Superconducting states of the quasi-2D Holstein model: Effects of vertex
  and non-local corrections
7 Superconducting states of the quasi-2D Holstein
model: Effects of vertex and non-local corrections
J.P.Hague
Dept. of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH
Abstract. I investigate superconducting states in a quasi-2D Holstein
model using the dynamical cluster approximation (DCA). The effects of
spatial fluctuations (non-local corrections) are examined and approximations
neglecting and incorporating lowest-order vertex corrections are computed. The
approximation is expected to be valid for electron-phonon couplings of less than
the bandwidth. The phase diagram and superconducting order parameter are
calculated. Effects which can only be attributed to theories beyond Migdal–
Eliashberg theory are present. In particular, the order parameter shows
momentum dependence on the Fermi-surface with a modulated form and s-
wave order is suppressed at half-filling. The results are discussed in relation
to Hohenberg’s theorem and the BCS approximation. [Published as: J. Phys.:
Condens. Matter vol. 17 (2005) 5663-5676]
PACS numbers: 71.10.-w, 71.38.-k, 74.20.-z, 74.62.-c
1. Introduction
The discovery of large couplings between electrons and the lattice in the cuprate
superconductors has led to a call for more detailed theoretical studies of electron-
phonon systems in low dimensions [1, 2, 3]. One of the best-known traditional
approaches to the electron-phonon problem is attributed to Migdal and Eliashberg
[4, 5]. In a bulk 3D system, the perturbation theory may be sharply truncated at
1st order and momentum dependence neglected if the phonon frequency is much less
than the Fermi energy [4]. In physical terms, Migdal’s approach requires that there
is a very high probability that emitted phonons are reabsorbed in a last-in-first-out
order. The typical materials of interest at the time were bulk metallic superconductors
where electron-phonon coupling is relatively weak, and the phonon frequency small
compared to the Fermi energy. For this reason, the application of Migdal–Eliashberg
(ME) theory has been very successful and remains highly regarded.
Strong electron-phonon coupling and large phonon frequencies in low dimensional
systems are outside the limits of validity of the Migdal–Eliashberg approach.
Therefore, the aim of this paper is to evaluate and discuss the effects of both vertex
corrections (VC) and spatial fluctuations on the theory of coupled electron-phonon
systems in the superconducting state. This follows on from the work by Hague
treating the normal (non-superconducting) state of the Holstein model using DCA
[6]. Initial attempts to include vertex corrections were carried out by Engelsberg and
Schrieffer [7]. Other previous attempts to extend ME theory include the introduction
of vertex corrections into the Eliashberg equations by Grabowski and Sham [8], and
http://arxiv.org/abs/0704.0241v1
Superconducting states of the quasi-2D Holstein model 2
an expansion to higher order in the Migdal parameter by Kostur and Mitrović to
investigate the 2D electron-phonon problem [9]. Grimaldi et al. generalised the
Eliashberg equations to include momentum dependence and vertex corrections [10].
An anomalous hardening of the phonon mode was seen by Alexandrov and Schrieffer
[11]. A discussion of the applicability of these and other approximations to the vertex
function can be found in reference [12].
The current paper uses DCA to introduce a fully self-consistent momentum-
dependent self-energy. DCA extends DMFT by introducing short-range fluctuations
in a controlled manner [13]. It is particularly good at describing the electron-phonon
problem, due to the limited momentum dependence of the self-energy, and in this
case, the self-consistent DCA can be viewed as an expansion about the Eliashberg
equations (in which momentum dependence is effectively coarse grained in a manner
similar to DMFT) [5]. In contrast to the Eliashberg equations, the full form of
the Green’s function is considered here, rather than the renormalised weak coupling
Green’s function (which has the form G−1(ǫk, iωn) = Ziωnσ0 − (ǫk + χ)σ3 −∆σ1).
Two approximations for the electron and phonon self energies are applied in this
paper. The first neglects vertex corrections, but incorporates non-local fluctuations.
The second incorporates lowest order vertex and non-local corrections. The vertex
corrections allow the sequence of phonon absorption and emission to be reordered
once, and therefore introduce exchange effects. The DCA result is compared to the
corresponding DMFT result and in this way low-dimensional effects are isolated. It
should be noted that in the extreme strong coupling limit, the Holstein model forms
a bipolaronic ground state, and perturbative methods in the electron and phonon
Green’s functions break down [14]. In the dilute limit, the Holstein model forms
a polaronic liquid. There are significant differences between the weak and strong
coupling limits of the polaron problem. In the strong-coupling limit, the Lang–Firsov
approximation may be applied, and physical properties have very different behaviour.
For example the effective mass is reduced by an exponential factor of coupling. Exact
numerical results show that the crossover between weak and strong coupling regimes
occurs rapidly at λ ∼ 1 [15]. For this reason, the present approximation should be
considered valid for |U | < W .
The paper is organised as follows. In section 2, the DCA is introduced. In
section 3, the Holstein model of electron-phonon interactions is described, and the
perturbation theory and the full algorithm used in this work are detailed. In section
4, the results are presented. The momentum dependence of the superconducting
order parameter is examined through the density of superconducting pairs. The phase
diagram is then computed and comparison is made with analytical results. A summary
of the major findings of this research is provided in section 5.
2. The dynamical cluster approximation
The dynamical cluster approximation [13, 16] is an extension to the dynamical mean-
field theory. DMFT has been applied as an approximation to models of 3D materials
[17, 18, 19]. However, application of DMFT to one- and two-dimensional models
gives an incomplete description of the physics. An example of significant differences
between two- and three-dimensional physics comes from quantum spin-systems. In 3D,
the Heisenberg model orders at a transition temperature, TN . Significant non-local
fluctuations in two dimensions reduce the Néel temperature to zero (Mermin–Wagner
theorem), and the mean-field approach fails completely.
Superconducting states of the quasi-2D Holstein model 3
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Figure 1. A schematic representation of the reciprocal-space coarse graining
scheme for a 4 site DCA. Within the shaded areas, the self-energy is assumed to
be constant. There is a many to one mapping from the crosshatched areas to the
points at the centre of those areas. The coarse-graining procedure corresponds
to the mapping to a periodic cluster in real space, with spatial extent N
Also shown are the high symmetry points Γ, W and X, and lines connecting
the high symmetry points. An infinite number of k states are involved in the
coarse-graining step, so the approximation is in the thermodynamic limit. DMFT
corresponds to NC = 1.
Conceptually, DCA is similar to DMFT. The Brillouin zone is divided up into
NC subzones consistent with the lattice symmetry (see figure 1). Within each of these
zones, the self-energy is assumed to be momentum independent. For a system in the
normal state, the Green’s function is determined as,
G(Ki, z) =
Di(ǫ) dǫ
z + µ− ǫ− Σ(Ki, z)
where Di(ǫ) is the non-interacting Fermion density of states for subzone i, and the
vectors Ki represent the average k for each subzone (plotted as the large dots in
figure 1). The theory deals with the thermodynamic limit, and introduces non-local
fluctuations with a characteristic length scale of N
C . For NC = 1, DCA is equivalent
to DMFT.
Since superconducting states are to be considered, DCA is extended within the
Nambu formalism [20] in a similar manner to DMFT [17]. Green’s functions and self-
energies are described by 2 × 2 matrices, with off-diagonal anomalous terms relating
to the superconducting states. Note that in the following equations 4-vectors are
used, i.e. K ≡ (iωn,K). The Green’s function and self-energy matrices have the
components,
G(K) =
G(K) F (K)
F ∗(K) −G(−K)
Σ(K) =
Σ(K) φ(K)
φ∗(K) −Σ(−K)
The coarse graining step is generalised to the superconducting state as,
G(K, iωn) =
Di(ǫ)(ζ(Ki, iωn)− ǫ)
|ζ(Ki, iωn)− ǫ|2 + φ(Ki, iωn)2
Superconducting states of the quasi-2D Holstein model 4
Figure 2. Diagrammatic representation of the approximation used in this paper.
Series (a) represents the vertex-neglected theory which corresponds to the Migdal–
Eliashberg approach. This is valid when there is a high probability that the last
emitted phonon is the first to be reabsorbed, which is true if the phonon energy
ω0 and electron-phonon coupling U are small compared to the Fermi energy.
Series (b) represents additional diagrams for the vertex corrected theory. The
inclusion of the lowest order vertex correction allows the order of absorption and
emission of phonons to be swapped once. For moderate phonon frequency and
electron-phonon coupling, these additions to the theory, in combination with non-
local corrections are expected to improve the theory to sufficient accuracy. The
phonon self energies are labeled with Π, and Σ denotes the electron self-energies.
Lines represent the full electron Green’s function and wavy lines the full phonon
Green’s function.
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).
The symmetry of the problem was constrained using the pm3m planar point
group suitable for a 2D square lattice [21]. The partial DOS used in the self-consistent
condition were calculated using the analytic tetrahedron method to ensure very high
accuracy [22].
3. The Holstein model
A simple, yet non-trivial, model of electron-phonon interactions treats phonons as
nuclei vibrating in a time-averaged harmonic potential (representing the interactions
between all nuclei) i.e. only one frequency ω0 is considered. The phonons couple
to the local electron density via a momentum-independent coupling constant g. The
resulting Holstein Hamiltonian [23] is written as,
H = −
t<ij>σc
iσcjσ +
niσ(gri−µ)+
Mω20r
The first term in this Hamiltonian represents a tight binding model with hopping
parameter t. Its Fourier transform takes the form ǫk = −2t
i=1 cos(ki). The second
term connects the local ion displacement, ri to the local electron density. Finally
the last term can be identified as the bare phonon Hamiltonian, which is a simple
harmonic oscillator. The creation and annihilation of electrons is represented by c
and ci respectively, pi is the ion momentum and M the ion mass. t = 0.25 in this
paper, corresponding to a bandwidth of W = 2. A small interplanar hopping of
Superconducting states of the quasi-2D Holstein model 5
t⊥ = 0.01 is included to reduce the strength of the logarithmic singularity at ǫ = 0
in Dπ,0(ǫ) and D0,π(ǫ) and stabilise the solution. This is only expected to modify the
results at very low temperature for large clusters, and gives the problem a quasi-2D
character.
It is possible to find an expression for the effective interaction between electrons by
integrating out phonon degrees of freedom [24]. In Matsubara space, this interaction
has the form,
U(iωs) =
ω2s + ω
Here, ωs = 2πsT represent the Matsubara frequencies for Bosons and s is an integer. A
variable U = −g2/Mω20 is defined to represent the effective electron-electron coupling
in the remainder of this paper.
When phonon frequency and coupling are small, Migdal’s theorem applies.
Migdal’s approach allows vertex corrections to be neglected and becomes exact when
U → 0−, ω0 → 0
+ and is 1st order in U . In the limit of huge phonon frequency,
the model maps onto an attractive Hubbard model, so the weak coupling limit of
the Holstein model is only obtained by considering all second-order diagrams in U ,
and ME theory fails. The vertex-corrected theory described in this paper has the
appropriate weak coupling behaviour for both large and small ω0.
In this paper, perturbation theory to 2nd order in U is used [19] (figure 2).
The derivation of the perturbation theory in Ref. [19] made use of the conserving
approximations of Bahm and Kadanoff [25, 24], which Miller et al. then simplified by
applying the dynamical mean-field theory (or local approximation). Here the theory
has been extended to include partial momentum dependence through the application
of the DCA. The electron self-energy has two terms, ΣME(K, iωn) neglects vertex
corrections (figure 2(a)), and ΣVC(K, iωn) corresponds to the vertex corrected case
(figure 2(b)). ΠME(K, iωs) and ΠVC(K, iωs) correspond to the equivalent phonon self
energies. The diagrams translate as follows:
ΣME(K) = UT
G(Q)D(K−Q) (8)
φME(K) = −UT
F (Q)D(K−Q) (9)
ΠME(K) = −2UT
[G(Q)G(K+Q)− F (Q)F ∗(K+Q)] (10)
ΣVC(K) = (UT )
Q1,Q2
[G(Q1)G(Q2)G(K−Q2 −Q1)
− F (Q1)G(Q2)F
∗(K−Q2 −Q1)
− F ∗(Q1)G(Q2)F (K−Q2 −Q1)
−G∗(Q1)F (Q2)F
∗(K−Q2 −Q1)]
×D(K−Q2)D(Q1 −Q2) (11)
φVC(K) = (UT )
Q1,Q2
[F ∗(Q1)F (Q2)F (K−Q2 +Q1)
−G(Q1)F (Q2)G(K−Q2 +Q1)
Superconducting states of the quasi-2D Holstein model 6
−G∗(Q1)F (Q2)G
∗(K−Q2 +Q1)
− F (Q1)G(Q2)G
∗(K−Q2 +Q1)]
×D(K−Q2)D(Q1 −Q2) (12)
ΠVC(K) = −(UT )
Q1,Q2
Tr {σ3G(Q2 +K)σ3G(Q2)σ3G(Q1)σ3G(K+Q1)}
×D(Q2 −Q1) (13)
where σ3 is the third Pauli matrix. Σ = ΣME +ΣVC and Π = ΠME + ΠVC.
The coarse-grained phonon propagator D(K, iωs) is calculated from,
D(K, iωs) =
ω2s + ω
0 −Π(K, iωs)
since the bare dispersion of the Holstein model is flat.
The time taken to perform the double integration over momentum and Matsubara
frequencies is the main barrier to performing vertex-corrected calculations, and this
limits the cluster size. Since the Holstein model with ω0, |U | ≪ W (W is the
bandwidth) has fluctuations which are almost momentum independent, the DCA has
especially fast convergence in NC for the parameter regime where ω0, |U | < W , and
calculations with relatively small cluster size accurately reflect the physics [6]. In this
respect, finite size calculations take too long to compute, and the application of DCA
to this problem is essential.
4. Results
In this section, I discuss results from the self-consistent scheme. Calculations are
carried out along the Matsubara axis, with sufficient Matsubara points for an accurate
calculation. The vertex corrected self-energies drop off more quickly with Matsubara
frequency, so it is possible to increase efficiency by calculating for less frequencies.
Typically, 256 Matsubara frequencies are used for the vertex neglected diagrams, and
64 for the vertex corrected diagrams, which reach asymptotic behaviour at smaller
Matsubara frequencies. The scheme was iterated until the normal and anomalous
self-energies had converged to an accuracy of approximately 1 part in 104. This
corresponds to a very high accuracy for the Green’s function.
Obtaining superconducting solutions involves an additional step, which is not
obvious at the outset. Since the anomalous Green’s function is proportional to
the anomalous self energy, initialising the problem with the non-interacting Green’s
function leads to a non-superconducting (normal) state. Also, the non-interacting
Green’s function is consistent with an ungapped state and opening a gap in the
electron spectrum can lead to limit cycles during self-consistency, which are damped
in the normal way [17].
To induce superconductivity, a constant superconducting field is applied to the
whole system, leading to a non-zero anomalous Green’s function, and automatically
opening a gap in the normal-state Green’s function. The procedure of applying a
fictitious superconducting field is analogous to the application of a magnetic field
to a spin system to induce a moment (the order parameter in that case). The
superconducting field is applied by adding a constant term to the anomalous self-
energy in equation 9. With the field applied, equations 4,5 and 8-14 are solved self-
consistently until convergence is reached. Once satisfactory convergence is reached,
Superconducting states of the quasi-2D Holstein model 7
 0.05
 0.15
 0.25
-2 -1.5 -1 -0.5  0  0.5  1  1.5  2
n=1.12
(0,0)
(π,0)
(π,π)
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
-2 -1.5 -1 -0.5  0  0.5  1  1.5  2
n=1.54
(0,0)
(π,0)
(π,π)
 0.01
 0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
-2 -1.5 -1 -0.5  0  0.5  1  1.5  2
n=1.12, VC
(0,0)
(π,0)
(π,π)
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
 0.035
-2 -1.5 -1 -0.5  0  0.5  1  1.5  2
n=1.54, VC
(0,0)
(π,0)
(π,π)
Figure 3. Real part of the anomalous self-energy at various fillings: (a) n = 1.12,
no vertex corrections (b) n = 1.54, no vertex corrections (c) n = 1.12, vertex
corrections, (d) n = 1.54, vertex corrections. Calculations were carried out at
T = 0.005 with U = 0.6 and ω0 = 0.4. Momentum dependence corresponding to
non-local corrections is clearly visible at half-filling, but drops off as the edge of
the superconducting phase is reached. Vertex corrections are also most important
at half-filling. There is a dip in the anomalous self-energy because the vertex
corrections drop off more quickly in ωn, with opposite sign to φME, indicating
that the approximation is close to breakdown at half-filling. The slight increase
in the anomalous self-energy at n = 1.54 due to vertex corrections arises from a
change in the form of the electronic Green’s function. For half filling, the Green’s
function at the van-Hove points is pure imaginary, whereas for the dilute system,
it is mostly real, so sums over products of Green’s functions in the vertex can
change sign.
the fictitious field is completely removed. Iteration then continues until the true
superconducting state is reached. This procedure corresponds to initialising the self-
consistent cycle with a superconducting solution; note that similar techniques are used
for obtaining Mott insulating solutions in the Hubbard model using DMFT [17].
By following this procedure, a superconducting state may be found below the
transition temperature, TC . Green’s functions and self-energies computed in the
superconducting state can then be used to initialise the self-consistent equations
for similar couplings, fillings, temperatures (with an appropriate rescaling of the
Matsubara frequencies) and phonon frequencies. Above the transition temperature,
the magnitude of the anomalous Green’s function tends to zero during self-consistency
as expected.
It is possible to see the generic effects of vertex and non-local corrections by
examining the anomalous self energy. In figure 3, the anomalous self energy is shown
for n = 1.12 and n = 1.54 for a cluster size of NC = 4 with parameters of U = 0.6,
T = 0.005 and ω0 = 0.4 with and without vertex corrections. The panels are (a)
n = 1.12, no vertex corrections (b) n = 1.54, no vertex corrections (c) n = 1.12,
vertex corrections, (d) n = 1.54, vertex corrections. In panel (a), the momentum
dependence of the vertex neglected theory is clearly visible, and φ(K, iωn) has a much
Superconducting states of the quasi-2D Holstein model 8
larger value at the (π, 0) point. Momentum dependence is significantly reduced as
the system moves away from half-filling (panel b), indicating that Midgal-Eliashberg
theory is more accurate in dilute systems. This is expected, since in very dilute
systems, the electron density is sufficiently low that electrons meet very infrequently,
and therefore the crossed diagrams of figure 2b have extremely small contributions. By
scanning vertically, the effect of including vertex corrections can be seen. Corrections
are strongest close to half-filling, and drop off as the edge of the superconducting
phase is reached. Migdal–Eliashberg theory is clearly quite accurate for dilute 2D
systems, but it consistently fails close to half-filling. Initially, it seems as though
vertex corrections are larger than the vertex neglected results at n = 1.12. In fact,
this is not the case. As discussed in ref. [6], at half-filling vertex corrections act to
reduce the magnitude of the phonon self-energy, so there is much less renormalisation
of the phonon propagator. A smaller phonon propagator means that the effective
coupling is smaller, stabilising the expansion in λeff . There is a dip in the anomalous
self-energy because the vertex corrections drop off more quickly in ωn, with opposite
sign to φME, which is an indication that the approximation is close to breakdown at
half-filling. The slight increase in the anomalous self-energy at n = 1.54 due to vertex
corrections comes about from a change in form of the electronic Green’s function. For
half filling, the Green’s function at the van-Hove points is pure imaginary, whereas for
the dilute system, it is mostly real, so sums over products of Green’s functions can
change sign with respect to the Migdal–Eliashberg result. This sign change is also
seen in DMFT simulations of the 3D Holstein model [19].
At this stage, it is appropriate to examine the size of the parameter λeff that
defines the vertex correction expansion. Since the expansion in this case is in the full
phonon Green’s function, the expansion parameter is renormalised by the phonons,
and reads λeff = UD(µ)D(iωs = 0), where D(iωs = 0) is the phonon propagator at
zero Matsubara frequency. For dilute systems, D(µ) is typically small, and so λeff is
small (N.B. Unlike in 3D, D(µ) is never zero in 2D, because of the discontinuity in
the band edge of the non-interacting DOS). Close to half-filling, the DOS in 2D is
divergent, and this parameter is expected to be large. In the current approximation,
a small interplanar hopping was applied to stabilise the solution, so λeff is smaller
than expected in a pure 2D system. As noted in ref. [6], D(0) is reduced by vertex
corrections as compared to the Migdal–Eliashberg result. For most energies, the bare
density of states in 2D is smaller than the bare density of states in 3D, since the
divergence drops off logarithmically close to half filling. Therefore, λeff is only really
large for n = 1 within the current parameter range. For the mid to dilute limits,
the relative magnitude of the second order vertex correction goes like λ2 ∼ 0.04. At
half filling, with the current parameters, λ2 ∼ 0.5, so the approximation can only
be considered to be qualitatively correct. Nonetheless, the current approximation has
features appropriate to Hohenberg’s theorem (discussed later) and the bare DOS drops
off so quickly moving away from half-filling, that results are expected to be accurate
for most n.
How do the differences in the self-energy relate to observable quantities? One
of the big questions in unconventional superconductivity concerns the possible forms
that the order parameter can take, and a large discussion has grown up around issues
such as the existence of unconventional order parameters such as extended s-wave
and higher harmonics. To examine this idea, I demonstrate the evolution of the shape
of the anomalous pairing density (ns(k) = |T
n F (k, iωn)|), which is related to the
order parameter. In this paper, the superconducting order parameter is treated in
Superconducting states of the quasi-2D Holstein model 9
 0.05
 0.15
U=0.6, ω0=0.4, T=0.005, Nc=1
 0.05
 0.15
 0.05
 0.15
U=0.6, ω0=0.4, T=0.005, Nc=4
 0.05
 0.15
 0.05
 0.15
U=0.6, ω0=0.4, T=0.005, Nc=64
 0.05
 0.15
 0.14
 0.15
 0.16
 0.17
 0.18
 0.19
 0.21
 0.22
(0,π)(π,0)
NC= 1
NC= 4
NC=64
Figure 4. Variation of superconducting (anomalous) pairing density across the
Brillouin zone. U = 0.6, ω0 = 0.4, n = 1 and T = 0.005. Cluster sizes are
increased from NC = 1 to NC = 64. Pairing occurs between electrons close to
the “Fermi-surface” at k and the opposite face of the surface at −k. Also shown
is the pairing density at the Fermi surface for the 3 different cluster sizes (bottom
right). For a cluster size of Nc = 1 corresponding to DMFT, the pairing is
uniform around the “Fermi-surface”, demonstrating that momentum dependence
has been neglected. Momentum dependence favours additional pairing along the
kx = ky line, and a peak can clearly be seen. The expansion in spherical harmonics
contains even momentum states with m = 0. For Nc = 64, additional peaks
can be seen, suggesting that the order parameter also contains extra higher-order
harmonics. The Fermi-surface is not very clearly defined, with the mobile electrons
spread out over a significant range of momentum states. T = 0.005 << W , so
the spread should be very small in all locations in the Brillouin zone, except at
the van Hove points, (π, 0) and (0, π), indicating that the spreading is due to the
low dimensionality.
a fully self-consistent manner within the lattice symmetry and no assumptions have
been made in advance about its form.
Figure 4 shows the variation of superconducting pairing across the Brillouin zone.
In all of the panels, U = 0.6, ω0 = 0.4, n = 1 and T = 0.005. A range of cluster sizes
is shown. In the dynamical mean-field theory which corresponds to the Eliashberg
solution (cluster size of Nc = 1) the pairing is uniform around the Fermi-surface,
as is expected when momentum-dependence is neglected. The inclusion of non-local
momentum dependent fluctuations has a small, but significant effect on the ordering.
Pairing is reduced most at the (π, 0) and (0, π) points, leading to a visible peak at
(π/2, π/2). This demonstrates that the order parameter must necessarily include
higher harmonics. For Nc = 64 additional peaks are also seen. The additional
features may be examined by determining the parameters of an expansion in spherical
harmonics, clm =
(2π)3
Ylm(θ, φ)ns(k) (figure 5). This shows that the anomalous
density can be thought of as m = 0 harmonics with s, d, g... character, and additional
harmonics with m = ±4 in the g channel. The harmonics can be quite large, especially
away from half-filling, and undoubtedly need to be included if the superconductivity
is to be described correctly.
Superconducting states of the quasi-2D Holstein model 10
-0.025
-0.02
-0.015
-0.01
-0.005
 0.005
 0.01
 1  1.1  1.2  1.3  1.4  1.5  1.6  1.7
s, m=0
d, m=0
g, m=0
g, m=4,-4
Figure 5. Harmonic decomposition of the anomalous density computed for
Nc = 64 as chemical potential is varied. It can be seen that pure s-wave states
are the largest contributors to the anomalous density, followed by g and then d
states with m = 0. Owing to the hump at the (π/2, π/2) point, there are also
g states with m = ±4. The m = ±4 states have equal magnitude, so mtot = 0.
Note that the relative contribution of higher harmonics is greatest away from half
filling.
 0.05
 0.15
U=0.9, ω0=0.05, T=0.005, Nc=4
 0.05
 0.15
 0.05
 0.15
U=0.9, ω0=0.4, T=0.005, Nc=4
 0.05
 0.15
 0.05
 0.15
U=0.6, ω0=0.05, T=0.005, Nc=4
 0.05
 0.15
 0.05
 0.15
U=0.6, ω0=0.4, T=0.005, Nc=4
 0.05
 0.15
Figure 6. Variation of superconducting (anomalous) pairing density across the
Brillouin zone. T = 0.005 and NC = 4. Changes in the order parameter are
shown as coupling and phonon frequency are changed. As the phonon frequency
is increased, the momentum dependence also increases, and the Fermi-surface
is less well defined. For U = 0.9, ω0 = 0.05, the order parameter is almost
flat along the Fermi-surface, indicating that DMFT is a good approximation for
those parameters. For the largest coupling and phonon frequencies (at the edge
of applicability for the current approximation), the Fermi-surface is practically
destroyed.
Superconducting states of the quasi-2D Holstein model 11
Figure 6 shows the variation of superconducting (anomalous) pairing density
across the Brillouin zone as coupling and phonon frequency are changed. T = 0.005,
n = 1 and NC = 4 with vertex corrections excluded. As the phonon frequency is
increased, the momentum dependence also increases. For U = 0.9, ω0 = 0.05, the
order parameter is almost flat along the Fermi-surface, indicating that DMFT is a
good approximation for those parameters. Typically, additional coupling makes the
order more uniform in the Brillouin zone. Note that for very strong coupling, the
DMFT solution is expected to become exact, even in 2D, since the bare dispersion
is then essentially flat, and the problem is completely local. For weak coupling and
phonon frequency, there is a well defined Fermi-surface. For the largest coupling and
phonon frequencies (at the edge of applicability for the current approximation), the
Fermi-surface is practically destroyed.
It is of clear interest to map the phase diagram associated with superconducting
order. First, the superconductivity arising from DMFT is investigated in the absence
of vertex corrections. Figure 7 shows the resulting phase diagram. Note that in
the DMFT solution, the superconductivity is strongest at half-filling and the order
drops off monotonically as the filling increases. Assuming a form for the density of
states in 2D (with small interplane hoppnig) of D(ǫ) = (1− t log[(ǫ2 + t2⊥)/16t
2])/tπ2
(for |ǫ| < 4t) [26], the BCS result may be calculated using the expression TC(n) =
2ω0 exp(−1/|U |D(µ(n)))/π, with the chemical potential taken from the self-consistent
solution for a given n. This result also drops off monotonically. Results in the dilute
limit are in good agreement with the BCS result. Closer to half-filling, the DMFT
result is significantly smaller than the BCS result (which predicts TC(n = 1) > 0.07).
The difference in results between the two mean-field theories at half-filling is due to
the self-consistency in the DMFT. For small U , the self-consistent equations converge
on the first iteraction, but for larger U , the phonon and electron Green’s functions are
significantly renormalised, thus reducing the transition temperature.
To show the differences induced by spatial fluctuations, the phase diagram is
computed for a cluster size of NC = 4. Figure 8 shows the total density of
superconducting pairs for U = 0.6, ω0 = 0.4 and various temperatures and fillings,
without vertex corrections. Of most interest is an anomalous bump centred about
n = 1.25, indicating that the strongest superconductivity occurs away from half filling,
and that this is due to non-local fluctuations in 2D. Assuming a material with a non-
interacting band width of 1eV, the highest transition temperature would correspond
to 145K. This value is higher that that eventually expected in real materials. For
instance, the effect of a Coulomb pseudopotential UC will be a reduction of the
transition temperature. The standard BCS result is modified by Coulomb repulsion
in the following way, TC = 2ω0 exp(−1/(λ− UC)))/π.
In addition to the TC reduction due to Coulomb repulsion, a fundamental limit
on the transition temperature in pure 2D is the Hohenberg theorem [27]. This is
closely related to the effects of spatial fluctuations. The basis of Hohenberg’s proof is
the divergence of certain quantities (which are known to be finite) in d ≤ 2 at k = 0
when anomalous expectation values (e.g. the superconducting order parameter) are
non zero. In the DMFT solution, there are no specific k = 0 states due to the
coarse graining, and so the divergence in the correlation functions that led Hohenberg
to determine that the order parameter must be zero for zero momentum pairing
in d ≤ 2 is washed out, leading to a finite transition temperature for the 2D local
approximation. In DCA, partial momentum dependence is restored. Therefore, the
effects of the washed out divergences are stronger. There is still a finite transition
Superconducting states of the quasi-2D Holstein model 12
 0.02
 0.04
 0.06
 0.08
 0.01
 0.02
 0.03
 0.04
 1  1.2  1.4  1.6  1.8
 0.02
 0.04
 0.06
 0.08
U=0.6, ω0=0.4, Nc=1
Figure 7. Superconducting phase diagram showing the total number of
superconducting states. U = 0.6, ω0 = 0.4 and various T . A cluster size of Nc = 1
has been used, and no vertex corrections are included. The superconductivity
is strongest at half-filling and the order drops off monotonically as the filling
increases. Results in the dilute limit are in good agreement with the BCS result
(the transition temperature from BCS is shown as the line with points in the
ns = 0 plane). Closer to half-filling, the DMFT result is significantly smaller
than the BCS result (which predicts TC(n = 1) > 0.07). The difference in results
between the two mean-field theories at half-filling is due to the self-consistency in
the DMFT.
temperature, but it is reduced wherever there is strong momentum dependence. This
is demonstrated by the drop in superconducting order at and close to half-filling in
figure 8, where the momentum dependence is strongest. As the number of cluster
points increases, the momentum resolution becomes superior, and the divergences of
Hohenberg’s theorem are expected to emerge in a systematic manner. In real materials
with quasi-2D character, some interplane hopping remains. In that case, the results
from small cluster DCA are expected to be more reliable.
Finally, I demonstrate the effects of vertex corrections on the superconducting
phase diagram. Figures 9 and 10 show the total number of superconducting states
as a function of filling and temperature for cluster sizes of NC = 1 and NC = 4
respectively. An electron-phonon coupling of U = 0.6 and phonon frequency of
ω0 = 0.4 have been used. Vertex corrections do not appear to make a large difference
to the DMFT result in figure 9. For NC = 4, the bulge that was seen in the non
vertex-corrected phase diagram is very clearly enhanced. Superconductivity at half-
filling is completely suppressed in the vertex corrected solution. This is the precursor
to Hohenberg’s theorem applying across the entire phase diagram, and both the effects
of spatial fluctuations and the lowest order vertex correction were essential to obtain
that agreement.
Superconducting states of the quasi-2D Holstein model 13
 0.02
 0.04
 0.06
 0.08
U=0.6, ω0=0.4, Nc=4
 0.01
 0.02
 0.03
 0.04
 1  1.2  1.4  1.6  1.8
 0.02
 0.04
 0.06
 0.08
Figure 8. Superconducting phase diagram showing the total number of
superconducting states. U = 0.6, ω0 = 0.4 and various T . A cluster size of Nc = 4
has been used, and no vertex corrections are included. There is an anomalous
bump centred about n = 1.25, indicating that the strongest superconductivity
occurs away from half filling. The highest transition temperature occurs for
T = 0.025. The reduction in the transition temperature close to half-filling shows
the onset of Hohenberg’s theorem. The largest superconductivity coincides with
the increase in components without pure s-wave character (see figure 5).
 0.02
 0.04
 0.06
 0.08
 0.12
U=0.6, ω0=0.4, Nc=1, VC
 0.01
 0.02
 0.03
 0.04
 1  1.2  1.4  1.6  1.8
 0.02
 0.04
 0.06
 0.08
 0.12
Figure 9. Superconducting phase diagram showing the total number of
superconducting states. U = 0.6, ω0 = 0.4 and various T . A cluster size of
Nc = 1 has been used, and vertex corrections are included. As in figure 7, the
DMFT result falls off monotonically with increased filling.
Superconducting states of the quasi-2D Holstein model 14
 0.02
 0.04
 0.06
 0.08
U=0.6, ω0=0.4, Nc=4, VC
 0.01
 0.02
 0.03
 0.04
 1  1.2  1.4  1.6  1.8
 0.02
 0.04
 0.06
 0.08
Figure 10. Superconducting phase diagram showing the total number of
superconducting states. U = 0.6, ω0 = 0.4 and various T . A cluster size of
Nc = 4 has been used, and vertex corrections are included. Superconducting
states are suppressed at half-filling, and there is a significant bulge away from
half-filling, with a maximum transition temperature of 0.015W. It is significant
that the transition temperature is reduced to zero at half-filling and supressed
close to half filling, since reduction of transition temperatures is expected in 2D
due to Hohenberg’s theorem.
5. Summary
In this paper I have carried out DCA calculations of a quasi-2D Holstein model in
the superconducting state with large in plane hopping and small out of plane hopping
(t = 0.25, t⊥ = 0.01). Several approximations to the self-energy were considered,
including the neglect of vertex corrections (which corresponds to a momentum-
dependent extension to the Eliashberg theory), the inclusion of vertex corrections
as a corrected approximation for stronger couplings, and the introduction of spatial
fluctuations. The anomalous self energy, superconducting order parameter and phase
diagram were calculated.
The superconducting order parameter was found to modulate around the fermi
surface, and is not pure s-wave. Analysis of the harmonics showed that the state is
a conbination of s, d, f etc. states with m = 0, and other states with integer value of
m = ±4n. The total angular momentum is always zero. The contribution of the m 6= 0
states is considerably larger away from half filling. Increases in bare phonon frequency
tended to increase the strength of the superconducting order, and contributed to a
degeneration of the Fermi-surface.
The phase diagram was analysed for small cluster sizes of NC = 1, 4. For NC = 1,
the phase diagram was shown to agree qualitatively with the BCS theory. When
spatial fluctuations are included, the superconducting order is suppressed at half-
filling, leading to a characteristic hump at a doping of approximately δn = 0.25. Vertex
corrections completely suppress superconductivity at half-filling, which is believed to
be a manifestation of Hohenberg’s theorem. In particular, the states with the largest
momentum dependence showed the strongest reduction in the transition temperature,
Superconducting states of the quasi-2D Holstein model 15
indicating that spatial fluctuations as well as vertex corrections contribute to the
supression of superconducting order in pure 2D materials.
Acknowledgments
JPH would like to thank the University of Leicester for hospitality and use of facilities
while carrying out this research.
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http://arxiv.org/abs/cond-mat/0404055
	Introduction
	The dynamical cluster approximation
	The Holstein model
	Results
	Summary
