On Migdal’s theorem and the pseudogap
arXiv:cond-mat/0301228v2 [cond-mat.str-el] 21 Mar 2003
P. Monthoux
Cavendish Laboratory, University of Cambridge
Madingley Road, Cambridge CB3 0HE, United Kingdom
(October 29, 2018)
Abstract
We study a model of quasiparticles on a two-dimensional square lattice
coupled to Gaussian distributed dynamical molecular fields. We consider two
types of such fields, a vector molecular field that couples to the quasiparticle
spin-density and a scalar field coupled to the quasiparticle number density.
The model describes quasiparticles coupled to spin or charge fluctuations,
and is solved by a Monte Carlo sampling of the molecular field distributions.
When the molecular field correlations are sufficiently weak, the corrections to
the self-consistent Eliashberg theory do not bring about qualitative changes
in the quasiparticle spectrum. But for a range of model parameters near the
magnetic boundary, we find that Migdal’s theorem does not apply and the
quasiparticle spectrum is qualitatively different from its Eliashberg approximation. In the range of model parameters studied, we find the transverse
spin-fluctuation modes play a key role. While a pseudogap opens when quasiparticles are coupled to antiferromagnetic fluctuations, such a pseudogap is
not observed in the corresponding charge-fluctuation case for the range of parameters studied, where vertex corrections are found to effectively reduce the
strength of the interaction. This suggests that one has to be closer to the border of long-range order to observe pseudogap effects in the charge-fluctuation
1
case than for a spin-fluctuation induced interaction under otherwise similar
conditions.
An important feature of the magnetic pseudogap found in the present
calculations is that it is strongly anisotropic. It vanishes along the diagonal
of the Brillouin zone and is large near the zone boundary. In the case of
ferromagnetic fluctuations, we also find a range of model parameters with
qualitative changes in the quasiparticle spectral function not captured by the
one-loop approximation, in that the quasiparticle peak splits into two. We find
that one needs to be closer to the magnetic boundary to observe the pseudogap
effects in the nearly ferromagnetic case relative to the nearly antiferromagnetic
one, under otherwise similar conditions. We provide intuitive arguments to
explain the physical origin of the breakdown of Migdal’s theorem.
PACS Nos. 71.27.+a
Typeset using REVTEX
2
I. INTRODUCTION
The polarizer-analyser analogy provides an intuitive description of the effective interaction between quasiparticles in a quantum many-body system. The first quasiparticle
polarizes the medium in which it travels and the second quasiparticle, the analyser, feels the
disturbance induced by the first one. In a strongly correlated system one can expect this
induced polarization of the medium to be very complex, and in practice simplifying assumptions are made. A commonly used approximation is the description of the polarization effects
by the appropriate linear response function of the material. Furthermore, one typically only
considers the interaction channel for which the linear response of the system is the largest.
On the border of long-range magnetic order for instance, the spin-spin correlation function
is the most enhanced and it is plausible that the dominant interaction channel is of magnetic
origin and depends on the relative spin orientations of the interacting quasiparticles.
It has been shown that such a magnetic interaction, treated in the self-consistent Eliashberg approximation, can produce anomalous normal state properties and superconducting
instabilities to anisotropic pairing states. It correctly predicted the symmetry of the Cooper
state in the copper oxide superconductors [1] and is consistent with spin-triplet p-wave pairing in superfluid 3 He [for a recent review see, e.g., Ref. [2]]. One also gets the correct order of
magnitude of the superconducting and superfluid transition temperature Tc when the model
parameters are inferred from experiments in the normal state of the above systems. For the
case of a nearly half-filled single band, the calculations showed that the Eliashberg superconducting transition temperature Tc is higher for the tetragonal (quasi-2D) than cubic (3D)
lattice [3–6]. Particularly striking is the comparison between the cubic antiferromagnetic
metal CeIn3 and the closely related compound CeCoIn5 [7]. Superconductivity is found
to extend over a much wider range in both temperature and pressure in CeCoIn5 than in
CeIn3 . These findings and the growing evidence that the pairing symmetry in CeCoIn5 is
d-wave in character, were correctly anticipated by the magnetic interaction model.
While there have been a number of examples of superconductivity on the border of anti3
ferromagnetism, the corresponding phenomenon on the edge of metallic ferromagnetism has
only been found recently. This result is not surprising within the framework of the magnetic
interaction model. For otherwise equivalent conditions, the superconducting transition temperature is typically much higher on the border of antiferromagnetism than on the border
of ferromagnetism [5,6,8]. An intuitive understanding of this finding pointed to candidate
systems in which superconductivity on the border of ferromagnetism would be more likely
to be observed. In particular, one ought to look for systems with strong spin anisotropy,
i.e, with stong spin-orbit coupling and/or in a weakly spin-polarized state. This suggested
a more detailed investigation of UGe2 at high pressure which satisfied the above conditions
and could be prepared in high purity form. This material proved to be the first example of
the co-existence of superconductivity and itinerant-electron ferromagnetism [9].
The self-consistent Eliashberg treatment of the magnetic interaction model can produce
strongly damped quasiparticles, but the electronic spectral function one obtains always shows
a quasiparticle peak as one approaches the Fermi level. This is in qualitative disagreement
with photoemission experiments on underdoped cuprate superconductors which show a near
absence of a quasiparticle peak near the (π, 0) point in the Brillouin zone [10]. This depletion of quasiparticle states, or pseudogap, is also seen in thermodynamic measurements
[11]. The phenomenon may not be specific to the underdoped cuprates. For instance, the
possible existence of a pseudogap in the heavy fermion compound CeCoIn5 has recently
been reported [12].
Is the disagreement between theory and experiment the manifestation of a fundamental
flaw in the approach or does it simply reflect the inadequacy of the approximations used in
the solution of the model? Questions regarding the validity of the Eliashberg treatment of the
magnetic interaction model have been raised [13]. The one-loop approximation effectively
assumes that quasiparticles behave as test particles. There must therefore be corrections to
the simple theory, referred to as vertex corrections, coming from the fact that real and test
particles behave differently. One can expect these vertex corrections to produce quantitative
changes to the self-consistent Eliashberg theory as one approaches the border of long-range
4
magnetic order. In Ref. [14] it was shown that for optimally doped cuprates, vertex corrections did not bring about significant changes to the single spin-fluctuation approximation,
producing an enhancement of the spin-fluctuation interaction of the order of 20%. The
sign of this correction [14] is opposite to that expected in the case of a phonon-mediated
interaction [15] and in the spin-density-wave phase [13]. This enhancement of the fermion
spin-fluctuation vertex in the paramagnetic state is due to the transverse spin-fluctuation
modes.
It has been argued, in particular by Schrieffer [13], that as the antiferromagnetic correlations got stronger, the system should display characteristics akin to the antiferromagnetic
insulating state and that the behavior of the quasiparticles should become qualitatively different from that of a simple metal. The results presented here show that this physical insight
is correct in that Midgal’s theorem can qualitatively break down when the antiferromagnetic
correlations become strong enough and a different state emerges.
The validity of Baym-Kadanoff many-body theories such as the fluctuation exchange
approximation [16], which is in many ways similar to the Eliashberg theory of the magnetic
interaction model, has been extensively studied by Tremblay and collaborators in the context of the Hubbard model [17–19]. They find that close to the magnetic boundary, Migdal’s
theorem qualitatively breaks down, in that a critical-fluctuation-induced pseudogap (or precursor pseudogap) is observed in the Quantum Monte Carlo simulations but is not found in
the fluctuation exchange approximation. Similar results about the role of vertex corrections
were reported in the case of the attractive Hubbard model [20–24], where in this case the
precursor pseudogap is caused by critical pairing fluctuations.
In this paper, we examine corrections to the single spin-fluctuation exchange approximation in two dimensions using a non-perturbative formulation of the magnetic interaction
model which is amenable to computer simulation [25]. We find that as the antiferromagnetic
correlations become strong enough for vertex corrections to produce a qualitative change relative to the one-loop approximation, a pseudogap opens in the quasiparticle spectrum. In
this respect, our results are similar to those obtained for the Hubbard model [17–19]. The
5
pseudogap we find is strongly anisotropic in that it vanishes along the diagonal of the Brillouin zone and is large near the zone boundary. We demonstrate that, in the range of model
parameters studied here, the transverse spin-fluctuation modes are key to the appearance
of the pseudogap by considering the case of commensurate charge-fluctuations with a spectrum identical to that of the paramagnons. While a pseudogap opens when quasiparticles
are coupled to magnetic fluctuations, such a pseudogap is not observed in the corresponding
charge-fluctuation case for the range of parameters studied, where vertex corrections are
found to effectively reduce the strength of the interaction. This suggests that one has to be
closer to the border of long-range order to observe pseudogap effects in the charge-fluctuation
case than for a spin-fluctuation induced interaction under otherwise similar conditions. In
the case of nearly ferromagnetic systems, as magnetic correlations get stronger, we also find
qualitative changes in the quasiparticle spectral function not captured by the one-loop approximation. The quasiparticle peak splits into two distinct peaks with a lowering of the
tunneling density of states at the Fermi level. For the range of parameters studied, this
suppression of the tunneling density of states at the Fermi level is weaker than in the case
of nearly antiferromagnetic systems for otherwise similar magnetic and electronic spectrum
parameters.
The paper is organized as follows. In the next section we describe the model and the class
of vertex corrections considered. Section III contains the results of the numerical simulations.
In section IV, we give intuitive arguments for the physical origin of the pseudogap. Finally
we give a summary and outlook. Most of the technical details are included in an appendix.
II. MODEL
There are essentially two different ways to describe quantum many-body systems. In
the Newtonian or Hamiltonian approach, one considers particles interacting with each other
via pairwise interaction potentials. This is the point of view commonly adopted in the
perturbation-theoretic approach to the non-relativistic many-electron problem. In the rela6
tivistic version of the theory, however, one adopts a Maxwellian point of view in which the
interactions between electrons are mediated by a field, the quantized electromagnetic field.
The Maxwellian approach is also widely used to carry out numerical simulations of interacting systems based on the Feynman path integral. In such functional integrals, bosonic
fields are represented by c-numbers. But fermion fields must be represented by anticommuting, or Grassmann, numbers that are not easily handled by digital computers. When
a dynamical molecular (or Hubbard-Stratonovich) field is introduced to mediate the interactions between the fermions, the problem is reduced to that of non-interacting particles
in a fluctuating field. The integrals over the anticommuting variables can then be evaluated exactly, at least formally, thus eliminating the troublesome Grassmann numbers from
the problem. In the following, we adopt the Maxwellian point of view and use dynamical
molecular fields to mediate the interactions between quasiparticles.
In general, the distribution of Hubbard-Stratonovich fields is very complex and usually
leads to intractable problems due to the infamous ”fermion sign problem”. In a many-body
system, the presence of other particles produces changes in the effective interaction between
two particles through screening effects and induces arbitrarily complex self-interactions of
the dynamical molecular fields.
In this paper, we assume that the renormalization of the effective two-body interaction
can be accounted for by a redefinition of the parameters entering the bare interaction. And
we ignore all the self-interactions of the Hubbard-Stratonovich fields. The model was introduced in Ref. [25] and bears some resemblance to the ”quenched approximation” of lattice
gauge theories introduced by Marinari et al., who, incidentally, chose the name by analogy
to condensed matter physics [26]. A recent application of the quenched approximation to
the pseudogap problem for static Hubbard-Stratonovich fields and a clear exposition of the
formalism is given by Posazhennikova and Coleman [27]. But in the present work, however,
we use dynamical rather than static Hubbard-Stratonovich fields and a non-separable form
of the molecular field correlation function.
Very close to the boundary of magnetic or charge long-range order, the self-interactions
7
of the dynamical molecular field ignored in the present work are known to be important for
low dimensional systems [28]. The approximations made here may not be appropriate in
a number of other cases. The virtue of the present approach is that it gives insights into
quantum many-body problems that are essentially non-perturbative. More importantly,
the results presented in this paper demonstrates that the simplest model already yields
interesting physics. Some of the simplifications made here can in principle be relaxed and
the theory extended accordingly.
To be more specific, we consider particles on a two-dimensional square lattice whose
Hamiltonian in the absence of interactions is
ĥ0 (τ ) = −
X
i,j,α
†
tij ψiα
(τ )ψjα (τ ) − µ
X
†
ψiα
(τ )ψiα (τ )
(2.1)
iα
†
where tij is the tight-binding hopping matrix, µ the chemical potential and ψiα
, ψiα respec-
tively create and annihilate a fermion of spin orientation α at site i. We take tij = t if sites
i and j are nearest-neighbors and tij = t′ if sites i and j are next-nearest-neighbors.
To introduce interactions between the particles, we couple them to a dynamical molecular
(or Hubbard-Stratonovich) field. It is instructive to consider two different types of molecular
fields. In the first instance, we consider a vector Hubbard-Stratonovich field that couples
locally to the fermion spin density. This is the case considered in Ref. [25] and describes
quasiparticles coupled to magnetic fluctuations. To illustrate the role played by transverse
spin-fluctuations we shall also consider the case of a scalar field that couples locally to
the fermion number density. This case corresponds to a coupling to charge-fluctuations or,
within the approximation we are using here, ”Ising”-like magnetic fluctuations where only
longitudinal modes are present. The Hamiltonians at imaginary time τ for particles coupled
to the fluctuating exchange or scalar dynamical field are then
g X
†
Mi (τ ) · ψiα
(τ )σαγ ψiγ (τ )
ĥ(τ ) = ĥ0 (τ ) − √
3 iαγ
(2.2)
ĥ(τ ) = ĥ0 (τ ) − g
(2.3)
X
†
(τ )ψiα (τ )
Φi (τ )ψiα
iα
where Mi (τ ) = (Mix (τ ), Miy (τ ), Miz (τ ))T and Φi (τ ) are the real vector exchange and scalar
8
Hubbard-Stratonovich fields respectively, and g the coupling constant. The reason for the
√
choice of an extra factor 1/ 3 in Eq. (2.2) will become clear later.
Since we ignore the self-interactions of the molecular fields, their distribution is Gaussian
and given by [25]
1
P[M] = exp
Z
Z=
Z
X
M(q, iνn ) · M(−q, −iνn )
−
2α(q, iνn )
q,νn
DM exp
!
(2.4)
X
M(q, iνn ) · M(−q, −iνn )
−
2α(q, iνn )
q,νn
!
(2.5)
in the case of a vector exchange molecular field and
1
P[Φ] = exp
Z
Z=
Z
X
Φ(q, iνn )Φ(−q, −iνn )
−
2α(q, iνn )
q,νn
DΦ exp
!
(2.6)
X
Φ(q, iνn )Φ(−q, −iνn )
−
2α(q, iνn )
q,νn
!
(2.7)
in the case of a scalar Hubbard-Stratonovich field. In both cases νn = 2πnT since the
dynamical molecular fields are periodic functions in the interval [0, β = 1/T ]. The Fourier
transforms of the molecular fields are defined as
MR (τ ) =
ΦR (τ ) =
X
(2.8)
q,νn
M(q, iνn ) exp − i[q · R − νn τ ]
(2.9)
q,νn
Φ(q, iνn ) exp − i[q · R − νn τ ]
X
We are considering the case where there is no long-range magnetic or charge order. The
average of the dynamical molecular fields must then vanish and their Gaussian distributions
Eqs. (2.4,2.6) are completely determined by their variance α(q, iνn ), which we take to be
α(q, iνn ) =
1 T
χ(q, iνn )
2N
if M(q, iνn ) or Φ(q, iνn ) complex
T
χ(q, iνn )
N
if M(q, iνn ) or Φ(q, iνn ) real
(2.10)
where N is the number of allowed wavevectors in the Brillouin zone. Then
D
E
T
χ(q, iνn )δq,−k δνn ,−Ωn δi,j
N
D
E
T
Φ(q, iνn )Φ(k, iΩn ) = χ(q, iνn )δq,−k δνn ,−Ωn
N
Mi (q, iνn )Mj (k, iΩn ) =
9
(2.11)
(2.12)
where < . . . > denotes an average over the probability distributions Eq. (2.4) and Eq. (2.6)
for the vector and scalar cases respectively. In order to compare the scalar and vector molecular fields, we take the same form for their correlation function χ(q, iνn ) and parametrize it
as in Refs. [5,8]. In what follows, we set the lattice spacing a to unity. For real frequencies,
we have
χ(q, ω) =
χ0 κ20
κ2 + qb2 − i η(ωbq )
(2.13)
where κ and κ0 are the correlation wavevectors or inverse correlation lengths in units of the
lattice spacing, with and without strong correlations, respectively. Let
2
= 4 ± 2(cos(qx ) + cos(qy ))
qb±
(2.14)
We shall consider commensuate charge-fluctuations and antiferromagnetic spinfluctuations, in which case the parameters qb2 and η(qb) in Eq. (2.13) are defined as
2
qb2 = qb+
η(qb) = T0 qb−
(2.15)
(2.16)
where T0 is a characteristic temperature.
We shall also consider the case of ferromagnetic spin-fluctuations, where the parameters
qb2 and η(qb) in Eq. (2.13) are given by
2
qb2 = qb−
η(qb) = T0 qb−
(2.17)
(2.18)
χ(q, iνn ) is related to the imaginary part of the response function Imχ(q, ω), Eq. (2.13), via
the spectral representation
χ(q, iνn ) = −
Z
+∞
−∞
dω Imχ(q, ω)
π iνn − ω
(2.19)
To get χ(q, iνn ) to decay as 1/νn2 as νn → ∞, as it should, we introduce a cutoff ω0 and take
Imχ(q, ω) = 0 for ω ≥ ω0 . A natural choice for the cutoff is ω0 = η(qb)κ20 .
10
In the approximation we are considering, the single particle Green’s function is the
average over the probability distributions P[M] (Eq. (2.4)) or P[Φ] (Eq. (2.6)) of the fermion
Green’s function in a dynamical vector or scalar field.
′ ′
G(iστ ; jσ τ ) =
G(iστ ; jσ ′ τ ′ ) =
Z
Z
DM P[M] G(iστ ; jσ ′ τ ′ |[M])
(2.20)
DΦ P[Φ] G(iστ ; jσ ′ τ ′ |[Φ])
(2.21)
where
†
′
G(iστ ; jσ ′ τ ′ |[M] or [Φ]) = −hTτ {ψiσ (τ )ψjσ
′ (τ )}i
(2.22)
is the single particle Green’s function in a dynamical molecular field and is discussed in
the appendix. In evaluating expressions Eqs. (2.20,2.21) one is summing over all Feynman
diagrams corresponding to spin or charge-fluctuation exchanges [25,27]. The diagrammatic
expansion of the Green’s function in a dynamical field, Eq. (2.22) and its average, Eq. (2.20)
are shown pictorially in Fig. 1. A non-perturbative study of quasiparticles coupled to static
magnetic fluctuations using a diagram summation technique was reported in Ref. [29]. In
this paper we do not resort to a diagrammatic expansion but rather evaluate the averages
in Eqs. (2.20,2.21) over the dynamical molecular fields by Monte Carlo sampling.
It is very instructive to compare the results of the Monte Carlo simulations with the
self-consistent Eliashberg calculations for the same model. If one only considers single spin
or charge-fluctuation exchange processes, the single particle Green’s function is given by:
Σ(p, iωn ) = g 2
T XX
χ(p − k, iωn − iΩn )G(k, iΩn )
N Ωn k
G(p, iωn ) =
1
iωn − (ǫp − µ) − Σ(p, iωn )
(2.23)
(2.24)
where Σ(p, iωn ) is the quasiparticle self-energy, G(p, iωn ) the one-particle Green’s function.
ǫp is the tight-binding dispersion relation obtained from Fourier transforming the hopping
√
matrix tij in Eq. (2.1) and µ the chemical potential. The choice of the factor 1/ 3 in Eq. (2.2)
means one obtains the same one-loop equations in the the case of an exchange molecular
11
field M and that of the scalar field Φ, thereby simplifying the comparison between the two
cases.
III. RESULTS
The quasiparticle dispersion relation for the two-dimensional square lattice is obtained
from Eq. (2.1). We measure all energies and temperatures in units of the nearest-neighbor
hopping parameter t. We set the next-nearest-neighbor hopping parameter t′ = −0.45t.
The chemical potential is adjusted so that the electronic band filling is n = 0.9. The
dimensionless parameters describing the molecular field correlations are g 2 χ0 /t, T0 /t, κ0
and κ. A complete exploration of the parameter space of the model is beyond the scope of
this preliminary study. We chose a representative value for κ20 = 12, and set T0 = 0.67t as in
our earlier work [5,8]. For an electronic bandwidth of 1eV , T0 ≈ 1000◦ K. We only consider
one value of the coupling constant g 2χ0 /t = 2. In the random phase approximation, the
magnetic instability would be obtained for a value of g 2 χ0 /t of the order of 10. We consider
what happens to the quasiparticle spectrum at a fixed temperature T = 0.25t as the inverse
correlation length κ changes.
The calculations were done on a 16 by 16 lattice, with 41 imaginary time slices, or
equivalently 41 Matsubara frequencies for the molecular fields, M(q, iνn ) and Φ(q, iνn ) (νn =
2πnT , with n = 0, ±1, . . . , ±20).
By analytic continuation of the single particle Green’s function G(k, τ ) or G(k, iωn ) one
can obtain the quasiparticle spectral function A(k, ω) = − π1 Im GR (k, ω) and the tunneling
density of states N(ω) =
1
N
P
k
A(k, ω), where GR (k, ω) is the retarded single particle Green’s
function. In the case of the one-loop approximation, G(k, iωn ) is analytically continued from
imaginary to real frequencies by means of Padé approximants [30]. The imaginary time
Monte Carlo data is analytically continued with the Maximum Entropy method [31]. We
have used 2000 MC samples binned in groups of 20 to make 100 measurements. We have
used three different versions of the Maximum Entropy method. The Classic MaxEnt and
12
two versions of Average MaxEnt, which is the method recommended in Ref. [31], where the
probability for the α parameter is either a constant or proportional to 1/α [31]. In all cases
we chose a flat default model. The results for the three different versions of the Maximum
Entropy method tried were nearly identical in the case of the vector molecular field. There
were slight differences between the Classic and Average MaxEnt solutions for the spectral
function in the scalar molecular field case, while the two versions of the Average MaxEnt
gave nearly identical results. The choice of the ∼ 1/α probability distribution in the Average
MaxEnt method gave a slightly better fit to the MC data and all the results shown here are
those obtained with this choice of probability distribution for the α parameter.
Fig. 2 shows the tunneling density of states N(ω) and the spectral function A(k, ω) =
− π1 ImGR (k, ω) for the one-loop Eliashberg approximation to the Green’s function,
Eqs. (2.23,2.24), in the case of commensurate charge-fluctuations and antiferromagnetic
spin-fluctuations. Figs. 2b,c show a strong anisotropy of the spectral function. A(k, ω) is
sharper when k is along the diagonal compared to the case where k is near a hot spot
(which is a point on the Fermi surface accessible from another via a momentum transfer of
Q = (π, π)). Also note the monotonic broadening of A(k, ω) as the parameter κ2 is reduced.
In Fig. 3 we show the tunneling density of states and quasiparticle spectral function one
obtains from the Monte Carlo calculation with a coupling to the scalar dynamical molecular
field Φ. By comparison to the one-loop self-consistent results of Fig. 2, the vertex corrections
give rise to a sharpening of the quasiparticle spectral function, except for κ2 = 4 and
k = (3π/8, 3π/8). For the range of values of model parameters considered here, the multiple
charge-fluctuation exchanges do not lead to a breakdown of the quasiparticle picture. Note
that contrary to the Eliashberg result, as κ2 is reduced from 4, the spectral function initially
sharpens before broadening again at the lower values of κ2 considered here. For κ2 = 4, 2,
the spectral function is sharper near the hot spot than along the diagonal, again in contrast
to the one-loop self-consistent result. It is approximately isotropic at κ2 = 1 and becomes
sharper along the diagonal than near the hot spot at the lower values of κ2 = 0.50, 0.25.
Quite generally Fig. 3 shows that when vertex corrections are included the spectral function
13
is less anisotropic than the Eliashberg result for the values of κ2 considered.
The Monte Carlo results for quasiparticles coupled to antiferromagnetic spin-fluctuations
are shown in Fig. 4. For κ2 ≤ 1, a pseudogap appears at the point k = (π, π/8) but not
along the diagonal, in qualitative agreement with experiments on underdoped cuprates [10].
The pseudogap also shows up in the tunneling density of states N(ω) which is suppressed at
the Fermi level for κ2 ≤ 1. By comparing the spectral function of Figs. 4b,c to the one-loop
self-consistent result, we see that for the values of κ2 where there isn’t a pseudogap, vertex
corrections lead to a broadening of the spectral function and a reduction of the momentum
anisotropy of the spectral function. The contrast between the non-perturbative calculations
for charge and spin-fluctuations and the comparison with the Eliashberg result is shown in
Fig. 5 for κ2 = 2. In this case it is clear that the effect of vertex corrections is to make
the spectral function sharper than its Eliashberg approximation in the charge-fluctuation
case and broader when the quasiparticles are coupled to magnetic fluctuations. For these
model parameters, corrections to the Eliashberg approximation reduce the effective chargemediated interaction and enhance the coupling to magnetic fluctuations.
In Fig. 6 the results of the one-loop calculations in the case of ferromagnetic spinfluctuation exchange are shown. As could be expected, the spectral function for quasiparticles near the Fermi level does not depend strongly on whether the momentum is near
the hot spot, Fig. 6b, or along the diagonal in the Brillouin zone, Fig. 6c, in contrast to the
nearly antiferromagnetic case. At the Eliashberg level, the spectral function broadens monotonically as the correlation wavevector κ is reduced and is broader than the corresponding
nearly antiferromagnetic case, Fig. 2b,c.
The Monte Carlo results for quasiparticles coupled to ferromagnetic spin-fluctuations
are shown in Fig. 7. The tunneling density of states at the Fermi level N(ω = 0) begins
to drop as the parameter κ2 ≤ 0.50. For κ2 = 0.25, the quasiparticle peak has effectively
split into two peaks, and a precursor of this effect can be seen at κ2 = 0.50. This phenomenon is qualitatively different from what one can obtain at the one-loop level, Fig. 6,
just as the pseudogap seen in Fig. 4, when the quasiparticle interactions are mediated by
14
an antiferromagnetically correlated exchange field. For the larger value of κ2 where the
splitting of the quasiparticle peak is not observed, a comparison of the Monte Carlo results
with the one-loop self-consistent calculations shown in Fig. 6 shows that vertex corrections
bring about a broadening of the quasiparticle spectral function. Therefore, just as in the
antiferromagnetic case, vertex corrections enhance the magnetically-mediated interaction,
albeit to a lesser degree, as a close look at Figs. 2,4 and 6,7 indicates. This could explain
why qualitative changes from the Eliashberg solution are seen for smaller values of κ2 for
coupling to ferromagnetic than antiferromagnetic fluctuations.
IV. DISCUSSION
For the model considered here, at the one-loop level the exchange of commensurate
charge-fluctuations and antiferromagnetic spin-fluctuations yield the same quasiparticle
properties. Once vertex corrections are included, the differences between the two cases
are evident. In all but one case, k = (3π/8, 3π/8) and κ2 = 4, corrections to the one-loop
approximation make the quasiparticle peak sharper than the Eliashberg result for chargefluctuations. In the case of antiferromagnetic spin-fluctuations, the quasiparticle peak gets
broader as vertex corrections are included and a pseudogap appears for the lower values of κ2
studied. The difference between the scalar and vector molecular fields appears at the leading
vertex correction to the one-loop approximation [14] shown in Fig. 8. The frequency and
momentum integrals are the same in both cases since we assume the commensurate chargefluctuations and antiferromagnetic spin-fluctuations have the same spectrum. However, the
spin sums in the two cases are not identical. In the case of a coupling of the molecular
field to the quasiparticle spin-density, one gets a factor coming from the Pauli matrices at
each vertex in the diagram. For the diagram shown in Fig. 8, this factor is
P
i,j
σiσj σiσj .
One can split the sum into the i = j and i 6= j terms, and use the fact that σ i σ i = 1 and
σ i σ j = −σ j σ i if i 6= j. Then
P
i,j
σi σj σiσj =
P
i
σi σiσiσi −
P
i6=j
σ i σ i σ j σ j = 3 − 6 = −3.
Note that the longitudinal spin-fluctuations contribute a term σ z σ z σ z σ z = 1 and thus the
15
change in sign is caused by the presence of transverse magnetic modes. The corresponding
factor in the case of charge-fluctuations is 1, just as in the case without transverse magnetic modes. Coupling the quasiparticles to the spin-density instead of the number density
produces a leading vertex correction with the opposite sign. The process depicted in Fig. 8
enhances the magnetic interaction [14] while this same diagram leads to a suppression of the
effective interaction in the case of charge-fluctuations. For the range of parameters studied, the Monte Carlo simulations show that this qualitative difference between coupling to
magnetic or charge fluctuations persists to all orders, save one case, k = (3π/8, 3π/8) and
κ2 = 4, where vertex corrections enhance the charge-mediated interaction.
As noted above, for large values of κ2 , vertex corrections do not produce qualitative
changes, but merely quantitative ones. One might therefore ask whether one can obtain the
non-perturbative results with a one-loop calculation provided the parameters of the theory
are renormalized. To illustrate the point, let us focus on the case of antiferromagnetic
paramagnons with κ2 = 2. One essentially has two parameters one can renormalize, the
dimensionless coupling constant g 2χ0 /t and the inverse correlation length κ. Since vertex
corrections make the spectral function broader and the quasiparticle residue smaller, one
could attempt to fit the non-perturbative calculations with a one-loop theory with a larger
coupling constant g 2χ0 /t or smaller κ2 , or a combination of both. If the corrections to
the one-loop theory are purely local, it is possible to absorb them in a redefinition of the
coupling constant g 2χ0 /t. It turns out that increasing the coupling constant does make the
quasiparticle residue smaller but the one-loop spectral function remains too sharp relative
to the non-perturbative calculation. By the time g 2 χ0 /t is large enough for the quasiparticle
lifetime to be approximately that obtained with the Monte Carlo simulation, the one-loop
quasiparticle residue is then too small. One has more success with making κ2 smaller and
Fig. 9 compares the spectral function of the one-loop calculation with κ2 = 0.25 with the
A(k, ω) obtained from the Monte Carlo simulations at κ2 = 2. One can get a decent fit near
the hot spot at k = (π, π/8) but the one-loop spectral function is always more anisotropic
in momentum than the non-perturbative A(k, ω) for this value of κ2 .
16
The most important feature of the calculations presented here is the qualitative change
in the quasiparticle spectral function that occurs as the magnetic interaction gets stronger,
either on the border of long-range antiferromagnetic or ferromagnetic order. The appearance
of a pseudogap in the quasiparticle spectrum near a second-order phase transition as the
temperature approaches the critical transition temperature from above was demonstrated
for the half-filled Hubbard model [17–19], for a superconducting instability [18,20–23,33,34]
and for a Peierls-CDW transition [32]. By contrast, in the model studied in this paper, the
quadratic actions of the dynamical molecular fields do not exhibit a phase transition at all
(with κ2 > 0). It may therefore be somewhat surprising that a pseudogap is observed in
the present calculations as the correlation wavevector κ becomes of order one in the nearly
antiferromagnetic case and of order one-half for nearly ferromagnetic systems.
The physical origin of the pseudogap was explained in Refs. [17–19] and later in Ref. [32].
Quasiparticles only remain coherent for a finite amount of time. When the distance they
can travel during that time becomes shorter than the correlation length of the molecular
field, quasiparticles effectively see long-range order. In the presence of long-range antiferromagnetic order, the spin-density-wave quasiparticle spectral function consists of two
peaks. In the case of long-range ferromagnetic order the spin-up quasiparticles have an
energy shifted downwards, say, relative to the paramagnetic quasiparticle energy while the
energy of a spin-down quasiparticle is shifted upwards relative to its energy in the absence of
long-range ferromagnetic order. The shifts in energy give rise to corresponding shifts in the
quasiparticle peak in the spectral function. Upon averaging over the two spin orientations,
the quasiparticle spectral function would then also consist of two peaks, the up-spin and
down-spin quasiparticle peaks. In our calculations, when the quasiparticles remain coherent
for such a short time that they effectively see long-range order, the spectral functions have
charactersitics akin to that of the ordered state. For instance, in Fig. 4b, for a coupling
of quasiparticles to antiferromagnetic spin-fluctuations, the spectral function A(k, ω) for
κ2 = 0.25 looks like that of a broadened spin-density wave. And in the case of a ferromagnetically correlated molecular field, in Figs. 7b,c, A(k, ω) is ”spin-split” as κ2 is reduced, a
17
feature which can be understood if, during their short lifetime, the quasiparticles effectively
see ferromagnetic order, where the moment is equally likely to point up or down, since we
are still in the paramagnetic phase.
Quasiparticles only remain coherent for a finite time due to thermal and quantum fluctuations. The question is what is that time scale, or the associated characteristic length scale
to be compared against the magnetic correlation length. In the half-filled Hubbard model
studied in Refs. [17–19], the renormalized classical regime for the spin-fluctuations always
precedes the zero temperature phase transition, and in that regime thermal fluctuations
dominate and the relevant charactersitic length scale is the thermal quasiparticle de Broglie
wavelength ξth = vF /T [17–19]. Thermal fluctuations are dominant near the Peierls-CDW
transition in two dimensions and the relevant quasiparticle length scale for the onset of the
pseudogap also turns out to be ξth [32]. For the model considered in this paper, however,
thermal fluctuations do not appear to dominate. Indeed, the onset of the pseudogap for
coupling of quasiparticles to antiferromagnetic fluctuations and the qualitative change seen
in the spectral function for coupling to ferromagnetic fluctuations occur for ξth κ > 1. The
departure from the criterion ξth κ ≈ 1 for the breakdown of the Midgal approximation is
larger in the ferromagnetic case. The above suggests that another, shorter, length scale is
relevant in the present case. A possible candidate length scale is the quasiparticle mean-free
path.
One can attempt to make this more quantitative by extracting quasiparticle lifetimes
and mean-free paths from our numerical results. In order to do this we fit the quasiparticle
peak of the spectral function with a Lorentzian AL (k, ω) =
zk Γk
1
π (ω−Ek )2 +Γ2k
which corresponds
to a quasiparticle approximation for the retarded Green’s function GR (k, ω) =
zk
ω−Ek −iΓk
describing the propagation of quasiparticles of energy Ek . Γk is related to the quasiparticle
lifetime τk through Γk =
1
.
2τk
If one ignores the momentum dependence of the self-energy,
the quasiparticle residue zk is related to the ratio of the band to the effective mass zk ≈
m
.
m∗
(0)
We define the bare velocity vk =
q
∂ǫk 2
∂ǫk 2
) + ( ∂k
) where ǫk is the band dispersion
( ∂k
x
y
(0)
relation and the renormalized velocity vk = zk vk . The quasiparticle mean-free path is then
18
approximately lk = vk τk . The values of lk one obtains for charge-fluctuations as well as
antiferromagnetic and ferromagnetic spin-fluctuations are shown in Fig. 10. In the case of
a coupling of quasiparticles to charge-fluctuations, Fig. 10a, lk ≫ 1/κ for all the values of κ
considered. In this case, the quasiparticles travel far enough during their lifetime to see there
is no long-range charge order. In the case of antiferromagnetic spin-fluctuations, Fig. 10b,
one sees that at k = (π, π/8) lk ∼ 1/κ at κ2 = 1, and that is where the pseudogap begins
to appear in the spectral function, Fig. 4b. At momentum k = (3π/8, 3π/8), the mean-free
path lk > 1/κ at κ2 = 1 and one wouldn’t expect a pseudogap. lk becomes less than the
correlation length at the lowest value of κ2 and one can see hints of a developing pseudogap
in the spectral function, Fig. 4c. In the ferromagnetic case, Fig. 10c, the appearance of the
two peaks in the spectral function is broadly consistent with the condition lk < 1/κ. Note
that we are using a simple criterion, in an attempt to capture the essential aspects of the
problem, to understand the emergence of a pseudogap in our calculations. The crossover to
the new state is likely to depend on other details not taken into account by our criterion and
is therefore not expected to occur exactly at lk = 1/κ. Moreover, the tails in most of the
spectral functions are not Lorentzians and thus our definition of the quasiparticle lifetime is
clearly approximate.
We have also extracted mean-free paths for the one-loop approximation using the same
methodology (results not shown). For low enough values of κ2 , one can get in the regime
where lk ≤ 1/κ, but no pseudogap in the spectral function is observed. What is the oneloop approximation missing? A candidate explanation, based on an analogy, is the following. When treating the potential scattering of a particle in a momentum space basis, to
lowest order of perturbation theory (Born approximation), one assumes the wavefunction is
unchanged, i.e remains a plane-wave. In order to study bound states, in which the wavefunction of the particle is qualitatively different since it is localized, one must treat the scattering
events to all orders. By analogy, the Eliashberg approximation does not seem to allow for
a change in the quasiparticle wavefunctions which essentially remain plane-waves. In the
pseudogap state, the quasiparticle wavefunctions must be qualitiatively different and one
19
must allow the quasiparticles to scatter multiple times against the locally ordered molecular
field to produce the required changes in their wavefunctions. It seems one would need to
sum an infinite set of spin-fluctuation exchanges.
The ”quenched approximation” formulation of the magnetic interaction model actually
goes beyond a diagrammatic perturbation expansion. The Green’s function, Eq. (2.20),
depends on the position, spin, and imaginary time, but also on the parameters of the theory,
λ ≡ g 2 χ0 /t, κ2 , etc., G = G(iστ ; jσ ′ τ ′ |λ, κ2 , . . .). (To make the dependence on λ = g 2 χ0 /t
explicit, one simply needs to introduce scaled molecular fields m = gM. The variance of
the new variables is then g 2 α(q, iνn ) ∝ g 2 χ(q, iνn )). It is clear from Eqs. (2.20,2.4,2.5)
that the theory does not make sense for λ < 0, since the Gaussian distributions of the
molecular fields would have a negative variance. This observation means that it is not
possible to analytically continue G to negative values of λ, implying an essential singularity
at λ = 0. The perturbation expansion in powers of λ = 0 is then an asymptotic rather
than convergent series [35]. This opens the possibility for phenomena that lie outside of
diagrammatic perturbation theory. Whether the pseudogap state found in the numerical
simulations reported on in this paper is precisely one such phenomenon is not presently
known to the author.
The results presented here do not imply one couldn’t get a pseudogap state when coupling
to charge fluctuations. On the basis of the arguments presented above, if one were to increase
the strength of the charge correlations or the coupling of quasiparticles to the molecular field
Φ, one would get in the regime lk ≪ 1/κ and the spectral function ought to resemble that
of a broadened charge-density-wave state. Our calculations simply show that one would
have to be closer to the ordered state in the case of charge fluctuations than for magnetic
fluctuations, under otherwise similar conditions.
20
V. OUTLOOK
We studied a non-perturbative formulation of the magnetic interaction model, in which
quasiparticles are coupled to a Gaussian distributed dynamical molecular exchange field.
Far from the magnetic boundary, the type of vertex corrections considered here do not
bring about qualitative changes to the quasiparticle spectrum. But as one gets closer to
the border of long-range magnetic order, we find, for a range of model parameters, that
Migdal’s theorem does not apply and the quasiparticle spectrum is qualitatively different
from its Eliashberg approximation. The physical origin of the phenomenon is that if the
distance quasiparticles can travel during their lifetime becomes shorter than the molecular
field correlation length, these quasiparticles effectively see long-range order. When the
molecular field correlations are antiferromagnetic, the quasiparticle spectral function has
the two-peak structure of a spin-density-wave state, even though there is no spontaneous
symmetry breaking. We find that the associated pseudogap is strongly anisotropic in that
it vanishes along the diagonal of the Brillouin zone and is large near the zone boundary, in
qualitative agreement with photoemission experiments on underdoped cuprates [10]. The
anisotropy of the pseudogap found in our calculations simply reflects the anisotropy of the
quasiparticle mean-free path. For coupling to ferromagnetic fluctuations, we also find a
range of parameters were the quasiparticle spectral function becomes qualitatively different
from its one-loop self-consistent approximation. The local ferromagnetic order also leads to
a splitting of the quasiparticle peak into two. These pseudogap effects are found to be weaker
for nearly ferromagnetic systems than for their nearly antiferromagnetic counterparts, under
otherwise similar conditions.
In the standard theory of quantum critical phenomena [28], if d + z < 4 where d is the
spatial dimensionality and z the dynamical exponent, the mode-mode coupling parameter
diverges upon renormalization as one approaches the instability and the critical exponents
are different from their mean-field values. Therefore, if d + z < 4, one would certainly
expect Migdal’s theorem to qualitatively break down close enough to the quantum critical
21
point. However, when d + z > 4, the critical exponents take their mean-field values, as in a
one-loop calculation. In that case, one could therefore expect, and it is often assumed, that
the Eliashberg approximation is at least qualitatively correct. For the magnetic spectrum
considered here, Eq. (2.13), z = 2 for antiferromagnetic fluctuations and z = 3 for ferromagnetic fluctuations. Hence d + z = 4, the marginal dimension in the case of antiferromagnetic
spin-fluctuations but crucially, d + z = 5 in the case of ferromagnetic spin-fluctuations. One
could thus have expected that in the latter case, the one-loop self-consistent calculations
should be at least qualitatively correct since d + z > 4. This is at variance with our results, which show qualitative differences bewteen the non-perturbative calculations and the
Eliashberg predictions. Note that the qualitative break down of Migdal’s theorem occurs for
smaller value of κ2 in the ferromagnetic case than for antiferromagnetic spin-fluctuations.
In that sense the effect of vertex corrections is weaker when d + z = 5 compared to the case
d + z = 4, which is what is expected.
To summarize, an often assumed criterion for the qualitative applicability of the Eliashberg theory near an instability, namely d + z > 4, is clearly a necessary condition but the
calculations presented here show that it is not a sufficient one. This result may point to
certain limitations of the standard theory of quantum critical phenomena [28], which has
recently been criticized by Anderson [36].
The crucial role of dimensionality for pseudogap phenomena has been emphasized by
Tremblay and collaborators [17,18,21] and by Preosti et al. [37]. Since critical fluctuations
responsible for the precursor pseudogap are much stronger in two dimensions than for three
dimensional systems, they find that pseudogap phenomena are much weaker in three dimensions. One would expect similar results for the magnetic interaction model studied in the
present paper. The role of lattice anisotropy is already important at the Eliashberg level
[3–6], because of the increased phase space of soft magnetic fluctuations in lower dimensional
systems. This phase space argument leads one to expect weaker vertex corrections to the
Eliashberg theory of the magnetic interaction model, and hence weaker pseudogap effects in
isotropic three dimensional systems.
22
The results presented here raise a number of obvious questions, which we hope to answer
in the future. For instance, what are the corrections to the Eliashberg theory for the superconducting transition temperature to the dx2 −y2 pairing state? We have studied corrections
to quasiparticle spectral properties, and found that vertex corrections can produce a pseudogap in the quasiparticle spectrum. When the magnetic correlations are weak enough vertex
corrections do not bring about qualitative changes but nevertheless reduce the quasiparticle
lifetime. The above effects are expected to suppress magnetic pairing. But one must also
include in the calculation of the superconducting transition temperature the corresponding
corrections to the pairing interaction. In leading order, it was shown [14] that vertex corrections lead to a stronger pairing interaction in the d-wave channel. There may therefore
be some cancellation of errors, at least in some range of model parameters. At this stage,
however, one can only speculate about the effect of vertex corrections on the Eliashberg
theory of the superconducting transition temperature.
Transport and thermodynamic properties in the quenched approximation would also be of
great interest. The calculations in Ref. [32] showed that the pseudogap in the single particle
density of states also appeared in the two-particle Green’s functions such as the optical
conductivity and the uniform Pauli susceptibility. I therefore expect that the pseudogap
observed in the quasiparticle spectral function will show up in the thermodynamic and
transport properties of the model studied here. The extent to which the model is able to
explain the normal state experimental data on the underdoped cuprates is an open question.
Our simplifying assumptions, which were born out of the necessity to carry out the calculations in a reasonable amount of time, should be relaxed. The dynamical molecular field
correlation function which enters their distribution was taken to be of the same functional
form for all the calculations. In Nature, however, the distribution function of the molecular field experienced by one quasiparticle is determined self-consistently by all of the other
quasiparticles in the system. It is therefore expected to change as the nature of the quasiparticle spectrum changes. One should also study the effect of mode-mode coupling terms
which were ignored in the present study. Monien has recently carried out such a study for
23
a model of the one-dimensional Peierls-CDW [38].
There is no doubt that the extension of the theory to deal with effects ignored here will
bring about quantitative changes to our results. It will be of great interest to figure out if
and for what model parameters our results are modified qualitatively. I would like to think
that the physical origin of the emergence of a pseudogap in the quasiparticle spectrum will
turn out to be independent of the details of the model.
VI. ACKNOWLEDGMENTS
I would like to thank P. Coleman, J.R. Cooper, P.B. Littlewood, G.G. Lonzarich, J.
Loram, and D. Pines for discussions on this and related topics. We acknowledge the support
of the EPSRC, the Newton Trust and the Royal Society.
VII. APPENDIX: GREEN’S FUNCTIONS
In this appendix we derive the mathematical formulas for the single-particle Green’s
functions in a fluctuating exchange or scalar field. We also give the algorithm for the
numerically stable and efficient calculation of such Green’s functions. We shall make use
of fermion coherent states and will derive the path-integral for an anti-normal ordered [39]
Hamiltonian quadratic in the fermion field operators.
We first need an identity for expressing the exponential of the quadratic form
Q̂A =
X
ψi Ai,j ψj†
(7.1)
i,j
in terms of fermion coherent states. In the above expression, A = A† is a Hermitian matrix
of dimension N and {ψ}, {ψ †} are fermion operators. One has:
exp
X
i,j
ψi Ai,j ψj†
=
Z Y
N
[
dξi∗dξi ]|ξ > exp
i=1
X
i,j
ξi (eA )i,j ξj∗ < ξ|
(7.2)
where {ξ ∗}, {ξ} are Grassmann numbers. This identity is proved by going into a basis in
which the matrix A is diagonal. Let U be the unitary transformation that diagonalizes A
24
U † AU = Λ ≡ diag(λ1 , . . . , λN )
(7.3)
In terms of the new fermion fields {Φ}, {Φ† }
X
ψi =
∗
Ui,j
Φj
(7.4)
Ui,j Φ†j
(7.5)
Uj,i ψj
(7.6)
∗ †
Uj,i
ψj
(7.7)
j
X
ψi† =
j
X
Φi =
j
X
Φ†i =
j
the quadratic form Eq. (7.1) becomes Q̂A =
P
i
λi Φi Φ†i . Since (Φi Φ†i )n = Φi Φ†i for n =
1, 2, . . . , ∞, and the terms with different indices i commute with one another, one has
exp(Q̂A ) =
Y
exp(λi Φi Φ†i ) =
Yn
i
i
1 + [exp(λi ) − 1]Φi Φ†i
o
(7.8)
Expanding the product into sums yields
exp(Q̂A ) = 1 +
X
i1
X
+
i1 <i2 <i3
+
X
[exp(λi1 ) − 1]Φi1 Φ†i1 +
X
i1 <i2
[exp(λi1 ) − 1][exp(λi2 ) − 1]Φi1 Φ†i1 Φi2 Φ†i2
[exp(λi1 ) − 1][exp(λi2 ) − 1][exp(λi3 ) − 1]Φi1 Φ†i1 Φi2 Φ†i2 Φi3 Φ†i3 + . . .
i1 <...<iN
[exp(λi1 ) − 1] . . . [exp(λiN ) − 1]Φi1 Φ†i1 . . . ΦiN Φ†iN
(7.9)
Since in the various sums one has i1 6= i2 6= i3 . . . 6= iN , one can straightforwardly anti-normal
order the terms
Φi1 Φ†i1 Φi2 Φ†i2 = Φi2 Φi1 Φ†i1 Φ†i2
(7.10)
Φi1 Φ†i1 Φi2 Φ†i2 Φi3 Φ†i3 = Φi3 Φi2 Φi1 Φ†i1 Φ†i2 Φ†i3
(7.11)
Φi1 Φ†i1 Φi2 Φ†i2 . . . ΦiN Φ†iN = ΦiN ΦiN−1 . . . Φi1 Φ†i1 Φ†i2 . . . Φ†iN
(7.12)
We can now insert a resolution of the identity between the annihilation and creation operators. For instance
Φi3 Φi2 Φi1 Φ†i1 Φ†i2 Φ†i3
= Φi3 Φi2 Φi1
Z Y
N
[
i=1
dξi∗ dξi ]|ξ > exp −
25
X
i
ξi∗ ξi < ξ|Φ†i1 Φ†i2 Φ†i3
=
Z Y
N
i=1
=
dξi∗ dξi ξi3 ξi2 ξi1 |ξ > exp −
Z Y
N
[
i=1
X
ξi∗ ξi < ξ|ξi∗1 ξi∗2 ξi∗3
i
dξi∗dξi ] ξi1 ξi∗1 ξi2 ξi∗2 ξi3 ξi∗3 exp −
X
i
ξi∗ ξi |ξ >< ξ|
Applying the above operation to every term in the sum in Eq. (7.9) one obtains
exp(Q̂A ) =
Z Y
N
[
dξi∗ dξi ]|ξ
i=1
=
Z Y
N
[
i=1
=
Z Y
N
[
> exp −
dξi∗ dξi ]|ξ > exp −
X
i
X
[1 + (eλi − 1)ξi ξi∗ ] < ξ|
i=1
ξi∗ ξi exp
X
i
(eλi − 1)ξi ξi∗ < ξ|
eλi ξi ξi∗ < ξ|
i
i=1
N
Y
i
X
dξi∗ dξi ]|ξ > exp
ξi∗ ξi
(7.13)
Going back to the original representation in which the matrix A is not diagonal yields
Eq. (7.2). If we have another quadratic form
Q̂B =
X
ψi Bi,j ψj†
(7.14)
ij
where B = B † is a Hermitian matrix of dimension N, it is a simple exercise in Grassmann
integration to show that
exp(Q̂A ) exp(Q̂B ) =
Z Y
N
[
dξi∗ dξi ]|ξ > exp
i=1
X
ξi (eA eB )i,j ξj∗ < ξ|
ij
(7.15)
To prove the above identity one uses the coherent state representation Eq. (7.2) for exp(Q̂A )
and exp(Q̂B )
exp(Q̂A ) =
Z Y
N
dξi∗ dξi ]|ξ > exp
[
dηi∗ dηi ]|η > exp
[
X
ij
i=1
exp(Q̂B ) =
Z Y
N
ξi (eA )i,j ξj∗ < ξ|
i=1
X
(7.16)
ηi (eB )i,j ηj∗ < η|
ij
(7.17)
Integrating over the Grassmann variables {ξi∗ } and {ηi } yields Eq. (7.15).
The imaginary time single-particle Green’s function in a time dependent field is given by
the usual expression
†
′
G(iστ ; jσ ′ τ ′ ) = −hTτ {ψiσ (τ ) ψjσ
′ (τ )}i
1 h n
= − T r Tτ exp −
Z
h
Z = T r Tτ exp −
Z
0
β
Z
0
β
ĥ(s)ds
26
†
′
ĥ(s)ds ψiσ (τ )ψjσ
′ (τ )
i
oi
(7.18)
(7.19)
where the time dependent Hamiltonian ĥ(τ ) is given by Eq. (2.2) and Eq. (2.3) for coupling
to a fluctuating exchange or scalar field respectively. The trace is over the fermion fields. For
ease of notation, we have dropped the explicit dependence of G(iστ ; jσ ′ τ ′ ) on the molecular
field. Because of the time ordering operator, one must distinguish the two cases τ > τ ′ and
τ ≤ τ ′.
"
n
Z β
o
n
Z τ
o
1
†
′
G(iστ ; jσ τ ) = − T r Tτ exp −
ĥ(s)ds ψiσ (τ ) Tτ exp −
ĥ(s)ds ψjσ
′ (τ )
Z
τ
τ′
′ ′
n
Z
n
Z
× Tτ exp −
τ
ĥ(s)ds
0
#
o
τ > τ′
(7.20)
"
o
n
Z τ′
o
n
Z β
1
†
′
ĥ(s)ds ψiσ (τ )
−
ĥ(s)ds ψjσ
G(iστ ; jσ τ ) = − T r Tτ exp −
′ (τ ) Tτ exp
Z
τ′
τ
′ ′
× Tτ exp −
0
τ′
ĥ(s)ds
o
#
τ ≤ τ′
(7.21)
The imaginary times τ and τ ′ are discretized such that τ = ∆τ m m = 0, 1, . . . Lt ,
τ ′ = ∆τ m′
m′ = 0, 1, . . . Lt , where ∆τ =
β
.
Lt
One then uses the Trotter break-
up for the Hamiltonian in an exchange, Eq. (2.2), or scalar field, Eq. (2.3), and hence
exp(−∆τ ĥ(m∆τ )) ≈ exp(−∆τ ĥ1 (m∆τ )) exp(−∆τ ĥ0 (m∆τ )). Up to an unimportant constant, the anti-normal ordered hopping Hamiltonian ĥ0 , Eq. (2.1) is written as
ĥ0 =
X
†
ψiσ Tiσ,jσ′ ψjσ
′
(7.22)
i,σ;jσ′
Tiσ,jσ′ = (tij + µδi,j )δσ,σ′
(7.23)
Similarly, up to constant terms that drop out of the expression for the single particle Green’s
function, the Hamiltonian for the coupling to the fluctuating field is written as
ĥ1 (m∆τ ) =
X
†
ψiσ Aiσ,jσ′ (m)ψjσ
′
(7.24)
iσjσ′
h
g
Aiσ,jσ′ (m) = √ δi,j Miz (m∆τ )δσ,↑ δσ′ ,↑ − Miz (m∆τ )δσ,↓ δσ′ ,↓
3
i
+ (Mix (m∆τ ) + iMiy (m∆τ ))δσ,↑ δσ′ ,↓ + (Mix (m∆τ ) − iMiy (m∆τ ))δσ,↓ δσ′ ,↑ (7.25)
Aiσ,jσ′ (m) = gδi,j δσ,σ′ Φi (m∆τ )
(7.26)
where Eq. (7.25) and Eq. (7.26) are for a fluctuating exchange and scalar field respectively.
Making use of Eqs. (7.2,7.15), one has
27
exp(−∆τ ĥ(m∆τ )) ≈ exp(−∆τ ĥ1 (m∆τ )) exp(−∆τ ĥ0 (m∆τ ))
=
Z
dµ(ξ)|ξ > exp
X
∗
< ξ|
ξiσ [Bm ]iσ,jσ′ ξjσ
′
iσjσ′
(7.27)
Bm = exp(−∆τ A(m)) exp(−∆τ T )
(7.28)
where from now on we use dµ(ξ) to denote the Grassmann integration measure
Y
dµ(ξ) ≡ [
∗
dξiσ
dξiσ ]
(7.29)
iσ
In the case of coupling to a scalar field
[exp(−∆τ A(m))]iσ,jσ′ = δi,j δσ σ′ exp(−g∆τ Φi (m∆τ ))
(7.30)
The case of the exchange field is slightly more complicated:
(7.31)
[exp(−∆τ A(m))]iσ,jσ′ = δi,j [exp(−∆τ ai (m))]σ,σ′
c − sMzi (m∆τ ) −sM+
i (m∆τ )
exp(−∆τ ai (m)) =
−
z
−sMi (m∆τ ) c + sMi (m∆τ )
√
c = cosh(g∆τ |Mi (m∆τ )|/ 3)
√
s = sinh(g∆τ |Mi (m∆τ )|/ 3)
(7.32)
(7.33)
(7.34)
Miz (m∆τ )
|Mi (m∆τ )|
Mix (m∆τ ) ± iMiy (m∆τ )
M±
(m∆τ
)
=
i
|Mi (m∆τ )|
Mzi (m∆τ ) =
|Mi (m∆τ )| =
(7.35)
(7.36)
q
[Mix (m∆τ )]2 + [Miy (m∆τ )]2 + [Miz (m∆τ )]2
(7.37)
Matrix multiplication by exp(−∆τ A(m)) is most easily done in real space. We have implemented the matrix multiplication by exp(−∆τ T ) by means of the Fast Fourier transform
instead of the usual checkerboard decomposition. In the discretized imaginary time version,
Tτ exp −
Z
0
β
ĥ(s)ds ≈ exp(−∆τ ĥ(∆τ Lt )) . . . exp(−∆τ ĥ(∆τ ))
=
Z
dµ(ξ)|ξ > exp
X
iσjσ′
∗
ξiσ [BLt BLt −1 . . . B1 ]iσ,jσ′ ξjσ
< ξ|
′
(7.38)
Note that we choose to approximate the imaginary time integral of a function in an
interval of length ∆τ by ∆τ times the value of the function at the upper limit of the
28
interval. The calculation of the normalization factor Z, Eq. (7.19), then becomes a simple
exercise in Grassmann integration
Z=
Z
dµ(η)dµ(ξ) < −η|ξ > exp −
= Det[1 +
B1T B2T
. . . BLTt ]
X
∗
ηiσ
ηiσ +
iσ
= Det[1 +
X
∗
ξiσ [BLt BLt −1 . . . B1 ]iσ,jσ′ ξjσ
< ξ|η >
′
iσjσ′
B0T B1T
. . . BLTt −1 ]
(7.39)
T
where Bm
is the transpose of the matrix Bm and in the last line we have used the periodicity
of the exchange and scalar fields which implies BLt = B0 and multiplied the expression by
1 = Det[B0T B0T
−1
].
The remainder of the calculation of the Green’s function proceeds along similar lines.
Consider the case m > m′ . One has
Tτ exp −
Z
β
τ
ĥ(s)ds ≈ exp(−∆τ ĥ(∆τ Lt )) . . . exp(−∆τ ĥ(∆τ (m + 1)))
=
Z
dµ(ξ)|ξ > exp
X
iσjσ′
with similar expressions for Tτ exp −
Rτ
τ′
ĥ(s)ds and Tτ exp −
forward algebraic manipulations, one arrives at:
1
G(iσm, jσ m ) = −
Z
′
′
exp
Z
dµ(ξ)dµ(η)dµ(ρ)dµ(φ) exp −
X
∗
ξiσ [BLt BLt −1 . . . Bm+1 ]iσ,jσ′ ξjσ
< ξ| (7.40)
′
iσ
exp −
∗
ηiσ
ρiσ
X
iσjσ′
exp
X
ρ∗iσ φiσ
iσ
iσ
ρiσ ρ∗jσ′
ρ
∗
exp −
ρiσ Miσ,jσ
′ ρjσ ′
M η = BLt BLt −1 . . . Bm+1
X
X
R τ′
0
ĥ(s)ds . After straight-
∗
ξiσ
ξiσ exp −
exp −
X
iσ,j,σ′
∗
ξiσ
ηiσ
iσ
η
∗
ηiσ Miσ,jσ
′ ηjσ ′
φ
∗
φiσ Miσ,jσ
′ φjσ ′
iσjσ′
X
(7.41)
(7.42)
M ρ = Bm Bm−1 . . . Bm′ +1
(7.43)
M φ = Bm′ Bm′ −1 . . . B1
(7.44)
The integral over the quadratic form is easily computed. The calculation in the case m ≤ m′
proceeds along similar lines and we just quote the final answer. The single particle Green’s
function is given by the following matrix expressions
′
G(m, m ) =
T
−Bm+1
. . . BLTt B1T
BT
h
T
T
T
T
T
. . . Bm
′ 1 + Bm′ +1 . . . BL B1 . . . Bm′
t
h
T
T
T
T
T
T
m+1 Bm+2 . . . Bm′ 1 + Bm′ +1 . . . BLt B1 . . . Bm′
29
i−1
i−1
if m > m′
if m ≤ m′
(7.45)
Note that these expressions differ from those of Ref. [40] since these authors are using
the standard formulation of the path integral with normal ordered operators, instead of the
anti-normal ordering used here. The numerical calculation of the single particle Green’s
function, Eq. (7.45), is complicated by the fact that the matrix products entering the above
expressions are seriously ill-conditioned [41]. I have found that the Gram-Schmidt matrix
factorization algorithm of Ref. [41] can be unstable at sufficiently low temperatures and
for strongly fluctuating molecular fields. I therefore use an alternative way to compute the
Green’s function which I have found to be particularly stable. It is based on the matrix
identities:
h
(1 + A1 A2 )−1 = T2 1 − T1 − T2 + 2T1 T2
h
ih
i−1
T1
A2 (1 + A1 A2 )−1 = 1 − T2 1 − T1 − T2 + 2T1 T2
Ti = (1 + Ai )−1
(7.46)
i−1
T1
(7.47)
(7.48)
In practice we calculate G(m, 0) and G(0, m′ ). The above matrix identities can be used
recursively to calculate these Green’s functions in a stable manner. The matrices A1 , A2 are
taken as products of B T matrices. The number of B T matrices in a ”block” is chosen to be
the maximum number such that (1 + A1 ), (1 + A2 ) can be inverted without significant loss
of precision. One can then obtain the Green’s functions G(m, 0) and G(0, m′ ) at a subset
of values m, m′ . The Green’s function for the other values of m, m′ can be obtained by
T
propagating forward or backward in time. For example, G(m + 1, 0) = [Bm+1
]−1 G(m, 0).
We leave it to the reader to fill in the details.
30
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Y. Takahashi and K. Ueda, J. Phys. Soc. Jpn. 52, 2905 (1990). P. Monthoux, A.V.
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D. Braithweite, J. Flouquet, Nature 406, 587 (2000).
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D.S. Dessau, D.S. Marshall, C.H. Park, P. Fournier, A. Kapitulnik, Science, 273, 325
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59, 2091 (1998).
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31
Thompson Phys. Rev. Lett. 89, 157004 (2002).
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Morr, Phys. Rev. B 56, 7789 (1997).
[15] D.J. Scalapino, in ”Superconductivity”, edited by R.D. Parks, Marcel Dekker, New
York, (1969).
[16] N.E. Bickers and D.J. Scalapino, Ann. Phys. (N.Y.) 193, 206 (1989).
[17] Y.M. Vilk and A.-M. S. Tremblay, Europhysics Lett. 33, 159 (1996).
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Phys. Chem. Solids, 59, 1873 (1998).
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Lett. 83, 4128 (1999).
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[27] A. Posazhennikova and P. Coleman, cond-mat/0209014.
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32
[29] J. Schmalian, D. Pines and B.P. Stojkovic, Phys. Rev. B 60, 667 (1999).
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[35] J.W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley (1988).
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[39] I learned this formulation of functional integrals from Patrick Lee.
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[41] S.R. White, D.J. Scalapino, R.L. Sugar, E.Y. Loh, J.E. Gubernatis, R.T. Scalettar,
Phys. Rev. B 40, 506 (1989).
33
FIGURES
FIG. 1. Diagrammatic expansion for the single particle Green’s function. The dashed line
connected at one end only to the solid line represents the interaction of the fermions with the
dynamical molecular field and the brackets the average over the Gaussian distribution of these
fields. The averaging over the distribution of molecular fields is carried out by pairing the dashed
lines in all possible ways, each pairing giving a factor proportional to the two-point correlation
function of the molecular field, the dynamical susceptibility χ(q, iνn ) according to Eqs. (2.11,2.12).
The lower part of the figure shows the pairings one obtains up to two spin or charge-fluctuation
exchanges.
34
Eliashberg
0.5
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(a)
N(ω)t
0.4
0.3
0.2
0.1
0
−4
−2
0
2
4
ω/t
35
6
8
10
Eliashberg ; k = (π,π/8)
10
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(b)
A(k,ω)t
8
6
4
2
0
−1
−0.5
0
ω/t
36
0.5
1
Eliashberg ; k = (3π/8,3π/8)
12
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(c)
10
A(k,ω)t
8
6
4
2
0
−1
−0.5
0
ω/t
0.5
1
FIG. 2. The one-loop approximation for the quasiparticle properties for both commensurate
charge-fluctuations and antiferromagnetic spin-fluctuations. The tunneling density of states N (ω)
is shown in (a) while (b) and (c) show the quasiparticle spectral function A(k, ω) for momenta just
below the Fermi level. (b) shows A(k, ω) for a wavevector close to the Van Hove singularity and
(c) shows A(k, ω) for a wavevector along the diagonal of the Brillouin zone.
37
Charge−fluctuations
0.5
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(a)
N(ω)t
0.4
0.3
0.2
0.1
0
−4
−2
0
2
4
ω/t
38
6
8
10
Charge−fluctuations ; k = (π,π/8)
20
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(b)
A(k,ω)t
15
10
5
0
−0.6
−0.4
−0.2
ω/t
39
0.0
0.2
Charge−fluctuations ; k = (3π/8,3π/8)
15
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(c)
A(k,ω)t
10
5
0
−0.6
−0.4
−0.2
ω/t
0.0
0.2
FIG. 3. Quasiparticle properties for a coupling to a scalar dynamical molecular field Φ. The
tunneling density of states N (ω) is shown in (a) while (b) and (c) show the quasiparticle spectral
function A(k, ω) for momenta just below the Fermi level. (b) shows A(k, ω) for a wavevector close
to the Van Hove singularity and (c) shows A(k, ω) for a wavevector along the diagonal of the
Brillouin zone.
40
AF Spin−fluctuations
0.4
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(a)
N(ω)t
0.3
0.2
0.1
0
−4
−2
0
2
4
ω/t
41
6
8
10
AF Spin−fluctuations ; k = (π,π/8)
2
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(b)
A(k,ω)t
1.5
1
0.5
0
−4
−2
0
ω/t
42
2
4
AF Spin−fluctuations ; k = (3π/8,3π/8)
2.5
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(c)
A(k,ω)t
2
1.5
1
0.5
0
−3
−2
−1
0
1
2
3
4
ω/t
FIG. 4. Quasiparticle properties for a coupling to an exchange vector dynamical molecular field
M with antiferromagnetic correlations. The tunneling density of states N (ω) is shown in (a) while
(b) and (c) show the quasiparticle spectral function A(k, ω) for momenta just below the Fermi
level. (b) shows A(k, ω) for a wavevector close to the Van Hove singularity and (c) shows A(k, ω)
for a wavevector along the diagonal of the Brillouin zone.
43
k = (π,π/8)
20
(a)
2
κ = 2.00
Eliashberg
Charge−fluct.
Spin−fluct.
A(k,ω)t
15
10
5
0
−0.8 −0.6 −0.4 −0.2
ω/t
44
0
0.2
0.4
k = (3π/8,3π/8)
15
2
κ = 2.00
(b)
Eliashberg
Charge−fluct.
Spin−fluct.
A(k,ω)t
10
5
0
−0.8 −0.6 −0.4 −0.2
ω/t
0.0
0.2
0.4
FIG. 5. Comparison of the one-loop self-consistent versus the non-perturbative calculation of
charge- and spin-fluctuation exchanges. (a) shows the spectral function A(k, ω) for a wavevector
close to the Van-Hove singularity and (b) shows A(k, ω) for a wavevector along the diagonal of the
Brillouin zone.
45
Eliashberg
0.7
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(a)
0.6
N(ω)t
0.5
0.4
0.3
0.2
0.1
0
−4
−2
0
2
4
ω/t
46
6
8
10
Eliashberg ; k = (π,π/8)
4
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(b)
A(k,ω)t
3
2
1
0
−2
−1
0
ω/t
47
1
2
Eliashberg ; k = (3π/8,3π/8)
5
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(c)
A(k,ω)t
4
3
2
1
0
−2
−1
0
ω/t
1
2
FIG. 6. The one-loop approximation for the quasiparticle properties for ferromagnetic
spin-fluctuations. The tunneling density of states N (ω) is shown in (a) while (b) and (c) show
the quasiparticle spectral function A(k, ω) for momenta just below the Fermi level. (b) shows
A(k, ω) for a wavevector close to the Van Hove singularity and (c) shows A(k, ω) for a wavevector
along the diagonal of the Brillouin zone.
48
FE Spin−fluctuations
0.5
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(a)
N(ω)t
0.4
0.3
0.2
0.1
0
−4
−2
0
2
4
ω/t
49
6
8
10
FE Spin−fluctuations ; k = (π,π/8)
1.6
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(b)
A(k,ω)t
1.2
0.8
0.4
0
−4
−2
0
ω/t
50
2
4
FE Spin−fluctuations ; k = (3π/8,3π/8)
2
2
κ = 4.00
2
κ = 2.00
2
κ = 1.00
2
κ = 0.50
2
κ = 0.25
(c)
A(k,ω)t
1.5
1
0.5
0
−3
−2
−1
0
1
2
3
4
ω/t
FIG. 7. Quasiparticle properties for a coupling to an exchange vector dynamical molecular field
M with ferromagnetic correlations. The tunneling density of states N (ω) is shown in (a) while (b)
and (c) show the quasiparticle spectral function A(k, ω) for momenta just below the Fermi level.
(b) shows A(k, ω) for a wavevector close to the Van Hove singularity and (c) shows A(k, ω) for a
wavevector along the diagonal of the Brillouin zone.
51
FIG. 8. First order vertex correction to the one-loop self-energy. The way spin is carried
through the diagram depends on the kind of molecular field. A Pauli matrix is associated with each
vertex in the case of a coupling of quasiparticles to magnetic fluctuations. In the case of exchange
of charge fluctuations, each vertex simply carries a unit matrix, namely the spin orientation is
unchanged at each vertex of the diagram.
52
k = (π,π/8)
1.2
(a)
1
2
Eliashberg ; κ = 0.25
2
Monte Carlo ; κ = 2.00
A(k,ω)t
0.8
0.6
0.4
0.2
0
−4
−3
−2
−1
ω/t
53
0
1
2
k = (3π/8,3π/8)
3
2
Eliashberg ; κ = 0.25
2
Monte Carlo ; κ = 2.00
(b)
A(k,ω)t
2
1
0
−2
−1
0
ω/t
1
2
FIG. 9. Comparison of the single antiferromagnetic spin-fluctuation exchange approximation
for the quasiparticle spectral function with a renormalized correlation wavevector κ2ef f = 0.35
and the non-perturbative result with κ2 = 2. In both cases the dimensionless coupling constant
g2 χ0 /t = 2. (a) and (b) show the quasiparticle spectral function A(k, ω) for momenta just below
the Fermi level. (a) shows A(k, ω) for a wavevector close to the Van Hove singularity and (b) shows
A(k, ω) for a wavevector along the diagonal of the Brillouin zone.
54
Charge−fluctuations
40
(a)
lk = vkτk
30
20
k = (π,π/8)
k = (3π/8,3π/8)
10
0
0
1
2
2
κ
55
3
4
AF Spin−fluctuations
6
k = (π,π/8)
k = (3π/8,3π/8)
(b)
5
lk = vkτk
4
3
2
1
0
0
1
2
2
κ
56
3
4
FE Spin−fluctuations
5
k = (π,π/8)
k = (3π/8,3π/8)
(c)
lk = vkτk
4
3
2
1
0
0
1
2
2
κ
3
4
FIG. 10. Approximate quasiparticle mean-free paths lk for k = (π, π/8) and k = (3π/8, 3π/8)
obtained from Lorentzian fits to the numerical results for the spectral function. (a) shows lk for
coupling to charge-fluctuations, (b) to antiferromagnetic spin-fluctuations and (c) to ferromagnetic
spin-fluctuations.
57