Optical properties of theHolstein-t-Jmodel fromdynamicalmean-field theory
E. Cappelluti a,b,∗, S. Ciuchi c, S. Fratini d
aDipartimento di Fisica, Università “La Sapienza”, P.le A. Moro 2, 00185 Rome, Italy
bSMC Research Center and ISC, INFM-CNR, v. dei Taurini 19, 00185 Rome, Italy
cINFM and Dipartimento di Fisica, Università dell’Aquila, via Vetoio, I-67010 Coppito-L’Aquila, Italy
dInstitut Néel - CNRS & Université Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France
Abstract
We employ dynamical mean-field theory to study the optical conductivity σ(ω) of one hole in the Holstein-t-J model. We provide
an exact solution for σ(ω) in the limit of infinite connectivity. We apply our analysis to Nd2−xCexCuO4. We show that our model
can explain many features of the optical conductivity in this compounds in terms of magnetic/lattice polaron formation.
Key words: magnetic/lattice polarons, spin fluctuations, optical conductivity, cuprates.
PACS: 71.10.Fd, 71.38.-k, 78.20.Bh, 75.30.Ds.
The problem of a single hole in the t-J model interact-
ing also with the lattice degrees of freedom has attracted
recently a notable interest in connection with the physical
properties of the underdoped high-T
cuprates [1,2,3,4]. An
important issue in this regime is the formation of lattice
or magnetic polarons (or both of them) and their mutual
interaction. Along this line, the one-particle properties (as
the effective mass, spectral function, etc.) have been widely
investigated with different techniques. Much less effort has
been however paid to the study of the optical properties.
On the analytical ground, the definition of the optical con-
ductivity (OC) in the single hole is a delicate matter which
needs particular care even for the pure t-J or Holstein
model [5,6]. On the other hand, numerical calculations on
clusters are limited by finite size effects [7]. As a general
rule, thus, the choice of a particular theoretical approach
depends on which property is under examination and on
its feasibility to investigate it.
In this paper we summarize the main results of our work
based on the dynamical mean-field theory (DMFT). Tech-
nical details will be presented in a forthcoming longer pub-
lication [8]. In the infinite coordination number limit z →
∞, we provide an exact solution for σ(ω) as a functional
of the local one-particle Green’s function at finite temper-
ature. It should be stressed that, due to the classical treat-
ment of the magnetic background, the DMFT solution for
∗ Corresponding author. Tel: (+39) 06-49937453 fax: (+39) 06-
49937440
Email address: emmcapp@roma1.infn.it (E. Cappelluti).
0 1 2 3 4
Ref. [7]
this work
λ=1, J/t=0.4, ω
Fig. 1. Comparison between the optical conductivity σ(ω) obtained
by our DMFT solution and Lanczos diagonalization in two dimen-
sions on a finite cluster (Ref. [7]).
z → ∞ is purely local so that it cannot describe the coher-
ent propagation of holes due to the spin fluctuations, nor
the metallic Drude-like peak in σ(ω). On the other hand,
the local properties (as the average number of phonons, size
of the magnetic polaron, etc.) are well captured by this ap-
proach, [9] as well as the incoherent contributions to the
OC. We can explicitly show this feature by comparing in
Fig. 1 our DMFT results with numerical calculations using
Lanczos diagonalization for a single hole in the 2DHolstein-
t-J model on a
10 cluster [7].
The remarkably good agreement of the overall shape as-
sesses the feasibility of our approach to investigate the in-
coherent contributions to the finite frequency OC. This is-
sue is particularly important in light of the intensive de-
bate about the origin of the mid-infrared (MIR) band in the
underdoped high-T
cuprates. Different interpretations for
this feature have been discussed in the literature, involving
Preprint submitted to Elsevier 29 October 2018
http://arxiv.org/abs/0704.0333v1
charge/spin fluctuations, stripe ordering, and other mecha-
nisms. This spread of differentmechanisms reflects the pres-
ence in this doping regime of several actors, which makes
it difficult to isolate each effect from the others. A simpler
and ideal situation is the case of electron-doped cuprates, as
Nd2−xCexCuO4. In these compounds, the long-range anti-
ferromagnetic (AF) order extents up to x ≃ 0.14, so that
the low doping regime x . 0.1 we are interested in, lies well
within the AF phase. On the experimental side, in addi-
tion, a detailed and exhaustive study of the optical conduc-
tivity as a function of temperature T and of the doping x
was recently provided in Ref. [10]. In that work the authors
showed that the low doping OC spectra are characterized
at low temperature by a MIR pseudogap, with an absorp-
tion band edge which varies from EMIR ≃ 0.5− 0.6 for x =
0.05 to EMIR ≃ 0.3− 0.4 for x = 0.1, and is barely distin-
guishable for x = 0.125. Quite interestingly, increasing the
temperature leads to a filling of the pseudogap, rather than
a closing of it. Also remarkable is the temperature depen-
dence of the MIR spectral weight which does not present
any signature at the long-range Néel temperature TN but
rather a kink to a higher “pseudogap” temperature T ∗.
We show here that our approach is able to describe all
these features, and in particular the MIR band edge, in
terms of an optical gap due to the formation of a mag-
netic/lattice polaron. We define T ∗ as the temperature
where the size of the spin polaron becomes larger than
the AF correlation length, that is the maximum tempera-
ture where an injected charge actually probes the magnetic
background. In this perspective we can identify T ∗ with the
mean field Néel temperature of our model, which represents
the temperature above which the system is described by
a paramagnetic state (rather than the onset of long range
order). From Ref. [10] we get for instance T ∗ = 440 K at
x = 0.05 and T ∗ = 200 K at x = 0.125. Using the Curie-
Weiss relation T ∗
= J/4 we estimate respectively J = 152
meV (J/t = 0.126) and J = 69 meV (J/t = 0.057). Note
that such values of J do not represent the bare exchange
interaction but rather the effective spin-exchange coupling
which is reduced by hole doping. We also set ω0 = 84 meV,
consistent with the energy window of the optical phonons
in the cuprates. The electron-phonon (el-ph) coupling con-
stant is fixed to λ = 0.75 in order to reproduce the exper-
imental MIR band edge ≈ 0.5− 0.6 eV in the optical con-
ductivity at x = 0.05, and we assume λ to be independent
of the doping x. Note that with these choices no more free
adjustable parameters remain.
In Fig. 2 we show the temperature evolution of the MIR
optical conductivity for the representative cases x = 0.05
and x = 0.125 (note that in order to compare with the ex-
perimental data of Ref. [10] the tail of a Drude-peak should
be superimposed). Most remarkable is the behavior of σ(ω)
at low temperature, which shows a well defined gap for
x = 0.05 while no gap is found for x = 0.125. This fea-
ture reflects the formation of the lattice polaron and its in-
terplay with the spin degrees of freedom. While the el-ph
coupling λ = 0.75 alone is not strong enough at x = 0.125
0 0.5 1
ω  [eV]
0.5 1 1.5
ω  [eV]
0 200 400
T  [K]
x=0.05
x=0.125
T=50K
T=440K
T=540K
T=540K
T=340K
T=50K
T=190K
Fig. 2. Temperature dependence of the optical conductivity σ(ω) for
x = 0.05 and x = 0.125. Solid lines are used for T ≤ T ∗, dashed
lines for T > T ∗. Inset: loss of the MIR spectral weight ∆Neff , as
defined in Ref. [10], as function of T for x = 0.05 (filled circles) and
x = 0.125 (empty squares). Arrows mark the corresponding T ∗.
(J/t = 0.057) to establish a spin/lattice polaron, the lo-
calization effects induced by the larger exchange coupling
J/t = 0.126 at x = 0.05 favor the lattice polaron forma-
tion. This leads thus to the opening of an optical gap in
σ(ω) (this key point will be extensively discussed in a forth-
coming publication[8]). Increasing T reduces the localiza-
tion effects induced by the magnetic ordering. This makes
the positive interplay with the el-ph coupling less effective,
leading to a progressive filling of the pseudogap. Note that
this effect disappears in the disordered magnetic case for
T > T ∗, and further increasing of T leads to a reduction of
the MIR optical conductivity which is spread on a larger
energy window. This is reflected in the characteristic tem-
perature behavior of the MIR spectral weight ∆Neff , as de-
fined in Ref. [10], which presents a kink at T ∗ (inset of Fig.
2)[11].
References
[1] A.S. Mishchenko and and N. Nagaosa, Phys. Rev. Lett. 93
(2004) 0236402; Phys. Rev. B 73 (2006) 092502.
[2] O. Rösch and O. Gunnarsson, Phys. Rev. Lett. 92 (2004) 146403;
Eur. Phys. J. B 43 (2005) 11.
[3] O. Gunnarsson and O. Rösch, Phys. Rev. B 73 (2006) 174521.
[4] P. Prelovšek, R. Zeyher, and P. Horsch, Phys. Rev. Lett. 96
(2006) 086402.
[5] M.P.H. Stumpf and D.E. Logan, Eur.Phys.J.B, 8 (1999) 377.
[6] S. Fratini and S. Ciuchi, Phys. Rev. B 74 (2006) 075101.
[7] B. Bäuml et al., Phys. Rev. B 58 (1998) 3663.
[8] E. Cappelluti, S. Ciuchi and S. Fratini, in preparation (2007).
[9] E. Cappelluti and S. Ciuchi, Phys. Rev. B 66 (2002) 165102.
[10] Y. Onose et., Phys. Rev. B 69 (2004) 024504.
[11] Since we do not find any isosbestic point in our calculations,
we use the experimental energy windows of Ref. [10] to define
∆Neff , namely ωmin = 0.12 eV, ωmax = 0.42 eV for x = 0.05
and ωmax = 0.21 eV for x = 0.125.
	References

  We employ dynamical mean-field theory to study the optical conductivity
$\sigma(\omega)$ of one hole in the Holstein-t-J model. We provide an exact
solution for $\sigma(\omega)$ in the limit of infinite connectivity. We apply
our analysis to Nd$_{2-x}$Ce$_x$CuO$_4$. We show that our model can explain
many features of the optical conductivity in this compounds in terms of
magnetic/lattice polaron formation.

Optical properties of theHolstein-t-Jmodel fromdynamicalmean-field theory
E. Cappelluti a,b,∗, S. Ciuchi c, S. Fratini d
aDipartimento di Fisica, Università “La Sapienza”, P.le A. Moro 2, 00185 Rome, Italy
bSMC Research Center and ISC, INFM-CNR, v. dei Taurini 19, 00185 Rome, Italy
cINFM and Dipartimento di Fisica, Università dell’Aquila, via Vetoio, I-67010 Coppito-L’Aquila, Italy
dInstitut Néel - CNRS & Université Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France
Abstract
We employ dynamical mean-field theory to study the optical conductivity σ(ω) of one hole in the Holstein-t-J model. We provide
an exact solution for σ(ω) in the limit of infinite connectivity. We apply our analysis to Nd2−xCexCuO4. We show that our model
can explain many features of the optical conductivity in this compounds in terms of magnetic/lattice polaron formation.
Key words: magnetic/lattice polarons, spin fluctuations, optical conductivity, cuprates.
PACS: 71.10.Fd, 71.38.-k, 78.20.Bh, 75.30.Ds.
The problem of a single hole in the t-J model interact-
ing also with the lattice degrees of freedom has attracted
recently a notable interest in connection with the physical
properties of the underdoped high-T
cuprates [1,2,3,4]. An
important issue in this regime is the formation of lattice
or magnetic polarons (or both of them) and their mutual
interaction. Along this line, the one-particle properties (as
the effective mass, spectral function, etc.) have been widely
investigated with different techniques. Much less effort has
been however paid to the study of the optical properties.
On the analytical ground, the definition of the optical con-
ductivity (OC) in the single hole is a delicate matter which
needs particular care even for the pure t-J or Holstein
model [5,6]. On the other hand, numerical calculations on
clusters are limited by finite size effects [7]. As a general
rule, thus, the choice of a particular theoretical approach
depends on which property is under examination and on
its feasibility to investigate it.
In this paper we summarize the main results of our work
based on the dynamical mean-field theory (DMFT). Tech-
nical details will be presented in a forthcoming longer pub-
lication [8]. In the infinite coordination number limit z →
∞, we provide an exact solution for σ(ω) as a functional
of the local one-particle Green’s function at finite temper-
ature. It should be stressed that, due to the classical treat-
ment of the magnetic background, the DMFT solution for
∗ Corresponding author. Tel: (+39) 06-49937453 fax: (+39) 06-
49937440
Email address: emmcapp@roma1.infn.it (E. Cappelluti).
0 1 2 3 4
Ref. [7]
this work
λ=1, J/t=0.4, ω
Fig. 1. Comparison between the optical conductivity σ(ω) obtained
by our DMFT solution and Lanczos diagonalization in two dimen-
sions on a finite cluster (Ref. [7]).
z → ∞ is purely local so that it cannot describe the coher-
ent propagation of holes due to the spin fluctuations, nor
the metallic Drude-like peak in σ(ω). On the other hand,
the local properties (as the average number of phonons, size
of the magnetic polaron, etc.) are well captured by this ap-
proach, [9] as well as the incoherent contributions to the
OC. We can explicitly show this feature by comparing in
Fig. 1 our DMFT results with numerical calculations using
Lanczos diagonalization for a single hole in the 2DHolstein-
t-J model on a
10 cluster [7].
The remarkably good agreement of the overall shape as-
sesses the feasibility of our approach to investigate the in-
coherent contributions to the finite frequency OC. This is-
sue is particularly important in light of the intensive de-
bate about the origin of the mid-infrared (MIR) band in the
underdoped high-T
cuprates. Different interpretations for
this feature have been discussed in the literature, involving
Preprint submitted to Elsevier 29 October 2018
http://arxiv.org/abs/0704.0333v1
charge/spin fluctuations, stripe ordering, and other mecha-
nisms. This spread of differentmechanisms reflects the pres-
ence in this doping regime of several actors, which makes
it difficult to isolate each effect from the others. A simpler
and ideal situation is the case of electron-doped cuprates, as
Nd2−xCexCuO4. In these compounds, the long-range anti-
ferromagnetic (AF) order extents up to x ≃ 0.14, so that
the low doping regime x . 0.1 we are interested in, lies well
within the AF phase. On the experimental side, in addi-
tion, a detailed and exhaustive study of the optical conduc-
tivity as a function of temperature T and of the doping x
was recently provided in Ref. [10]. In that work the authors
showed that the low doping OC spectra are characterized
at low temperature by a MIR pseudogap, with an absorp-
tion band edge which varies from EMIR ≃ 0.5− 0.6 for x =
0.05 to EMIR ≃ 0.3− 0.4 for x = 0.1, and is barely distin-
guishable for x = 0.125. Quite interestingly, increasing the
temperature leads to a filling of the pseudogap, rather than
a closing of it. Also remarkable is the temperature depen-
dence of the MIR spectral weight which does not present
any signature at the long-range Néel temperature TN but
rather a kink to a higher “pseudogap” temperature T ∗.
We show here that our approach is able to describe all
these features, and in particular the MIR band edge, in
terms of an optical gap due to the formation of a mag-
netic/lattice polaron. We define T ∗ as the temperature
where the size of the spin polaron becomes larger than
the AF correlation length, that is the maximum tempera-
ture where an injected charge actually probes the magnetic
background. In this perspective we can identify T ∗ with the
mean field Néel temperature of our model, which represents
the temperature above which the system is described by
a paramagnetic state (rather than the onset of long range
order). From Ref. [10] we get for instance T ∗ = 440 K at
x = 0.05 and T ∗ = 200 K at x = 0.125. Using the Curie-
Weiss relation T ∗
= J/4 we estimate respectively J = 152
meV (J/t = 0.126) and J = 69 meV (J/t = 0.057). Note
that such values of J do not represent the bare exchange
interaction but rather the effective spin-exchange coupling
which is reduced by hole doping. We also set ω0 = 84 meV,
consistent with the energy window of the optical phonons
in the cuprates. The electron-phonon (el-ph) coupling con-
stant is fixed to λ = 0.75 in order to reproduce the exper-
imental MIR band edge ≈ 0.5− 0.6 eV in the optical con-
ductivity at x = 0.05, and we assume λ to be independent
of the doping x. Note that with these choices no more free
adjustable parameters remain.
In Fig. 2 we show the temperature evolution of the MIR
optical conductivity for the representative cases x = 0.05
and x = 0.125 (note that in order to compare with the ex-
perimental data of Ref. [10] the tail of a Drude-peak should
be superimposed). Most remarkable is the behavior of σ(ω)
at low temperature, which shows a well defined gap for
x = 0.05 while no gap is found for x = 0.125. This fea-
ture reflects the formation of the lattice polaron and its in-
terplay with the spin degrees of freedom. While the el-ph
coupling λ = 0.75 alone is not strong enough at x = 0.125
0 0.5 1
ω  [eV]
0.5 1 1.5
ω  [eV]
0 200 400
T  [K]
x=0.05
x=0.125
T=50K
T=440K
T=540K
T=540K
T=340K
T=50K
T=190K
Fig. 2. Temperature dependence of the optical conductivity σ(ω) for
x = 0.05 and x = 0.125. Solid lines are used for T ≤ T ∗, dashed
lines for T > T ∗. Inset: loss of the MIR spectral weight ∆Neff , as
defined in Ref. [10], as function of T for x = 0.05 (filled circles) and
x = 0.125 (empty squares). Arrows mark the corresponding T ∗.
(J/t = 0.057) to establish a spin/lattice polaron, the lo-
calization effects induced by the larger exchange coupling
J/t = 0.126 at x = 0.05 favor the lattice polaron forma-
tion. This leads thus to the opening of an optical gap in
σ(ω) (this key point will be extensively discussed in a forth-
coming publication[8]). Increasing T reduces the localiza-
tion effects induced by the magnetic ordering. This makes
the positive interplay with the el-ph coupling less effective,
leading to a progressive filling of the pseudogap. Note that
this effect disappears in the disordered magnetic case for
T > T ∗, and further increasing of T leads to a reduction of
the MIR optical conductivity which is spread on a larger
energy window. This is reflected in the characteristic tem-
perature behavior of the MIR spectral weight ∆Neff , as de-
fined in Ref. [10], which presents a kink at T ∗ (inset of Fig.
2)[11].
References
[1] A.S. Mishchenko and and N. Nagaosa, Phys. Rev. Lett. 93
(2004) 0236402; Phys. Rev. B 73 (2006) 092502.
[2] O. Rösch and O. Gunnarsson, Phys. Rev. Lett. 92 (2004) 146403;
Eur. Phys. J. B 43 (2005) 11.
[3] O. Gunnarsson and O. Rösch, Phys. Rev. B 73 (2006) 174521.
[4] P. Prelovšek, R. Zeyher, and P. Horsch, Phys. Rev. Lett. 96
(2006) 086402.
[5] M.P.H. Stumpf and D.E. Logan, Eur.Phys.J.B, 8 (1999) 377.
[6] S. Fratini and S. Ciuchi, Phys. Rev. B 74 (2006) 075101.
[7] B. Bäuml et al., Phys. Rev. B 58 (1998) 3663.
[8] E. Cappelluti, S. Ciuchi and S. Fratini, in preparation (2007).
[9] E. Cappelluti and S. Ciuchi, Phys. Rev. B 66 (2002) 165102.
[10] Y. Onose et., Phys. Rev. B 69 (2004) 024504.
[11] Since we do not find any isosbestic point in our calculations,
we use the experimental energy windows of Ref. [10] to define
∆Neff , namely ωmin = 0.12 eV, ωmax = 0.42 eV for x = 0.05
and ωmax = 0.21 eV for x = 0.125.
	References
