arXiv:1001.0764v2  [cond-mat.str-el]  13 Jan 2010
Optical Integral and Sum Rule Violation
Saurabh Maiti, Andrey V. Chubukov
Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA
(Dated: November 9, 2018)
The purpose of this work is to investigate the role of the latt ice in the optical Kubo sum rule in
the cuprates. We compute conductivities, optical integral s W , and ∆ W between superconducting
and normal states for 2-D systems with lattice dispersion ty pical of the cuprates for four diﬀerent
models – a dirty BCS model, a single Einstein boson model, a ma rginal Fermi liquid model, and a
collective boson model with a feedback from superconductiv ity on a collective boson. The goal of
the paper is two-fold. First, we analyze the dependence of W on the upper cut-oﬀ ( ωc) placed on
the optical integral because in experiments W is measured up to frequencies of order bandwidth.
For a BCS model, the Kubo sum rule is almost fully reproduced a t ωc equal to the bandwidth. But
for other models only 70%-80% of Kubo sum rule is obtained up t o this scale and even less so for
∆ W , implying that the Kubo sum rule has to be applied with cautio n. Second, we analyze the sign
of ∆ W . In all models we studied ∆ W is positive at small ωc, then crosses zero and approaches a
negative value at large ωc, i.e. the optical integral in a superconductor is smaller th an in a normal
state. The point of zero crossing, however, increases with t he interaction strength and in a collective
boson model becomes comparable to the bandwidth at strong co upling. We argue that this model
exhibits the behavior consistent with that in the cuprates.
I. INTRODUCTION
The analysis of sum rules for optical conductivity has a
long history. Kubo, in an extensive paper 1 in 1957, used
a general formalism of a statistical theory of irreversible
processes to investigate the behavior of the conductivity
in electronic systems. For a system of interacting elec-
trons, he derived the expression for the integral of the real
part of a (complex) electric conductivityσ(Ω) and found
that it is independent on the nature of the interactions
and reduces to
∫ ∞
0
Re σ(Ω) dΩ = π
2
ne2
m (1)
Here n is the density of the electrons in the system and
m is the bare mass of the electron. This expression is
exact provided that the integration extends truly up to
inﬁnity, and its derivation uses the obvious fact that at
energies higher than the total bandwidth of a solid, elec-
trons behave as free particles.
The independence of the r.h.s. of Eq. (1) on temper-
ature and the state of a solid (e.g., a normal or a super-
conducting state – henceforth referred to as NS and SCS
respectively) implies that, while the functional form of
σ(Ω) changes with, e.g., temperature, the total spectral
weight is conserved and only gets redistributed between
diﬀerent frequencies as temperature changes. This con-
servation of the total weight ofσ(Ω) is generally called a
sum rule.
One particular case, studied in detail for conventional
superconductors, is the redistribution of the spectral
weight between normal and superconducting states. This
is known as Ferrel-Glover-Tinkham (FGT) sum rule:2,3
∫ ∞
0+
Re σNS (Ω) =
∫ ∞
0+
Re σsc(Ω) + πnse2
2m (2)
where ns is the superﬂuid density, and πnse2/(2m) is
the spectral weight under the δ-functional piece of the
conductivity in the superconducting state.
In practice, the integration up to an inﬁnite frequency
is hardly possible, and more relevant issue for practical
applications is whether a sum rule is satisﬁed, at least ap-
proximately, for a situation when there is a single electron
band which crosses the Fermi level and is well separated
from other bands. Kubo considered this case in the same
paper of 1957 and derived the expression for the “band”,
or Kubo sum rule
∫ ‘∞′
0
Re σ(Ω) dΩ = WK = πe2
2N
∑
⃗k
∇2
⃗kx
ε⃗k n⃗k (3)
where n⃗k is the electronic distribution function and ε⃗k is
the band dispersion. Prime in the upper limit of the inte-