Article
Reference
Optical integral in the cuprates and the question of sum-rule violation
NORMAN, M., et al.
Abstract
Much attention has been given to a possible violation of the optical sum rule in the cuprates
and the connection this might have to kinetic energy lowering. The optical integral is
composed of a cutoff-independent term (whose temperature dependence is a measure of the
sum-rule violation), plus a cutoff-dependent term that accounts for the extension of the Drude
peak beyond the upper bound of the integral. We find that the temperature dependence of the
optical integral in the normal state of the cuprates can be accounted for solely by the latter
term, implying that the dominant contribution to the observed sum-rule violation in the normal
state is due to the finite cutoff. This cutoff-dependent term is well modeled by a theory of
electrons interacting with a broad spectrum of bosons.
NORMAN, M., et al. Optical integral in the cuprates and the question of sum-rule violation.
Physical review. B, Condensed matter and materials physics, 2007, vol. 76, no. 22, p.
220509
DOI : 10.1103/PhysRevB.76.220509
Available at:
http://archive-ouverte.unige.ch/unige:24157
Disclaimer: layout of this document may differ from the published version.
1 / 1
Optical integral in the cuprates and the question of sum-rule violation
M. R. Norman,
1
A. V. Chubukov,
2
E. van Heumen,
3
A. B. Kuzmenko,
3
and D. van der Marel
3
1
Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
2
Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA
3
University of Geneva, 24, Quai E.-Ansermet, Geneva 4, Switzerland
共Received 21 September 2007; revised manuscript received 27 November 2007; published 28 December 2007
兲
Much attention has been given to a possible violation of the optical sum rule in the cuprates and the
connection this might have to kinetic energy lowering. The optical integral is composed of a cutoff-
independent term 共whose temperature dependence is a measure of the sum-rule violation兲, plus a cutoff-
dependent term that accounts for the extension of the Drude peak beyond the upper bound of the integral. We
find that the temperature dependence of the optical integral in the normal state of the cuprates can be accounted
for solely by the latter term, implying that the dominant contribution to the observed sum-rule violation in the
normal state is due to the finite cutoff. This cutoff-dependent term is well modeled by a theory of electrons
interacting with a broad spectrum of bosons.
DOI: 10.1103/PhysRevB.76.220509 PACS number共s兲: 74.72.⫺h, 74.20.⫺z, 74.25.Gz
The integral of the real part of the optical conductivity
with respect to frequency up to infinity is known as the f sum
rule. It is proportional to n / m and is preserved by charge
conservation.
1
In experiments, however, the conductivity is
measured up to a certain frequency cutoff. In a situation
when the system has a single band of low-energy carriers,
separated by an energy gap from other high-energy bands 共as
in the cuprates兲, the exact f sum rule reduces to the single-
band sum rule of Kubo,
2
W =
冕
0
c
Re
共
兲d
= f共
c
兲
pl
2
8
⬅ f共
c
兲
e
2
a
2
2ប
2
V
E
K
. 共1兲
Here a is the in-plane lattice constant, V the unit cell volume,
c
an ultraviolet cutoff,
pl
the bare plasma frequency, and
E
K
=
2
a
2
N
兺
k
2
⑀
k
k
x
2
n
k
, 共2兲
with N the number of k vectors,
⑀
k
the bare dispersion as
defined by the effective single band Hamiltonian, and n
k
the
momentum distribution function. For a Hamiltonian with
near-neighbor hopping,
3
E
K
is equivalent to minus the kinetic
energy, E
kin
⬅
2
N
兺
k
⑀
k
n
k
, but in general these two quantities
differ.
4
The cutoff
c
is assumed to be larger than the band-
width of the low-energy band, but smaller than the gap of
other bands. For this reason, its value in cuprates is typically
chosen to be 1 –1.25 eV. f共
c
兲 accounts for the cutoff depen-
dence, which arises from the presence of Drude spectral
weight beyond
c
共Ref. 5兲 and is unity if we formally set
c
to infinity while ignoring the interband transitions.
Although the f sum rule is preserved, W共
c
,T兲 in general
is not a conserved quantity since both f共
c
兲共Ref. 5兲 and n
k
共Ref. 6兲 depend on T. The T variation of W has been termed
the “sum-rule violation.” In conventional superconductors,
the sum rule is preserved within experimental accuracy.
7
In
the cuprates, however, the c -axis conductivity indicates a
pronounced violation of the sum rule.
8
More recently, similar
behavior was found for the in-plane conductivity.
9–12
The reported violation takes two forms. First, as the tem-
perature is lowered in the normal state, the optical integral
increases in magnitude. Then, below T
c
, there is an addi-
tional change in the optical integral compared to that of the
extrapolated normal state. For overdoped compounds, this
change is a decrease,
13,14
but for underdoped and optimal
doped compounds, two groups
9–12
found an increase, though
other groups found either no additional effect
15,16
or a
decrease.
17
The finite cutoff was taken into account in several theo-
retical analyses of the T dependence of the optical integral—
for instance, work based on the Hubbard model,
18
the t-J
model,
14
and the d-density-wave model.
19
In Ref. 5, the ef-
fect of the cutoff was considered in the context of electrons
coupled to phonons. The goal of the present paper is to study
the influence of the cutoff on the optical integral for a model
of electrons interacting with a broad spectrum of bosons that
two of us have used previously to model optics data.
20
In a Drude model,
共
兲=共
pl
2
/ 4
兲/ 共1/
−i
兲. From Eq.
共1兲, we see that the true sum-rule violation is encoded in the
T dependence of
pl
. Although
pl
is well known to be a
strong function of doping,
21
the question of its T dependence
is more subtle because of the presence of f共
c
兲. Integrating
over
and expanding for
c
1, we obtain W共
c
兲
=
pl
2
8
f共
c
兲, where f共
c
兲=
共
1−
2
1
c
兲
. For infinite cutoff, f共
c
兲
=1 and W =
pl
2
/ 8, but for a finite cutoff,f共
c
兲 is a constant
minus a term proportional to 1 /
c
.If1/
changes with T,
then one obtains a sum-rule violation even if
pl
is T
independent.
5
In general, the optical integral changes due to E
K
共i.e.,
pl
兲 and f共
c
兲 are both present, and the difference of optical
integrals at two different temperatures, ⌬W =W共T
1
兲−W共T
2
兲,
goes as
⌬W =
␣
⌬E
K
+

⌬f共
c
兲, 共3兲
where
␣
and

are constants. The issue then is which term
contributes more to the sum-rule violation at a given
c
.If
the variation predominantly comes from E
K
, it would be a
true sum-rule violation, related to the variation of the kinetic
PHYSICAL REVIEW B 76, 220509共R兲共2007兲
RAPID COMMUNICATIONS
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energy. The increase of W with decreasing T would then
imply that the kinetic energy decreases with decreasing T.If
the change of W comes from f共
c
兲, the sum-rule violation
would be a cutoff effect, unrelated to the behavior of the
kinetic energy.
In this paper, for simplicity, we concentrate on the tem-
perature variation of the optical integral in the normal state.
We find that the data can be fit by Eq. 共3兲 with ⌬E
K
=0.
Based on the accuracy of the fit, we estimate that the true
sum-rule violation ⌬E
K
must be smaller than ⬃20% of ⌬W.
Moreover, we find that the temperature variation due to the
second term in Eq. 共3兲, and its dependence on the cutoff, is
well modeled by a theory of fermions interacting with a
broad spectrum of bosons.
We considered two models for the bosonic spectrum. The
first is a “gapped” marginal Fermi liquid, where the spectrum
is flat in frequency up to an upper cutoff
2
,
22
␣
2
F共⍀兲
GMFL
=
⌫
2
−
1
⌰共⍀ −
1
兲⌰共
2
− ⍀兲, 共4兲
with a lower cutoff
1
put in by hand to prevent divergences.
The second is a Lorentzian spectrum typical for overdamped
spin and charge fluctuations,
23
␣
2
F共⍀兲
Lor
=
⌫⍀
␥
2
+ ⍀
2
. 共5兲
The computational procedure is straightforward:
␣
2
F is
used to calculate the single-particle self-energy and, from
this, the current-current response function to obtain the con-
ductivity. The computational procedure can be simplified, as
shown by Allen,
24
by presenting
共
兲 in a generalized Drude
form
共
兲 =
pl
2
4
1
1/
共
兲 − i
m
*
共
兲
共6兲
and approximating 1 /
共
兲 by
25
1/
共
兲 =2⌫
i
+
1
冕
0
⬁
d⍀
␣
2
F共⍀兲
冋
2
coth
冉
⍀
2T
冊
− 共
+ ⍀兲coth
冉
+ ⍀
2T
冊
+ 共
− ⍀兲coth
冉
− ⍀
2T
冊
册
,
共7兲
where 2⌫
i
is the impurity contribution.
26
For electrons inter-
acting with a broad spectrum of bosons, this approximation
is essentially identical to the exact Kubo result.
20
The optical
mass m
*
共
兲 can then be determined by a Kramers-Kronig
transformation of 1 /
共
兲.
One can show quite generally that for an arbitrary
form of
␣
2
F共⍀兲, W共
c
,T兲 asymptotically approaches
W共⬁兲 as W共
c
,T兲= W共⬁兲共1−共8/
兲A共T兲/
c
兲, where A 共T兲
=兰
0
⬁
d⍀
␣
2
F共⍀兲n
B
共⍀兲 with n
B
the Bose function. At high T,
therefore, W共
c
,T兲− W共⬁兲 scales as T for arbitrary
c
. This
asymptotic behavior, however, sets in for
c
much larger
than the upper cutoff in
␣
2
F共⍀兲. This behavior would ad-
equately describe the data if the bosonic spectrum sits at
low frequencies as for phonons,
5
but this does not appear
to be the case in the cuprates, where the inferred bosonic
spectrum from the infrared data extends to quite high
frequencies.
20,27,28
For comparison with experiment, there-
fore, we need to know W共
c
,T兲 not only for arbitrary T, but
also for
c
which are only a few times larger than the energy
range of
␣
2
F.
We start with the gapped marginal Fermi-liquid model.
The parameters ⌫,
1
, and
2
were chosen
20
so as to fit the
data of Ref. 29 at one particular temperature. We do not
optimize them for the data we compare to here, in order to
demonstrate the generality of our arguments.
2
is essen-
tially equal to the peak frequency in the real part of the
optical self-energy, 关m
*
共
兲−1兴
, whereas ⌫ is set by the
overall size of the optical self-energy. We treat
␣
2
F and
pl
as T independent, so as to concentrate exclusively on the
effect of
c
, though the actual quantities may depend on
T.
27,28
As a consequence, the only thermal effects which en-
ter are the coth factors in 1 /
共
兲 in Eq. 共7兲. In Fig. 1共a兲,we
show the variation of the calculated optical integral with T
for two different values of ⌫
i
and compare this to the data of
Ref. 9. The results above T
c
are consistent with a behavior
that goes as a constant minus a T
2
term in both the data and
the theory. Moreover, the T
2
slopes are identical 共the relative
shift in W can be matched by small changes in either ⌫
i
or
pl
兲. In Fig. 1共b兲, we show the difference of the calculated
optical integrals at two different T versus
c
. After an initial
rise 共due to the fact that the Drude peak is narrower at lower
T兲, the difference decays. Unlike the simple Drude model
where this decay goes as 1 /
c
, the decay appears to be more
like 1/
冑
c
for cutoffs ranging from 0.1 eV to 1 eV. To ob-
tain more insight, we show in Fig. 2共b兲 the logarithmic de-
rivative of ⌬W versus
c
. The approximate −1/ 2 power is an
intermediate-frequency result, and one can see the approach
to the asymptotic power of −1 for very large cutoffs.
The behavior for large
c
can be also studied analytically.
To start, we rewrite the optical integral as
0
0.02
0.04
0.06
0.08
0.1
0 0.2 0.4 0.6 0.8 1
Γ
i
=26.5
7K-260K
ω
c
-1/2
∆
W (eV
2
)
ω
c
(eV)
(b)
0.49
0.5
0.51
0.52
0.53
0.54
0210
4
410
4
610
4
810
4
W (eV
2
) @ 1.25eV
T
2
(K)
(a)
Γ
i
=26.5
Γ
i
=67.5
FIG. 1. 共Color online兲共a兲 Optical integral W for a 1.25-eV cut-
off versus T
2
for the gapped marginal Fermi-liquid model
共⌫= 270.5 meV,
1
=15.5 meV,
2
=301 meV兲 for two different
values of the impurity scattering ⌫
i
, with
pl
=2.4 eV. The curves
are fits to the calculated solid circles using the T dependence of Eq.
共11兲. The middle curve is the experimental W of Ref. 9 for a
Bi
2
Sr
2
CaCu
2
O
8
sample with a T
c
of 88 K. The open circles are the
⌫
i
=26.5 meV case with
pl
=2.356 eV instead. 共b兲 Difference of
the calculated optical integrals 共T = 7 K minus T =260 K兲 versus the
cutoff for ⌫
i
=26.5 meV. The dashed line is a 1 /
冑
c
fit 共with
⌬E
K
⫽0兲.
NORMAN et al. PHYSICAL REVIEW B 76, 220509共R兲共2007兲
RAPID COMMUNICATIONS
220509-2
W共
c
,T兲 = W共⬁兲 −
冕
c
⬁
Re
共
,T兲d
, 共8兲
where W共⬁兲=
pl
2
/ 8. We then note
20
that for
larger than the
upper cutoff
2
of the gapped marginal Fermi-liquid model,
1/
=1/
high
=2⌫
i
+
⌫
2
−
1
冢
4T ln
sinh
2
2T
sinh
1
2T
−
2
2
−
1
2
冣
.
共9兲
For T
2
and
1
2
共which are always satisfied兲, this
reduces to
1/
high
=1/
0
−
4⌫T
2
ln共1−e
−
1
/T
兲, 共10兲
where 1/
0
=2⌫
i
+2⌫− ⌫
2
/
. Ignoring the frequency de-
pendence of 1/
0
and setting m
*
to 1 共
c
2
兲,
30
we then
obtain ⌬W共
c
兲=
pl
2
4
⌬关tan
−1
共
c
high
兲兴 where again ⌬W共
c
兲
=W共
c
,T
1
兲−W共
c
,T
2
兲. Expanding in ⌬T, we obtain
⌬W共
c
兲 =
pl
2
2
2
c
*
1+共
c
*
兲
2
⌬关T ln共1−e
−
1
/T
兲兴, 共11兲
where
c
*
=
c
/ 共2⌫兲. In Fig. 2共a兲, we plot Eq. 共11兲 versus the
calculated optical integral difference and see that they match
for cutoffs beyond 0.7 eV. Moreover, the T dependence of
Eq. 共11兲 matches the T evolution of the optical integral, as
can be seen in Fig. 1共a兲, and so a T
2
behavior is only ap-
proximate. The true dependence is T ln共1−e
−
1
/T
兲 as in Eq.
共11兲; however, this is very close to T
2
over a wide range of
temperatures.
The above analysis can also be performed for the Lorent-
zian model 共the numerical results are similar to Fig. 1, and
we do not present them here兲. Extending the analysis in Ref.
20 to finite temperatures, we obtain
1/
共T兲 =1/
0
+4⌫
冕
0
⬁
xdx
x
2
+1
1
e
x/T
*
−1
, 共12兲
where 1 /
0
=2⌫ ln
冑
⍀
c
2
+
␥
2
␥
, with T
*
=T/
␥
, and ⍀
c
is an upper
cutoff for
␣
2
F
Lor
. Assuming that ⍀
c
␥
, we have ⌬W共
c
兲
=
pl
2
4
⌬关tan
−1
共
c
兲兴. Expanding around T =0, we obtain
W共
c
,T兲⬇W共
c
,T =0兲 −
pl
2
4
C共T
*
兲
2
, 共13兲
where
C =
2
6
c
⌫ ln
2
共⍀
c
/
␥
兲
1
1+
冉
c
2⌫ ln共⍀
c
/
␥
兲
冊
2
. 共14兲
This time, we find a truly quadratic behavior in T,
31
which is
a consequence of the fact that
␣
2
F
Lor
is linear in
at small
. The dependence of ⌬W on the frequency cutoff is the
same as the gapped marginal Fermi-liquid model, except that
the quantity
c
*
in Eq. 共11兲 is now
c
/ 关2⌫ ln共⍀
c
/
␥
兲兴.
We now return to experiment. In Fig. 1共a兲, we plot the
experimental optical integral for a 1.25-eV cutoff from the
data of Ref. 9 versus our calculations. The magnitude and T
variation of W are essentially equivalent to these calcula-
tions, which assumed a T-independent
pl
.
32
In Fig. 3共a兲,we
show the difference between the measured optical integrals
for two temperatures versus the frequency cutoff from the
data of Ref. 12.A1/
冑
c
dependence, with a zero offset, fits
the data quite well, as with the theory in Fig. 1共b兲. This is
further demonstrated by the logarithmic derivative, as plotted
in Fig. 3共b兲. From these observations, we conclude that the
dominant contribution to the T dependence of the optical
integral in the normal state can be attributed to the finite
cutoff. The true sum-rule violation term ⌬E
K
is estimated to
be no larger than ⬃20% of ⌬W, as noted above. Although
we do not expect our analysis to be the entire story, in that
there is experimental evidence that
␣
2
F is T dependent,
27
even though this dependence is weak in the normal state,
28
still, based on the good agreement of the calculations with
experiment, we would argue that the bulk of the observed T
dependence in the normal state is related to the finite cutoff.
The above analysis is nontrivial to extend to below T
c
,as
this requires some assumptions about the pairing kernel,
since one needs to construct the anomalous Green’s function
F in order to evaluate the current-current response function.
The additional increase of W共
c
,T兲 below T
c
in optimal and
-1
-0.5
0
0.5
1
0 0.5 1 1.5 2 2.5
dln(
∆W)/dln(
ω
c
)
ω
c
(eV)
(b)
0
0.02
0.04
0.06
0.08
0.1
0 0.5 1 1.5 2 2.5
7K-260K
asymptotic
∆W (eV
2
)
ω
c
(eV)
(a)
FIG. 2. 共Color online兲共a兲 Optical integral difference from Fig.
1共b兲 versus the cutoff as compared to the asymptotic expression of
Eq. 共11兲. 共b兲 Logarithmic derivative of ⌬W versus the cutoff.
-1
-0.5
0
0.5
1
0 0.2 0.4 0.6 0.8 1
dln(
∆W)/dln(
ω
c
)
ω
c
(eV)
(b)
0
0.02
0.04
0.06
0.08
0
.
1
0 0.2 0.4 0.6 0.8 1
100K-290K
ω
c
-1/2
∆W (eV
2
)
ω
c
(eV)
(a)
FIG. 3. 共Color online兲共a兲 Optical integral difference for a
HgBa
2
CuO
4
sample with a T
c
of 97 K 共data from Ref. 12兲. The
dashed line is a 1 /
冑
c
fit 共with ⌬E
K
⫽0兲. 共b兲 logarithmic derivative
of ⌬W versus
c
.
OPTICAL INTEGRAL IN THE CUPRATES AND THE… PHYSICAL REVIEW B 76, 220509共R兲共2007兲
RAPID COMMUNICATIONS
220509-3
underdoped cuprates, reported in Refs. 9–12, could be due to
the strong decrease in 1/
observed by a variety of probes.
On the other hand, strong coupling calculations cast doubt on
a cutoff explanation, as the influence of f共
c
兲 would be to
give rise to a negative W
sc
−W
n
for cutoffs near 1 eV.
33
Moreover, similar strong coupling calculations of the varia-
tion of E
K
between the normal and superconducting states
yield a positive E
K
sc
−E
K
n
⬃1 meV for the underdoped case,
34
which is consistent both in sign and magnitude with the re-
sults of Refs. 9–12. This implies that there may be a true
sum-rule violation below T
c
.
The authors would like to thank Nicole Bontemps for sug-
gesting this study and Frank Marsiglio for useful conversa-
tions. M.N. was supported by the U.S. Department of En-
ergy, Office of Science, under Contract No. DE-AC02-
06CH11357, and A.C. by Grant No. NSF-DMR 0604406.
The work at the University of Geneva is supported by the
Swiss NSF through Grant No. 200020-113293 and the Na-
tional Center of Competence in Research 共NCCR兲 Materials
with Novel Electronic Properties-MaNEP. M.N. and A.C.
thank the Aspen Center for Physics where this work was
completed.
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pl
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Corrections from the optical mass lead to higher-order terms in W
of the form ln共
c
兲/
c
3
.
31
This T
2
behavior holds for T
*
⬍1 / 3. For higher T, the behavior of
W共
c
,T兲 crosses over to a linear-T behavior.
32
If we allowed for a realistic 共i.e., non-free-electron兲 band disper-
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⌬E
K
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NORMAN et al. PHYSICAL REVIEW B 76, 220509共R兲共2007兲
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