutral OP only which has discrete symmetry and thus is not at variance with general theorems. Recall, that according to the equations obtained above both these temperatures, what is important, directly depend on the carrier density in the system.
Critical temperature T ρ can be found from (21)-(24) by putting ρ ch = 0 (what in accordance with derivation of these equations corresponds, in fact, to the mean-field approximation). As a result, a 2D metal with temperature decreasing passes from normal phase (T > T ρ ) to another one where averaged homogeneous (charged) OP < φ(x, y) >= 0, or, what is the same, superconductivity is absent, but chargeless OP ρ ch = 0. It is evident that the pseudogap is formed just in the temperature region T BKT < T < T ρ , because, as follows from the above formulas (see, for instance, (21)-(24)), ρ ch = ρ ch (T ) enters all spectral characteristics of 2D metal in the same way as the superconducting gap ∆(T ) enters into corresponding expressions for ordinary superconductors. It justifies why this new phase can be called the "abnormal normal" phase or better pseudogap one. The density of states near ǫ F in the pseudogap is definitely less than in the normal phase, but is not to be equal zero as in superconducting one. The latter has to be checked by direct calculation of the one-particle fermion Green function what is most likely a separate problem which we do not touched upon here.
The phase diagram of the system can be found from the equations (21)- (24). There are different behaviours of T ρ (n f ) and T BKT (n f ) for various correlation between interaction constants.
This case has been partly analyzed in Ref. [15]. It corresponds to fermion-fermion pairing due to the local interaction. Note (see (23)) that in the case of resulting attraction between fermions fermion-antifermion (or electron-hole) insulating pairing channel is absent, i.e. ρ ins = 0. The phase diagram for this case is presented in fig.1. It shows that the abnormal normal phase exists at any concentration value n f and the temperature width of this phase region weakly increases with n f increasing, and BKT phase always begin to form when ρ ch (T BKT ) is finite.
At ǫ F → 0 the temperature of BKT phase formation is defined by equality T BKT = ǫ f /2, and T ρ as function of n f can be found from the equation which follows from (23) (W is the conduction band width).
The situation here is almost the same as previous one. The presence of the indirect interaction leads to the effective growth of the effective 4F-interaction constant g * 4F > g 4F (see (23)). The latter in one's turn simply results in increasing of the region between T ρ and T BKT (fig.2) and keeps the form shown on fig.1. 3) g 4F = 0, g ph = 0 (a pure indirect interaction). This is one of the most interesting cases because it corresponds to the widely accepted electron-phonon (or BCS-Bogolyubov-Eliashberg) model of superconductivity. The numerical calculations of the phase diagram is presented in fig.2. This diagram shows that comparatively large region with the abnormal normal (pseudogap) phase exists at rather low carrier concentrations only and its temperature area shrinks at n f → ∞. Such a behaviour qualitatively agrees with that one which takes place in real HTSCs samples [5,6,7,8] demonstrating that pseudogap (and spin gap also) region is observed in underdoped samples.
Indeed, it is not difficult to make certain that the assymptotics for T ρ (n f ) and T BKT (n f ) have the following forms: i) when ratio ǫ F /ω 0 ≪ 1 (very low fermion density, or local pair case) the first of them satisfies the equation T ρ ln(T ρ /ǫ F ) = ω 0 exp(−4π/g 2 ph m) which immediatedly results in ∂T ρ (n f )/∂n f | n f →0 → ∞. At the same time the temperature T BKT at n f → 0 has another carrier density dependence and as above T BKT = ǫ F /2 what simply means that here it again is equal to the number of composite bosons; in this density region T ρ /T BKT ≫ 1 (such an inequality is also correct for the pure 4F-interaction).
ii) in the case ǫ F /ω 0 ≫ 1 (very large fermion density, or Cooper pair case) one easily arrives to the standard BCS value: T ρ = (2γω 0 /π) exp(−2π/g 2 ph m) ≡ T M F BCS = (2γ/π)∆ BCS (∆ BCS is the usual one-particle BCS gap at T = 0). In other words, the temperature T ρ in this limit becomes equal to its BCS value 3 . The T BKT asymptotics for this case is not so evident and requires more detailed consideration.
First of all, it is naturally to suppose that for large n f value T BKT → T ρ . Then it is necessary to check the dependence of ρ on T as T → T ρ . For that the equation (23) can be transformed to: (where it was used that in this concentration region, the ratio µ/2T ρ ≃ ǫ F /2T ρ ≫ 1 because of µ ≃ ǫ F [10,11,12,16]).
On account of usually ω 0 /2T ρ ≫ 1, only very small x give the main contribution to the integral (25) (it is seen from the limit ρ/2T ρ → 0 when ǫ F /ω 0 → ∞). Therefore the latter expression takes the approximate form: 2π On the other hand, the condition ρ ch = 0 in (26) leads to the equation for T ρ . From (26) and (27) it directly follows that Then using the approximation one directly comes to the expression needed: Recall that the generally accepted 3D result is ∆ BCS (T ) = 3.06T M F BCS T M F BCS /T − 1 [17] and this small difference can be explained by the above approximation what, however, is suitable for the following below qualitative discussion (see next Section).
The dependence (28) has to be substituted in equation (22). And again because of µ/2T BKT ≃ ǫ F /2T BKT ≫ 1 and ρ ch (T BKT )/2T BKT ≪ 1 when T BKT → T ρ this equation can be written as At last, using expansion in ρ ch /2T BKT in integral (29), the latter can be transformed to where the numerical constant Combining now (28) and (30) one comes to the final simple relation between T ρ ch and T BKT for the large carrier density: i.e. T BKT as a function of n f really approaches T ρ (or T M F BCS ) (see fig.2). As to crossover region defined by the equality µ ≃ 0 it is easy to convince from the equations (21)- (24) and fig.2 that it corresponds to the densities when the temperatures T ρ are essentially different. It is important that because of relatively low for the phonon case value of the bound pair-state energy and so very small region of negative n f [16], the behaviour T BKT (n f ) ∼ ǫ F hardly corresponds to Bose-Einstein condensation and in fact takes place at µ > 0. 4) g 4F < 0, g ph = 0, but g 2 ph >> |g 4F |. This condition provides the total fermion-fermion attraction channel only, so ρ ch = 0 (see below). As it was said above the local 4F-repulsion qualitatively can correspond to the screened Coulomb repulsion. In this situation the cut parameter must be introduced to avoid the divergence in (23). It is interesting to consider two situations: i) the cut parameter (the boundary Coulomb frecuency ω c ) goes to infinity (i.e. small concentration or local pairing) and ii) when ω c is large but finite ω c >> ω 0 but g ph >> |g 4F |. Probably it is the situation what is intimately related to the real HTSCs (and superconductors at all).
i) The case 1) is restored in general but the effective 4F-constant g * * 4F < 0. It is important that T BKT preserves its linear assymptotics at small n f . ii) In this case g 4F in (23) can be substituted by g 4F D(iω n ) with ω 0 = ω c (the propagator D is defined by (2)). The situation is similar to the case 3) but the effective coupling constant is smaller. Such a decreasing leads to the narrowing of the abnormal normal phase region because of lowering of the temperature T ρ . In particular, it is not difficult to obtain the well known Tolmachev logarithmic correction to T ρ (see, for example, [22]): where µ * = g 4F N(0)/(1 + g 4F N(0) ln(ǫ F /ω 0 )) (N(0) is the density of states at the Fermi surface).

Conclusion
The model proposed to describe the possible two-stage superconducting phase transition in 2D (and quasi-2D) metallic systems is in fact very simplified in order to investigate their most typical and general features. All the more surprisingly that it catches some essential details which are characteristic for underdoped HTSC copper oxides. In particular, the experimental data demonstrates [28,29] that i) indeed for low n f the critical temperature T c is proportional to n f (what is simply ǫ F ), ii) T c shows saturation when n f approaches so called "optimal doping" (i.e. carrier concentration when T c as function of n f reaches its highest possible in given compound value), iii) the ratio T c /ǫ F in these and other "exotic" superconductors is as high as 10 −2 − 10 −1 what independently points out on rather small Fermi energy, etc (for details see [29]).
One would think that the pecularities mentioned receive their natural explanation on the basis of the model of metal with indirect inter-fermion interaction if the temperature T BKT is implied as critical one T c (this is justified for pure 2D systems [30]). In quasi-2D model because of the third spatial direction and the phase fluctuation stabilization the true temperature of ordinary homogeneous ordering arises [29,31] (see also [11]).
As regards the second temperature, T ρ , it usually introduced by empirically as some temperature point T * where observable spectral (or magnetic) properties of HTSC begin to deviate appreciably from their standard for normal metallic state behaviour [5,6,7,8,9]. As a rule such a deviation is connected with appearance of fluctuating (short-living) pairs. We, however, showed that some finite number of these pairs does exist and is to formed at definite temperature due to phase transition between normal and pseudogap (also normal) phases.
The only difference with supposed dependence T * on the density of doped holes consists of assymptotics at n f decreasing: we have obtained that this temperature is also reduced while usually (see, for example, [29]) T * is presumably depicted as such one that incriases with n f decreases. It seems that the latter has no satisfactory grounds. Nevertheless it must be stressed that the above limit T ρ (n f ) → 0 when n f → 0 can not be also considered as  Phase T -n f diagram of 2D metal with indirect inter-carrier attraction. Similarly to the