d-wave superconductivity from electron-phonon interactions
J.P.Hague
Dept. of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH and
Dept. of Physics, Loughborough University, Loughborough, LE11 3TU
(Dated: 4th May 2005)
I examine electron-phonon mediated superconductivity in the intermediate coupling and phonon
frequency regime of the quasi-2D Holstein model. I use an extended Migdal–Eliashberg theory which
includes vertex corrections and spatial fluctuations. I find a d-wave superconducting state that is
unique close to half-filling. The order parameter undergoes a transition to s-wave superconductivity
on increasing filling. I explain how the inclusion of both vertex corrections and spatial fluctuations is
essential for the prediction of a d-wave order parameter. I then discuss the effects of a large Coulomb
pseudopotential on the superconductivity (such as is found in contemporary superconducting ma-
terials like the cuprates), which results in the destruction of the s-wave states, while leaving the
d-wave states unmodified. Published as: Phys. Rev. B 73, 060503(R) (2006)
PACS numbers: 71.10.-w, 71.38.-k, 74.20.-z
The discovery of high transition temperatures and a
d-wave order parameter in the cuprate superconductors
are remarkable results and have serious implications for
the theory of superconductivity. The presence of large
Coulomb interactions in the cuprates which have the po-
tential to destroy conventional s-wave BCS states has
prompted the search for new mechanisms that can give
rise to superconductivity. However, electron-phonon me-
diated superconductivity is still not well understood, es-
pecialy in lower dimensional systems. In particular, the
electron-phonon problem is particularly difficult at inter-
mediate couplings with large phonon frequency (such as
found in the cuprates) and the electron-phonon mech-
anism cannot be fully ruled out. It is therefore of
paramount importance to develop new theories to under-
stand electron-phonon mediated superconductivity away
from the BCS limit.
The assumption that electron-phonon interactions can-
not lead to high transition temperatures and unusual or-
der parameters was made on the basis of calculations
from BCS theory, which is a very-weak-coupling mean-
field theory (although of course highly successful for
pre-1980s superconductors)1. In the presence of strong
Coulomb interaction, the BCS s-wave transition temper-
ature is vastly reduced. However, the recent measure-
ment of large couplings between electrons and the lat-
tice in the cuprate superconductors means that exten-
sions to the conventional theories of superconductivity
are required2,3,4. In particular, low dimensionality, in-
termediate dimensionless coupling constants of ∼ 1 and
large and active phonon frequencies of ∼ 75meV mean
that BCS or the more advanced Migdal–Eliashberg (ME)
theory cannot be applied. In fact, the large coupling con-
stant and a propensity for strong renormalization in 2D
systems, indicate that the bare unrenormalized phonon
frequency could be several times greater than the mea-
sured 75 meV5.
Here I apply the dynamical cluster approxima-
tion (DCA) to introduce a fully self-consistent
momentum-dependent self-energy to the electron-phonon
problem5,6,7,8. Short ranged spatial fluctuations and low-
est order vertex corrections are included, allowing the
sequence of phonon absorption and emission to be re-
ordered once. In particular, the theory used here is
second order in the effective electron-electron coupling
U = −g2/Mω20, which provides the correct weak coupling
limit from small to large phonon frequencies18. In this
paper, I include symmetry broken states in the anoma-
lous self energy to investigate unconventional order pa-
rameters such as d-wave. No assumptions are made in
advance about the form of the order parameter.
DCA6,8,9 is an extension to the dynamical mean-field
theory for the study of low dimensional systems. To ap-
ply the DCA, the Brillouin zone is divided into NC sub-
zones within which the self-energy is assumed to be mo-
mentum independent, and cluster Green functions are
determined by averaging over the momentum states in
each subzone. This leads to spatial fluctuations with
characteristic range, N
c . In this paper, Nc = 4 is used
throughout. This puts an upper bound on the strength of
the superconductivity, which is expected to be reduced
in larger cluster sizes10. To examine superconducting
states, DCA is extended within the Nambu formalism7,8.
Green functions and self-energies are described by 2 × 2
matrices, with off diagonal terms relating to the super-
conducting states. The self-consistent condition is:
G(K, iωn) =
Di(ǫ)(ζ(Ki, iωn)− ǫ)
|ζ(Ki, iωn)− ǫ|2 + φ(Ki, iωn)2
F (K, iωn) = −
Di(ǫ)φ(Ki, iωn)
|ζ(Ki, iωn)− ǫ|2 + φ(Ki, iωn)2
where ζ(Ki, iωn) = iωn + µ−Σ(Ki, iωn), µ is the chem-
ical potential, ωn are the Fermionic Matsubara frequen-
cies, φ(K, iω) is the anomalous self energy and Σ(K, iω)
is the normal self energy. G(K, iωn) must obey the lat-
tice symmetry. In contrast, it is only |F (K, iωn)| which
is constrained by this condition, since φ is squared in
the denominator of Eqn. 1. Therefore the sign of φ
http://arxiv.org/abs/0704.0633v1
FIG. 1: Diagrammatic representation of the current approx-
imation. Series (a) represents the vertex-neglected theory
which corresponds to the Migdal–Eliashberg approach, valid
when the phonon energy ω0 and electron-phonon coupling
U are small compared to the Fermi energy. Series (b) repre-
sents additional diagrams for the vertex corrected theory. The
phonon self energies are labeled with Π, and Σ denotes the
electron self-energies. Lines represent the full electron Green
function and wavy lines the full phonon Green function.
can change. For instance, if the anomalous self energy
has the rotational symmetry φ(π, 0) = −φ(0, π), the
on-diagonal Green function, which represents the elec-
tron propagation retains the correct lattice symmetry
G(π, 0) = G(0, π). Therefore, only inversion symmetry
is required of the anomalous Green function representing
superconducting pairs and the anomalous self energy.
Here I examine the Holstein model11 of electron-
phonon interactions. It treats phonons as nuclei vibrat-
ing in a time-averaged harmonic potential (representing
the interactions between all nuclei), i.e. only one fre-
quency ω0 is considered. The phonons couple to the local
electron density via a momentum-independent coupling
constant g11.
H = −
<ij>σ tc
iσcjσ +
iσ niσ(gri − µ)
The first term in this Hamiltonian represents hopping of
electrons between neighboring sites and has a dispersion
ǫk = −2t
i=1 cos(ki). The second term couples the lo-
cal ion displacement, ri to the local electron density. The
last term is the bare phonon Hamiltonian, i.e. a sim-
ple harmonic oscillator. The creation and annihilation of
electrons is represented by c
i (ci), pi is the ion momen-
tum and M the ion mass. The effective electron-electron
interaction is,
U(iωs) =
ω2s + ω
where, ωs = 2πsT , s is an integer and U = −g
2/Mω20
represents the magnitude of the effective electron-
electron coupling. D = 2 with t = 0.25, resulting in
a non-interacting band width W = 2. A small interpla-
nar hopping t⊥ = 0.01 is included. This is necessary to
stabilise superconductivity, which is not permitted in a
pure 2D system12.
Perturbation theory in the effective electron-electron
interaction (Fig. 1) is applied to second order in U , us-
ing a skeleton expansion. The electron self-energy has
two terms, ΣME(ω,K) neglects vertex corrections (Fig.
1(a)), and ΣVC(ω,K) corresponds to the vertex corrected
case (Fig. 1(b)). ΠME(ω,K) and ΠVC(ω,K) correspond
to the equivalent phonon self energies. At large phonon
frequencies, all second order diagrams including ΣV C are
essential for the correct description of the weak coupling
limit.
The phonon propagator D(z,K) is calculated from,
D(iωs,K) =
ω2s + ω
0 −Π(iωs,K)
and the Green function from equations 1 and 2. Σ =
ΣME+ΣVC and Π = ΠME+ΠVC. Details of the transla-
tion of the diagrams in Fig. 1 and the iteration procedure
can be found in Ref. 7. Calculations are carried out along
the Matsubara axis, with sufficient Matsubara points for
an accurate calculation. The equations were iterated un-
til the normal and anomalous self-energies converged to
an accuracy of approximately 1 part in 103.
Since the anomalous Green function is proportional
to the anomalous self energy, initializing the problem
with the non-interacting Green function leads to a non-
superconducting (normal) state. A constant supercon-
ducting field with d-wave symmetry was applied to the
system to induce superconductivity. The external field
was then completely removed. Iteration continued with-
out the field until convergence. This solution was then
used to initialize self-consistency for other similar val-
ues of the parameters. The symmetry conditions used in
Refs 5 and 7 have been relaxed to reflect the additional
breaking of the anomalous lattice symmetry in the d-wave
state. This does not affect the normal state Green func-
tion, but does affect the anomalous state Green function.
In Fig. 2, the anomalous self energy is examined for
n = 1.0 (half-filling). The striking feature is that sta-
ble d-wave superconductivity is found. This is mani-
fested through a change in sign of the anomalous self
energy, which is negative at the (π, 0) point and positive
at the (0, π) point. The electron Green function (equa-
tion 1) depends on φ2, so causality and lattice symmetry
are maintained. Since the gap function φ(iωn)/Z(iωn)
is directly proportional to φ(iωn), and Z(iωn,K(π,0)) =
Z(iωn,K(0,π)), then the sign of the order parameter i.e.
the sign of the superconducting gap changes under 90o
rotation. Z(iωn) = 1− Σ(iωn)/iωn.
Figure 3 shows the variation of superconducting pair-
ing across the Brillouin zone. ns(k) = T
n F (iωn,k).
U = 0.6, ω0 = 0.4, n = 1 and T = 0.005. The d-wave
order can be seen very clearly. The largest anomalous
densities are at the (π, 0) and (0, π) points, with a node
situated at the (π/2, π/2) point and a sign change on 90o
rotation. Pairing clearly occurs between electrons close
to the Fermi surface.
So far, the model has been analyzed at half filling.
Figure 4 demonstrates the evolution of the order param-
-0.025
-0.02
-0.015
-0.01
-0.005
 0.005
 0.01
 0.015
 0.02
 0.025
-2 -1.5 -1 -0.5  0  0.5  1  1.5  2
(0,0)
(π,0)
(0,π)
(π,π)
FIG. 2: Anomalous self-energy at half-filling. The anomalous
self energy is real. It is clear that φ(π, 0) = −φ(0, π). This
is characteristic of d-wave order. Similarly, the electron self
energy has the correct lattice symmetry Σ(π, 0) = Σ(0, π),
which was not imposed from the outset. The gap function is
related to the anomalous self energy via φ(iωn)/Z(iωn).
-0.25
 0.25
-0.25
 0.25
ns(k)
FIG. 3: Variation of superconducting (anomalous) pairing
density across the Brillouin zone. ns(k) = T
F (iωn,k).
U = 0.6, ω0 = 0.4, n = 1 and T = 0.005. The d-wave order
can be seen very clearly, with a change in sign on 90o rota-
tion and a node situated at the (π/2, π/2) point. The largest
anomalous (superconducting) densities are at the (π, 0) and
(0, π) points.
eter as the number of holes is first increased, and then
decreased. The total magnitude of the anomalous den-
sity, ns =
|ns(Ki)| is examined. When the number
of holes is increased, stable d-wave order persists to a
filling of n = 1.18, while decreasing monotonically. At
the critical point, there is a spontaneous transition to s-
wave order. Starting from a high filling, and reducing the
number of holes, there is a spontaneous transition from
s to d-wave order at n = 1.04. There is therefore hys-
teresis associated with the self-consistent solution. It is
reassuring that the d-wave state can be induced without
the need for the external field. As previously established,
s-wave order does not exist at half-filling as a mainfesta-
 0.02
 0.04
 0.06
 0.08
 1  1.05  1.1  1.15  1.2  1.25  1.3
External
field
FIG. 4: Hysteresis of the superconducting order parameters.
|ns(Ki)|. Starting from a d-wave state at half-
filling, increasing the chemical potential increases the filling
and decreases the d-wave order. Eventually, at n = 1.18 the
system changes to an s-wave state. On return from large
filling, the s-wave superconductivity is persistent to a low
filling of n = 1.04, before spontaneously reverting to a d-
wave state. The system is highly susceptible to d-wave order,
and application of a very small external superconducting field
to an s-wave state results in a d-wave state. Note that d-
and s-wave channels are coupled in the higher order theory,
so the transition can take place spontaneously, unlike in the
standard gap equations.
tion of Hohenberg’s theorem7, so the computed d-wave
order at half-filling is the ground state of the model. It
is interesting that the d- and s-channels are able to co-
exist, considering that the BCS channels are separate on
a square lattice. This is due to the vertex corrections,
since the self consistent equations are no longer linear in
the gap function (the 1st order gap equation vanishes in
the d-wave case, leaving 2nd order terms as the leading
contribution).
I finish with a brief discussion of Coulomb effects.
In the Eliashberg equations, a Coulomb pseudopotential
may be added to the theory as,
φC = UCT
F (iωn,K) (6)
It is easy to see the effect of d-wave order on this term.
Since the sign of the anomalous Green function is mod-
ulated, the average effect of d-wave order is to nullify
the Coulomb contribution to the anomalous self-energy
(i.e. φCd = 0). This demonstrates that the d-wave
state is stable to Coulomb perturbations, presumably be-
cause the pairs are distance separated. In contrast, the
s-wave state is not stable to Coulomb interaction, with
a corresponding reduction of the transition temperature
(TC = 0 for λ < µC). Thus, such a Coulomb filter selects
the d-wave state (see e.g. Ref. 13). Since large local
Coulomb repulsions are present in the cuprates (and in-
deed most transition metal oxides), then this mechanism
seems the most likely to remove the hysteresis. Without
the Coulomb interactions, it is expected that the s-wave
state will dominate for n > 1.04, since the anomalous
order is larger.
I note that a further consequence of strong Coulomb
repulsion is antiferromagnetism close to half-filling. Typ-
ically magnetic fluctuations act to suppress phonon medi-
ated superconducting order. As such, one might expect a
suppression of superconducting order close to half-filling,
with a maximum away from half filling. The current the-
ory could be extended to include additional anomalous
Green functions related to antiferromagnetic order. This
would lead to a 4x4 Green function matrix. A full anal-
ysis of antiferromagnetism and the free energy will be
carried out at a later date.
a. Summary In this paper I have carried out simu-
lations of the 2D Holstein model in the superconducting
state. Vertex corrections and spatial fluctuations were
included in the approximation for the self-energy. The
anomalous self energy and superconducting order param-
eter were calculated. Remarkably, stable superconduct-
ing states with d-wave order were found at half-filling.
d-wave states persist to n = 1.18, where the symmetry
of the parameter changes to s-wave. Starting in the s-
wave phase and reducing the filling, d-wave states spon-
taneously appear at n = 1.04. The spontaneous appear-
ance of d-wave states in a model of electron-phonon in-
teractions is of particular interest, since it may negate
the need for novel pairing mechanisms in the cuprates19.
The inclusion of vertex corrections and spatial fluctua-
tions was essential to the emergence of the d-wave states
in the Holstein model, which indicates why BCS and ME
calculations do not predict this phenomenon. For very
weak coupling, the off diagonal Eliashberg self-energy
has the form −UT
Q,n F (iωn,Q)D0(iωs − iωn), so it
is clear (for the same reasons as the Coulomb pseudopo-
tential) that this diagram has no contribution in the d-
wave phase (the weak coupling phonon propagator is mo-
mentum independent for the Holstein model). Therefore,
vertex corrections are the leading term in the weak cou-
pling limit. Furthermore, I have discussed the inclusion
of Coulomb states to lowest order, which act to desta-
bilize the s-wave states, while leaving the d-wave states
unchanged. Since the Coulomb pseudopotential has no
effect then it is possible that electron-phonon interactions
are the mechanism inducing d-wave states in real mate-
rials such as the cuprates. The Coulomb filtering mech-
anism works for p-wave symmetry and higher, so it is
possible that electron-phonon interactions could explain
many novel superconductors. Certainly, such a mecha-
nism cannot be ruled out. The doping dependence of the
order qualitatively matches that of La2−xSrxCuO4 (here
order extends to x = 0.18, in the Cuprate to x = 0.3).
Antiferromagnetism is only present in the cuprate very
close to half filling (up to approx x = 0.02), and on a
mean-field level does not interfere with the d-wave su-
perconductivity at larger dopings.
It has been determined experimentally that strong
electron-phonon interactions and high phonon frequen-
cies are clearly visible in the electron and phonon band
structures of the cuprates, and are therefore an essential
part of the physics3,4. Similar effects to those observed
in the cuprates are seen in the electron and phonon band
structures of the 2D Holstein model in the normal phase5.
It is clearly of interest to determine whether other fea-
tures and effects in the cuprate superconductors could be
explained with electron-phonon interactions alone.
b. Acknowledgments I thank the University of Le-
icester for hospitality while carrying out this work. I
thank E.M.L.Chung for useful discussions. I am currently
supported under EPSRC grant no. EP/C518365/1.
1 J.Bardeen, L.N.Cooper, and J.R.Schrieffer, Phys. Rev. B
108, 1175 (1957).
2 G.M.Zhao, M.B.Hunt, H.Keller, and K.A.Müller, Nature
385, 236 (1997).
3 A.Lanzara, P.V.Bogdanov, X.J.Zhou, S.A.Kellar,
D.L.Feng, E.D.Lu, T.Yoshida, H.Eisaki, A.Fujimori,
K.Kishio, et al., Nature 412, 6846 (2001).
4 R.J.McQueeney, Y.Petrov, T.Egami, M.Yethiraj,
G.Shirane, and Y.Endoh, Phys. Rev. Lett. 82, 628
(1999).
5 J.P.Hague, J. Phys. Condens. Matt 15, 2535 (2003).
6 M.H.Hettler, A.N.Tahvildar-Zadeh, M.Jarrell,
T.Pruschke, and H.R.Krishnamurthy, Phys. Rev. B
58, R7475 (1998).
7 J.P.Hague, J. Phys.: Condens. Matter 17, 5663 (2005).
8 T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler, Rev.
Mod. Phys 77, 1027 (2005).
9 M.H.Hettler, M.Mukherjee, M.Jarrell, and
H.R.Krishnamurthy, Phys. Rev. B 61, 12739 (2000).
10 M.Jarrell, Th.Maier, C.Huscroft, and S.Moukouri, Phys.
Rev. B 64, 195130 (2001).
11 T.Holstein, Ann. Phys. 8, 325 (1959).
12 P.C.Hohenberg, Phys. Rev. 158, 383 (1967).
13 J.F.Annett, Superconductivity, Superfluidity and Conden-
sates (Oxford University Press, 2004).
14 C.Grimaldi, L.Pietronero, and S.Strässler, Phys. Rev. Lett.
75, 1158 (1995).
15 A.A.Abrikosov, Physica C 244, 243 (1995).
16 A.A.Abrikosov, Phys. Rev. B 52, R15738 (1995).
17 R. J. Birgeneau,and G. Shirane, in Physical Properties
of High Temperature Superconductors I, edited by D.
M.Ginsberg (World Scientific, Singapore, 1989).
18 I also note the extensions to Eliashberg theory carried out
by Grimaldi et al.14.
19 On the basis of a screened electron-phonon interaction,
Abrikosov claims to have found stable d-wave states in a
BCS like theory15,16. However with an unscreened Holstein
potential, the transition temperature it the d-wave chan-
nel given by the standard theory is zero. Also, the assumed
order parameter in his work does not clearly have d-wave
symmetry.

  I examine electron-phonon mediated superconductivity in the intermediate
coupling and phonon frequency regime of the quasi-2D Holstein model. I use an
extended Migdal-Eliashberg theory which includes vertex corrections and spatial
fluctuations. I find a d-wave superconducting state that is unique close to
half-filling. The order parameter undergoes a transition to s-wave
superconductivity on increasing filling. I explain how the inclusion of both
vertex corrections and spatial fluctuations is essential for the prediction of
a d-wave order parameter. I then discuss the effects of a large Coulomb
pseudopotential on the superconductivity (such as is found in contemporary
superconducting materials like the cuprates), which results in the destruction
of the s-wave states, while leaving the d-wave states unmodified.

d-wave superconductivity from electron-phonon interactions
J.P.Hague
Dept. of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH and
Dept. of Physics, Loughborough University, Loughborough, LE11 3TU
(Dated: 4th May 2005)
I examine electron-phonon mediated superconductivity in the intermediate coupling and phonon
frequency regime of the quasi-2D Holstein model. I use an extended Migdal–Eliashberg theory which
includes vertex corrections and spatial fluctuations. I find a d-wave superconducting state that is
unique close to half-filling. The order parameter undergoes a transition to s-wave superconductivity
on increasing filling. I explain how the inclusion of both vertex corrections and spatial fluctuations is
essential for the prediction of a d-wave order parameter. I then discuss the effects of a large Coulomb
pseudopotential on the superconductivity (such as is found in contemporary superconducting ma-
terials like the cuprates), which results in the destruction of the s-wave states, while leaving the
d-wave states unmodified. Published as: Phys. Rev. B 73, 060503(R) (2006)
PACS numbers: 71.10.-w, 71.38.-k, 74.20.-z
The discovery of high transition temperatures and a
d-wave order parameter in the cuprate superconductors
are remarkable results and have serious implications for
the theory of superconductivity. The presence of large
Coulomb interactions in the cuprates which have the po-
tential to destroy conventional s-wave BCS states has
prompted the search for new mechanisms that can give
rise to superconductivity. However, electron-phonon me-
diated superconductivity is still not well understood, es-
pecialy in lower dimensional systems. In particular, the
electron-phonon problem is particularly difficult at inter-
mediate couplings with large phonon frequency (such as
found in the cuprates) and the electron-phonon mech-
anism cannot be fully ruled out. It is therefore of
paramount importance to develop new theories to under-
stand electron-phonon mediated superconductivity away
from the BCS limit.
The assumption that electron-phonon interactions can-
not lead to high transition temperatures and unusual or-
der parameters was made on the basis of calculations
from BCS theory, which is a very-weak-coupling mean-
field theory (although of course highly successful for
pre-1980s superconductors)1. In the presence of strong
Coulomb interaction, the BCS s-wave transition temper-
ature is vastly reduced. However, the recent measure-
ment of large couplings between electrons and the lat-
tice in the cuprate superconductors means that exten-
sions to the conventional theories of superconductivity
are required2,3,4. In particular, low dimensionality, in-
termediate dimensionless coupling constants of ∼ 1 and
large and active phonon frequencies of ∼ 75meV mean
that BCS or the more advanced Migdal–Eliashberg (ME)
theory cannot be applied. In fact, the large coupling con-
stant and a propensity for strong renormalization in 2D
systems, indicate that the bare unrenormalized phonon
frequency could be several times greater than the mea-
sured 75 meV5.
Here I apply the dynamical cluster approxima-
tion (DCA) to introduce a fully self-consistent
momentum-dependent self-energy to the electron-phonon
problem5,6,7,8. Short ranged spatial fluctuations and low-
est order vertex corrections are included, allowing the
sequence of phonon absorption and emission to be re-
ordered once. In particular, the theory used here is
second order in the effective electron-electron coupling
U = −g2/Mω20, which provides the correct weak coupling
limit from small to large phonon frequencies18. In this
paper, I include symmetry broken states in the anoma-
lous self energy to investigate unconventional order pa-
rameters such as d-wave. No assumptions are made in
advance about the form of the order parameter.
DCA6,8,9 is an extension to the dynamical mean-field
theory for the study of low dimensional systems. To ap-
ply the DCA, the Brillouin zone is divided into NC sub-
zones within which the self-energy is assumed to be mo-
mentum independent, and cluster Green functions are
determined by averaging over the momentum states in
each subzone. This leads to spatial fluctuations with
characteristic range, N
c . In this paper, Nc = 4 is used
throughout. This puts an upper bound on the strength of
the superconductivity, which is expected to be reduced
in larger cluster sizes10. To examine superconducting
states, DCA is extended within the Nambu formalism7,8.
Green functions and self-energies are described by 2 × 2
matrices, with off diagonal terms relating to the super-
conducting states. The self-consistent condition is:
G(K, iωn) =
Di(ǫ)(ζ(Ki, iωn)− ǫ)
|ζ(Ki, iωn)− ǫ|2 + φ(Ki, iωn)2
F (K, iωn) = −
Di(ǫ)φ(Ki, iωn)
|ζ(Ki, iωn)− ǫ|2 + φ(Ki, iωn)2
where ζ(Ki, iωn) = iωn + µ−Σ(Ki, iωn), µ is the chem-
ical potential, ωn are the Fermionic Matsubara frequen-
cies, φ(K, iω) is the anomalous self energy and Σ(K, iω)
is the normal self energy. G(K, iωn) must obey the lat-
tice symmetry. In contrast, it is only |F (K, iωn)| which
is constrained by this condition, since φ is squared in
the denominator of Eqn. 1. Therefore the sign of φ
http://arxiv.org/abs/0704.0633v1
FIG. 1: Diagrammatic representation of the current approx-
imation. Series (a) represents the vertex-neglected theory
which corresponds to the Migdal–Eliashberg approach, valid
when the phonon energy ω0 and electron-phonon coupling
U are small compared to the Fermi energy. Series (b) repre-
sents additional diagrams for the vertex corrected theory. The
phonon self energies are labeled with Π, and Σ denotes the
electron self-energies. Lines represent the full electron Green
function and wavy lines the full phonon Green function.
can change. For instance, if the anomalous self energy
has the rotational symmetry φ(π, 0) = −φ(0, π), the
on-diagonal Green function, which represents the elec-
tron propagation retains the correct lattice symmetry
G(π, 0) = G(0, π). Therefore, only inversion symmetry
is required of the anomalous Green function representing
superconducting pairs and the anomalous self energy.
Here I examine the Holstein model11 of electron-
phonon interactions. It treats phonons as nuclei vibrat-
ing in a time-averaged harmonic potential (representing
the interactions between all nuclei), i.e. only one fre-
quency ω0 is considered. The phonons couple to the local
electron density via a momentum-independent coupling
constant g11.
H = −
<ij>σ tc
iσcjσ +
iσ niσ(gri − µ)
The first term in this Hamiltonian represents hopping of
electrons between neighboring sites and has a dispersion
ǫk = −2t
i=1 cos(ki). The second term couples the lo-
cal ion displacement, ri to the local electron density. The
last term is the bare phonon Hamiltonian, i.e. a sim-
ple harmonic oscillator. The creation and annihilation of
electrons is represented by c
i (ci), pi is the ion momen-
tum and M the ion mass. The effective electron-electron
interaction is,
U(iωs) =
ω2s + ω
where, ωs = 2πsT , s is an integer and U = −g
2/Mω20
represents the magnitude of the effective electron-
electron coupling. D = 2 with t = 0.25, resulting in
a non-interacting band width W = 2. A small interpla-
nar hopping t⊥ = 0.01 is included. This is necessary to
stabilise superconductivity, which is not permitted in a
pure 2D system12.
Perturbation theory in the effective electron-electron
interaction (Fig. 1) is applied to second order in U , us-
ing a skeleton expansion. The electron self-energy has
two terms, ΣME(ω,K) neglects vertex corrections (Fig.
1(a)), and ΣVC(ω,K) corresponds to the vertex corrected
case (Fig. 1(b)). ΠME(ω,K) and ΠVC(ω,K) correspond
to the equivalent phonon self energies. At large phonon
frequencies, all second order diagrams including ΣV C are
essential for the correct description of the weak coupling
limit.
The phonon propagator D(z,K) is calculated from,
D(iωs,K) =
ω2s + ω
0 −Π(iωs,K)
and the Green function from equations 1 and 2. Σ =
ΣME+ΣVC and Π = ΠME+ΠVC. Details of the transla-
tion of the diagrams in Fig. 1 and the iteration procedure
can be found in Ref. 7. Calculations are carried out along
the Matsubara axis, with sufficient Matsubara points for
an accurate calculation. The equations were iterated un-
til the normal and anomalous self-energies converged to
an accuracy of approximately 1 part in 103.
Since the anomalous Green function is proportional
to the anomalous self energy, initializing the problem
with the non-interacting Green function leads to a non-
superconducting (normal) state. A constant supercon-
ducting field with d-wave symmetry was applied to the
system to induce superconductivity. The external field
was then completely removed. Iteration continued with-
out the field until convergence. This solution was then
used to initialize self-consistency for other similar val-
ues of the parameters. The symmetry conditions used in
Refs 5 and 7 have been relaxed to reflect the additional
breaking of the anomalous lattice symmetry in the d-wave
state. This does not affect the normal state Green func-
tion, but does affect the anomalous state Green function.
In Fig. 2, the anomalous self energy is examined for
n = 1.0 (half-filling). The striking feature is that sta-
ble d-wave superconductivity is found. This is mani-
fested through a change in sign of the anomalous self
energy, which is negative at the (π, 0) point and positive
at the (0, π) point. The electron Green function (equa-
tion 1) depends on φ2, so causality and lattice symmetry
are maintained. Since the gap function φ(iωn)/Z(iωn)
is directly proportional to φ(iωn), and Z(iωn,K(π,0)) =
Z(iωn,K(0,π)), then the sign of the order parameter i.e.
the sign of the superconducting gap changes under 90o
rotation. Z(iωn) = 1− Σ(iωn)/iωn.
Figure 3 shows the variation of superconducting pair-
ing across the Brillouin zone. ns(k) = T
n F (iωn,k).
U = 0.6, ω0 = 0.4, n = 1 and T = 0.005. The d-wave
order can be seen very clearly. The largest anomalous
densities are at the (π, 0) and (0, π) points, with a node
situated at the (π/2, π/2) point and a sign change on 90o
rotation. Pairing clearly occurs between electrons close
to the Fermi surface.
So far, the model has been analyzed at half filling.
Figure 4 demonstrates the evolution of the order param-
-0.025
-0.02
-0.015
-0.01
-0.005
 0.005
 0.01
 0.015
 0.02
 0.025
-2 -1.5 -1 -0.5  0  0.5  1  1.5  2
(0,0)
(π,0)
(0,π)
(π,π)
FIG. 2: Anomalous self-energy at half-filling. The anomalous
self energy is real. It is clear that φ(π, 0) = −φ(0, π). This
is characteristic of d-wave order. Similarly, the electron self
energy has the correct lattice symmetry Σ(π, 0) = Σ(0, π),
which was not imposed from the outset. The gap function is
related to the anomalous self energy via φ(iωn)/Z(iωn).
-0.25
 0.25
-0.25
 0.25
ns(k)
FIG. 3: Variation of superconducting (anomalous) pairing
density across the Brillouin zone. ns(k) = T
F (iωn,k).
U = 0.6, ω0 = 0.4, n = 1 and T = 0.005. The d-wave order
can be seen very clearly, with a change in sign on 90o rota-
tion and a node situated at the (π/2, π/2) point. The largest
anomalous (superconducting) densities are at the (π, 0) and
(0, π) points.
eter as the number of holes is first increased, and then
decreased. The total magnitude of the anomalous den-
sity, ns =
|ns(Ki)| is examined. When the number
of holes is increased, stable d-wave order persists to a
filling of n = 1.18, while decreasing monotonically. At
the critical point, there is a spontaneous transition to s-
wave order. Starting from a high filling, and reducing the
number of holes, there is a spontaneous transition from
s to d-wave order at n = 1.04. There is therefore hys-
teresis associated with the self-consistent solution. It is
reassuring that the d-wave state can be induced without
the need for the external field. As previously established,
s-wave order does not exist at half-filling as a mainfesta-
 0.02
 0.04
 0.06
 0.08
 1  1.05  1.1  1.15  1.2  1.25  1.3
External
field
FIG. 4: Hysteresis of the superconducting order parameters.
|ns(Ki)|. Starting from a d-wave state at half-
filling, increasing the chemical potential increases the filling
and decreases the d-wave order. Eventually, at n = 1.18 the
system changes to an s-wave state. On return from large
filling, the s-wave superconductivity is persistent to a low
filling of n = 1.04, before spontaneously reverting to a d-
wave state. The system is highly susceptible to d-wave order,
and application of a very small external superconducting field
to an s-wave state results in a d-wave state. Note that d-
and s-wave channels are coupled in the higher order theory,
so the transition can take place spontaneously, unlike in the
standard gap equations.
tion of Hohenberg’s theorem7, so the computed d-wave
order at half-filling is the ground state of the model. It
is interesting that the d- and s-channels are able to co-
exist, considering that the BCS channels are separate on
a square lattice. This is due to the vertex corrections,
since the self consistent equations are no longer linear in
the gap function (the 1st order gap equation vanishes in
the d-wave case, leaving 2nd order terms as the leading
contribution).
I finish with a brief discussion of Coulomb effects.
In the Eliashberg equations, a Coulomb pseudopotential
may be added to the theory as,
φC = UCT
F (iωn,K) (6)
It is easy to see the effect of d-wave order on this term.
Since the sign of the anomalous Green function is mod-
ulated, the average effect of d-wave order is to nullify
the Coulomb contribution to the anomalous self-energy
(i.e. φCd = 0). This demonstrates that the d-wave
state is stable to Coulomb perturbations, presumably be-
cause the pairs are distance separated. In contrast, the
s-wave state is not stable to Coulomb interaction, with
a corresponding reduction of the transition temperature
(TC = 0 for λ < µC). Thus, such a Coulomb filter selects
the d-wave state (see e.g. Ref. 13). Since large local
Coulomb repulsions are present in the cuprates (and in-
deed most transition metal oxides), then this mechanism
seems the most likely to remove the hysteresis. Without
the Coulomb interactions, it is expected that the s-wave
state will dominate for n > 1.04, since the anomalous
order is larger.
I note that a further consequence of strong Coulomb
repulsion is antiferromagnetism close to half-filling. Typ-
ically magnetic fluctuations act to suppress phonon medi-
ated superconducting order. As such, one might expect a
suppression of superconducting order close to half-filling,
with a maximum away from half filling. The current the-
ory could be extended to include additional anomalous
Green functions related to antiferromagnetic order. This
would lead to a 4x4 Green function matrix. A full anal-
ysis of antiferromagnetism and the free energy will be
carried out at a later date.
a. Summary In this paper I have carried out simu-
lations of the 2D Holstein model in the superconducting
state. Vertex corrections and spatial fluctuations were
included in the approximation for the self-energy. The
anomalous self energy and superconducting order param-
eter were calculated. Remarkably, stable superconduct-
ing states with d-wave order were found at half-filling.
d-wave states persist to n = 1.18, where the symmetry
of the parameter changes to s-wave. Starting in the s-
wave phase and reducing the filling, d-wave states spon-
taneously appear at n = 1.04. The spontaneous appear-
ance of d-wave states in a model of electron-phonon in-
teractions is of particular interest, since it may negate
the need for novel pairing mechanisms in the cuprates19.
The inclusion of vertex corrections and spatial fluctua-
tions was essential to the emergence of the d-wave states
in the Holstein model, which indicates why BCS and ME
calculations do not predict this phenomenon. For very
weak coupling, the off diagonal Eliashberg self-energy
has the form −UT
Q,n F (iωn,Q)D0(iωs − iωn), so it
is clear (for the same reasons as the Coulomb pseudopo-
tential) that this diagram has no contribution in the d-
wave phase (the weak coupling phonon propagator is mo-
mentum independent for the Holstein model). Therefore,
vertex corrections are the leading term in the weak cou-
pling limit. Furthermore, I have discussed the inclusion
of Coulomb states to lowest order, which act to desta-
bilize the s-wave states, while leaving the d-wave states
unchanged. Since the Coulomb pseudopotential has no
effect then it is possible that electron-phonon interactions
are the mechanism inducing d-wave states in real mate-
rials such as the cuprates. The Coulomb filtering mech-
anism works for p-wave symmetry and higher, so it is
possible that electron-phonon interactions could explain
many novel superconductors. Certainly, such a mecha-
nism cannot be ruled out. The doping dependence of the
order qualitatively matches that of La2−xSrxCuO4 (here
order extends to x = 0.18, in the Cuprate to x = 0.3).
Antiferromagnetism is only present in the cuprate very
close to half filling (up to approx x = 0.02), and on a
mean-field level does not interfere with the d-wave su-
perconductivity at larger dopings.
It has been determined experimentally that strong
electron-phonon interactions and high phonon frequen-
cies are clearly visible in the electron and phonon band
structures of the cuprates, and are therefore an essential
part of the physics3,4. Similar effects to those observed
in the cuprates are seen in the electron and phonon band
structures of the 2D Holstein model in the normal phase5.
It is clearly of interest to determine whether other fea-
tures and effects in the cuprate superconductors could be
explained with electron-phonon interactions alone.
b. Acknowledgments I thank the University of Le-
icester for hospitality while carrying out this work. I
thank E.M.L.Chung for useful discussions. I am currently
supported under EPSRC grant no. EP/C518365/1.
1 J.Bardeen, L.N.Cooper, and J.R.Schrieffer, Phys. Rev. B
108, 1175 (1957).
2 G.M.Zhao, M.B.Hunt, H.Keller, and K.A.Müller, Nature
385, 236 (1997).
3 A.Lanzara, P.V.Bogdanov, X.J.Zhou, S.A.Kellar,
D.L.Feng, E.D.Lu, T.Yoshida, H.Eisaki, A.Fujimori,
K.Kishio, et al., Nature 412, 6846 (2001).
4 R.J.McQueeney, Y.Petrov, T.Egami, M.Yethiraj,
G.Shirane, and Y.Endoh, Phys. Rev. Lett. 82, 628
(1999).
5 J.P.Hague, J. Phys. Condens. Matt 15, 2535 (2003).
6 M.H.Hettler, A.N.Tahvildar-Zadeh, M.Jarrell,
T.Pruschke, and H.R.Krishnamurthy, Phys. Rev. B
58, R7475 (1998).
7 J.P.Hague, J. Phys.: Condens. Matter 17, 5663 (2005).
8 T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler, Rev.
Mod. Phys 77, 1027 (2005).
9 M.H.Hettler, M.Mukherjee, M.Jarrell, and
H.R.Krishnamurthy, Phys. Rev. B 61, 12739 (2000).
10 M.Jarrell, Th.Maier, C.Huscroft, and S.Moukouri, Phys.
Rev. B 64, 195130 (2001).
11 T.Holstein, Ann. Phys. 8, 325 (1959).
12 P.C.Hohenberg, Phys. Rev. 158, 383 (1967).
13 J.F.Annett, Superconductivity, Superfluidity and Conden-
sates (Oxford University Press, 2004).
14 C.Grimaldi, L.Pietronero, and S.Strässler, Phys. Rev. Lett.
75, 1158 (1995).
15 A.A.Abrikosov, Physica C 244, 243 (1995).
16 A.A.Abrikosov, Phys. Rev. B 52, R15738 (1995).
17 R. J. Birgeneau,and G. Shirane, in Physical Properties
of High Temperature Superconductors I, edited by D.
M.Ginsberg (World Scientific, Singapore, 1989).
18 I also note the extensions to Eliashberg theory carried out
by Grimaldi et al.14.
19 On the basis of a screened electron-phonon interaction,
Abrikosov claims to have found stable d-wave states in a
BCS like theory15,16. However with an unscreened Holstein
potential, the transition temperature it the d-wave chan-
nel given by the standard theory is zero. Also, the assumed
order parameter in his work does not clearly have d-wave
symmetry.
