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arXiv:cond-mat/0305278v2 [cond-mat.supr-con] 18 Oct 2003

Superconductivity in the Pseudogap State in “Hot – Spots”

model: Ginzburg – Landau Expansion

E.Z.Kuchinskii, M.V.Sadovskii, N.A.Strigina

Institute for Electrophysics,

Russian Academy of Sciences, Ural Branch,

Ekaterinburg, 620016, Russia

E-mail: kuchinsk@iep.uran.ru, sadovski@iep.uran.ru, strigina@iep.uran.ru

Abstract

We analyze properties of superconducting state (for both s-wave and d-wave

pairing), appearing on the “background” of the pseudogap state, induced by

fluctuations of “dielectric” (AFM(SDW) or CDW) short – range order in

the model of the Fermi surface with “hot spots”. We present microscopic

derivation of Ginzburg – Landau expansion, taking into account all Feynman

diagrams of perturbation theory over electron interaction with this short –

range order fluctuations, leading to strong electronic scattering in the vicinity

of “hot spots”. We determine the dependence of superconducting critical tem-

perature on the effective width of the pseudogap and on correlation length of

short – range order fluctuations. We also find similar dependences of the main

characteristics of such superconductor close to transition temperature. It is

shown particularly, that specific heat discontinuity at the transition tempera-

ture is significantly decreased in the pseudogap region of the phase diagram.

PACS numbers: 74.20.Fg, 74.20.De

Typeset using REVTEX

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I. INTRODUCTION

Pseudogap state observed in a wide region of the phase diagram of high – temperature

superconducting cuprates is characterized by numerous anomalies of their properties, both

in normal and superconducting states [1,2]. Apparently, the most probable scenario of pseu-

dogap state formation in HTSC – oxides can be based [2] on the picture of strong scattering

of current carriers by fluctuations of short – range order of “dielectric” type (e.g. antiferro-

magnetic (AFM(SDW)) or charge density wave (CDW)) existing in this region of the phase

diagram. In momentum space this scattering takes place in the vicinity of characteristic

scattering vector Q = (π

a,π

a) (a – lattice constant), corresponding to doubling of lattice

period (e.g. vector of antiferromagnetism), being a “precursor” of spectrum transformation,

appearing after the establishment of AFM(SDW) long – range order. Correspondingly, there

appears an essentially non Fermi – liquid like renormalization of electronic spectrum in the

vicinity of the so called “hot spots” on the Fermi surface [2]. Recently there appeared a

number of experiments giving rather convincing evidence for precisely this scenario of pseu-

dogap formation [3–5]. Within this picture it is possible to formulate simplified “nearly

exactly” solvable model of the pseudogap state, describing the main properties of this state

[2], and taking into account all Feynman diagrams of perturbation theory for scattering

by (Gaussian) fluctuations of (pseudogap) short – range order with characteristic scattering

vectors from the area of Q, with the width of this area defined by the appropriate correlation

length ξ [6,7].

Up to now the majority of theoretical papers is devoted to the studies of models of

the pseudogap state in the normal phase for T > Tc. In Refs. [8–11] we have analyzed

superconductivity in the simplified model of the pseudogap state, based on the assumption of

existence of “hot” (flat) patches on the Fermi surface. Within this model we have constructed

Ginzburg – Landau expansion for different types (symmetries) of Cooper pairing [8,10] and

also studied the main properties of superconducting state for T < Tc, solving the appropriate

Gor’kov’s equations [9–11]. At first stage, we have considered greatly simplified (“toy”)

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model of Gaussian fluctuations of short – range order with an infinite correlation length,

where it is possible to obtain an exact (analytic) solution for the pseudogap state [8,9], further

analysis of more realistic case of finite correlation lengths was performed both for “nearly

exactly” solvable model of Ref. [10] (assuming the self – averaging nature of superconducting

order parameter over pseudogap fluctuations) and for a simplified exactly solvable model

[11], where we have been able to analyze the effects due to the absence of self – averaging

[9,11,12].

The aim of the present paper is to analyze the main properties of superconducting

state (for different types of pairing), appearing on the “background” of the pseudogap of

“dielectric” nature in more realistic model of “hot spots” on the Fermi surface. Here we

shall limit ourselves to the region close to superconducting temperature Tcand perform an

analysis based on microscopic derivation of Ginzburg – Landau expansion, assuming the self

– averaging nature of superconducting order parameter and generalizing similar approach

used earlier for “hot patches” model in Ref. [10].

II. “HOT – SPOTS” MODEL AND PAIRING INTERACTION.

In the model of “nearly antiferromagnetic” Fermi – liquid, which is actively used to

describe the microscopic nature of high – temperature superconductivity [13,14], it is usually

assumed that the effective interaction of electrons with spin fluctuations of antiferromagnetic

(AFM(SDW)) short – range order is of the following form:

Veff(q,ω) =

g2ξ2

1 + ξ2(q − Q)2− iω

ωsf

(1)

where g – is some interaction constant, ξ – correlation length of spin fluctuations, Q =

(π/a,π/a) – vector of antiferromagnetic ordering in dielectric phase, ωsf – characteristic

frequency of spin fluctuations. Both dynamic spin susceptibility and effective interaction

(1) are peaked in the region of q ∼ Q), which leads to the appearance of “two types” of

quasiparticles – “hot” one, with momenta in the vicinity of “hot spots” on the Fermi surface

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(Fig. 1), and “cold” one, with momenta close to the parts of the Fermi surface, surrounding

diagonals of the Brillouin zone [6]. This is due to the fact, that quasiparticles from the

vicinity of the “hot spots” are strongly scattered with the momentum transfer of the order

of Q, due to interaction with spin fluctuations (1), while for quasiparticles with momenta

far from these “hot spots” this interaction is relatively weak.

For high enough temperatures πT ≫ ωsf, we can neglect spin dynamics [6], limiting

ourselves to static approximation in (1). Considerable simplification, allowing to analyze

higher – order contributions, can be achieved by substitution of (1) by model – like static

interaction of the following form [7]:

Veff(q) = W2

2ξ−1

ξ−2+ (qx− Qx)2

2ξ−1

ξ−2+ (qy− Qy)2

(2)

where W is an effective parameter with dimension of energy. In the following, as in Refs.

[6,7], we consider parameters W as ξ phenomenological (to be determined from the exper-

iment). Anyhow, Eq. (2) is qualitatively quite similar to the static limit of (1) and almost

indistinguishable from it in most interesting region of |q − Q| < ξ−1, determining scattering

in the vicinity of “hot spots”.

The spectrum of “bare” (free) quasiparticles can be taken as [6]:

ξp= −2t(cospxa + cospya) − 4t

′cospxacospya − µ(3)

where t is the transfer integral between nearest neighbors, while t′is the transfer integral

between second nearest neighbors on the square lattice, a is the lattice constant, µ – chemical

potential. This expression gives rather good approximation to the results of band structure

calculations of real HTSC – system, e.g. for Y Ba2Cu3O6+δwe have t = 0.25eV , t′= −0.45t

[6]. Chemical potential µ is fixed by concentration of carriers.

The least justified is an assumption of the static nature of fluctuations, which can be

valid only for rather high temperatures [6,7]. For low temperatures, particularly in su-

perconducting phase, spin dynamics can become quite important, e.g. for microscopics of

Cooper pairing in the model of “nearly antiferromagnetic” Fermi liquid [13,14]. However,

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we assume here that the static approximation is sufficient for qualitative understanding of

the influence of pseudogap formation upon superconductivity, which will be modelled within

phenomenological BCS – like approach.

In the limit of infinite correlation length ξ → ∞ this model acquires an exact solution [15].

For finite ξ we can construct “nearly exact” solution [7], generalizing the one – dimensional

approach, proposed in Ref. [16]. Then we can (approximately) sum the whole diagrammatic

series for the one – particle electronic Green’s function.

For the contribution of an arbitrary diagram for electronic self – energy, in the N-th

order over the interaction (2), we write down the following Ansatz [7,16]:

Σ(N)(εnp) = W2N

2N−1

?

j=1

G0kj(εnp),

G0kj(εnp) =

1

iεn− ξkj(p) + ikjvkjκ

(4)

where κ = ξ−1, kj – is the number of interaction lines, surrounding the j-th (from the

beginning) electronic line in a given diagram, εn= 2πT(n + 1/2) (assuming εn> 0).

ξk(p) =

ξp+Q for odd k

ξp

for even k

(5)

vk=

|vx(p + Q)| + |vy(p + Q)| for odd k

|vx(p)| + |vy(p)|for even k

(6)

where v(p) =

∂ξp

∂p– velocity of a “bare” (free) quasiparticle.

In this approximation the contribution of an arbitrary diagram is determined, in fact, by

the set of integers kj. Any diagram with intersection of interaction lines is actually equal to

some diagram of the same order without intersections and the contribution of all diagrams

with intersections can be accounted with the help of combinatorial factors s(kj) attributed

to interaction lines on diagrams without intersections [16,7,6].

Combinatorial factor:

s(k) = k (7)

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in the case of commensurate fluctuations with Q = (π/a,π/a) [16], if we neglect their spin

structure [6]) (i.e. limiting ourselves to CDW – type fluctuations). Taking into account the

spin structure of interaction within the model of “nearly antiferromagnetic” Fermi – liquid

(spin – fermion model of Ref. [6]), we obtain more complicated combinatorics of diagrams. In

particular, spin – conserving scattering leads to the formally commensurate combinatorics,

while the spin – flip scattering is described by diagrams of incommensurate case (“charged”

random field in terms of Ref. [6]). As a result the Ansatz (4) for one – particle Green’s

function is conserved, but the combinatorial factor s(k) takes the form [6]:

s(k) =

k+2

3

for odd k

k

3

for even k

(8)

However, in the case of two – particle processes and in the analysis of the pseudogap influence

on superconductivity, the use of spin – fermion model leads to significant complications. In

particular, in this model there is a significant difference between vertexes, corresponding to

spin dependent and charge interactions, and combinatorics of diagrams for spin dependent

vertex is different from that of (8) [6].

In spin – fermion model [6] the spin dependent part of interaction is usually described by

isotropic Heisenberg model. If for this interaction we assume the Ising like form, only spin –

conserving processes remain and commensurate combinatorics of diagrams (7) remains valid

both for one – particle Green’s function and for spin and charge vertexes. For this reason,

in the present work we shall limit ourselves to the analysis of commensurate (7) “Ising

like” spin – fluctuations only (AFM, SDW), as well as commensurate charge fluctuations

(CDW). Details concerning incommensurate fluctuations of CDW – type can be found in

Refs. [7,15,16].

As a result we obtain the following recurrence procedure for the one – particle Green’s

function G(εnp) (continuous fraction representation) [16,7,6]:

Gk(εnp) =

1

iεn− ξk(p) + ikvkκ − W2s(k + 1)Gk+1(εnp)

(9)

and “physical” Green’s function is determined as G(εnp) ≡ G0(εnp).

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Ansatz (4) for the contribution of an arbitrary diagram of N-th order is not exact in

general case [7]. However, for two – dimensional system we were able to show that for

certain topologies of the Fermi surface (4) is actually an exact representation [7], in other

cases it can be shown [7], that this approximation overestimates (in some sense) the effects

of the finite values of correlation length ξ for the given order of perturbation theory. For

one – dimensional case, when this problem is most serious [7], the values of the density of

states found with the help of (4) in the case of incommensurate fluctuations are in almost

ideal quantitative correspondence [17] with the results of an exact numerical modelling of

this problem in Refs. [18,19]. In the limit of ξ → ∞ Ansatz (4) reduces to an exact solution

of Ref. [15], while in the limit of ξ → 0 and fixed value of W it reduces to the trivial case of

free electrons.

This model describes non Fermi – liquid like spectral density in the vicinity of “hot

spots” on the Fermi surface and “smooth” pseudogap in the density of states [6,7].

To analyze superconductivity in such a system with well developed fluctuations of short

– range order we shall assume that superconducting pairing is determined by attractive BCS

– like interaction (between electrons with opposite spins) of the following simplest possible

form:

Vsc(p,p′) = −V e(p)e(p′), (10)

where e(p) is given by:

e(p) =

1( s-wave pairing)

cos(pxa) − cos(pya) ( dx2−y2-wave pairing)

sin(pxa)sin(pya)( dxy-wave pairing)

cos(pxa) + cos(pya) ( anisotropic s-wave pairing)

.(11)

Interaction constant V is assumed, as usual, to be non zero in some interval of the width of

2ωcaround the Fermi level (ωc– characteristic frequency of quanta responsible for attractive

interaction of electrons). Then in general case the superconducting gap is anisotropic and

given by: ∆(p) = ∆e(p).

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In Refs. [8,9] we have analyzed the peculiarities of superconducting state in an exactly

solvable model of the pseudogap state, induced by short – range order fluctuations with

infinite correlation length (ξ → ∞). In particular, in Ref. [9] it was shown that these

fluctuations may lead to strong fluctuations of superconducting order parameter (energy

gap ∆), which break the standard assumption of the self – averaging nature of the gap

[20–22], which allows to perform independent averaging (over the random configurations of

static fluctuations of short – range order) of superconducting order parameter ∆ and different

combinations of electronic Green’s functions, entering the main equations. Usual argument

for the possibility of such an independent averaging goes as follows [20,22]: the value of ∆

significantly changes on the length scale of the order of coherence length ξ0 ∼ vF/∆0 of

BCS theory, while the Green’s functions oscillate on much shorter length scales of the order

of interatomic spacing. Naturally, this assumption becomes invalid with the appearance

(in electronic system) of a new characteristic length scale ξ → ∞. However, when this

correlation length of short – range order ξ ≪ ξ0(i.e. when fluctuations are correlated on the

length scale shorter than the characteristic size of Cooper pairs), the assumption of self –

averaging of ∆ is apparently still valid, being broken only in case of ξ > ξ0

1. Thus, below we

perform all the analysis under the usual assumption of self – averaging energy gap over the

fluctuations of short – range order, which allows us to use standard methods of the theory

of disordered superconductors (mean field approximation in terms of Ref. [9]).

III. COOPER INSTABILITY. RECURRENCE PROCEDURE FOR THE VERTEX

PART.

It is well known that critical temperature of superconducting transition can be deter-

mined from the equation for Cooper instability of normal phase:

1The absence of self – averaging of superconducting gap even for the case of ξ < ξ0, obtained in

Ref. [11], is apparently due to rather special model of short – range order used in this paper.

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1 − V χ(0;T) = 0(12)

where the generalized Cooper susceptibility is determined by diagram shown in Fig. 2 and

equal to:

χ(q;T) = −T

?

εn

?

p,p′e(p)e(p′)Φp,p′(εn,−εn,q) (13)

where Φp,p′(εn,−εn,q) is two – particle Green’s function in Cooper channel, taking into

account scattering by fluctuations of short – range order.

Let us consider first the case of charge (CDW) fluctuations, when electron – fluctuation

interaction is spin independent. In case of isotropic s and dxy-wave pairing superconducting

gap does not change after scattering by Q, i.e. e(p + Q) = e(p) and e(p′) ≈ e(p). In case

of anisotropic s and dx2−y2 pairings superconducting gap changes sign after scattering by Q,

i.e. e(p + Q) = −e(p), thus e(p′) ≈ e(p) for p′≈ p and e(p′) ≈ −e(p) for p′≈ p + Q.

Thus, for diagrams with even number of interaction lines, connecting the upper (εn) and

lower (−εn) electronic lines in Fig. 2, we have p′≈ p and the appropriate contribution to

susceptibility is the same for both isotropic s and dxy-wave pairing. However, for diagrams

with odd number of such interaction lines we obtain contributions differing by sign. This

sign change can be accounted by changing the sign of interaction line, connecting the upper

and lower lines in the loop in Fig. 2. Then for the generalized susceptibility we obtain:

χ(q;T) = −T

?

εn

?

p

G(εnp + q)G(−εn,−p)e2(p)Γ±(εn,−εn,q) (14)

where Γ±(εn,−εn,q) is “triangular” vertex, describing electron interaction with fluctuations

of short – range order, while the superscript ± denotes the abovementioned change of signs

for interaction lines connecting the upper and lower electronic lines.

Consider now the case of scattering by spin (AFM(SDW)) fluctuations. In this case

interaction line, describing the longitudinal Szpart of interaction, surrounding the vertex

changing spin direction, should be attributed an extra factor of (−1) [6]. Because of this

factor, in case of electron interaction with spin fluctuations contributions to generalized

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susceptibility for different types of pairing considered above just “change places”

2and

susceptibility for the case of isotropic s and dxy-wave pairing is determined by “triangular”

vertex Γ−, while for anisotropic s and dx2−y2-wave case by “triangular” vertex Γ+.

Thus we come to the problem of calculation of “triangular” vertex parts, describing inter-

action with “dielectric” (pseudogap) fluctuations. For one – dimensional case and a similar

problem (for real frequencies, T = 0) the appropriate recursion procedure was formulated,

for the first time, in Ref. [23]. For a two – dimensional model of the pseudogap with “hot

spots” on the Fermi surface the generalization of this procedure was performed in Ref. [24]

and used to calculate optical conductivity. All the details of appropriate derivation can also

be found there. The generalization to the case of Matsubara frequencies, of interest to us

here, can be made directly. Below, for definiteness we assume εn> 0. Finally, for “trian-

gular” vertex we obtain the recurrence procedure, described by diagrams of Fig. 3 (where

the wavy line denotes interaction with pseudogap fluctuations), and having the following

analytic form:

Γ±

k−1(εn,−εn,q) =

?

= 1 ± W2s(k)Gk¯Gk

?

1 +

2ikvkκ

2iεn− vkq − W2s(k + 1)(Gk+1−¯Gk+1)

Γ±

k(εn,−εn,q)(15)

where Gk = Gk(εnp + q) and¯Gk = Gk(−εn,−p) are calculated according to (9), vk is

defined by (6), and vkare given by:

vk=

v(p + Q) for odd k

v(p) for even k

(16)

“Physical” vertex is defined as Γ±(εn,−εn,q) ≡ Γ±

0(εn,−εn,q).

To determine Tcwe need vertices at q = 0. Then¯Gk= G∗

kand vertex parts Γ+

kand Γ−

k

become real significantly simplifying our procedures (15). For ImGkand ReGkwe obtain

the following system of recurrence equations:

2This is due to the fact of spin projections change in the vertex, describing “interaction” with

superconducting gap (restricting to the case of singlet pairing).

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ImGk= −εn+ kvkκ − W2s(k + 1)ImGk+1

Dk

ReGk= −ξk(p) + W2s(k + 1)ReGk+1

Dk

(17)

where Dk= (ξk(p)+W2s(k +1)ReGk+1)2+(εn+kvkκ−W2s(k +1)ImGk+1)2, and vertex

parts at q = 0 are determined by:

Γ±

k−1= 1 ∓ W2s(k)

ImGk

εn− W2s(k + 1)ImGk+1Γ±

k

(18)

Going to numerical calculations we have to define characteristic energy (temperature)

scale associated with superconductivity in the absence of pseudogap fluctuations (W = 0).

In this case the equation for superconducting critical temperature Tc0 becomes standard

BCS – like (for the general case of anisotropic pairing):

1 =2V T

π2

¯ m

?

n=0

?π

0

dpx

?π

0

dpy

e2(p)

ξ2

p+ ε2

n

(19)

where ¯ m = [

ωc

2πTc0] is the dimensionless cutoff for the sum over Matsubara frequencies. All

calculations were performed for typical quasiparticle spectrum in HTSC given by (3) with

µ = −1.3t and t′/t = −0.4.Choosing, rather arbitrarily, the value of ωc = 0.4t and

Tc0= 0.01t we can easily determine the appropriate values of pairing constant V in (19),

giving this value of Tc0for different types of pairing listed in (11). In particular, for the usual

isotropic s-wave pairing we obtain

V

ta2 = 1, while for dx2−y2 pairing we get

V

ta2 = 0.55. For

other types of pairing from (11) the values of pairing constant for this choice of parameters

are found to be unrealistically large and we shall not quote the the results of numerical

calculations for these symmetries3.

3Of course, our description, based on BCS equations of weak coupling theory, is not pretending

to be realistic also for the cases of s-wave and dx2−y2-wave pairing.We only need to define

characteristic scale of Tc0and express all temperatures below in units of this temperature, assuming

certain universality of all dependences with respect to this scale.

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Typical results of numerical calculations of superconducting transition temperature Tc

for the system with pseudogap obtained using our recursion relations directly from (12) are

shown in Figs. 4,5. We can see that in all cases pseudogap (“dielectric”) fluctuations lead

to significant reduction of superconducting transition temperature. This reduction stronger

in case of dx2−y2-wave pairing than for isotropic s-wave case. At the same time, reduction

of the correlation length ξ (growth of κ) of pseudogap fluctuations leads to the growth of

Tc. These results are similar to those obtained earlier in the “hot patches” model [8,10].

However, significant qualitative differences also appear. From Fig. 4 it is seen, that for

the case of isotropic s-wave pairing and scattering by charge (CDW) fluctuations, as well

as for dx2−y2 pairing and scattering by spin (AFM(SDW)) fluctuations4(i.e. in cases when

the upper sign is “operational” in Eqs. (15) and (18), leading to recursion procedure for

the vertex with same signs) there appears characteristic plateau in the dependence of Tc

on the width of the pseudogap W in the region of W < 10Tc0, while significant suppres-

sion of Tctakes place on the scale of W ∼ 50Tc0. Qualitative differences appear also for

the case of s-wave pairing and scattering by spin (AFM(CDW)) fluctuations, as well as for

the case of dx2−y2 pairing and scattering by charge fluctuations. From Fig. 5 we can see

that in this case (when lower sign is “operational” in Eqs. (15) and (18), i.e. when we

have recursion procedure for the vertex with alternating signs) reduction of Tcis an order

of magnitude faster. In the case of dx2−y2pairing, for the values of W/Tc0corresponding

to almost complete suppression of superconductivity, our numerical procedures become un-

reliable. In particular, we can observe here characteristic non single valued dependence of

Tcon W, which may signify the existence of a narrow region of phase diagram with “reen-

trant” superconductivity5. This behavior of Tcresembles similar dependences, appearing in

4This last case is apparently realized in copper oxides.

5Our calculations show that manifestations of such behavior of Tcbecome stronger for the case

of scattering by incommensurate pseudogap fluctuations.

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superconductors with Kondo impurities [25]. Alternative possibility is the appearance here

of the region, where superconducting transition becomes first order, similarly to situation in

superconductors with strong paramagnetic effect in external magnetic field [26]. However,

it should be stressed that our calculations mostly show the probable appearance of some

critical value of W/Tc0, corresponding to the complete suppression of superconductivity. In

any case, the observed behavior requires special studies in the future and below we only give

the results, corresponding to the region of single – valued dependences of Tc.

IV. GINZBURG – LANDAU EXPANSION.

In Ref. [8] Ginzburg – Landau expansion was derived for an exactly solvable model of

the pseudogap with infinite correlation length of short – range order fluctuations. In Ref.

[10] these results were extended for the case of finite correlation lengths. In these papers the

analysis was, in fact, done only for the case of charge fluctuations and simplified model of

pseudogap state, based on the picture of “hot” (flat) parts (patches) on the Fermi surface.

Also in this model the signs of superconducting gap after the transfer by vector Q were

assumed to be the same, both for s-wave and d-wave pairings [10]. Here we shall make

appropriate generalization for the present more realistic model of “hot spots” on the Fermi

surface.

Ginzburg – Landau expansion for the difference of free energy density of superconducting

and normal states can be written in the usual form:

Fs− Fn= A|∆q|2+ q2C|∆q|2+B

2|∆q|4,(20)

where ∆qis the amplitude of the Fourier component of the order parameter, which can be

written for different types of pairing as: ∆(p,q) = ∆qe(p). In fact (refGiLa) is determined

by diagrams of loop expansion of free energy in the field of random fluctuation of the order

parameter (denoted by dashed lines) with small wave vector q [8], shown in Fig. 6.

Coefficients of Ginzburg – Landau expansion can be conveniently expressed as:

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A = A0KA;C = C0KC;B = B0KB, (21)

where A0, C0and B0denote expressions of these coefficients (derived in the Appendix) in

the absence of pseudogap fluctuations (W = 0) for the case of an arbitrary quasiparticle

spectrum ξpand different types of pairing:

A0= N0(0)T − Tc

Tc

< e2(p) >;C0= N0(0)7ζ(3)

32π2T2

c

< |v(p)|2e2(p) >;

B0= N0(0)7ζ(3)

8π2T2

c

< e4(p) >,(22)

where angular brackets denote the usual averaging over the Fermi surface: < ... >=

1

N0(0)

?

pδ(ξp)..., and N0(0) is the density of states at the Fermi level for free electrons.

Now all peculiarities of the model under consideration, due to the appearance of the

pseudogap, are contained within dimensionless coefficients KA, KCand KB. In the absence

of pseudogap fluctuations all these coefficients are equal to 1.

Coefficients KAand KC, according to Fig. 6(a) are completely determined by the gen-

eralized Cooper susceptibility [8,10] χ(q;T), shown in Fig. 2:

KA=χ(0;T) − χ(0;Tc)

A0

(23)

KC= lim

q→0

χ(q;Tc) − χ(0;Tc)

q2C0

(24)

Generalized susceptibility, as was shown above, can be found from (14), where “triangular”

vertices are determined by recurrence procedures (15), allowing to perform direct numerical

calculations of the coefficients KAand KC.

Situation with coefficient B is, in general case, more complicated. Significant simplifi-

cations arise if we limit ourselves (in the order |∆q|4), as usual, to the case of q = 0, and

define the coefficient B by diagram shown in Fig. 6(b). Then for coefficient KBwe get:

KB=Tc

B0

?

εn

?

p

e4(p)(G(εnp)G(−εn,−p))2(Γ±(εn,−εn,0))4

(25)

It should be noted that Eq. (25) immediately leads to positively defined coefficient B. This

is clear from G(−εn,−p) = G∗(εnp), so that G(εnp)G(−εn,−p) is real and accordingly

Γ±(εn,−εn,0), defined by recurrence procedure (18), is also real.

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V. PHYSICAL PROPERTIES OF SUPERCONDUCTORS IN THE PSEUDOGAP

STATE.

It is well known that Ginzburg – Landau equations define two characteristic lengths —

coherence length and penetration depth. Coherence length at a given temperature ξ(T)

determines characteristic length scale of inhomogeneities of the order parameter ∆:

ξ2(T) = −C

A.

(26)

In the absence of the pseudogap:

ξ2

BCS(T) = −C0

A0

(27)

Thus, in our model:

ξ2(T)

ξ2

BCS(T)=KC

KA. (28)

For the penetration depth of magnetic field we have:

λ2(T) = −

c2

32πe2

B

AC

(29)

Then, analogously to (28), we obtain:

λ(T)

λBCS(T)=

?

KB

KAKC

?1/2

. (30)

Close to Tcthe upper critical field Hc2is defined via Ginzburg – Landau coefficients as:

Hc2=

φ0

2πξ2(T)= −φ0

2π

A

C,

(31)

where φ0= cπ/|e| is the magnetic flux quantum. Then the derivative (slope) of the upper

critical field close to Tcis given by:

?????

dHc2

dT

?????Tc

=

16πφ0< e2(p) >

7ζ(3) < |v(p)|2e2(p) >TcKA

KC. (32)

Specific heat discontinuity at the transition point is defined as:

15

Page 16

(Cs− Cn)Tc=Tc

B

?

A

T − Tc

?2

, (33)

where Csand Cnare specific heats of superconducting and normal states. At temperature

Tc0(in the absence of the pseudogap, W = 0):

(Cs− Cn)Tc0= N(0)8π2Tc0< e2(p) >2

7ζ(3) < e4(p) >.

(34)

Then the relative specific heat discontinuity in our model can be written as:

∆C ≡(Cs− Cn)Tc

(Cs− Cn)Tc0

=Tc

Tc0

K2

KB.

A

(35)

Numerical calculations of KA, KB, KCwere performed for the same typical parameters of

the model as calculation of Tc described above. The values of these coefficients, are not

of great interest and we drop these data6presenting in Figs. 7–12 appropriate W/Tc0 –

dependences of physical characteristics, defined by Eqs. (26) – (35). In accord with typical

behavior of Tcdescribed above, here we can also observe two qualitatively different types of

behavior cue to either constant or alternating signs in recurrence equations for the vertex

part (upper or lower signs in (15), charge or spin fluctuations). Results of our calculations of

physical characteristics for the first type of behavior (s-wave pairing and scattering by charge

(CDW) fluctuations, dx2−y2-wave pairing and scattering by spin (AFM(SDW)) fluctuations)

are shown in Figs. 7 – 10.We can see that with the growth of pseudogap width W

coherence length ξ(T) is reduced, while penetration depth λ(T) grows, as compared to the

appropriate values of BCS theory. Dependence of both characteristic lengths on parameter

κa is rather weak and in Figs. 7, 8 we present data for only one value of κa = 0.2. The slope

(derivative) of the upper critical field at T = Tc initially grows, then starts to diminish.

Most characteristic is the suppression of specific heat discontinuity (as compared to BCS

value) shown in Fig. 10, which is in direct qualitative accord with experimental data [27].

6Typical dependences of these coefficients on the parameter W/Tc0are functions rather rapidly

decreasing from 1 in the region of existence of superconducting state.

16

Page 17

Note that for specific heat discontinuity we also have noticeable plateau in the region of

W/Tc0< 10, analogous to that observed above in similar dependence of Tc.

Behavior of physical characteristics of superconductor for the case of s-wave pairing and

scattering by spin (AFM(CDW)) fluctuations as well as for dx2−y2 pairing and scattering by

charge (CDW) fluctuations are shown in Figs. 11, 12. Data on characteristic lengths are

not given, as both coherence length ξ(T), and penetration depth λ(T) are practically the

same as in BCS theory in almost all region of existence of superconducting state (except

very narrow region where Tccan be non single valued and disappear, where both lengths are

rapidly growing). As to the slope of an upper critical field and specific heat discontinuity

at superconducting transition, both are rapidly suppressed with the growth of W/Tc0, up

to the region of non single valued Tc(the region of possible “reentrant” behavior or type I

transition) (cf. Fig. 11, 12), where both are characterized by non monotonous behavior.

VI. CONCLUSION

In this paper we have analyzed peculiarities of superconducting state forming on the

“background” of the pseudogap induced by electron scattering by fluctuations of “dielectric”

short – range order in the model of “hot spots” on the Fermi surface. Our analysis is based on

microscopic derivation of Ginzburg – Landau expansion and takes into account all high order

contributions of perturbation theory for scattering by pseudogap fluctuations. “Condensed”

phase of such superconductor should be described by appropriate Gor’kov’s equations for a

superconductor with the pseudogap [10], which will be the task for the future work.

Our main conclusion is that superconductivity is suppressed by pseudogap fluctuations of

CDW, AFM(SDW) type with two qualitatively different model of such suppression, related

to either constant or alternating signs in recurrence equations for the vertex part (upper

or lower signs in (15, charge or spin fluctuations). While the case of scattering by spin

fluctuations and pairing with dx2−y2 – symmetry is apparently realized in high – temper-

ature superconducting copper oxides, we are not aware of systems with unusual behavior

17

Page 18

obtained above for the case of s-wave pairing and scattering by spin (AFM(CDW)) fluctua-

tions, as well as for the case of dx2−y2 pairing and scattering by charge (CDW) fluctuations.

Experimental search of systems with this behavior may be of significant interest.

The most important problem in the description of the pseudogap state in real HTSC –

cuprates is an explanation of doping dependences of the main physical characteristics. In

our model, these dependences should be expressed via corresponding doping dependences of

the effective pseudogap width W and correlation length ξ. Unfortunately, these dependences

are determined from rather indirect experimental data and are poorly defined [1,2]7. As a

very crude statement, we can claim that correlation length ξ does not change significantly

in rather wide region of doping concentrations, while the pseudogap width W drops linearly

with concentration growth from the values of the order of 103K close to the insulating

phase, to the values of the order of Tcat optimal doping, becoming zero at slightly higher

concentrations of current carriers (Cf. Fig.6 in the review paper [2], which is based of Fig.4

of Ref. [3], with some compilation of appropriate data for Y BCO). Using such doping

dependence we can immediately reexpress W – dependences given above as dependences on

concentration of doping impurity. In the oversimplfied version of our model with an infinite

correlation length and Fermi surface with complete nesting, assuming also linear doping

dependence for Tc0, such calculations were given in a recent paper [28]. It was shown that

even within this simplest approach the qualitative form of the phase diagram of cuprates can

be nicely reproduced. However, the obvious crudeness of the model, as well as the absence

of reliable data on concentration dependences of W, ξ and Tc0, do not allow any serious

“improvement” of these qualitative conclusions.

Among shortcomings of the present model, besides many times mentioned neglect of

dynamics of short – range order fluctuations and limitation to Gaussian statistics of these

7Concentartion dependence of Tc0 may also be quite important, but in fact it is completely

unknown.

18

Page 19

fluctuations, we also note the simplified analysis of spin structure of interaction, which

assumed its Ising like nature. It will be of great interest to perform similar analysis for the

general case of Heisenberg interaction.

This work is partly supported by RFBR grant 02-02-16031, as well as by the program

of fundamental research of Presidium of the RAS “Quantum macrophysics” and program of

the Division of Physical Sciences of the RAS “Strongly correlated electrons in semiconduc-

tors, metals, superconductors and magnetic materials”, and by the research project of the

Russian Ministry of Science and Industries “The studies of collective and quantum effects

in condensed matter”.

19

Page 20

APPENDIX A: GINZBURG – LANDAU COEFFICIENTS FOR ANISOTROPIC

SUPERCONDUCTOR IN THE ABSENCE OF THE PSEUDOGAP.

In the absence of pseudogap fluctuations (W = 0) the generalized Cooper susceptibility

define by diagram of Fig. 2 takes the following form:

χ0(q;T) = −T

?

εn

?

p

e2(p)

1

iεn− ξp+q

1

−iεn− ξp

(A1)

Then for susceptibility at q = 0, defining the coefficient A0, we get:

χ0(0;T) = −T

?

εn

?

p

e2(p)

1

ε2

n+ ξ2

p

= −T

?

?

εn

pδ(ξp)e2(p)

N0(0)

?∞

−∞dξ

1

ε2

n+ ξ2

?

p

δ(ξ − ξp)e2(p) ≈

≈ −N0(0)T

?

εn

?∞

−∞dξ

1

ε2

n+ ξ2

= χBCS(0;T) < e2(p) >(A2)

where angular brackets denote averaging over the Fermi surface and we introduced the

standard susceptibility of BCS model for isotropic s-wave pairing χBCS(0;T).

As a result, we obtain coefficient A0as:

A0= χ0(0;T) − χ0(0;Tc) = ABCS< e2(p) > (A3)

where

ABCS= χBCS(0;T) − χBCS(0;Tc) = N0(0)T − Tc

Tc

(A4)

is the standard expression for coefficient A for the case of isotropic s-wave pairing.

Coefficient C0of Ginzburg – Landau expansion is defined by susceptibility (A1) at small

q:

C0= lim

q→0

χ0(q;Tc) − χ0(0;Tc)

q2

(A5)

Expanding expression (A1) for χ0(q;Tc) in powers of q we get:

χ0(q;Tc) = χ0(0;Tc) + Tc

?

εn

?

p

3ε2

4(ε2

n− ξ2

n+ ξ2

p

p)3e2(p)(v(p)q)2

(A6)

so that coefficient C0is found to be:

20

Page 21

C0= Tc

?

εn

?∞

−∞dξ

3ε2

4(ε2

n− ξ2

n+ ξ2)3

3ε2

4(ε2

?

p

δ(ξ − ξp)e2(p)|v(p)|2cos2(φ) ≈

≈ Tc

?

εn

?∞

−∞dξ

n− ξ2

n+ ξ2)3

= N0(0)7ζ(3)

?

p

δ(ξp)e2(p)|v(p)|2cos2(φ) =

16π2T2

c

< e2(p)|v(p)|2cos2(φ) >(A7)

where φ is an angle between vectors v(p) and q, ζ(3) =?∞

For square lattice both the Fermi surface and |v(p)| are symmetric with respect to

n=1

1

n3≈ 1,202

rotation over an angle

π

2, the same symmetry is valid for e(p) (for all types of pairing,

mentioned above). Thus we easily find:

< e2(p)|v(p)|2cos2(φ) >=1

2< e2(p)|v(p)|2(1 + cos(2φ)) >=1

2< e2(p)|v(p)|2>(A8)

because cos(2φ) changes sign after the rotation of p by an angleπ

2. In fact, the direction of

the velocity v(p) after this rotation changes to a perpendicular one, accordingly cos(2φ) →

−cos(2φ). As a result we obtain for the coefficient C0an isotropic expression:

C0= N0(0)7ζ(3)

32π2T2

c

< |v(p)|2e2(p) >,(A9)

which for the case of isotropic s-wave pairing and spherical Fermi surface becomes the

standard one:

CBCS= N0(0)7ζ(3)v2

F

32π2T2

c

(A10)

Coefficient B defined by the diagram shown in Fig. 6(b), in the absence of pseudogap

fluctuations (W = 0) and for q = 0 takes the following form:

B0= Tc

?

εn

?

p

1

(ε2

n+ ξ2

p)2e4(p) = Tc

?

1

n+ ξ2)2

εn

?∞

−∞dξ

1

(ε2

n+ ξ2)2

pδ(ξp)e4(p)

N0(0)

?

= BBCS< e4(p) >

p

δ(ξ − ξp)e4(p) ≈

≈ N0(0)Tc

?

εn

?∞

−∞dξ

(ε2

?

(A11)

where

BBCS= N0(0)7ζ(3)

8π2T2

c

(A12)

is the standard expression for B in case of isotropic s-wave pairing.

21

Page 22

FIGURES

FIG. 1. Fermi surface with “hot spots” connected by scattering vector of the order of

Q = (π

a,π

a).

22

Page 23

FIG. 2. Diagaram for the generalized susceptibility χ(q) in Cooper channel.

23

Page 24

FIG. 3. Recurrence equations for the vertex part.

24

Page 25

? ?? ????

???

???

???

???

???

???

??

?????

??????

??

??

??

FIG. 4. Dependence of superconducting transition temperature Tc/Tc0on the effective width

of the pseudogap W/Tc0for s-wave pairing and scattering by charge (CDW) fluctuations (curves

s1 and s2) and for dx2−y2-wave pairing and scattering by spin (AFM(SDW)) fluctuations (curves

d1 and d2). Data are given for the values of inverse correlation length κa = 0.2 (s1 and d1) and

κa = 0.5 (s2 and d2).

25

Page 26

?????????

???

???

???

???

???

???

??

?????

??????

??

??

??

FIG. 5. Dependence of superconducting transition temperature Tc/Tc0on the effective width of

the pseudogap W/Tc0for s-wave pairing and scattering by spin (AFM(SDW)) fluctuations (curves

s1 and s2) and for dx2−y2-wave pairing and scattering by charge (CDW) fluctuations (curves d1

and d2). Data are given for the values of inverse correlation length κa = 0.2 (s1 and d1) and

κa = 1.0 (s2 and d2).

26

Page 27

FIG. 6. Diagrammatic representation of Ginzburg – Landau expansion.

27

Page 28

???????

???

???

???

???

???

???

?????

??????

???

?

?

FIG. 7. Dependence of the square of the coherence length ξ2/ξ2

BCSon the effective width of the

pseudogap W/Tc0for s-wave pairing and scattering by charge (CDW) fluctuations (full curve) and

for dx2−y2-wave pairing and scattering by spin (AFM(SDW)) fluctuations (dotted curve). Data are

given for the value of inverse correlation length κa = 0.2.

28

Page 29

???????

???

???

???

???

???

???

???

???

???

?

?????

λ?λ???

?

FIG. 8. Dependence of penetration depth λ/λBCS on the effective width of the pseudogap

W/Tc0 for s-wave pairing and scattering by charge (CDW) fluctuations (full curve) and for

dx2−y2-wave pairing and scattering by spin (AFM(SDW)) fluctuations (dashed curve). Data are

given for the value of inverse correlation length κa = 0.2.

29

Page 30

???????

???

???

???

???

???

??

?????

?????????

??

??

??

FIG. 9. Dependence of the derivative (slope) of the upper critical field (normalized by its

value for W = 0) on the effective width of the pseudogap for s-wave pairing and scattering by

charge (CDW) fluctuations (curves s1 and s2) and for dx2−y2-wave pairing and scattering on spin

(AFM(SDW)) fluctuations (curves d1 and d2). Data are given for the values of inverse correlation

length κa = 0.2 (s1 and d1) and κa = 0.5 (s2 and d2).

30

Page 31

? ???? ??

???

???

???

???

???

???

??

?????

??

??

??

??

FIG. 10. Dependence of specific heat discontinuity at superconducting transition on the effec-

tive width of the pseudogap W/Tc0for s-wave pairing and scattering by charge (CDW) fluctuations

(curves s1 and s2) and for dx2−y2-wave pairing and scattering by spin (AFM(SDW)) fluctuations

(curves d1 and d2). Data are given for the values of inverse correlation length κa = 0.2 (s1 and

d1) and κa = 0.5 (s2 and d2).

31

Page 32

????? ??

???

???

???

???

???

???

??

?????

?????????

??

??

??

FIG. 11. Dependence of the derivative (slope) of the upper critical field (normalized by its value

for W = 0) on the effective width of the pseudogap W/Tc0for s-wave pairing and scattering by

spin (AFM(CDW)) fluctuations (curves s1 and s2) and for dx2−y2-wave pairing and scattering by

charge (CDW) fluctuations (curves d1 and d2). Data are given for the values of inverse correlation

length κa = 0.2 (s1 and d1) and κa = 1.0 (s2 and d2).

32

Page 33

???????

???

???

???

???

???

???

??

?????

∆?

??

??

??

FIG. 12. Dependence of specific heat discontinuity at superconducting transition on the ef-

fective width of the pseudogap W/Tc0 for s-wave pairing and scattering by spin (AFM(CDW))

fluctuations (curves s1 and s2) and for dx2−y2-wave pairing and scattering on charge (CDW) fluc-

tuations (curves d1 and d2). Data are given for the values of inverse correlation length κa = 0.2

(s1 and d1) and κa = 1.0 (s2 and d2).

33

Page 34

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