Tuning correlation effects with electron-phonon
interactions
J.P.Hague and N.d’Ambrumenil
Department of Physics, University of Warwick, CV4 7AL, U.K.
We investigate the effect of tuning the phonon energy on the correlation ef-
fects in models of electron-phonon interactions using DMFT. In the regime
where itinerant electrons, instantaneous electron-phonon driven correlations
and static distortions compete on similar energy scales, we find several in-
teresting results including (1) A crossover from band to Mott behavior in the
spectral function, leading to hybrid band/Mott features in the spectral func-
tion for phonon frequencies slightly larger than the band width. (2) Since the
optical conductivity depends sensitively on the form of the spectral function,
we show that such a regime should be observable through the low frequency
form of the optical conductivity. (3) The resistivity has a double kondo peak
arrangement [Published as J. Low. Temp. Phys. 140 pp77-89 (2005)].
PACS numbers: 71.10.Fd, 71.27.+a, 71.38.-k
1. Introduction
Mounting experimental evidence from high-Tc cuprates 1, nickelates 2,
manganites 3,4 and other interesting materials suggests that large electron-
phonon interactions may play a more important role in the physics of strongly
correlated electron systems than previously thought. Migdal-Eliashberg and
BCS theories have proved extremely successful in describing the effects of
phonons in many materials. However, if the coupling between electrons and
the underlying lattice is large, and/or the phonons can not be treated within
an adiabatic approximation, conventional approaches fail.
The Holstein model contains most of the fundamental physics of the
electron-phonon problem 5. Tight-binding electrons are coupled to the lat-
tice through a local interaction with Einstein modes. For large phonon
frequencies, electrons interact with a strongly correlated Hubbard-like at-
traction, while for small phonon frequencies the lattice gives rise to a static
http://arxiv.org/abs/0704.0030v1
potential which is essentially uncorrelated. Between these two extreme lim-
its of correlated and uncorrelated behavior, levels of correlation are tuned
by the size of the phonon frequency and novel physics is expected. In par-
ticular, it is normally the strength of interaction which is said to tune the
correlation in e.g. the Hubbard model, whereas in the Holstein model, it can
be seen that both interaction strength and phonon frequency may compete
with each other to play this role.
The dynamical mean-field theory (DMFT) approach has proved suc-
cessful in treating the Holstein and other models 6,7,8. DMFT treats the
self-energy as a momentum-independent quantity and is accurate as long as
the variation across the Brillouin zone is small. For many aspects of the
electron-phonon problem in 3D, correlations are short ranged and DMFT
can be successfully applied. The weak coupling phase diagram was studied
by Ciuchi et al. where competing charge-order (CO) and superconducting
states were found 7. Freericks et al. developed a quantum Monte-Carlo
(QMC) algorithm 8,9 and examined the applicability of several perturba-
tion theory based techniques to the electron-phonon problem 10,11,12. The
prediction of measurable quantities away from certain well-defined limits is
severely restricted owing to difficulties inherent in the analytic continuation.
Dynamic properties such as spectral functions can be computed in the case
of static phonons 13, and close to the static limit 14. Alternatively, the limit
of high phonon frequency (attractive Hubbard model) has been studied with
a QMC algorithm 15.
In the current study we are concerned with the behavior of dynamical
properties that could be measured directly with experiment. We use the iter-
ated perturbation theory approximation, which has been demonstrated to be
accurate for the Hubbard model, and use maximum entropy to analytically
continue the results. We compare the resulting single-particle spectral func-
tions over a wide range of electron-phonon coupling strengths and phonon
frequencies. The results obtained using iterated perturbation theory (IPT)
are promising and capture generic weak and strong coupling behavior for
all phonon frequencies. At intermediate phonon frequencies, we find that
electron-phonon interactions produce a spectral function which is simulta-
neously characteristic of both uncorrelated band (static) and strongly cor-
related Mott/Hubbard regimes. We also find that the competition between
band-like and correlated states causes unusual structures in the optical con-
ductivity and resistivity. Provided a material with high enough phonon
frequency can be identified, it is possible that such a state could be observed
experimentally.
This paper is organized as follows. First, we introduce the Holstein
model, the dynamical mean-field theory and analytic continuation techniques
(1a) (2a)
(2b) (2c)
Fig. 1. Second order contributions to the self-energy. Straight lines repre-
sent electron Green’s functions of the host and wavy lines phonon Green’s
functions.
(section 2.). In section 3., we use IPT to determine the spectral functions of
the Holstein model. We compare IPT with exactly known results in the static
limit. This, in conjunction with the conclusions of Ref. 11 leads us to argue
that IPT is a reasonable approximation for the calculation of dynamical
properties in the intermediate phonon frequency regime. We compute the
density of states, optical conductivity and resistivity, and give a heuristic
explanation for their behavior.
2. Formalism
The Holstein Hamiltonian is written as,
H = −t
<ij>σ
iσcjσ +
(gxi − µ)niσ +
Mω20x
The first term in this Hamiltonian represents a tight binding model with
hopping parameter t. The second term couples the local ion displacement,
xi to the local electron density. The final term can be identified as the
non-interacting phonon Hamiltonian. c
i (ci) create (annihilate) electrons at
site i, pi is the ion momentum, M the ion mass, µ the chemical potential
and g the electron-phonon coupling. The phonons are dispersionless with
frequency ω0.
The perturbation theory of this model may be written down in terms
of electrons interacting via phonons with the effective interaction,
U(iωs) = −
M(ω2s + ω
Here, ωs = 2πsT are the Matsubara frequencies for bosons and s is an
integer. Taking the limit ω0 → ∞, g → ∞, while keeping the ratio g/ω0
finite, leads to an attractive Hubbard model with a non-retarded on-site
interaction U = −g2/Mω20 . Iterated-perturbation theory (IPT) is known to
be a reasonable approximation to the half-filled Hubbard model 16,17. Taking
the opposite limit (ω0 → 0, M → ∞, keeping Mω20 ≡ κ finite) the phonon
kinetic energy term vanishes, and the phonons depend on a static variable
xi. As such, the model may be considered as uncorrelated.
We solve the Holstein model using dynamical mean-field theory (DMFT).
DMFT freezes spatial fluctuations, leading to a theory which is completely
momentum independent, while fully including dynamical effects of excita-
tions. In spite of this simplification, DMFT predicts non-trivial (correlated)
physics and may be used as an approximation to 3D models 18. As discussed
in Ref. 6, DMFT involves the solution of a set of coupled equations which
are solved self-consistently. The Green’s function for the single site problem,
G(iωn) can be written in terms of the self-energy Σ(iωn) as,
G−1(iωn) = G−10 (iωn)− Σ(iωn), (3)
where Σ is a functional of G0, the Green’s function for the host of a single
impurity model. Here ωn = 2πT (n+ 1/2) are the usual Matsubara frequen-
cies. The assumptions of DMFT are equivalent to taking the self-energy of
the original lattice problem to be local, hence G is also given by,
G(iωn) =
dǫD(ǫk)
iωn + µ− Σ(iωn)− ǫk
where D(ǫ) the density of states (DOS) of the non-interacting problem (in
our case g = 0). We work with a Gaussian DOS which corresponds to a
hypercubic lattice 18, D(ǫ) = exp(−ǫ2/2t2)/t
2π. Equations (3) and (4)
are solved according to the following self-consistent procedure: Compute
the Green’s function from equation (4) and the host Green’s function of the
effective impurity problem, G0, from equation (3); then calculate a new self-
energy from the host or full Green’s functions. In the following we will take
the hopping parameter t = 0.5, which sets the energy scale.
Once the algorithm has converged, and after analytically continuing to
the real axis, response functions can be computed. We use the MAXENT
method for the determination of spectral functions from Matsubara axis
data. MAXENT treats the analytic continuation as an inverse problem 19.
The Green’s function, G(z), is given by the integral transform,
G(z) =
z − x
dx (5)
where ρ(x) is the spectral function (ρ(ω) = Im[G(ω + iη)]/π). The problem
of finding ρ is therefore one of inverting the integral transform. Since the
data for Gn are incomplete and noisy for any finite set of Matsubara fre-
quencies, the inversion of the kernel of the discretised problem is ill-defined.
The MAXENT method selects the distribution ρ(x) which assumes the least
structure consistent with the calculated or measured data. These methods
have been extensively reviewed in the context of the inversion of the kernel
in Refs. 19,20. The applicability to the current problem has been thoroughly
tested, and is found to be accurate.
Within the DMFT formalism, many response functions follow directly
from the one-electron spectral function and the electron self-energy (essen-
tially because of the neglect of all connected higher point functions apart
from G0). Here we will be interested in the conductivity 6:
Re[σ(ω)] =
dǫD(ǫ)
dν ρ(ǫ, ν)ρ(ǫ, ν + ω)[f(ν)− f(ν + ω)] (6)
where f(x) is the Fermi-Dirac distribution. Taking the limit, ω → 0, leads
to the DC conductivity. (The conductivity is in units of e2V/ha2, where a
is the lattice spacing, V the volume of the unit cell and e and h are the
electron charge and Planck’s constant respectively.)
3. Results
In this section, we examine the validity of an approximation to the self-
energy constructed from only first and second order terms with respect to the
spectral functions calculated at very high and very low phonon frequencies.
Finally, we calculate the optical conductivity and resistivity.
Spectral functions are shown in figures 2 and 3. The perturbation theory
is carried out in the host Green’s function (i.e. both electrons and phonons
are bare). All diagrams in fig. 1 are considered,
Σ1a(iωn) = −UT
G0(iωn − iωs)D0(iωs) (7)
Σ2a(iωn) = −2U2T 2
D20(iωn−m)G0(iωm)G0(iωs)G0(iωn−m+s) (8)
Σ2b(iωn) = U
D0(iωm−s)D0(iωn−m)G0(iωm)G0(iωs)G0(iωn−m+s)
Σ2c(iωn) = U
D0(iωn−m)D0(iωm−s)G20(iωm)G0(iωs) (10)
This also gives the correct weak coupling limit for the electronic Green’s
function.
We consider first the calculation of spectral functions close to the static
and Hubbard limits. In the instantaneous limit the perturbation theories
for the Holstein and Hubbard models are equivalent. It is well known that
second order perturbation theory in the host Green’s function provides a
good approximation to the Hubbard model 21. In the static limit, the exact
solution can easily be calculated 13. Figure 2 shows spectral functions from
the exact solution, computed for a hypercubic lattice and spectral functions,
computed using 2nd order perturbation theory at a temperature of T = 0.08.
The phonon frequency ω0 = T/20 was chosen so that the effects of the
phonon kinetic energy are negligible compared to thermal fluctuations. This
allows a direct comparison to be made between the exact and approximate
results. The comparison shows that the widths and positions of the major
features are closely related.
The results in Figure 2 for the static limit (ω0 → 0), together with the
fact that second order IPT is known to give reasonable results in the instan-
taneous limit (ω0 → ∞), suggest that the calculation of spectral functions
should also be reliable at intermediate frequencies. We note that Freericks et
al. also find a reasonable agreement between the IPT and QMC self-energies
at half-filling, and that this should lead to a good agreement in the Matsub-
ara axis Green’s function. We have therefore solved the IPT equations for
the spectral functions at intermediate frequencies. We show the results in
Fig 3.
The results of the IPT calculations in the regime of intermediate cou-
pling (Fig 3) are consistent with known results for the limiting cases. For
frequencies ω ≫ ω0, the system has the qualitative behavior of the static
limit: The original unperturbed density of states splits into two sub-bands
centered around ±U/2. For small frequencies (ω ≪ U,ω0) and interaction
strength, U , less than some critical value, the system behaves as an in-
teracting electron (Hubbard) model, since the retarded interaction between
particles U(iωs) (see equation 2) is effectively constant for ωs ≪ ω0. There is
then a narrow quasiparticle band at the Fermi energy with density of states
at the Fermi energy pinned at its non-interacting value 6. We also note that
the results for small coupling and small frequencies are in good agreement
with those calculated using ME theory in the metallic phase 22.
The recent renormalization group (NRG) calculations of Meyer et al.
23,24 also report the spectral function in the intermediate regime. The NRG
is in principle an exact method for solving the impurity problem onto which
the DMFT equations map. Our results are largely consistent with theirs
adding further support to the use of IPT in the intermediate regime. When
comparing with the results of Meyer et al 23, one should note that the Hamil-
tonian (1) is exactly the one considered in Ref. 23 but with the quantity
2Mω0 =
Uω0/2 denoted by g in Ref. 23 and with energies measured
in terms of the full bandwidth (instead of the half-bandwidth used here).
In this paper, we work with the Gaussian density of states for the non-
interacting electron DOS, whereas reference 23 uses the semi-elliptic DOS.
In general, we would expect the critical values for the opening of a gap to
be larger for the Gaussian case than for the semi-elliptic case. The critical
coupling for the parameters in Figure 3(c) lies just above U = 2.0, corre-
sponding in the units used in Ref. 23 to g = 1, compared with the critical
value found for the semi-elliptic DOS of g = 0.69 (note that because of the
different energy scales ω0 = 2 in our results corresponds to ω0 = 1 in the
units of 23). However, the shapes of the spectral functions are similar in
both cases, with a five-peaked structure below and four peaked structure
above the transition. The peaks are narrower in the IPT results than in the
NRG results and there is less weight in the high energy peaks. This may
reflect the different DOS, or inaccuracies in the NRG method at frequencies
far from the Fermi energy resulting from the logarithmic discretisation, but
more likely the limitations of the IPT method.
Using the method outlined in section 2. it is possible to calculate the
real-axis self-energy. The temperature evolution of the imaginary part of the
self-energy may be seen in figure 4 for U = 2 and ω0 = 2 The self-energy
at low temperatures and small frequencies shows a quadratic (Fermi-liquid
like) behavior consistent with the narrow quasiparticle peak seen in the spec-
tral function (Fig 3) and develops to a broad central peak at higher tem-
peratures. There are also peaks corresponding to the Hubbard sub-bands.
With increasing temperature these phonon-induced peaks move together and
merge into the single central maximum associated with incoherent on-site
scattering. This peak is naturally characterized within the framework of
the self-consistent impurity model formulation of the DMFT equations 6 in
terms of a Kondo resonance. In this formulation, the dynamical mean field
G0(ω) is written in terms of a hybridization ∆(ω) between the site orbital and
a bath of conduction electrons and is therefore equivalent to an Anderson
impurity model with the added complication that ∆ is frequency-dependent
and needs to be computed self-consistently. However, many of the properties
in the metallic state are similar to those of the Anderson impurity model.
In particular the central peak in the spectral function can be viewed as the
Kondo resonance of the impurity model.
As all connected point-functions with order higher than G0 are neglected
within DMFT, the computation of q = 0 response functions is straightfor-
ward. As an example we show in fig 5 the (real part of the) optical con-
ductivity for various temperatures with U = 2 and ω0 = 2. The structure
seen in the curves reflects the structure of the density of states. There is a
strong response at low frequencies as particle-hole pairs are excited within
the ‘Kondo-like’ quasiparticle resonance at the Fermi energy. The second
peak at frequencies ω ∼ 1 arises from excitations between the quasiparti-
cle resonance and the large satellite (Hubbard band), while the third peak
around ω ∼ 5.0 involves excitations between the satellites. The first dip
at ω = 0.5 is the signature of the small Mott bands close to the Kondo
resonance and is the feature most likely to be observable experimentally.
Also calculated is the resistivity as a function of temperature (figure 6).
The curves reflect the structure of the self-energy shown in Figure 4: At low
temperatures the resistivity rises quadratically as expected for interacting
electrons. The temperature scale is given by the quasiparticle bandwidth
(‘Kondo temperature’). Above this temperature the resistivity drops as the
on-site (Kondo) scattering amplitude for electrons reduces. There is a slight
second peak at higher temperatures. The structures in ρ can be traced back
to the behavior seen in the self-energy. This second peak is the result of an
increase in scattering off the phonons: these soften slowly with increasing
temperature and, around the second peak in the resistivity curve, outweigh
the reduction in Kondo-like scattering as the temperature increases. This
effect clearly involves a partial cancellation between two effects and hence
may be sensitive to the accuracy of the analytic continuation, which at higher
temperatures starts from reduced information (since the majority of Mat-
subara points simply show asymptotic behaviour).
4. Summary
We have discussed the result of changing the ratio of electron and
phonon energies as a method for tuning the amount of correlation in a model
of electron-phonon interactions. We use approximate schemes to solve for
the spectral functions of the Holstein model. On the basis that second-order
iterated perturbation theory predicts the correct qualitative behavior at a
range of couplings in the static limit as well as describing correctly the limit
of infinite phonon frequency, we have computed the spectral function at in-
termediate frequencies and couplings. We have used an adaptation of the
standard maximum entropy scheme to obtain the spectral function, the self-
energy and the conductivity of the model by analytic continuation. These
quantities had not previously been studied.
The results for the intermediate frequency regime are consistent with
what might be expected on the basis of the limiting cases (high and low
frequencies). At energy scales smaller than ω0, the system shows behav-
ior similar to that of the Hubbard model found in the instantaneous limit
ω0 → ∞: there is a narrow central ‘Kondo-resonance’ or quasiparticle band.
At large energies the model behaves as it does in the static regime with a
well-defined band splitting. At intermediate frequencies the picture is com-
plicated by the interplay of the loss of coherence in the quasiparticle band
and the effective renormalization of the phonon frequency as a function of
coupling and temperature. We suggest that if systems with anomalously
large phonon frequencies and couplings exist, then the optical conductivity
should bear the hallmark of the correlation tuned regime.
5. Acknowledgements
The authors would like to thank F.Essler and F.Gebhard for useful
discussions.
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-6 -4 -2 0 2 4 6
(a) Static
U=0.33
U=1.17
U=2.00
U=4.50
-6 -4 -2 0 2 4 6
(b) IPT
U=0.33
U=1.17
U=2.00
U=4.50
-6 -4 -2  0  2  4  6
(c) Conserving
U=0.33
U=1.17
U=2.00
U=4.50
Fig. 2. The spectral function in the static limit of the half-filled Holstein
model computed at temperature T = 0.08 (a) using the exact solution and
(b) using 2nd order IPT at a low frequency, ω0 = 0.004. The IPT solution
at this small non-zero frequency is quite close to the exact solution in the
static limit. In particular, the band splitting and the positions of the maxima
agree. To contrast, panel (c) shows the results of the approximation using
the full Green’s function (Diagram 2c from figure 1 is not included to avoid
overcounting)
-6 -4 -2 0 2 4 6
(a) ω0=0.056
U=0.33
U=1.17
U=2.00
U=4.50
-6 -4 -2 0 2 4 6
(b) ω0=0.500
U=0.33
U=2.00
U=4.50
-6 -4 -2 0 2 4 6
(c) ω0=2.000
U=0.33
U=2.00
U=4.50
Fig. 3. Spectral functions of the half-filled Holstein model for various
electron-phonon couplings U , approximated using 2nd order perturbation
theory at T = 0.02 and ω0 = 0.056 (top), ω0 = 0.5 (center) and ω0 = 2
(bottom). In the low frequency limit (ω0 = 0.125), the spectral functions
are similar to those in the static limit shown in Fig. 2, with only a small
effect from the non-zero phonon frequency. As the temperature is lower than
the phonon frequency, the central quasiparticle peak is clearly resolved for
U ≤ 2. For the intermediate frequencies (central panel) the peak around
ω = 0 is again clear and has a width ∼ ω0 at low coupling. In the gapped
phase at large couplings two band-splittings are visible. For ω ≫ ω0 the
band splits just as in the static limit, while for ω ≪ U there is a peak at
a renormalized phonon frequency (which is less than the bare phonon fre-
quency). In the ungapped phases for ω0 = 0.5 and 2, the low energy behavior
is similar to that found in the Hubbard model with a narrow quasiparticle
band forming near the Fermi energy with the value at the Fermi energy
pinned to its value in the non-interacting case.
-6 -4 -2 0 2 4 6
T=0.08
T=0.16
T=0.32
Fig. 4. Imaginary part of the self-energy of the half-filled Holstein model
when U = 2 and ω0 = 2 computed using IPT and analytically continued
using MAXENT. At low temperatures the low frequency behavior is Fermi-
liquid like (quadratic dependence on ω) down to quite low frequencies (at
very low frequencies and low temperatures there are some inaccuracies as-
sociated with the truncation in Matsubara frequencies). There are peaks at
the frequencies associated with the phonon energy and with U. As the tem-
perature increases the minimum at the Fermi energy (ω = 0) increases as
incoherent on-site scattering in the corresponding local impurity increases
(see text). At temperatures above the characteristic (Kondo-like) energy
scale the central peak subsides and disappears.
0 1 2 3 4 5 6
T=0.08
T=0.16
T=0.32
Fig. 5. The real part of the optical conductivity for a system with U = 2.0
and ω0 = 2.0 for a range of temperatures. The structure of the spectrum
reflects that in the density of states (see fig 3. At low frequencies, electrons
may be excited within the quasiparticle resonance. The second peak at
ω ∼ 2.0 represents excitations from the Kondo resonance to the large satellite
(Hubbard band), and the peak at ω ∼ 5.0 represents excitations between the
satellites.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
U=2.00
U=2.13
U=2.28
Fig. 6. The resistivity as a function of temperature for the Holstein model
for ω0 = 2 for varying electron-phonon coupling strengths. The resistivity is
in units of e2V/ha2 with V the unit cell volume and a the lattice cell spacing.
The behavior reflects what is seen in the self-energy. At low temperatures
the behavior is similar to that in a Kondo lattice. The resistivity rises
sharply with temperature for temperatures smaller than the quasiparticle
bandwidth. The resistivity then drops for temperatures larger than this
lattice coherence temperature. A simple logarithmic decay with temperature
is not visible because, in addition to the Kondo-like scattering processes,
the electrons are scattered from thermally excited phonons whose spectral
weight broadens and shifts towards lower frequencies as the temperature
rises. This leads to a second peak. In contrast, the second peak is not
visible for the Hubbard model, and indicates the presence of two energy
scales in the Holstein model.
	Introduction
	Formalism
	Results
	Summary
	Acknowledgements

  We investigate the effect of tuning the phonon energy on the correlation
effects in models of electron-phonon interactions using DMFT. In the regime
where itinerant electrons, instantaneous electron-phonon driven correlations
and static distortions compete on similar energy scales, we find several
interesting results including (1) A crossover from band to Mott behavior in the
spectral function, leading to hybrid band/Mott features in the spectral
function for phonon frequencies slightly larger than the band width. (2) Since
the optical conductivity depends sensitively on the form of the spectral
function, we show that such a regime should be observable through the low
frequency form of the optical conductivity. (3) The resistivity has a double
kondo peak arrangement

Introduction
Mounting experimental evidence from high-Tc cuprates 1, nickelates 2,
manganites 3,4 and other interesting materials suggests that large electron-
phonon interactions may play a more important role in the physics of strongly
correlated electron systems than previously thought. Migdal-Eliashberg and
BCS theories have proved extremely successful in describing the effects of
phonons in many materials. However, if the coupling between electrons and
the underlying lattice is large, and/or the phonons can not be treated within
an adiabatic approximation, conventional approaches fail.
The Holstein model contains most of the fundamental physics of the
electron-phonon problem 5. Tight-binding electrons are coupled to the lat-
tice through a local interaction with Einstein modes. For large phonon
frequencies, electrons interact with a strongly correlated Hubbard-like at-
traction, while for small phonon frequencies the lattice gives rise to a static
http://arxiv.org/abs/0704.0030v1
potential which is essentially uncorrelated. Between these two extreme lim-
its of correlated and uncorrelated behavior, levels of correlation are tuned
by the size of the phonon frequency and novel physics is expected. In par-
ticular, it is normally the strength of interaction which is said to tune the
correlation in e.g. the Hubbard model, whereas in the Holstein model, it can
be seen that both interaction strength and phonon frequency may compete
with each other to play this role.
The dynamical mean-field theory (DMFT) approach has proved suc-
cessful in treating the Holstein and other models 6,7,8. DMFT treats the
self-energy as a momentum-independent quantity and is accurate as long as
the variation across the Brillouin zone is small. For many aspects of the
electron-phonon problem in 3D, correlations are short ranged and DMFT
can be successfully applied. The weak coupling phase diagram was studied
by Ciuchi et al. where competing charge-order (CO) and superconducting
states were found 7. Freericks et al. developed a quantum Monte-Carlo
(QMC) algorithm 8,9 and examined the applicability of several perturba-
tion theory based techniques to the electron-phonon problem 10,11,12. The
prediction of measurable quantities away from certain well-defined limits is
severely restricted owing to difficulties inherent in the analytic continuation.
Dynamic properties such as spectral functions can be computed in the case
of static phonons 13, and close to the static limit 14. Alternatively, the limit
of high phonon frequency (attractive Hubbard model) has been studied with
a QMC algorithm 15.
In the current study we are concerned with the behavior of dynamical
properties that could be measured directly with experiment. We use the iter-
ated perturbation theory approximation, which has been demonstrated to be
accurate for the Hubbard model, and use maximum entropy to analytically
continue the results. We compare the resulting single-particle spectral func-
tions over a wide range of electron-phonon coupling strengths and phonon
frequencies. The results obtained using iterated perturbation theory (IPT)
are promising and capture generic weak and strong coupling behavior for
all phonon frequencies. At intermediate phonon frequencies, we find that
electron-phonon interactions produce a spectral function which is simulta-
neously characteristic of both uncorrelated band (static) and strongly cor-
related Mott/Hubbard regimes. We also find that the competition between
band-like and correlated states causes unusual structures in the optical con-
ductivity and resistivity. Provided a material with high enough phonon
frequency can be identified, it is possible that such a state could be observed
experimentally.
This paper is organized as follows. First, we introduce the Holstein
model, the dynamical mean-field theory and analytic continuation techniques
(1a) (2a)
(2b) (2c)
Fig. 1. Second order contributions to the self-energy. Straight lines repre-
sent electron Green’s functions of the host and wavy lines phonon Green’s
functions.
(section 2.). In section 3., we use IPT to determine the spectral functions of
the Holstein model. We compare IPT with exactly known results in the static
limit. This, in conjunction with the conclusions of Ref. 11 leads us to argue
that IPT is a reasonable approximation for the calculation of dynamical
properties in the intermediate phonon frequency regime. We compute the
density of states, optical conductivity and resistivity, and give a heuristic
explanation for their behavior.
2. Formalism
The Holstein Hamiltonian is written as,
H = −t
<ij>σ
iσcjσ +
(gxi − µ)niσ +
Mω20x
The first term in this Hamiltonian represents a tight binding model with
hopping parameter t. The second term couples the local ion displacement,
xi to the local electron density. The final term can be identified as the
non-interacting phonon Hamiltonian. c
i (ci) create (annihilate) electrons at
site i, pi is the ion momentum, M the ion mass, µ the chemical potential
and g the electron-phonon coupling. The phonons are dispersionless with
frequency ω0.
The perturbation theory of this model may be written down in terms
of electrons interacting via phonons with the effective interaction,
U(iωs) = −
M(ω2s + ω
Here, ωs = 2πsT are the Matsubara frequencies for bosons and s is an
integer. Taking the limit ω0 → ∞, g → ∞, while keeping the ratio g/ω0
finite, leads to an attractive Hubbard model with a non-retarded on-site
interaction U = −g2/Mω20 . Iterated-perturbation theory (IPT) is known to
be a reasonable approximation to the half-filled Hubbard model 16,17. Taking
the opposite limit (ω0 → 0, M → ∞, keeping Mω20 ≡ κ finite) the phonon
kinetic energy term vanishes, and the phonons depend on a static variable
xi. As such, the model may be considered as uncorrelated.
We solve the Holstein model using dynamical mean-field theory (DMFT).
DMFT freezes spatial fluctuations, leading to a theory which is completely
momentum independent, while fully including dynamical effects of excita-
tions. In spite of this simplification, DMFT predicts non-trivial (correlated)
physics and may be used as an approximation to 3D models 18. As discussed
in Ref. 6, DMFT involves the solution of a set of coupled equations which
are solved self-consistently. The Green’s function for the single site problem,
G(iωn) can be written in terms of the self-energy Σ(iωn) as,
G−1(iωn) = G−10 (iωn)− Σ(iωn), (3)
where Σ is a functional of G0, the Green’s function for the host of a single
impurity model. Here ωn = 2πT (n+ 1/2) are the usual Matsubara frequen-
cies. The assumptions of DMFT are equivalent to taking the self-energy of
the original lattice problem to be local, hence G is also given by,
G(iωn) =
dǫD(ǫk)
iωn + µ− Σ(iωn)− ǫk
where D(ǫ) the density of states (DOS) of the non-interacting problem (in
our case g = 0). We work with a Gaussian DOS which corresponds to a
hypercubic lattice 18, D(ǫ) = exp(−ǫ2/2t2)/t
2π. Equations (3) and (4)
are solved according to the following self-consistent procedure: Compute
the Green’s function from equation (4) and the host Green’s function of the
effective impurity problem, G0, from equation (3); then calculate a new self-
energy from the host or full Green’s functions. In the following we will take
the hopping parameter t = 0.5, which sets the energy scale.
Once the algorithm has converged, and after analytically continuing to
the real axis, response functions can be computed. We use the MAXENT
method for the determination of spectral functions from Matsubara axis
data. MAXENT treats the analytic continuation as an inverse problem 19.
The Green’s function, G(z), is given by the integral transform,
G(z) =
z − x
dx (5)
where ρ(x) is the spectral function (ρ(ω) = Im[G(ω + iη)]/π). The problem
of finding ρ is therefore one of inverting the integral transform. Since the
data for Gn are incomplete and noisy for any finite set of Matsubara fre-
quencies, the inversion of the kernel of the discretised problem is ill-defined.
The MAXENT method selects the distribution ρ(x) which assumes the least
structure consistent with the calculated or measured data. These methods
have been extensively reviewed in the context of the inversion of the kernel
in Refs. 19,20. The applicability to the current problem has been thoroughly
tested, and is found to be accurate.
Within the DMFT formalism, many response functions follow directly
from the one-electron spectral function and the electron self-energy (essen-
tially because of the neglect of all connected higher point functions apart
from G0). Here we will be interested in the conductivity 6:
Re[σ(ω)] =
dǫD(ǫ)
dν ρ(ǫ, ν)ρ(ǫ, ν + ω)[f(ν)− f(ν + ω)] (6)
where f(x) is the Fermi-Dirac distribution. Taking the limit, ω → 0, leads
to the DC conductivity. (The conductivity is in units of e2V/ha2, where a
is the lattice spacing, V the volume of the unit cell and e and h are the
electron charge and Planck’s constant respectively.)
3. Results
In this section, we examine the validity of an approximation to the self-
energy constructed from only first and second order terms with respect to the
spectral functions calculated at very high and very low phonon frequencies.
Finally, we calculate the optical conductivity and resistivity.
Spectral functions are shown in figures 2 and 3. The perturbation theory
is carried out in the host Green’s function (i.e. both electrons and phonons
are bare). All diagrams in fig. 1 are considered,
Σ1a(iωn) = −UT
G0(iωn − iωs)D0(iωs) (7)
Σ2a(iωn) = −2U2T 2
D20(iωn−m)G0(iωm)G0(iωs)G0(iωn−m+s) (8)
Σ2b(iωn) = U
D0(iωm−s)D0(iωn−m)G0(iωm)G0(iωs)G0(iωn−m+s)
Σ2c(iωn) = U
D0(iωn−m)D0(iωm−s)G20(iωm)G0(iωs) (10)
This also gives the correct weak coupling limit for the electronic Green’s
function.
We consider first the calculation of spectral functions close to the static
and Hubbard limits. In the instantaneous limit the perturbation theories
for the Holstein and Hubbard models are equivalent. It is well known that
second order perturbation theory in the host Green’s function provides a
good approximation to the Hubbard model 21. In the static limit, the exact
solution can easily be calculated 13. Figure 2 shows spectral functions from
the exact solution, computed for a hypercubic lattice and spectral functions,
computed using 2nd order perturbation theory at a temperature of T = 0.08.
The phonon frequency ω0 = T/20 was chosen so that the effects of the
phonon kinetic energy are negligible compared to thermal fluctuations. This
allows a direct comparison to be made between the exact and approximate
results. The comparison shows that the widths and positions of the major
features are closely related.
The results in Figure 2 for the static limit (ω0 → 0), together with the
fact that second order IPT is known to give reasonable results in the instan-
taneous limit (ω0 → ∞), suggest that the calculation of spectral functions
should also be reliable at intermediate frequencies. We note that Freericks et
al. also find a reasonable agreement between the IPT and QMC self-energies
at half-filling, and that this should lead to a good agreement in the Matsub-
ara axis Green’s function. We have therefore solved the IPT equations for
the spectral functions at intermediate frequencies. We show the results in
Fig 3.
The results of the IPT calculations in the regime of intermediate cou-
pling (Fig 3) are consistent with known results for the limiting cases. For
frequencies ω ≫ ω0, the system has the qualitative behavior of the static
limit: The original unperturbed density of states splits into two sub-bands
centered around ±U/2. For small frequencies (ω ≪ U,ω0) and interaction
strength, U , less than some critical value, the system behaves as an in-
teracting electron (Hubbard) model, since the retarded interaction between
particles U(iωs) (see equation 2) is effectively constant for ωs ≪ ω0. There is
then a narrow quasiparticle band at the Fermi energy with density of states
at the Fermi energy pinned at its non-interacting value 6. We also note that
the results for small coupling and small frequencies are in good agreement
with those calculated using ME theory in the metallic phase 22.
The recent renormalization group (NRG) calculations of Meyer et al.
23,24 also report the spectral function in the intermediate regime. The NRG
is in principle an exact method for solving the impurity problem onto which
the DMFT equations map. Our results are largely consistent with theirs
adding further support to the use of IPT in the intermediate regime. When
comparing with the results of Meyer et al 23, one should note that the Hamil-
tonian (1) is exactly the one considered in Ref. 23 but with the quantity
2Mω0 =
Uω0/2 denoted by g in Ref. 23 and with energies measured
in terms of the full bandwidth (instead of the half-bandwidth used here).
In this paper, we work with the Gaussian density of states for the non-
interacting electron DOS, whereas reference 23 uses the semi-elliptic DOS.
In general, we would expect the critical values for the opening of a gap to
be larger for the Gaussian case than for the semi-elliptic case. The critical
coupling for the parameters in Figure 3(c) lies just above U = 2.0, corre-
sponding in the units used in Ref. 23 to g = 1, compared with the critical
value found for the semi-elliptic DOS of g = 0.69 (note that because of the
different energy scales ω0 = 2 in our results corresponds to ω0 = 1 in the
units of 23). However, the shapes of the spectral functions are similar in
both cases, with a five-peaked structure below and four peaked structure
above the transition. The peaks are narrower in the IPT results than in the
NRG results and there is less weight in the high energy peaks. This may
reflect the different DOS, or inaccuracies in the NRG method at frequencies
far from the Fermi energy resulting from the logarithmic discretisation, but
more likely the limitations of the IPT method.
Using the method outlined in section 2. it is possible to calculate the
real-axis self-energy. The temperature evolution of the imaginary part of the
self-energy may be seen in figure 4 for U = 2 and ω0 = 2 The self-energy
at low temperatures and small frequencies shows a quadratic (Fermi-liquid
like) behavior consistent with the narrow quasiparticle peak seen in the spec-
tral function (Fig 3) and develops to a broad central peak at higher tem-
peratures. There are also peaks corresponding to the Hubbard sub-bands.
With increasing temperature these phonon-induced peaks move together and
merge into the single central maximum associated with incoherent on-site
scattering. This peak is naturally characterized within the framework of
the self-consistent impurity model formulation of the DMFT equations 6 in
terms of a Kondo resonance. In this formulation, the dynamical mean field
G0(ω) is written in terms of a hybridization ∆(ω) between the site orbital and
a bath of conduction electrons and is therefore equivalent to an Anderson
impurity model with the added complication that ∆ is frequency-dependent
and needs to be computed self-consistently. However, many of the properties
in the metallic state are similar to those of the Anderson impurity model.
In particular the central peak in the spectral function can be viewed as the
Kondo resonance of the impurity model.
As all connected point-functions with order higher than G0 are neglected
within DMFT, the computation of q = 0 response functions is straightfor-
ward. As an example we show in fig 5 the (real part of the) optical con-
ductivity for various temperatures with U = 2 and ω0 = 2. The structure
seen in the curves reflects the structure of the density of states. There is a
strong response at low frequencies as particle-hole pairs are excited within
the ‘Kondo-like’ quasiparticle resonance at the Fermi energy. The second
peak at frequencies ω ∼ 1 arises from excitations between the quasiparti-
cle resonance and the large satellite (Hubbard band), while the third peak
around ω ∼ 5.0 involves excitations between the satellites. The first dip
at ω = 0.5 is the signature of the small Mott bands close to the Kondo
resonance and is the feature most likely to be observable experimentally.
Also calculated is the resistivity as a function of temperature (figure 6).
The curves reflect the structure of the self-energy shown in Figure 4: At low
temperatures the resistivity rises quadratically as expected for interacting
electrons. The temperature scale is given by the quasiparticle bandwidth
(‘Kondo temperature’). Above this temperature the resistivity drops as the
on-site (Kondo) scattering amplitude for electrons reduces. There is a slight
second peak at higher temperatures. The structures in ρ can be traced back
to the behavior seen in the self-energy. This second peak is the result of an
increase in scattering off the phonons: these soften slowly with increasing
temperature and, around the second peak in the resistivity curve, outweigh
the reduction in Kondo-like scattering as the temperature increases. This
effect clearly involves a partial cancellation between two effects and hence
may be sensitive to the accuracy of the analytic continuation, which at higher
temperatures starts from reduced information (since the majority of Mat-
subara points simply show asymptotic behaviour).
4. Summary
We have discussed the result of changing the ratio of electron and
phonon energies as a method for tuning the amount of correlation in a model
of electron-phonon interactions. We use approximate schemes to solve for
the spectral functions of the Holstein model. On the basis that second-order
iterated perturbation theory predicts the correct qualitative behavior at a
range of couplings in the static limit as well as describing correctly the limit
of infinite phonon frequency, we have computed the spectral function at in-
termediate frequencies and couplings. We have used an adaptation of the
standard maximum entropy scheme to obtain the spectral function, the self-
energy and the conductivity of the model by analytic continuation. These
quantities had not previously been studied.
The results for the intermediate frequency regime are consistent with
what might be expected on the basis of the limiting cases (high and low
frequencies). At energy scales smaller than ω0, the system shows behav-
ior similar to that of the Hubbard model found in the instantaneous limit
ω0 → ∞: there is a narrow central ‘Kondo-resonance’ or quasiparticle band.
At large energies the model behaves as it does in the static regime with a
well-defined band splitting. At intermediate frequencies the picture is com-
plicated by the interplay of the loss of coherence in the quasiparticle band
and the effective renormalization of the phonon frequency as a function of
coupling and temperature. We suggest that if systems with anomalously
large phonon frequencies and couplings exist, then the optical conductivity
should bear the hallmark of the correlation tuned regime.
5. Acknowledgements
The authors would like to thank F.Essler and F.Gebhard for useful
discussions.
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-6 -4 -2 0 2 4 6
(a) Static
U=0.33
U=1.17
U=2.00
U=4.50
-6 -4 -2 0 2 4 6
(b) IPT
U=0.33
U=1.17
U=2.00
U=4.50
-6 -4 -2  0  2  4  6
(c) Conserving
U=0.33
U=1.17
U=2.00
U=4.50
Fig. 2. The spectral function in the static limit of the half-filled Holstein
model computed at temperature T = 0.08 (a) using the exact solution and
(b) using 2nd order IPT at a low frequency, ω0 = 0.004. The IPT solution
at this small non-zero frequency is quite close to the exact solution in the
static limit. In particular, the band splitting and the positions of the maxima
agree. To contrast, panel (c) shows the results of the approximation using
the full Green’s function (Diagram 2c from figure 1 is not included to avoid
overcounting)
-6 -4 -2 0 2 4 6
(a) ω0=0.056
U=0.33
U=1.17
U=2.00
U=4.50
-6 -4 -2 0 2 4 6
(b) ω0=0.500
U=0.33
U=2.00
U=4.50
-6 -4 -2 0 2 4 6
(c) ω0=2.000
U=0.33
U=2.00
U=4.50
Fig. 3. Spectral functions of the half-filled Holstein model for various
electron-phonon couplings U , approximated using 2nd order perturbation
theory at T = 0.02 and ω0 = 0.056 (top), ω0 = 0.5 (center) and ω0 = 2
(bottom). In the low frequency limit (ω0 = 0.125), the spectral functions
are similar to those in the static limit shown in Fig. 2, with only a small
effect from the non-zero phonon frequency. As the temperature is lower than
the phonon frequency, the central quasiparticle peak is clearly resolved for
U ≤ 2. For the intermediate frequencies (central panel) the peak around
ω = 0 is again clear and has a width ∼ ω0 at low coupling. In the gapped
phase at large couplings two band-splittings are visible. For ω ≫ ω0 the
band splits just as in the static limit, while for ω ≪ U there is a peak at
a renormalized phonon frequency (which is less than the bare phonon fre-
quency). In the ungapped phases for ω0 = 0.5 and 2, the low energy behavior
is similar to that found in the Hubbard model with a narrow quasiparticle
band forming near the Fermi energy with the value at the Fermi energy
pinned to its value in the non-interacting case.
-6 -4 -2 0 2 4 6
T=0.08
T=0.16
T=0.32
Fig. 4. Imaginary part of the self-energy of the half-filled Holstein model
when U = 2 and ω0 = 2 computed using IPT and analytically continued
using MAXENT. At low temperatures the low frequency behavior is Fermi-
liquid like (quadratic dependence on ω) down to quite low frequencies (at
very low frequencies and low temperatures there are some inaccuracies as-
sociated with the truncation in Matsubara frequencies). There are peaks at
the frequencies associated with the phonon energy and with U. As the tem-
perature increases the minimum at the Fermi energy (ω = 0) increases as
incoherent on-site scattering in the corresponding local impurity increases
(see text). At temperatures above the characteristic (Kondo-like) energy
scale the central peak subsides and disappears.
0 1 2 3 4 5 6
T=0.08
T=0.16
T=0.32
Fig. 5. The real part of the optical conductivity for a system with U = 2.0
and ω0 = 2.0 for a range of temperatures. The structure of the spectrum
reflects that in the density of states (see fig 3. At low frequencies, electrons
may be excited within the quasiparticle resonance. The second peak at
ω ∼ 2.0 represents excitations from the Kondo resonance to the large satellite
(Hubbard band), and the peak at ω ∼ 5.0 represents excitations between the
satellites.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
U=2.00
U=2.13
U=2.28
Fig. 6. The resistivity as a function of temperature for the Holstein model
for ω0 = 2 for varying electron-phonon coupling strengths. The resistivity is
in units of e2V/ha2 with V the unit cell volume and a the lattice cell spacing.
The behavior reflects what is seen in the self-energy. At low temperatures
the behavior is similar to that in a Kondo lattice. The resistivity rises
sharply with temperature for temperatures smaller than the quasiparticle
bandwidth. The resistivity then drops for temperatures larger than this
lattice coherence temperature. A simple logarithmic decay with temperature
is not visible because, in addition to the Kondo-like scattering processes,
the electrons are scattered from thermally excited phonons whose spectral
weight broadens and shifts towards lower frequencies as the temperature
rises. This leads to a second peak. In contrast, the second peak is not
visible for the Hubbard model, and indicates the presence of two energy
scales in the Holstein model.
	Introduction
	Formalism
	Results
	Summary
	Acknowledgements
