Doping a Mott insulator: Physics of high-temperature superconductivity
Patrick A. Lee
Department of Physics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA
Naoto Nagaosa
CREST, Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-8656, Japan and Correlated Electron Research Center, AIST,
Tsukuba Central 4, Tsukuba 305-8562, Japan
Xiao-Gang Wen
Department of Physics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA

Published 6 January 2006
This article reviews the physics of high-temperature superconductors from the point of view of the
doping of a Mott insulator. The basic electronic structure of cuprates is reviewed, emphasizing the
physics of strong correlation and establishing the model of a doped Mott insulator as a starting point.
A variety of experiments are discussed, focusing on the region of the phase diagram close to the Mott
insulator
the underdoped region where the behavior is most anomalous. The normal state in this
region exhibits pseudogap phenomenon. In contrast, the quasiparticles in the superconducting state
are well defined and behave according to theory. This review introduces Anderson’s idea of the
resonating valence bond and argues that it gives a qualitative account of the data. The importance of
phase fluctuations is discussed, leading to a theory of the transition temperature, which is driven by
phase fluctuations and the thermal excitation of quasiparticles. However, an argument is made that
phase fluctuations can only explain pseudogap phenomenology over a limited temperature range, and
some additional physics is needed to explain the onset of singlet formation at very high temperatures.
A description of the numerical method of the projected wave function is presented, which turns out
to be a very useful technique for implementing the strong correlation constraint and leads to a number
of predictions which are in agreement with experiments. The remainder of the paper deals with an
analytic treatment of the t-J model, with the goal of putting the resonating valence bond idea on a
more formal footing. The slave boson is introduced to enforce the constraint againt double occupation
and it is shown that the implementation of this local constraint leads naturally to gauge theories. This
review follows the historical order by first examining the U
1 formulation of the gauge theory. Some
inadequacies of this formulation for underdoping are discussed, leading to the SU
2 formulation.
Here follows a rather thorough discussion of the role of gauge theory in describing the spin-liquid
phase of the undoped Mott insulator. The difference between the high-energy gauge group in the
formulation of the problem versus the low-energy gauge group, which is an emergent phenomenon, is
emphasized. Several possible routes to deconfinement based on different emergent gauge groups are
discussed, which leads to the physics of fractionalization and spin-charge separation. Next the
extension of the SU
2 formulation to nonzero doping is described with a focus on a part of the
mean-field phase diagram called the staggered flux liquid phase. It will be shown that inclusion of the
gauge fluctuation provides a reasonable description of the pseudogap phase. It is emphasized that
d-wave superconductivity can be considered as evolving from a stable U
1 spin liquid. These ideas are
applied to the high-T
c
cuprates, and their implications for the vortex structure and the phase diagram
are discussed. A possible test of the topological structure of the pseudogap phase is described.
DOI: 10.1103/RevModPhys.78.17 PACS number
s: 74.20.Mn, 74.72.
h, 71.27.a
CONTENTS
I. Introduction 18
II. Basic Electronic Structure of the Cuprates 21
III. Phenomenology of the Underdoped Cuprates 23
A. Pseudogap phenomenon in the normal state 23
B. Neutron scattering, resonance, and stripes 28
C. Quasiparticles in the superconducting state 30
IV. Introduction to the Resonating Valence Bond and a
Simple Explanation of the Pseudogap 33
V. Phase Fluctuation versus Competing Order 34
A. A theory of T
c
34
B. Cheap vortices and the Nernst effect 36
C. Two kinds of pseudogaps 38
VI. Projected Trial Wave Functions and Other Numerical
Results 38
A. The half-filled case 39
B. The doped case 40
C. Properties of projected wave functions 40
D. Improvement of projected wave functions, effect of
t


, and the Gutzwiller approximation 41
VII. The Single-Hole Problem 42
REVIEWS OF MODERN PHYSICS, VOLUME 78, JANUARY 2006
0034-6861/2006/78
1/17
69/$50.00 2006 The American Physical Society17

VIII. Slave-Boson Formulation of the t-J Model and
Mean-Field Theory 43
IX. U
1 Gauge Theory of the Uniform RVB State 46
A. Effective gauge action and non-Fermi-liquid
behavior 46
B. Ioffe-Larkin composition rule 49
C. Ginzburg-Landau theory and vortex structure 50
D. Confinement-deconfinement problem 52
E. Limitations of the U
1 gauge theory 54
X. SU
2 Slave-Boson Representation for Spin Liquids 54
A. Where does the gauge structure come from? 54
B. What determines the gauge group? 56
C. From U
1 to SU
2 56
D. A few mean-field Ansätze for symmetric spin liquids 57
E. Physical properties of the symmetric spin liquids at
mean-field level 58
F. Classical dynamics of the SU
2 gauge fluctuations 59
1. Trivial SU
2 flux 59
2. Collinear SU
2 flux 60
3. Noncollinear SU
2 flux 61
G. The relation between different versions of
slave-boson theory 61
H. The emergence of gauge bosons and fermions in
condensed-matter systems 62
I. The projective symmetry group and quantum order 64
XI. SU
2 Slave-Boson Theory of Doped Mott Insulators 64
A. SU
2 slave-boson theory at finite doping 64
B. The mean-field phase diagram 65
C. Simple properties of the mean-field phases 66
D. Effect of gauge fluctuations: Enhanced
 , spin
fluctuations in the pseudogap phase 66
E. Electron spectral function 68
1. Single-hole spectrum 68
2. Finite-hole density: pseudogap and Fermi
arcs 69
F. Stability of algebraic spin liquids 71
XII. Application of Gauge Theory to the High-T
c
Superconductivity Problem 73
A. Spin liquid, quantum critical point, and the
pseudogap 73
B. -model effective theory and new collective modes
in the superconducting state 75
C. Vortex structure 76
D. Phase diagram 77
E. Signature of the spin liquid 78
XIII. Summary and Outlook 79
Acknowledgments 80
References 81
I. INTRODUCTION
The discovery of high-temperature superconductivity
in cuprates
Bednorz and Müller, 1986 and the rapid
raising of the transition temperature to well above the
melting point of nitrogen
Wu et al., 1987 ushered in an
era of great excitement for the condensed-matter-
physics community. For decades prior to this discovery,
the highest T
c
had been stuck at 23 K. Not only was the
old record T
c
shattered, but the fact that high-T
c
super-
conductivity was discovered in a rather unexpected ma-
terial, a transition-metal oxide, made it clear that some
novel mechanism must be at work. The intervening
years have seen great strides in high-T
c
research. First
and foremost, the growth and characterization of cu-
prate single crystals and thin films have advanced to the
point where sample quality and reproducibility prob-
lems which plagued the field in the early days are no
longer issues. At the same time, basically all conceivable
experimental tools have been applied to cuprates. In-
deed, the need for more refined data has spurred the
development of experimental techniques such as angle-
resolved photoemission spectroscopy
ARPES and low-
temperature scanning tunneling microscopy
STM.To-
day the cuprate is arguably the best studied material
outside of the semiconductor family and a great many
facts are known. It is also clear that many of the physical
properties are unusual, particularly in the metallic state
above the superconductor. Superconductivity is only one
aspect of a rich phase diagram which must be under-
stood in its totality.
While there are hundreds of high-T
c
compounds, they
all share a layered structure made up of one or more
copper-oxygen planes. They all fit into a universal phase
diagram shown in Fig. 1. We start with the so-called par-
ent compound, in this case La
2
CuO
4
. There is now gen-
eral agreement that the parent compound is an insulator,
and should be classified as a Mott insulator. The concept
of Mott insulation was introduced many years ago

Mott, 1949 to describe a situation where a material
should be metallic according to band theory, but is insu-
lating due to strong electron-electron repulsion. In our
case, in the copper-oxygen layer there is an odd number
of electrons per unit cell. More specifically, the copper
ion is doubly ionized and is in a d
9
configuration so that
there is a single hole in the d shell per unit cell. Accord-
ing to band theory, the band is half-filled and must be
metallic. Nevertheless, there is a strong repulsive energy
cost when putting two electrons
or holes on the same
ion, and when this energy
commonly called U domi-
nates over the hopping energy t, the ground state is an
insulator due to strong correlation effects. It also follows
that the Mott insulator should be an antiferromagnet
FIG. 1. Schematic phase diagram of high-T
c
superconductors
showing hole doping
right side and electron doping
left
side. From Damascelli et al., 2003.
18
Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-¼
Rev. Mod. Phys., Vol. 78, No. 1, January 2006