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: email@example.com, firstname.lastname@example.org, email@example.com
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
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  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) (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 . 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
, 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”)
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.  (assuming the self – averaging nature of superconducting
order parameter over pseudogap fluctuations) and for a simplified exactly solvable model
, where we have been able to analyze the effects due to the absence of self – averaging
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. .
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:
1 + ξ2(q − Q)2− iω
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
(Fig. 1), and “cold” one, with momenta close to the parts of the Fermi surface, surrounding
diagonals of the Brillouin zone . 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 , 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 :
Veff(q) = W2
ξ−2+ (qx− Qx)2
ξ−2+ (qy− Qy)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 :
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
. 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,
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 .
For finite ξ we can construct “nearly exact” solution , generalizing the one – dimensional
approach, proposed in Ref. . 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
iεn− ξkj(p) + ikjvkjκ
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).
ξp+Q for odd k
for even k
|vx(p + Q)| + |vy(p + Q)| for odd k
|vx(p)| + |vy(p)|for even k
where v(p) =
∂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].
s(k) = k (7)
in the case of commensurate fluctuations with Q = (π/a,π/a) , if we neglect their spin
structure ) (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. ), 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. ). As a result the Ansatz (4) for one – particle Green’s
function is conserved, but the combinatorial factor s(k) takes the form :
for odd k
for even k
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) .
In spin – fermion model  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
As a result we obtain the following recurrence procedure for the one – particle Green’s
function G(εnp) (continuous fraction representation) [16,7,6]:
iεn− ξk(p) + ikvkκ − W2s(k + 1)Gk+1(εnp)
and “physical” Green’s function is determined as G(εnp) ≡ G0(εnp).
Ansatz (4) for the contribution of an arbitrary diagram of N-th order is not exact in
general case . However, for two – dimensional system we were able to show that for
certain topologies of the Fermi surface (4) is actually an exact representation , in other
cases it can be shown , 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 , 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  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. , while in the limit of ξ → 0 and fixed value of W it reduces to the trivial case of
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
Vsc(p,p′) = −V e(p)e(p′), (10)
where e(p) is given by:
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)
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).
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.  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. ).
III. COOPER INSTABILITY. RECURRENCE PROCEDURE FOR THE VERTEX
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. , is apparently due to rather special model of short – range order used in this paper.
1 − V χ(0;T) = 0(12)
where the generalized Cooper susceptibility is determined by diagram shown in Fig. 2 and
χ(q;T) = −T
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
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) . Because of this
factor, in case of electron interaction with spin fluctuations contributions to generalized
susceptibility for different types of pairing considered above just “change places”
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. . For a two – dimensional model of the pseudogap with “hot
spots” on the Fermi surface the generalization of this procedure was performed in Ref. 
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
= 1 ± W2s(k)Gk¯Gk
2iεn− vkq − W2s(k + 1)(Gk+1−¯Gk+1)
where Gk = Gk(εnp + q) and¯Gk = Gk(−εn,−p) are calculated according to (9), vk is
defined by (6), and vkare given by:
v(p + Q) for odd k
v(p) for even k
“Physical” vertex is defined as Γ±(εn,−εn,q) ≡ Γ±
To determine Tcwe need vertices at q = 0. Then¯Gk= G∗
kand vertex parts Γ+
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).
ImGk= −εn+ kvkκ − W2s(k + 1)ImGk+1
ReGk= −ξk(p) + W2s(k + 1)ReGk+1
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:
k−1= 1 ∓ W2s(k)
εn− W2s(k + 1)ImGk+1Γ±
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
where ¯ m = [
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
ta2 = 1, while for dx2−y2 pairing we get
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.
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.
superconductors with Kondo impurities . 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 . 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.  Ginzburg – Landau expansion was derived for an exactly solvable model of
the pseudogap with infinite correlation length of short – range order fluctuations. In Ref.
 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 . Here we shall make
appropriate generalization for the present more realistic model of “hot spots” on the Fermi
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
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 , shown in Fig. 6.
Coefficients of Ginzburg – Landau expansion can be conveniently expressed as:
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
< e2(p) >;C0= N0(0)7ζ(3)
< |v(p)|2e2(p) >;
< e4(p) >,(22)
where angular brackets denote the usual averaging over the Fermi surface: < ... >=
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)
χ(q;Tc) − χ(0;Tc)
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:
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.
V. PHYSICAL PROPERTIES OF SUPERCONDUCTORS IN THE PSEUDOGAP
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
In the absence of the pseudogap:
BCS(T) = −C0
Thus, in our model:
For the penetration depth of magnetic field we have:
λ2(T) = −
Then, analogously to (28), we obtain:
Close to Tcthe upper critical field Hc2is defined via Ginzburg – Landau coefficients as:
where φ0= cπ/|e| is the magnetic flux quantum. Then the derivative (slope) of the upper
critical field close to Tcis given by:
16πφ0< e2(p) >
7ζ(3) < |v(p)|2e2(p) >TcKA
Specific heat discontinuity at the transition point is defined as:
T − Tc
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) >.
Then the relative specific heat discontinuity in our model can be written as:
∆C ≡(Cs− Cn)Tc
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 .
6Typical dependences of these coefficients on the parameter W/Tc0are functions rather rapidly
decreasing from 1 in the region of existence of superconducting state.
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.
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 , 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
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 , which is based of Fig.4
of Ref. , 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 . 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
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”.
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
Then for susceptibility at q = 0, defining the coefficient A0, we get:
χ0(0;T) = −T
δ(ξ − ξp)e2(p) ≈
= 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)
ABCS= χBCS(0;T) − χBCS(0;Tc) = N0(0)T − Tc
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
χ0(q;Tc) − χ0(0;Tc)
Expanding expression (A1) for χ0(q;Tc) in powers of q we get:
χ0(q;Tc) = χ0(0;Tc) + Tc
so that coefficient C0is found to be:
δ(ξ − ξp)e2(p)|v(p)|2cos2(φ) ≈
< 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
rotation over an angle
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
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:
< |v(p)|2e2(p) >,(A9)
which for the case of isotropic s-wave pairing and spherical Fermi surface becomes the
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:
p)2e4(p) = Tc
= BBCS< e4(p) >
δ(ξ − ξp)e4(p) ≈
is the standard expression for B in case of isotropic s-wave pairing.
FIG. 1. Fermi surface with “hot spots” connected by scattering vector of the order of
Q = (π
FIG. 2. Diagaram for the generalized susceptibility χ(q) in Cooper channel.
FIG. 3. Recurrence equations for the vertex part.
? ?? ????
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).
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).
FIG. 6. Diagrammatic representation of Ginzburg – Landau expansion.
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.
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.
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).
? ???? ??
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).
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).
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).
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