Dynamic spin-response function of the high-temperature Bi
2
Sr
2
CaCu
2
O
8+
superconductor from
angle-resolved photoemission spectra
U. Chatterjee,
1,2
D. K. Morr,
1
M. R. Norman,
2
M. Randeria,
3
A. Kanigel,
1
M. Shi,
1,4
E. Rossi,
1
A. Kaminski,
5
H. M. Fretwell,
5
S. Rosenkranz,
2
K. Kadowaki,
6
and J. C. Campuzano
1,2
1
Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607, USA
2
Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
3
Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
4
Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen, Switzerland
5
Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
6
Institute of Materials Science, University of Tsukuba, Ibaraki 305-3573, Japan
Received 13 June 2006; published 17 May 2007
We introduce a formalism for calculating dynamic response functions using experimental single-particle
Green’s functions. As an illustration of this procedure, we estimate the dynamic spin-response of the cuprate
superconductor Bi
2
Sr
2
CaCu
2
O
8+
. We find good agreement with superconducting state neutron data, in par-
ticular, the , resonance with its unusual “hourglass” shaped dispersion. We anticipate that our formalism
will also be useful in interpreting results from other spectroscopies, such as optical and Raman responses.
DOI: 10.1103/PhysRevB.75.172504 PACS numbers: 74.25.Jb, 74.25.Ha, 74.72.Hs, 79.60.Bm
The linear response to an external probe as a function of
momentum and frequency is of great importance in elucidat-
ing the properties of complex materials. Examples include
various two-particle correlation functions involving spin,
current, and charge as measured by inelastic neutron scatter-
ing INS, nuclear magnetic resonance NMR, optical con-
ductivity, and Raman scattering experiments. On the other
hand, angle-resolved photoemission spectroscopy
1
ARPES
directly gives information about single-particle excitations of
a system. The response function of a system can be ex-
pressed in terms of a two-particle correlation function of the
observable to which the external probe couples. The goal of
this Brief Report is to develop an approach to use single-
particle spectroscopy data to gain insight into two-particle
correlation functions. In particular, we focus here on using
the Green’s functions obtained from superconducting state
ARPES data in the high T
c
cuprates to compute the dynamic
spin susceptibility, which we then compare with INS data.
2
From a theoretical point of view, dynamic response func-
tions are difficult to calculate in general and many different
approximate formalisms exist in the literature. For instance,
there are two rather different approaches for computing the
dynamic spin-response for the high T
c
cuprate superconduct-
ors. The first is based on the random-phase approximation
3
RPA and related diagrammatic formulations.
4
This ap-
proach not only assumes that momentum is a good quantum
number but also that the spin and charge degrees of freedom
are coupled. The second is based on spin ladders separated
by one-dimensional domain walls known as stripes. In this
formalism, spatial inhomogeneity is important, and the
charge sector is assumed to be secondary when calculating
the spin response.
5
Despite the quite different physics under-
lying these two schemes, the results for the calculated spin-
response function of the cuprates are similar—one of the
current dilemmas facing the field of high T
c
superconductiv-
ity. It is thus important to go beyond a purely theoretical
approach and directly employ information obtained from one
experiment ARPES to make progress on interpreting the
dynamic susceptibility measured by another INS.
We use a formalism based on a diagrammatic k-space
approach which goes beyond RPA in that it uses fully
dressed Green’s function obtained from ARPES data on
Bi
2
Sr
2
CaCu
2
O
8+
Bi2212. We compare the calculated su-
perconducting state susceptibility with INS data. We obtain
the , resonance seen in many cuprates,
2
including
Bi2212,
6
and also its unusual “hourglass” shaped dispersion
as observed in YBa
2
Cu
3
O
7-
YBCORefs. 7 and 8 and
more recently in Bi2212.
9
We also find that the magnetic
dispersion is sensitive to the momentum dependence of the
effective interaction used to calculate the susceptibility.
We use ARPES spectra from a near-optimal sample T
c
=90 K of Bi2212, the data having been presented
previously.
10,11
While a resonance peak was observed in this
material some time ago,
6
a more detailed study with results
similar to the much more extensive INS data for YBCO has
appeared only recently.
9
Quite generally, two-particle correlation functions can be
written in terms of single-particle Green’s functions and ver-
tex parts.
12
The lowest-order term contributing to the spin
susceptibility the bare polarization bubble in the supercon-
ducting state can be written as
13
0
q, =
1
2
k
-
ddIm Gk, Im Gk + q,
+ Im Fk, Im Fk + q,
n
F
- n
F
+ - + i
=
0
G
+
0
F
, 1
where Im denotes the imaginary part of the normal and
anomalous Green’s functions G and F, respectively, and
0
G
and
0
F
denote the GG and FF contributions to
0
, respec-
tively.
We next describe in detail how Im G is extracted from
ARPES data and return later to the question of estimating the
PHYSICAL REVIEW B 75, 172504 2007
1098-0121/2007/7517/1725044 ©2007 The American Physical Society 172504-1