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-Tc 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 Tc 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 Society 17

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 Ans��tze 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-Tc 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 M��ller, 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 Tc had been stuck at 23 K. Not only was the old record Tc shattered, but the fact that high-Tc 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-Tc 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-Tc 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 La2CuO4. 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 d9 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-Tc 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-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

because when neighboring spins are oppositely aligned one can gain an energy 4t2 /U by virtual hopping. This is called the exchange energy J. The parent compound is indeed an antiferromagnetic insulator. The ordering temperature TN 300 K shown in Fig. 1 is in fact mis- leadingly low because it is governed by a small interlayer coupling, which is, furthermore, frustrated in La2CuO4 see Kastner et al., 1998 . The exchange energy J is in fact extraordinarily high, of order 1500 K, and the par- ent compound shows strong antiferromagnetic correla- tion much above TN. The parent compound can be doped by substituting some of the trivalent La by divalent Sr. The result is that x holes are added to the Cu-O plane in La2-xSrxCuO4, which is called hole doping. In the compound Nd2-xCexCuO4 Tokura et al., 1989 , the reverse happens in that x electrons are added to the Cu-O plane, which is called electron doping. As we can see from Fig. 1, on the hole-doping side the antiferromagnetic order is rapidly suppressed and is gone by a 3���5 % hole concentration. Almost immediately after the suppression of the antifer- romagnet, superconductivity appears, ranging from x =6���25 %. The dome-shaped Tc is characteristic of all hole-doped cuprates, even though the maximum Tc var- ies from about 40 K in the La2-xSrxCuO4 LSCO family to 93 K and higher in other families such as YBa2Cu3O6+y YBCO and Ba2Sr2CaCu2O8+y Bi-2212 . On the electron-doped side, the antiferromagnet is more robust and survives up to x=0.14, beyond which a region of superconductivity arises. One view is that the carriers are more prone to be localized on the electron-doped side so that electron doping is closer to dilution by non- magnetic ions, which is less effective in suppressing an- tiferromagnetic order than itinerant carriers. Another possibility is that the next-neighbor hopping term favors the antiferromagnet on the electron-doped side Singh and Ghosh, 2002 . It is as though a more robust antifer- romagnetic region is covering up the more interesting phase diagram revealed on the hole-doped side. In this review we shall focus on the hole-doped materials, even though we shall address the issue of the particle-hole asymmetry of the phase diagram from time to time. The region in the phase diagram with doping x less than that of the maximum Tc is the underdoped region. The metallic state above Tc has been under intense study and exhibits many unusual properties not encoun- tered before in any other metal. This region of the phase diagram has been called the pseudogap phase. It is not a well-defined phase in that a definite finite-temperature phase boundary has never been found. The line drawn in Fig. 1 should be regarded as a crossover. Since we view the high-Tc problem as synonymous with that of the doping of a Mott insulator, the underdoped region is where the battleground between Mott insulator and su- perconductivity is drawn and this is what we shall con- centrate on in this review. The region of the normal state above the optimal Tc also exhibits unusual properties. The resistivity is linear in T and the Hall coefficient is temperature dependent see Chien et al., 1991 . These were cited as examples of non-Fermi-liquid behavior since the early days of high Tc. Beyond optimal doping the overdoped region , san- ity gradually returns. The normal state behaves more normally in that the temperature dependence of the re- sistivity resembles T2 over a temperature range which increases with further overdoping. The anomalous re- gion above optimal doping is sometimes referred to as the ���strange-metal��� region. We offer a qualitative de- scription of this region in Sec. IX, but the understanding of the strange metal is even more rudimentary than that of the pseudogap. A popular notion is that the strange metal is characterized by a quantum critical point lying under the superconducting dome Castellani et al., 1997 Varma, 1997 Tallon and Loram, 2000 . In our view, un- less the nature of the ordered side of a quantum critical point is classified, the simple statement of quantum criti- cality does not teach us too much about the behavior in the critical region. For this reason, we prefer to concen- trate on the underdoped region and leave the strange- metal phase for future studies. Contrary to the experimental situation, the develop- ment of high-Tc theory follows a rather tortuous path and people often have the impression that the field is highly contentious and without a clear direction or con- sensus. We do not agree with this assessment and would like to clearly state our point of view from the outset. Our starting point is that the physics of high-Tc super- conductivity is the physics of the doping of a Mott insu- lator. Strong correlation is the driving force behind the phase diagram. We believe that there is a general con- sensus on this starting point. The simplest model which captures the strong-correlation physics is the Hubbard model and its strong-coupling limit, the t-J model. Our view is that one should focus on understanding these simple models before adding various elaborations. For example, further neighbor hopping certainly is signifi- cant and, as we shall discuss, plays an important role in understanding the particle-hole asymmetry of the phase diagram. Electron-phonon coupling can generally be ex- pected to be strong in transition-metal oxides, and we shall discuss their role in affecting spectral line shape. However, these discussions must be presented in the context of strong correlation. The logical step is to first understand whether simple models such as the t-J model contain enough physics to explain the appearance of su- perconductivity and pseudogaps in the phase diagram. The strong-correlation viewpoint was put forward by Anderson 1987 , who revived his earlier work on a pos- sible spin-liquid state in a frustrated antiferromagnet. This state, called the resonating valence bond RVB , has no long-range antiferromagnetic order and is a unique spin-singlet ground state. It has spin-1/2 fermi- onic excitations which are called spinons. The idea is that when doped with holes, the RVB is a singlet state with coherent mobile carriers and is indistinguishable in terms of symmetry from a singlet BCS superconductor. The process of hole doping was further developed by Kivelson et al. 1987 , who argued that the combination of the doped hole with the spinon forms a bosonic exci- 19 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

tation. This excitation, called the holon, carries charge but no spin whereas the spinon carries spin 1/2 but no charge, and the notion of spin-charge separation was born. Meanwhile, a slave-boson theory was formulated by Baskaran et al. 1987 . Many authors contributed to the development of the mean-field theory, culminating in the paper by Kotliar and Liu 1988 , who found that the superconducting state should have d symmetry and that a state with spin-gap properties should exist above the superconducting temperature in the underdoped re- gion. The possibility of d-wave superconductivity was discussed in terms of the exchange of spin fluctuations Emery, 1983, 1986 Miyake et al., 1986 Scalapino et al., 1986, 1987 Monthonx and Pines, 1993 . These discus- sions were either based on phenomenological coupling between spins and fermions or via the random-phase- approximation treatment for the Hubbard model, which is basically a weak-coupling expansion. In contrast, the slave-boson theory was developed in the limit of strong repulsion. Details of the mean-field theory will be dis- cussed in Sec. VIII. At about the same time, the proposal by Anderson 1987 of using projected mean-field states as trial wave functions was implemented on the computer by Gros 1988, 1989 . The idea was to remove by hand on a com- puter all components of the mean-field wave function with doubly occupied sites and to use this as a varia- tional wave function for the t-J model. Gros 1988, 1989 concluded that the projected d-wave superconductor was the variational ground state for the t-J model over a range of doping. The projected wave-function method remains one of the best numerical tools for tackling the t-J or Hubbard model and is reviewed in Sec. VI. It was soon realized that inclusion of fluctuations about the mean field invariably leads to gauge theory Baskaran and Anderson, 1988 Ioffe and Larkin, 1989 Nagaosa and Lee, 1990 . The gauge-field fluctuations can be treated at a Gaussian level and these early devel- opments together with some of the difficulties are re- viewed in Sec. IX. In hindsight, the slave-boson mean-field theory and the projected wave-function studies contain many of the qualitative aspects of the hole-doped phase diagram. It is indeed quite remarkable that the main tools for treat- ing the t-J model, i.e., projected trial wave function, slave-boson mean-field, and gauge theory, were in place a couple of years after the discovery of high Tc. In some ways the theory was ahead of its time because the ma- jority view in the early days was that the pairing symme- try was s wave, and the pseudogap phenomenology re- mained to be discovered. The first hint came from Knight-shift measurements in 1989 shown in Fig. 4 a . Some of the early history and recent extensions are re- viewed by Anderson et al. 2004 . The gauge-theory approach is a difficult one to pursue systematically because it is a strong-coupling problem. One important development is the realization that the original U 1 gauge theory should be extended to SU 2 in order to make a smooth connection to the under- doped limit Wen and Lee, 1996 . This is discussed in Secs. XI and XII. More generally, it was gradually real- ized that the concepts of confinement and deconfine- ment, which are central to QCD, also play a key role here except that the presence of fermions and bosons in addition to gauge fields makes this problem even more complex. Since gauge theories are not so familiar to condensed-matter physicists, these concepts are dis- cussed in some detail in Sec. X. One of the notable re- cent advances is that the notion of the spin liquid and its relation to deconfinement in gauge theory has been greatly clarified and several soluble models and candi- dates based on numerical exact diagonalization have been proposed Misguich et al., 1999 LiMing et al., 2000 Misguich and Lhuillier, 2004 . It remains true, however, that so far no two-dimensional spin liquid has been con- vincingly realized experimentally. We would like to men- tion two promising examples. The first is the organic compound 2 - BEDT-TTF 2 Cu2 CN 3 . This material is a S= 1 2 system on an approximate triangular lattice just on the insulating side of the Mott transition and shows no spin order down to mK, while the spin susceptibility reaches a finite constant Shimizu et al., 2003 Kawamoto et al., 2004 Kurosaki et al., 2005 . Motrunich 2005 has interpreted this as an example of a spin liquid with a spinon Fermi surface which is stabilized by ring ex- change, while Morita et al. 2002 and Lee and Lee 2005a have proposed that the Hubbard model on a triangular lattice may support this spin liquid near the Mott transition. A second example is the nuclear spin of a 3 He solid layer adsorbed on graphite surfaces Masu- tomo et al., 2004 . Our overall philosophy is that the RVB idea of a spin liquid and its relation to superconductivity contains the essence of the physics and gives a qualitative description of the underdoped phase diagram. The goal of our re- search is to put these ideas on a more quantitative foot- ing. Given the strong-coupling nature of the problem, the only way progress can be made is for theory to work in consort with experiment. Our aim is to make as many predictions as possible, beyond saying that the pseudogap is a RVB spin liquid, and challenge the ex- perimentalists to perform tests. Ideas along these lines are reviewed in Sec. XII. High-Tc research is an enormous field and we cannot hope to be complete in our references. Here we refer to a number of excellent review articles on various aspects of the subject. Imada et al. 1998 reviewed the general topic of the metal-insulator transition. Orenstein and Millis 2000 and Norman and Pepin 2003 have pro- vided highly readable accounts of experiments and gen- eral theoretical approaches. Early numerical work was reviewed by Dagotto 1994 . Kastner et al. 1998 sum- marized the earlier optical and magnetic neutron- scattering data mainly on La2-xSrxCuO4. Major reviews of angle-resolved photoemission data ARPES have been provided by Campuzano et al. 2003 and Damas- celli et al. 2003 . Optics measurements on underdoped materials were reviewed by Timusk and Statt 1999 and Basov and Timusk 2005 . The volume edited by Ginz- berg 1989 contains excellent reviews of early NMR 20 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

work by C. P. Slichter and early transport measurements by N. P. Ong among others. Discussions of stripe physics were recently given by Carlson et al. 2003 and Kivelson et al. 2003 . A discussion of spin-liquid states is given by Sachdev 2003 , with an emphasis on dimer order and by Wen 2004 , with an emphasis on quantum order. For an account of experiments and early RVB theory, see the book by Anderson 1997 . II. BASIC ELECTRONIC STRUCTURE OF THE CUPRATES It is generally agreed that the physics of high-Tc su- perconductivity is that of the copper-oxygen layer, as shown in Fig. 2. In the parent compound such as La2CuO4, the formal valence of Cu is 2+, which means that its electronic state is in the d9 configuration. The copper is surrounded by six oxygens in an octahedral environment the apical oxygen lying above and below Cu are not shown in Fig. 2 . The distortion from a per- fect octahedron due to the shift of the apical oxygens splits the eg orbitals so that the highest partially occu- pied d orbital is x2 -y2. The lobes of this orbital point directly to the p orbital of the neighboring oxygen, form- ing a strong covalent bond with a large hopping integral tpd. As we shall see, the strength of this covalent bonding is responsible for the unusually high energy scale for the exchange interaction. Thus the electronic state of the cuprates can be described by the so-called three-band model, where in each unit cell we have the Cu dx2-y2 orbital and two oxygen p orbitals Emery, 1987 Varma et al., 1987 . The Cu orbital is singly occupied while the p orbitals are doubly occupied, but these are admixed by tpd. In addition, admixtures between the oxygen orbitals may be included. These tight-binding parameters may be obtained by fits to band-structure calculations Mat- theiss, 1987 Yu et al., 1987 . However, the largest energy in the problem is the correlation energy for doubly oc- cupying the copper orbital. To describe these correlation energies, it is more convenient to refer to the hole pic- ture. The Cu d9 configuration is represented by energy level Ed occupied by a single hole with S= 1 2 . The oxygen p orbital is empty of holes and lies at energy Ep, which is higher than Ed. The energy to doubly occupy Ed lead- ing to a d8 configuration is Ud, which is very large and can be considered infinity. The lowest-energy excitation is the charge-transfer excitation in which the hole hops from d to p with amplitude -tpd. If Ep -Ed is sufficiently large compared with tpd, the hole will form a local mo- ment on Cu. This is referred to as a charge-transfer in- sulator in the scheme of Zaanen et al. 1985 . Essentially, Ep -Ed plays the role of the Hubbard U in the one-band model of the Mott insulator. Experimentally an energy gap of 2.0 eV is observed and interpreted as the charge- transfer excitation see Kastner et al., 1998 . Just as in the one-band Mott-Hubbard insulator in which virtual hopping to doubly occupied states leads to an exchange interaction JS1 ��S2, where J=4t2 /U, in the charge-transfer insulator the local moments on nearest- neighbor Cu prefer antiferromagnetic alignment be- cause both spins can virtually hop to the Ep orbital. Ig- noring the Up for doubly occupying the p orbital with holes, the exchange integral is given by J = tpd4 Ep - Ed 3 . 1 The relatively small size of the charge-transfer gap means that we are not deep in the insulating phase and the exchange term is expected to be large. Indeed ex- perimentally the insulator is found to be in an antiferro- magnetic ground state. By fitting Raman scattering to two magnon excitations Sulewsky et al., 1990 , the ex- change energy is found to be J=0.13 eV. This is one of the largest exchange energies known. It is even larger in the ladder compounds which involve the same Cu-O bonding. This value of J is confirmed by fitting the spin- wave energy to theory, in which an additional ring ex- change term is found Coldea et al., 2001 . By substituting divalent Sr for trivalent La, the elec- tron count on the Cu-O layer can be changed in a pro- cess called doping. For example, in La2-xSrxCuO4, x holes per Cu are added to the layer. As seen in Fig. 2, due to the large Ud the hole will reside on the oxygen p orbital. The hole can hop via tpd, and due to transla- tional symmetry the holes are mobile and form a metal, unless localization due to disorder or some other phase transition intervenes. The full description of hole hop- ping in the three-band model is complicated, and a num- ber of theories consider this essential to the understand- ing of high-Tc superconductivity Emery, 1987 Varma et al., 1987 . On the other hand, there is strong evidence that the low-energy physics on a scale small compared FIG. 2. Color online Electronic structure of the cuprates. a Two-dimensional copper-oxygen layer left simplified to the one-band model right . b The copper d and oxygen p orbit- als in the hole picture. A single hole with S=1/2 occupies the copper d orbital in the insulator. 21 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

with tpd and Ep -Ed can be understood in terms of an effective one-band model, and we shall follow this route. The essential insight is that the doped hole resonates on the four oxygen sites surrounding a Cu and the spin of the doped hole combines with the spin on the Cu to form a spin singlet. This is known as the Zhang-Rice singlet Zhang and Rice, 1988 . This state is split off by an energy of order tpd 2 / Ep -Ed because the singlet gains energy by virtual hopping. On the other hand, the Zhang-Rice singlet can hop from site to site. Since the hopping is a two-step process, the effective hopping in- tegral t is also of order tpd 2 / Ep -Ed . Since t is the same parametrically as the binding energy of the singlet, the justification of this point of view relies on a large nu- merical factor for the binding energy, which is obtained by studying small clusters. By focusing on the low-lying singlet, the hole-doped three-band model simplifies to a one-band tight-binding model on the square lattice, with an effective nearest- neighbor hopping integral t given earlier and with Ep -Ed playing a role analogous to U. In the large Ep -Ed limit this maps onto the t-J model, H = P - ij , tijci ��� ci + J ij Si �� Sj - 1 4 ninj P. 2 Here the ci��� is the usual fermion creation operator on site i, ni = ci��� c is the number operator, and P is a projection operator restricting the Hilbert space to ex- clude double occupancy of any site. J is given by 4t2 /U and we can see that it is the same functional form as that of the three-band model described earlier. It is also pos- sible to dope with electrons rather than holes. The typi- cal electron-doped system is Nd2-xCexCuO4+ NCCO . The added electron corresponds to the removal of a hole from the copper site in the hole picture Fig. 2 , i.e., the Cu ion is in the d10 configuration. This vacancy can hop with a teff and the mapping to the one-band model is more direct than the hole-doped case. Note that in the full three-band model, the object which is hopping is the Zhang-Rice singlet for hole doping and the Cu d10 con- figuration for electron doping. These have rather differ- ent spatial structure and are physically quite distinct. For example, the strength of their coupling to lattice distor- tions may be quite different. When mapped to the one- band model, the nearest-neighbor hopping t has the same parametric dependence but could have a different numerical constant. As we shall see, the value of t de- rived from cluster calculations turns out to be surpris- ingly similar for electron and hole doping. For a bipar- tite lattice, the t-J model with nearest-neighbor t has particle-hole symmetry because the sign of t can be ab- sorbed by changing the sign of the orbital on one sub- lattice. Experimentally the phase diagram exhibits strong particle-hole asymmetry. On the electron-doped side, the antiferromagnetic insulator survives up to a much higher doping concentration up to x 0.2 and the superconducting transition temperature is quite low about 30 K . Many of the properties of the supercon- ductor resemble that of the overdoped region of the hole-doped side and pseudogap phenomenon, which is prominent in the underdoped region, is not observed with electron doping. It is as though the greater stability of the antiferromagnet has covered up any anomalous regime that might exist otherwise. Precisely why is not clear at the moment. One possibility is that polaron ef- fects may be stronger on the electron-doped side, lead- ing to carrier localization over a broader range of dop- ing. There has been some success in modeling the contrast in the single-hole spectrum by introducing further-neighbor coupling into the one-band model, which breaks the particle-hole symmetry Shih et al., 2004 . This will be discussed further below. We conclude that the electron correlation is strong enough to produce a Mott insulator at half-filling. Fur- thermore, the one-band t-J model captures the essence of the low-energy electronic excitations of the cuprates. Particle-hole asymmetry may be accounted for by in- cluding further-neighbor hopping t . This point of view has been tested extensively by Hybertson et al. 1990 who used ab initio local-density-functional theory to generate input parameters for the three-band Hubbard model and then solved the spectra exactly on finite clus- ters. The results were compared with the low-energy spectra of the one-band Hubbard model and the t-t -J model. They found an excellent overlap of the low-lying wave functions for both the one-band Hubbard and the t-t -J model and were able to extract effective param- eters. They found J to be 128��5 meV, in excellent agreement with experimental values. Furthermore, they found t 0.41 and 0.44 eV for electron and hole doping, respectively. The near particle-hole symmetry in t is sur- prising because the underlying electronic states are very different in the two cases, as already discussed. Based on their results, the commonly used parameter J/t for the t-J model is 1/3. They also found a significant next- nearest-neighbor t term, again almost the same for elec- tron and hole doping. More recently, Andersen et al. 1996 pointed out that in addition to the three-band model an additional Cu 4s orbital has a strong influence on further-neighbor hop- ping t and t , where t is the hopping across the diagonal and t is hopping to the next-nearest neighbor along a straight line. Recently Pavarini et al. 2001 emphasized the importance of the apical oxygen in modulating the energy of the Cu 4s orbital and found a sensitive depen- dence of t /t on the apical oxygen distance. They also pointed out an empirical correlation between optimal Tc and t /t. As we shall discuss in Secs. VI.D and VII, t may play an important role in determining Tc and in explaining the difference between electron and hole doping. However, in view of the fact that on-site repul- sion is the largest energy scale in the problem, it would make sense to begin our modeling of the cuprates with the t-J model and ask to what extent the phase diagram can be accounted for. As we shall see, even this is not a simple task and will constitute the major thrust of this review. 22 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

III. PHENOMENOLOGY OF THE UNDERDOPED CUPRATES The essence of the problem of doping into a Mott insulator is readily seen from Fig. 2. When a vacancy is introduced into an antiferromagnetic spin background, it would like to hop with amplitude t to lower its kinetic energy. However, after one hop its neighboring spin finds itself in a ferromagnetic environment, at an energy cost of 3 2 J if the spins are treated as classical S= 1 2 . It is clear that the holes are very effective in destroying the antiferromagnetic background. This is particularly so at t J when the hole is strongly delocalized. The basic physics is the competition between the exchange energy J and the kinetic energy, which is of order t per hole or xt per unit area. When xt J, we expect the kinetic en- ergy to win and the system would be a Fermi-liquid metal with a weak residual antiferromagnetic correla- tion. When xt J, however, the outcome is much less clear because the system would like to maintain the an- tiferromagnetic correlation while allowing the hole to move as freely as possible. Experimentally we know that the N��el order is destroyed with 3% hole doping, after which the d-wave superconducting state emerges as the ground state up to 30% doping. Exactly how and why superconductivity emerges as the best compromise is the centerpiece of the high-Tc puzzle, but we already see that the simple competition between J and xt sets the correct scale x=J/t= 1 3 for the appearance of nontrivial ground states. We shall focus our attention on the under- doped region where this competition rages most fiercely. Indeed it is known experimentally that the normal state above the superconducting Tc behaves differently from any other metallic state that we have known about up to now. Essentially an energy gap appears in some proper- ties and not others. This region of the phase diagram is referred to as the pseudogap region and is well docu- mented experimentally. Below we review some of the key properties. A. Pseudogap phenomenon in the normal state As seen in Fig. 3, the Knight-shift measurement in the YBCO 124 compound shows that while the spin suscep- tibility s is almost temperature independent between 700 and 300 K, as in an ordinary metal, it decreases be- low 300 K and by the time the Tc of 80 K is reached, the system has lost 80% of the spin susceptibility Curro et al., 1997 . To emphasize the universality of this phe- nomenon, we reproduce in Fig. 4 some old data on YBCO and LSCO. Figure 4 a shows the Knight-shift data from Alloul et al. 1989 . We have subtracted the orbital contribution which is generally agreed to be 150 ppm Takigawa et al., 1993 and drawn in the zero line to highlight the spin contribution to the Knight shift, which is proportional to s . The proportionality constant is known, which allows us to draw in the Knight shift, which corresponds to the two-dimensional square S= 1 2 Heisenberg antiferromagnet with J=0.13 eV Ding and FIG. 3. Knight shift for an underdoped YB2Cu4O8 with Tc =79 K. From Curro et al., 1997. FIG. 4. Color online Spin susceptibility data for a variety of doping. a Knight-shift data of YBCO from Alloul et al., 1989 . The zero reference level for the spin contribution is indicated by the arrow and the dashed line represents the pre- diction of the 2D S= 1 2 Heisenberg model for J=0.13 eV. b Uniform magnetic susceptibility for LSCO from Nakano et al., 1994 . The orbital contribution 0 is shown see text and the solid line represents the Heisenberg-model prediction. 23 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

Makivic, 1991 Sandvik et al., 1997 . The point of this exercise is to show that in the underdoped region the spin susceptibility drops below that of the Heisenberg model at low temperatures before the onset of super- conductivity. This trend continues even in the severely underdoped limit O0.53���O0.41 , showing that the s re- duction cannot simply be understood as a fluctuation towards the antiferromagnet. Note that the discrepancy is worse if J were replaced by a smaller Jeff due to dop- ing since s Jeff. -1 The data seen in this light strongly point to singlet formation as the origin of the pseudogap seen in the uniform spin susceptibility. It is worth noting that the trend shown in Fig. 4 a is not so apparent if one looks at the measured spin sus- ceptibility directly Tranquada et al., 1988 . This is be- cause the van Vleck part of the spin susceptibility is dop- ing dependent due to the changing chain contribution. This problem does not arise for LSCO, and in Fig. 4 b we show the uniform susceptibility data Nakano et al., 1994 . The zero of the spin part is determined by com- paring susceptibility measurements to O Knight-shift data Ishida et al., 1991 . Nakano et al. 1994 found an excellent fit for the x=0.15 sample see Fig. 9 of this reference and determined the orbital contribution for this sample to be 0 0.4 10-7 emu/g. This again al- lows us to plot the theoretical prediction for the Heisen- berg model. Just as for YBCO, s for the underdoped samples x=0.1 and 0.08 drops below that of the Heisenberg model. In fact, the behavior of s for the two systems is remarkably similar, especially in the under- doped region.1 A second indication of the pseudogap comes from the linear T coefficient of the specific heat, which shows a marked decrease below room temperature see Fig. 5 . Furthermore, the specific-heat jump at Tc is greatly re- duced with decreasing doping. It is apparent that the spins are forming into singlets and the spin entropy is gradually lost. On the other hand, as shown in Fig. 6, the frequency-dependent conductivity behaves very differ- ently depending on whether the electric field is in the ab plane ab or perpendicular to it c . At low frequencies below 500 cm-1 ab shows a typi- cal Drude-like behavior for a metal with a width which decreases with temperature, but with an area spectral weight which is independent of temperature Santander-Syro et al., 2002 . Thus there is no sign of the pseudogap in the spectral weight. This is surprising be- cause in other examples in which an energy gap appears in a metal, such as the onset of charge- or spin-density waves, there is a redistribution of the spectral weight from the Drude part to higher frequencies. An impor- tant observation concerning the spectral weight is that the integrated area under the Drude peak is found to be 1We note that a comparison of s for YBCO and LSCO was made by Millis and Monien 1993 . Their YBCO analysis was similar to ours. However, for LSCO they found a rather differ- ent 0 by matching the measurement above 600 K to that of the Heisenberg model. Consequently, their s looked different for YBCO and LSCO. FIG. 5. The specific-heat coefficient . a YBa2Cu3O6+y. b La2-xSrxCuO4. Curves are labeled by the oxygen content y in the top figure and by the hole concentration x in the bottom figure. Optimal and overdoped samples are shown in the inset. The jump in indicates the superconducting transition. Note the reduction of the jump size with underdoping. From Loram et al., 1993, 2001. FIG. 6. The frequency-dependent conductivity with electric field parallel to the plane a , top figure and perpendicular to the plane c , bottom figure in an underdoped YBCO crystal. From Uchida, 1997. 24 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

linear in x Orenstein et al., 1990 Uchida et al., 1991 Cooper et al., 1993 Padilla et al., 2005 . In the supercon- ducting state this weight collapses to form the delta- function peak, with the result that the superfluid density ns /m is also linear in x. It is as though only the doped holes contribute to the charge transport in the plane. In contrast, angle-resolved photoemission spectroscopy shows a Fermi surface at optimal doping very similar to that predicted by band theory, with an area correspond- ing to 1-x electrons see Fig. 7 d . With underdoping, this Fermi surface is partially gapped in an unusual man- ner, which we shall discuss next. In contrast to the metallic behavior of ab , Homes et al. 1993 discovered that below 300 K c is gradually reduced for frequencies below 500 cm-1 and a deep hole is carved out of c by the time Tc is reached. This is clearly seen in the lower panel of Fig. 6. Finally, angle-resolved photoemission spectroscopy shows that an energy gap in the form of a pulling back of the leading edge of the electronic spectrum from the Fermi energy is observed near momentum 0, . Note that the line shape is extremely broad and completely incoherent. The onset of superconductivity is marked by the appearance of a small coherent peak at this gap edge Fig. 7 . The size of the pullback of the leading edge is the same as the energy gap of the superconducting state as measured by the location of the coherence peak. As shown in Fig. 7, this gap energy increases with decreas- ing doping while the superconducting Tc decreases. This trend is also seen in tunneling data. It is possible to map out the Fermi surface by tracking the momentum of the minimum excitation energy in the superconducting state for each momentum direction. Along the Fermi surface the energy gap does exactly what is expected for a d-wave superconductor. It is maximal near 0, and vanishes along the line connec- tion 0,0 and , , where the excitation is often re- ferred to as nodal quasiparticles. Above Tc the gapless region expands to cover a finite region near the nodal point, beyond which the pseudogap gradually opens as one moves towards 0, . This unusual behavior is sometimes referred to as the Fermi arc Ding et al., 1996 Loeser et al., 1996 Marshall et al., 1996 . It is worth noting that unlike the antinodal direction near 0, , the line shape is relatively sharp along the nodal direc- tion even above Tc. From the width in momentum space, a lifetime which is linear in temperature has been ex- tracted for a sample near optimal doping Valla et al., 1999 . A narrow line shape in the nodal direction has also been observed in LSCO Yoshida et al., 2003 and in Na-doped Ca2CuO2Cl2 Ronning et al., 2003 . So the no- tion of relatively well-defined nodal excitations in the normal state is most likely a universal feature. As mentioned earlier, the onset of superconductivity is marked by the appearance of a sharp coherence peak near 0, . The spectral weight of this peak is small and gets even smaller with decreasing doping, as shown in Fig. 8 b . Note that this behavior is totally different from conventional superconductors. There the quasiparticles are well defined in the normal state and according to BCS theory the sharp peak pulls back from the Fermi energy and opens an energy gap in the superconducting state. FIG. 7. Color online Angle-resolved photoemission data for underdoped cuprates. a ��� c Spectra from underdoped Bi- 2212 Tc =85 K taken at different k points along the Fermi surface shown in d . Note the pullback of the spectrum from the Fermi surface as determined by the Pt reference shown by grey lines red online for T Tc. e Temperature dependence of the leading-edge midpoints from Norman et al., 1998 . The temperature T* where the pseudogap determined from the leading edge first appears plotted as a function of doping for Bi-2212 samples bottom . Triangles are determined from data such as shown in a and squares are lower-bound estimates. Circles show the energy gap measured at 0, at low tem- peratures. From Campuzano et al., 2003. 25 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

Yet another indication that the superconducting tran- sition is different from BCS theory comes from the mea- surement of the change in kinetic energy through the transition. In conventional BCS theory, pairing between quasiparticles leads to a gain in the attractive potential energy at the expense of increasing the kinetic energy since the Fermi distribution is smeared by the creation of the energy gap. By carefully monitoring the optical spectral weight above and below Tc, it was found that while optimally doped samples behave as expected for BCS superconductors, underdoped samples exhibit the opposite behavior in that the kinetic energy is lowered by the onset of superconductivity Molegraaf et al., 2002 Santander-Syro et al., 2002 Boris et al., 2004 Kuzmenko et al., 2005 Santander-Syro and Bontemps, 2005 . In the past few years, low-temperature STM data have become available, mainly on Bi-2212 samples. STM pro- vides a measurement of the local density of states E,r with atomic resolution. It is complementary to ARPES in that it provides real-space information but no direct momentum-space information. One important outcome is that STM reveals the spatial inhomogeneity of Bi-2212 on roughly a 50���100-�� length scale, which becomes more significant with underdoping. As shown in Fig. 9 f , spectra with different energy gaps are associated with different patches and with progressively more under- doping patches with large gaps become more predomi- nant. Since ARPES is measuring the same surface, it becomes necessary to reinterpret the ARPES data with inhomogeneity in mind. In particular, the decrease of the weight of the coherent peak shown in Fig. 8 b may sim- ply be due to a reduction of the fraction of the sample which has sharp coherent peaks. However, we should note that there are concerns as to whether the surface inhomogeneity may behave differently depending on the temperature at which the crystals are cleaved since the STM experiments shown here are cleaved at low tem- peratures, whereas ARPES and other STM experiments Maggio-Aprile et al., 1995 are typically cleaved at higher temperatures. A second remarkable observation by STM is that the low-lying density of states E,r for E 10���15 meV is remarkably homogeneous. This is clearly seen in Fig. 9 f . It is reasonable to associate this low-energy excita- tion with quasiparticles near the nodes. Indeed, low- lying quasiparticles exhibit interference effects due to scattering by impurities, which is direct evidence for their spatial coherence over long distances. Then the combined STM and ARPES data suggest a kind of phase separation in momentum space, i.e., the spectra in the antinodal region near 0, is highly inhomogeneous in space, whereas the quasiparticles near the nodal re- gion are homogeneous and coherent. The nodal quasi- particles must be extended and capable of averaging over the spatial homogeneity, while the antinodal quasi- particles appear more localized. In this picture the pseudogap phenomenon mainly has to do with the anti- nodal region. McElroy et al. 2005 have argued that there is a lim- iting spectrum the broadest curve in Fig. 9 f which characterizes the extreme underdoped region at zero temperature. It has no coherent peak at all, but shows a reduction of spectral weight up to a very high energy of 100���200 meV. Very recently, Hanaguri et al. 2004 have provided support for this point of view in their study of Na-doped Ca2CuO2Cl2. In this material the apical oxy- gen in the CuO4 cage is replaced by Cl and the crystal cleaves easily. For Na doping ranging from x=0.08 to 0.12, a tunneling spectrum very similar to the limiting spectrum for Bi-2212 is observed. This material appears free of the inhomogeneity which plagues the Bi-2212 FIG. 8. Photoemission spectra and spectral weight. a Doping dependence of the ARPES spectra at 0, at T Tc for overdoped OD , optimally doped OP , and underdoped UD materials labeled by their Tc���s. b The spectral weight of the coherent peak in a nor- malized to the background is plotted vs doping x. From Feng et al., 2000. 26 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

surface. ARPES experiments on these crystals are be- coming available Ronning et al., 2003 and the combi- nation of STM and ARPES should yield much informa- tion on the real- and momentum-space dependence of the electron spectrum. There is much excitement con- cerning the discovery of a static 4 4 pattern in this ma- terial and its relation to the incommensurate pattern seen in the vortex core of Bi-2212 Hoffman et al., 2002 and also reported in the absence of a mag- netic field, albeit in a much weaker form Howland et al., 2003 Vershinin et al., 2004 . How this spatial modulation is related to the pseudogap spectrum is a topic of current debate. In the literature, pseudogap behavior is often associ- ated with the anomalous behavior of the nuclear-spin relaxation rate 1/T1. In normal metals the nuclear spin relaxes by producing low-energy particle-hole excita- tions, leading to Koringa behavior, i.e., 1/T1T is tem- perature independent. In high-Tc materials, it is instead 1/T1 which is temperature independent, and the en- hanced relaxation relative to Koringa as the tempera- ture is reduced as ascribed to antiferromagnetic spin fluctuations. It was found that in underdoped YBCO, the nuclear-spin relaxation rate at the copper site reaches a peak at a temperature of T1 * and decreases rapidly below this temperature Warren et al., 1989 Ya- suoka et al., 1989 Takigawa et al., 1991 . The resistivity also shows a decrease below T1. * In some work in the literature T1 * is referred to as the pseudogap scale. How- ever, we note that T1 * is lower than the energy scale we have been discussing so far, especially compared with that for the uniform spin susceptibility and the c-axis conductivity. Furthermore, the gap in 1/T1 is not univer- sally observed in cuprates, e.g., it is not seen in LSCO. In YBa2Cu4O8, which is naturally underdoped, the gap in 1/T1T is wiped out by 1% Zn doping, while the Knight shift remains unaffected Zheng et al., 2003 . It is known from neutron scattering that low-lying spin exci- tations near , are sensitive to disorder. Since 1/T1 at the copper site is dominated by these fluctuations, it is reasonable that 1/T1 is sensitive as well. In contrast, the gaplike behavior we have described thus far in a variety of physical properties is universally observed across dif- ferent families of cuprates wherever data exist and are FIG. 9. Color online STM images showing the spatial distribution of energy gaps for a variety of samples which are progressively more underdoped from a to e . f The av- erage spectrum for a given energy gap. From McElroy et al., 2005. 27 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

robust. Thus we prefer not to consider T1 * as the pseudogap temperature scale. B. Neutron scattering, resonance, and stripes Neutron scattering provides a direct measure of the spin-excitation spectrum. Early work see Kastner et al., 1998 showed that with doping, the long-range N��el or- der gives way to short-range order with a progressively shorter correlation length so that at optimal doping the static spin-correlation length is no more than two or three lattice spacings. Much of the early work was fo- cused on the La2-xSrxCuO4 family because of the avail- ability of large single crystals. It was found that there is enhanced spin scattering at low energies, centered around the incommensurate positions q0 = �� /2, �� Cheong et al., 1991 . Yamada et al. 1998 found that increases systematically with doping, as shown in Fig. 10. Meanwhile it was noted that in the La2CuO4 family there is a marked suppression of Tc near x= 1 8 . This sup- pression is particularly strong with Ba doping, and Tc is completely destroyed if some Nd is substituted for La, as in La1.6-xNd0.4SrxCuO4 for x= 1 8 . Tranquada et al. 1995 discovered static spin-density-wave and charge-density- wave order in this system, which appears below about 50 K. The period of the spin- and charge-density waves are eight and four lattice constants, respectively. The static order is modeled by a stripe picture in which holes are concentrated in period-4 charge stripes separated by spin-ordered regions with antiphase domain walls. Re- cently, the same kind of stripe order was observed in La1.875Ba0.125CuO4 Fujita et al., 2004 . Note that in this model there is one hole per two sites along the charge stripe. It is tempting to interpret the low-energy spin- density wave observed in LSCO as a slowly fluctuating form of stripe order, even though the associated charge order presumably dynamical also has not yet been seen. The most convincing argument for this interpreta- tion comes from the observation that over a range of doping x=0.06 to x=0.125 the observed incommensura- bility is given precisely by the stripe picture, i.e., =x, while saturates at approximately 1 8 for x 0.125 see Fig. 10 . However, it must be noted that with this inter- pretation the charge stripe must be incompressible, i.e., it behaves as a charge insulator. Upon changing x, it is energetically more favorable to add or remove stripes and change the average stripe spacing, rather than to change the hole density on each stripe, which is pinned at 1 4 filling. It is difficult to reconcile this picture with the fact that LSCO is metallic and superconducting in the same doping range. An alternative interpretation of the incommensurate spin scattering is that it is due to Fermi- surface nesting Littlewood et al., 1993 Si et al., 1993 Tanamoto et al., 1993 . However, in this case the x de- pendence of requires some fine-tuning. Regardless of interpretation, it is clear that in the LSCO family there are low-lying spin-density-wave fluctuations which are almost ready to condense. At low temperatures, static spin-density-wave order is stabilized by Zn doping Kimura et al., 1999 near x= 1 8 Wakimoto et al., 1999 and in oxygen-doped systems Lee et al., 1999 . How- ever, in the latter case there is evidence from muon spin rotation Savici et al., 2002 that there may be micro- scopic phase separations in this material not too sur- prising in view of the STM data on Bi-2202 . It was also found that spin-density-wave order is stabilized in the vicinity of vortex cores Kitano et al., 2000 Lake et al., 2001 Khaykovich et al., 2002 . The key question then is whether the fluctuating stripe picture is special to the LSCO family or plays a significant role in all the cuprates. Outside of the LSCO family, the spin response is dominated by a narrow reso- nance at , . The resonance was first discovered at 41 meV for optimally doped YBCO Rossat-Mignod et al., 1991 Mook et al., 1993 . Careful subtraction of an accidentally degenerate phonon line reveals that the resonance appears only below Tc at optimal doping Fong et al., 1995 . Now it is known that with underdop- ing the resonance moves down in energy and survives into the pseudogap regime above Tc. The resonance moves smoothly to almost zero energy at the edge of the transition to N��el order in YBa2Cu3O6.35 Stock et al., 2005a and clearly plays the role of a soft mode at that transition. The resonance was interpreted as a spin-triplet exci- ton bound below 2 0 Fong et al., 1995 . This idea was elaborated upon by a number of random-phase- approximation calculations Liu et al., 1995 Bulut and Scalapino, 1996 Brinckmann and Lee, 1999, 2002 Kao et al., 2000 Norman, 2000, 2001 Abanov et al., 2002 Onufrieva and Pfeuty, 2002 . An alternative picture was proposed which made use of the particle-particle chan- nel Demler and Zhang, 1995 . However, as explained by Tchernyshyov et al. 2001 and by Norman and Pepin FIG. 10. Color online Plot of the incommensurability vs hole concentration x. From Matsuda et al., 2000. In the super- conducting state, the open circles denote the position of the fluctuating spin-density wave observed by neutron scattering. Data from Yamada et al., 1998. In the insulator the spin- density wave becomes static at low temperatures and its orien- tation is rotated by 45��. The dashed line =x is the prediction of the stripe model, which assumes a fixed density of holes along the stripe. 28 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

2003 , this theory predicts an antibound resonance above the two-particle continuum, which is not in accord with experiments. Further support of the triplet exciton idea comes from the observation that incommensurate branches extend below the resonance energy Bourges et al., 2000 . This behavior is predicted by random-phase-approximation- type theories Kao et al., 2000 Norman, 2000 Brinck- mann and Lee, 2002 Li and Gong, 2002 Onufrieva and Pfeuty, 2002 in that the gap in the particle-hole con- tinuum extends over a region near , , where the resonance can be formed. With further underdoping this incommensurate branch extends to lower energies see Fig. 11 . Now it becomes clear that the low-energy in- commensurate scattering previously reported for under- doped YBCO Mook et al., 2000 is part of this down- ward dispersing branch Pailhes et al., 2004 Stock et al., 2004 . It should be noted that while the resonance is promi- nent due to its sharpness, its spectral weight is actually quite small, of order 2% of the total spin moment sum rule for optimal doping and increasing somewhat with underdoping. There is thus considerable controversy over its significance in terms of its contribution to the electron self-energy and towards pairing see Norman and Pepin, 2003 . The transfer of this spectral weight from above to below Tc has been studied in detail by Stock et al. 2004 . These authors have emphasized that in the pseudogap state above Tc in YBa2Cu3O6.5 the scattering below the resonance is gapless and in fact in- creases in strength with decreasing temperature. This is in contrast to the sharp drop seen in 1/T1T below 150 K. Either a gap opens up at very low energy below 4 meV or the , spins fluctuating seen by neutrons are not the dominant contribution to the nuclear-spin relax- ation, i.e., the latter may be due to excitations which are smeared out in momentum space and undetected by neutrons. We note that a similar discrepancy between the neutron-scattering spectral weight and 1/T1T was noted for LSCO Aeppli et al., 1995 . This reinforces our view that the decrease in 1/T1T should not be consid- ered a signature of the pseudogap. We also note that an enhanced , scattering together with singlet forma- tion is just what is predicted by the SU 2 theory in Sec. XI.D. Recently, neutron scattering has been extended to en- ergies much above the resonance. It is found that very broad features disperse upward from the resonance, re- sulting in the ���hourglass��� structure shown in Fig. 11, which was first proposed by Bourges et al. 2000 Hay- den et al., 2004 Stock et al., 2004 . Interestingly, there has also been a significant evolution of the understand- ing of neutron scattering in the LSCO family. For a long time it was thought that the LSCO family does not ex- hibit the resonance which shows up prominently below Tc in YBCO and other compounds. However, neutron scattering does show a broad peak around 50 meV, which is temperature independent. Tranquada et al. 2004 have studied La1.875Ba0.125CuO4, which exhibits static charge and spin stripes below 50 K and a greatly suppressed Tc. Their data also exhibit an hourglass-type dispersion, remarkably similar to that of underdoped YBCO. In particular, the incommensurate scattering which was previously believed to be dispersionless now exhibits downward dispersion Fujita et al., 2004 . The same phenomenon is also seen in optimally doped La2-xSrxCuO4 Christensen et al., 2004 . It is remarkable that in these materials known to have static or dynamic stripes the incommensurate low-energy excitations are not spin waves emanating from /2�� as one might have expected, but instead are connected to the peak at , in the hourglass fashion. Tranquada et al. 2004 fit the k-integrated intensity to a model of a two-leg ladder. It is not clear how unique this fit is because one may expect high-energy excitations to be relatively insensi- tive to details of the model. What is emerging though is a picture of a universal hourglass-shaped spectrum, which is common to LSCO and YBCO families. The high-energy excitations appear common while the major difference seems to be in the rearrangement of spectral weight at low energy. In La1.875Ba0.125CuO4 significant FIG. 11. Neutron scattering from YBCO6.5. This sample has Tc =59 K and the experiment was performed at 6 K from Stock et al., 2005 . Top panel refers to in-phase fluctuations between the bilayer which shows a resonance located at , q=0 and at energy 33 meV. Incommensurate peaks disperse down from the resonance. Broad peaks also disperse upward from the resonance forming the hourglass pattern. The solid line is the spin-wave spectrum of the insulating parent com- pound. Bottom panel denotes out-of-phase fluctuations be- tween the bilayers. 29 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

weight has been transferred to the low-energy incom- mensurate scattering, as shown in Fig. 12, and is associ- ated with stripes. In our view the universality supports the picture that all the cuprates share the same short- distance and high-energy physics, which includes the pseudogap behavior. Stripe formation is a competing state which becomes prominent in the LSCO family, es- pecially near x= 1 8 , and may dominate the low-energy and low-temperature below 50 K physics. There is a school of thought which holds the opposite view see Carlson et al., 2003 , that fluctuating stripes are respon- sible for the pseudogap behavior and the appearance of superconductivity. From this point of view the same data have been interpreted as an indication that stripe fluc- tuations are also important in the YBCO family Tran- quada et al., 2004 . Clearly, this is a topic of much cur- rent debate. C. Quasiparticles in the superconducting state In contrast with the anomalous properties of the nor- mal state, the low-temperature properties of the super- conductor seem relatively normal. There are two major differences with conventional BCS superconductors, however. First, due to the proximity to the Mott insula- tor, the superfluid density of the superconductor is small and vanishes with decreasing hole concentration. Sec- ond, because the pairing is d wave, the gap vanishes on four points on the Fermi surface called gap nodes so that the quasiparticle excitations are gapless and affect the physical properties even at the lowest temperatures. We shall focus on these nodal quasiparticles in this sub- section. The nodal quasiparticles clearly contribute to the ther- mal dynamical quantities such as the specific heat. Be- cause their density of states vanish linearly in energy, they give rise to a T2 term, which dominates the low- temperature specific heat. In practice, disorder rounds off the linear density of states, giving instead an T + T3 behavior. An interesting effect in the presence of a magnetic field was proposed by Volovik 1993 . He ar- gued that in the presence of a vector potential A or superfluid flow, the quasiparticle dispersion E k = k - 2 + k 2 is shifted by EA k = E k + 1 2e - A �� jk, 3 where jk is the current carried by normal-state quasipar- ticles with momentum k and is usually taken to be -e k / k. Note that since the BCS quasiparticle is a su- perposition of a particle and a hole, the charge is not a good quantum number. However, the particle compo- nent with momentum k and the hole component with momentum -k each carry the same electrical current jk =-e k / k and it makes sense to consider this to be the current carried by the quasiparticle. Note that jk /e is very different from the group velocity E k / k. In a magnetic field which exceeds Hc1, vortices enter the sample. The superfluid flow 2 /r, where r is the distance to the vortex core. On average, 1 2 /R, where R= 0 /H 1/2 is the average spacing between vor- tices and 0 =hc/2e is the flux quantum. Volovik then predicted a shift of the quasiparticle spectra by evF H/ 0 1/2, which in turn gives a contribution to the specific heat proportional to H. This contribution was observed experimentally Moler et al., 1994 . Very re- cently Wen et al. 2005 have used this effect to extract v , the velocity of the nodal quasiparticle in the direc- tion of the maximum gap 0 . The result is consistent with that shown in Fig. 13, which will be discussed later. FIG. 12. Color online Neutron scattering from La1.875Ba0.125CuO4 at 12 K Tc from Tranquada et al., 2004 . b The hourglass pat- tern of the excitation spectrum cf. Fig. 11 . The solid line is a fit to a two-leg-ladder spin model. a The momentum-integrated scatter- ing intensity. The dashed line is a Lorenztian fit to the rising intensity at the incommensu- rate positions. The sharp peak at 40 meV could be a phonon. 30 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006

The quasiparticles contribute to the low-temperature transport properties as well. Lee 1993 considered the frequency-dependent conductivity due to quasipar- ticle excitations. In the low-temperature limit, he found that the low-frequency limit of the conductivity is uni- versal in the sense that it does not depend on impurity strength, but only on the ratio vF /v , i.e., ��� 0 =e2 vF /hv if vF v . This result was derived within the self-consistent t-matrix approximation and can easily be understood as follows. In the presence of impurity scat- tering, the density of states at zero energy becomes fi- nite. At the same time, the scattering rate is propor- tional to the self-consistent density of states. Since the conductivity is proportional to the density of states and inversely to the scattering rate, the impurity dependence cancels. The frequency-dependent is difficult to measure and it was realized that the thermal conductivity may provide a better test of the theory because according to the Wiedemann-Franz law, /T is proportional to the conductivity and should be universal. Unlike , ther- mal conductivity does not have a superfluid contribution and can be measured at dc. More detailed consider- ations by Durst and Lee 2000 showed that has two nonuniversal corrections: one due to backscattering effects, which distinguishes the transport rate from the impurity rate which enters the density of state, and a second one due to Fermi-liquid corrections. On the other hand, these corrections do not exist for thermal conductivity. Consequently, the Wiedemann-Franz law is violated but the thermal conductivity per layer is truly universal and is given by T = kB 2 3 c vF v + v vF . 4 We note that this result is obtained within the self- consistent t-matrix approximation, which is expected to break down if the impurity scattering is strong, leading to localization effects. The localization of nodal quasi- particles is a complex subject. Due to particle-hole mix- ing in the superconductor, zero energy is a special point and quasiparticle localization belongs to a different uni- versality class Senthil and Fisher, 1999 than the stan- dard ones. Senthil and Fisher also pointed out that since quasiparticles carry well-defined spin, the Wiedemann- Franz law for spin conductivity should hold and spin conductivity should be universal. We note that Durst and Lee 2000 argued that Fermi-liquid corrections en- ter the spin conductivity, but we now believe their argu- ment on this point is faulty. Thermal conductivity has been measured to mK tem- peratures in a variety of YBCO and Bi-2212 samples. The universal nature of /T has been demonstrated by studying samples with different Zn doping and showing that /T extrapolates to the same constant at low tem- peratures Taillefer et al., 1997 . A magnetic-field depen- dence analogous to the Volovik effect for the specific heat has also been observed Chiao et al., 2000 . Using Eq. 4 , the experimental data can be used to extract the ratio vF /v . In the case of Bi-2212 for which photoemis- sion data for vF and the energy gap are available, the extracted ratio vF /v is in excellent agreement with ARPES results assuming a simple d-wave extrapolation of the energy gap from the node to the maximum gap 0 . In particular, the trend that 0 increases with de- creasing doping x is directly observed as a decrease of vF /v extracted from /T. A summary of the data is shown in Fig. 13 Sutherland et al., 2003 . The results of such systematic studies strongly support the notion that in clean samples nodal quasiparticles behave exactly as one expects for well-defined quasiparticles in a d-wave superconductor. We should add that in LSCO the ratio vF /v extracted from /T seems anomalously small, sug- gesting that strong disorder may be playing a role here to invalidate Eq. 4 . Lee and Wen 1997 pointed out that nodal quasipar- ticles also manifest themselves in the linear T depen- dence of the superfluid density. They showed that by treating them as well-defined quasiparticles in the sense of Landau, a general expression of the linear T coeffi- cient can be written down, independent of the micro- scopic origin of the superconductivity. We have ns T m = ns 0 m - 2 ln 2 2 vF v T. 5 The only assumption made is that the quasiparticles carry an electric current, FIG. 13. Doping dependence of the superconducting gap 0 obtained from the quasiparticle velocity v using Eq. 4 filled symbols . Here we assume = 0 cos 2 so that 0 = kFv /2, and plot data for YBCO alongside Bi-2212 Chiao et al., 2000 and Tl-2201 Proust et al., 2002 . For comparison, a BCS gap of the form BCS=2.14kBTc is also plotted. The value of the en- ergy gap in Bi-2212, as determined by ARPES, is shown as measured in the superconducting state Campuzano et al., 1999 and the normal state Norman et al., 1998 open sym- bols . The thick dashed line is a guide to the eye. From Suth- erland et al., 2003. 31 Lee, Nagaosa, and Wen: Doping a Mott insulator: Physics of high-�� Rev. Mod. Phys., Vol. 78, No. 1, January 2006