Orbital Physics in Transition-Metal OxidesY. Tokura1,2 and N. Nagaosa1

An electron in a solid, that is, bound to or nearly localized on the specificatomic site, has three attributes: charge, spin, and orbital. The orbitalrepresents the shape of the electron cloud in solid. In transition-metaloxides with anisotropic-shaped d-orbital electrons, the Coulomb interac-tion between the electrons (strong electron correlation effect) is ofimportance for understanding their metal-insulator transitions and prop-erties such as high-temperature superconductivity and colossal magne-toresistance. The orbital degree of freedom occasionally plays an impor-tant role in these phenomena, and its correlation and/or order-disordertransition causes a variety of phenomena through strong coupling withcharge, spin, and lattice dynamics. An overview is given here on this“orbital physics,” which will be a key concept for the science and tech-nology of correlated electrons.

The quantum mechanical wave function of anelectron takes various shapes when bound toan atomic nucleus by Coulomb force. Con-sider a transition-metal atom in a crystal withperovskite structure. It is surrounded by sixoxygen ions, O22, which give rise to thecrystal field potential and hinder the freerotation of the electrons and quenches theorbital angular momentum by introducing thecrystal field splitting of the d orbitals. Wavefunctions pointing toward O22 ions havehigher energy in comparison with thosepointing between them. The former wavefunctions, dx22y2 and d3z22r2, are called eg

orbitals, whereas the latter, dxy, dyz, and dzx,are called t2g orbitals (Fig. 1).

When electrons are put into these wavefunctions, the ground state is determined bythe semiempirical Hund’s rule. As an exam-ple, consider LaMnO3, where Mn31 has a d4

configuration, i.e., four electrons in d orbit-als. Because of Hund’s rule, all of the spinsare aligned parallel, that is, S 5 2, and threespins are put to t2g orbitals and one spinoccupies one of the eg orbitals.

The relativistic correction gives rise to theso-called spin-orbit interaction Hspin-orbit 5lLW z SW , where LW is the orbital angular mo-mentum and SW is the spin angular momentum.This interaction plays an important role insome cases, especially for t2g electrons. How-ever, the coupling between spin and orbitaldegrees of freedom described below is notdue to this relativistic spin-orbit coupling.

Up to now, we have considered only onetransition-metal ion. However, in solids,there are periodic arrays of ions. There aretwo important aspects caused by this: one isthe magnetic interactions, i.e., exchange in-teractions, between the spins and the other is

the possible band formation and metallic con-duction of the electrons. Before explainingthese two, let us introduce the Mott insulatingstate. Band theory predicts an insulating statewhen all bands are fully occupied or empty,whereas a metallic state occurs under differ-ent conditions. However, it is possible thatthe system is insulating because of the Cou-lomb interaction when the electron number isan integer per atom, even if the band theorywithout the period doubling predicts a metal-lic state. This occurs when the kinetic energygain is smaller and blocked by the strongCoulomb repulsion energy U, and the elec-tron cannot hop to the other atom. This insu-lator is called a Mott insulator. The mostimportant difference from the usual band in-sulator is that the internal degrees of freedom,spin and orbital, still survive in the Mottinsulator. LaMnO3 is a Mott insulator withspin S 5 2 and the orbital degrees of free-dom. The spin S 5 2 can be represented bythe t2g spin 3/2 strongly coupled to the eg spin1/2 with ferromagnetic JH (Hund’s coupling).The two possible choices of the orbitals arerepresented by the pseudospin TW , whose zcomponent Tz 5 1/ 2 when dx22y2 is occu-pied and Tz 5 21/ 2 when d3z22r2 is occu-pied. Three components of this pseudospinsatisfy the similar commutation relation withthose of the spin operator, i.e., [Ta, Tb] 5iεabgTg.

There is an interaction between the spinand pseudospin, of SW and TW , between differ-ent ions. This exchange interaction is repre-sented by the following generalized Heisen-berg Hamiltonian (1):

H 5 Oij

@ Jij~TW i,TW j!SW i z SW j 1 Kij~TW i,TW j!# (1)

The exchange interactions Jij and Kij origi-nate from the quantum mechanical processwith intermediate virtual states (2, 3). Therotational symmetry in the spin space leads tothe inner product form of the interaction.

When more than two orbitals are involved, avariety of situations can be realized, and thisquantum mechanical process depends on theorbitals (4, 5). In this way, the spin SW and theorbital pseudospin TW are coupled. In moregeneral cases, the transfer integral tij dependson the direction of the bond ij and also on thepair of the two orbitals a, b 5 ( x2 2 y2) or(3z2 2 r2). This gives rise to the anisotropyof the Hamiltonian in the pseudospin space aswell as in the real space. For example, thetransfer integral between the two neighboringMn atoms in the crystal lattice is determinedby the overlaps of the d orbitals with the porbital of the O atom between them. Theoverlap between the dx22y2 and pz orbitals iszero because of the different symmetrywith respect to the holding in the xy plane.Therefore, the electron in the dx22y2 orbitalcannot hop along the z axis. This fact willbe important later in our discussion.

One can consider the long-range orderedstate of the orbital pseudospin TW as well as thespin SW . In many respects, analogies can bedrawn between SW and TW in spite of the aniso-tropy in TW space. However, there is one moreaspect that is special to TW —Jahn-Teller (JT)coupling (6–8). Because each orbital has dif-ferent anisotropy of the wave function, it iscoupled to the displacement of the O atomssurrounding the transition-metal ion. For ex-ample, when the two apical O atoms movetoward the ion, the energy of d3z22r2 becomeshigher than dx22y2 and the degeneracy is lift-ed. This is called the JT effect (6) and isrepresented by the following Hamiltonian fora single octahedron:

HJT 5 2g~TxQ2 1 TzQ3! (2)

where (Q2, Q3) are the coordinates for thedisplacements of O atoms surrounding thetransition-metal atom and g is the couplingconstant. When the crystal is considered,(Q2, Q3) should be generalized to (Qi2, Qi3)(i, the site index), which is represented as thesum of the phonon coordinates and the uni-form component (u2, u3). Here, (u2, u3)describes the crystal distortion as a whole.When the long-range orbital order exists, i.e.,^Tix& Þ 0 and/or ^Tiz& Þ 0, the JT distortionis always present.

Up to now, we have discussed the Mottinsulating state. Let us now consider thedoped carriers into a Mott insulator. High-transition-temperature superconductor cup-rates, e.g., La22xSrxCuO4, offer the mostdramatic example of this carrier doping.However, the two-dimensional (2D) natureof the lattice, as well as the larger coherent

1Department of Applied Physics, University of Tokyo,Bunkyo-ku, Tokyo 113-8656, Japan. 2Joint ResearchCenter for Atom Technology, Tsukuba 305-0046,Japan.

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R E V I E W

(F) JT distortion for the Cu21–O sheet,gives a large energy splitting betweend3z22r2 and dx22y2 orbitals, and only dx22y2 isrelevant (the orbital degrees of freedom arequenched). In the case of La12xSrxMnO3,Mn41 or holes with concentration x are doped,and still, the orbital degrees of freedom areactive. The most important and fundamentalinteraction in the doped case is the double-exchange interaction (9, 10). eg electrons areforced to be parallel to the localized t2g spins bythe strong JH. When an eg electron hops fromatom i to j, at each atom the spin wave functionux& of the eg electron is projected to uxi& and ux j&corresponding to the spin direction of each t2g

spin. Therefore, the effective transfer integral isgiven by tij 5 t ^xiux j&, which depends on therelative direction of the two spins; utiju 5tu^xiux j&u 5 t cos(uij/2) (uij, the angle be-tween the two spins) is maximum for par-allel spins and is zero for antiparallel spins.Therefore, the kinetic energy gain of thedoped holes is maximized for parallelspins, which gives the ferromagnetic inter-action between the spins. This is calleddouble-exchange interaction.

Manganese Oxides as a PrototypeAn active role of orbital degree of freedom inthe lattice and electronic response can bemost typically seen in manganese oxide com-pounds with perovskite structure. In this classof compounds, colossal magnetoresistance(CMR), i.e., a very large decrease of resis-tance, is observed upon the application of anexternal magnetic field and has attracted a lotof interest (11). The CMR phenomenon itselfis, as argued in the following, most relevantto the orbital ordering and correlation.

The orbital ordering gives rise to the an-isotropy of the electron-transfer interaction.This favors or disfavors the double-exchangeinteraction and the superexchange (ferromag-netic or antiferromagnetic) interaction in anorbital direction–dependent manner andhence gives a complex spin-orbital coupledstate. One typical example is the case ofLaMnO3 with no double-exchange carriers,in which the in-plane (ab plane) alternateordering of (3x2 2 r2)/(3y2 2 r2) orbitalscauses the in-plane ferromagnetic spin cou-pling. This A-type antiferromagnetic (AF)state is the manifestation of the anisotropicsuperexchange interactions, that is, ferromag-netic within the plane and AF between theplane, due to the orbital ordering.

The importance of the orbital and latticedegrees of freedom has long been recognizedtheoretically (12). The spin and orbital orderhas been studied for the realistic model Ham-iltonian Eq. 1 in the mean field approxima-tion. As for the eg electrons in 3D perovskitestructure, the A-type AF state is obtainedwith alternating ( z2 2 x2)/( y2 2 z2) orbitalswithin the plane. In addition to this superex-

change interaction, the Jahn-Teller interac-tion (JTI) also contributes to determining theorbital ordering. When JTI prefers the planarorbitals, such as ( z2 2 x2) and ( y2 2 z2), itdoes not contradict with the above orbitalordering; however, JTI could also prefer therod-type orbitals, such as (3z2 2 r2). In theformer case and/or when JTI is weak, theA-type AF state with alternating ( z2 2 x2)/( y2 2 z2) orbitals should be realized asobserved in KCuF3 (1). The A-type AF statewith alternating (3x2 2 r2)/(3y2 2 r2) inLaMnO3, on the other hand, is attributed tothe JT distortion (13–15).

In the hole-doped manganese oxides, inwhich the double-exchange interactionemerges with the strength being dependent onthe doping level, various orbital-ordered anddisordered states show up, accompanying therespective spin-ordering features (Fig. 2,top). Let us here take the case ofNd12xSrxMnO3 under ambient pressure (16–19). With appreciable doping on the parentcompound NdMnO3, the orbital order meltsinto a quantum-disordered state, and the com-pound shows the ferromagnetic-metallic (F)state for 0.3 , x , 0.5. When dopedfurther, the kinetic energy of carriers decreas-es, and the compound shows the 2D metallicstate with layered-type antiferromagnetic (A)state for 0.5 , x , 0.7. Doping above x 50.7 further alters the magnetic structure to thechain type (C). This rich phase diagram canbe reproduced and understood theoretically interms of the mean field approximation ap-plied to the generalized Hubbard model (20).The A state is realized as the compromisebetween the AF superexchange interactionbetween the t2g spins and the double-ex-change interaction through the ferromagnetic

(homogeneous) order of ( x2 2 y2) orbital(20). In cubic perovskite, the electron transferis almost prohibited along the c axis becauseof the ( x2 2 y2) orbital order, which is alsothe origin for the interplane AF coupling. Infact, the charge dynamics in this A-type AFstate is almost 2D (18).

The C-type AF state for x . 0.7 is ac-companied by the (3z2 2 r2) orbital. Thisstate is perhaps affected also by the chargeordering and shows an insulating feature(Fig. 2A). The large orbital polarization TW isindispensable for this rich phase diagram.The shape of the wave function is well de-fined, and the anistropy appears only in thiscase, i.e., the dimensional control by the or-bital occurs. Otherwise, it would become aboring phase diagram in which the ferromag-netic state dominates.

Instead of changing the carriers’ kinetic en-ergy with doping level, one can use the latticestrain as a biasing field on the orbital statethrough the JT channel; namely, the uniaxialstrain with respect to the MnO6 octahedron canserve as a pseudo magnetic field on thepseudospin TW. Figure 2B shows a schematicspin-orbital phase diagram in the moderatelydoped (0.3 , x , 0.7) manganese oxides on theplane of the uniaxial strain measured as theratio of lattice parameter c/a (or almost equiv-alently the ratio of the apical to equatorialMn–O bond length) and the doping x. Thephase diagram was based on the local densityfunctional calculation as well as the experimen-tal results for the epitaxial single-crystallinefilms of La12xSrxMnO3 with coherent latticestrain due to the lattice mismatching with thesubstrate (21). The entanglement of the dop-ing and the strain causes the slanted phaseboundaries for the F (orbital-disordered), A

t2g orbitals

eg orbitals

zx yz xy

3z2-r2 x2-y2

y

x

z

Fig. 1. Five d orbitals.In the cubic crystalfield, this fivefold de-generacy is lifted totwo eg orbitals [( x2 2y2) and (3z2 2 r2)]and three t2g orbitals[( xy), ( yz), and ( zx)].

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[( x2 2 y2)-ordered], and C [(3x2 2 r2)-ordered] states. As a general trend, the de-crease of hole doping enlarges the F region,whereas the increase (decrease) in the c/aratio stabilizes the C (A) state as expected. Infact, thin films of La12xSrxMnO3 ( x 5 0.5)epitaxially grown on three different perov-skite substrates show the respective groundstates (F, A, and C) and the similar transportproperties to those shown in the case ofNd12xSrxMnO3 (Fig. 2A).

The orbital ordering in the manganeseoxides occasionally accompanies the con-comitant charge ordering. The most prototyp-ical case, namely the CE type shown in Fig.3A, is realized at a doping level ( x) of 0.5. Inthe pseudo cubic perovskite, the ab planesare coupled antiferromagnetically whilekeeping the same in-plane charge and orbitalpattern (22, 23).

Theoretically, a band calculation with thelocal density approximation (LDA) combinedwith the on-site Coulomb interaction U hassuccessfully reproduced the observed spin/charge/orbital–ordered state for Pr1/2Ca1/2

MnO3 (24). It is an important issue to iden-

tify the major driving force of the spin/charge/orbital ordering. The experimental ob-servation in Pr1/2Ca1/2MnO3 that the transi-tion temperature of charge ordering is higherthan that of antiferromagnetism suggests thatthe former is the driving force.

Once the zigzag chain structure is as-sumed for the orbital ordering, the electronicstructure becomes 1D, and the sign alterna-tion of the transfer integrals due to the anti-symmetric combination of x2 and y2 in ( x2 2y2) plays some important roles (25, 26).

In the single-layered perovskiteLa12xSr11xMnO4 ( x 5 0.5), having theso-called K2NiF4 structure like La2CuO4, thesame charge-orbital pattern (Fig. 3A) wasconfirmed by the resonant x-ray scatteringmethod (27). In this compound, the chargeand orbital ordering occurs concomitantly atthe charge-ordering temperature TCO 5 220K, and then the CE-type spin-ordering tran-sition occurs at the AF temperature TN 5 150K. At temperatures above TCO, the averagestructure of the crystal is tetragonal, andhence, the optical property is isotropic in thelateral plane. Upon the orbital ordering, the

crystal is deformed to orthorhombic, thoughhardly detected by conventional diffractionmeasurements. However, the anisotropic or-dering of the orbital causes a fairly largein-plane anisotropy in the optical electronictransitions (28); hence, in the image withcross-polarized light, we can visualize theorbital-ordered domain (Fig. 3B). The orbit-al-disordered state above TCO is opticallyisotropic in the plane, giving the extinction ofcross-polarized reflection light, whereas weobserve the globally bright image for theorbital-ordered state below TCO, where theorthorhombic domain structure and domainwalls (dark stripes) are clearly visible. (Aperiodic structure of the domains perhapsarises from the slight residual strain intro-duced during the process of the crystalgrowth.)

The CE-type orbital/charge–ordered statein the manganese oxides is generally amena-ble to an application of a magnetic field. Thevariation of the orbital/charge–ordered stateis shown (Fig. 4) in the plane of magneticfield (H) and temperature (T ) for the x 50.5 perovskite manganites R0.5

31 A0.521MnO3,

with various combinations of (R, A) (R and Aare trivalent rare-earth and divalent alkaline-earth ions, respectively). The change in theaverage size of the (R, A) site controls adeviation of the Mn–O–Mn bond angle from180° and hence controls the eg electron-hop-ping interaction t through a change in Mn 3dand O 2p hybridization. With a decrease ofthe ionic radius, say from (Nd, Sr) to (Sm,Ca), the H-T region for the stable orbital/charge–ordered state is enlarged (Fig. 4).There are two types of orbital/charge/spinphase diagrams (29). In a relative wide-band-width system like Nd0.5Sr0.5MnO3, the ferro-magnetic ordering first occurs at the criticaltemperature Tc in the cooling process, andthen at a lower temperature, the orbital,charge, and spin (AF) ordering occurs con-comitantly at TCO 5 TN (type I). The type Icrystal undergoes the CE-type orbital/charge/spin–ordering transition only at the dopinglevel very close to x 5 1/ 2. In the smallerbandwidth system, say Pr0.5Ca0.5MnO3, firstthe orbital and charge–ordered state appearsconcomitantly at TCO > 250 K, and then theAF spin ordering takes place at a lower tem-perature TN (type II). The ferromagnetic andmetallic state is only realized under a mag-netic field in this type. The crossover fromtype I to type II shows a complicated feature(30), but near such a multicritical point forthe orders of orbital/charge and spin, a verylarge fluctuation and a critical field suppres-sion seem to appear as typically seen in theCMR behavior.

The aforementioned orbital-charge corre-lation is a source of the high-resistance stateabove the ferromagnetic transition tempera-ture TC, which causes the CMR upon the

Fig. 2. Spin-orbital phase diagram in the perovskite manganese oxide. The top panel shows theorbital and spin order realized in the hole-doped manganese oxides. (A) Temperature dependenceof resistivity in various magnetic fields m0H for Nd12xSrxMnO3 with the respective magneticphases (F, A, and C). The numbers in parentheses represent the uniaxial lattice strain, c/a ratio,indicating the coupling of the magnetism to the orbital order as shown in the top panel. (B) Theschematic phase diagram in the plane of lattice strain c/a and doping level x. The data labeled LAO,LSAT, and STO represent the results for the coherently strained epitaxial thin films ofLa12xSrxMnO3 grown on the perovskite single-crystal substrates of LaAlO3, (La,Sr)(Al,Ta)O3, andSrTiO3, respectively. LSMO-bulk and NSMO-bulk stand for the results for the bulk single crystals ofLa12xSrxMnO3 and Nd12xSrxMnO3, respectively.

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application of a magnetic field. Even with nolong-range orbital/charge ordering, the rem-nant of the short-range order can be clearlyseen, for example, in diffuse x-ray scatteringwith the broad incommensurate peak (31, 32)and Raman phonon spectral anomaly (33).Figure 5 displays temperature variation of theCMR-related behavior for the Sm12xSrx-MnO3 ( x 5 0.45) crystal on the verge of theaforementioned multicritical point betweentypes I and II (34, 35). When the ferromag-netic double-exchange interaction competeswith the charge/orbital ordering and extin-guishes the long-range ordering, as in thepresent case, the superlattice x-ray peak, say(2, 1/2, 0) in the orthorhombic setting, arisingfrom the CE-type orbital ordering (Fig. 3A),turns into incommensurate and diffuse scat-tering as exemplified in Fig. 5A. The diffusescattering intensity measured at (2, 1.7, 0)increases with lowering temperature down toTC and then suddenly drops below TC (Fig.5B). The temperature dependence compareswell with that of resistivity change throughTC (Fig. 5C). All of the results, including theRaman phonon spectral anomaly, the AF spincorrelation, and the in-plane expansion andc-axis compression of the lattice parametersabove TC (35), indicate that the orbital showsthe dynamical and short-range (but direction-ally long-range) ordering above TC that col-lapses immediately below TC or in a magnet-ic field.

The orbital fluctuation in the doped man-ganese oxides may also be a cause for thehighly incoherent charge dynamics, even inthe ferromagnetic metallic state with full spinpolarization (no spin fluctuation). Accordingto the recent studies (36–39), the ferromag-netic ground state shows the very small spec-tral weight of the quasi-particle peak at theFermi level in the photoemission spectrum aswell as the minimal Drude weight in theoptical conductivity spectrum. The Drudeweight is, in fact, an order of magnitude smallerthan expected from the results of the electronicspecific heat coefficient (39, 40), indicating theleast mass renormalization effect, and of thesmall Hall coefficient (39), indicating the me-tallic density of charge carriers.

Given the almost perfectly aligned spins,the only remaining degrees of freedom arethe charge and orbital. Considering that theorbital polarization is large in the orbital-ordered state, it is reasonable that it remainsso even in the ferromagnetic metallic statewithout the ordering. Therefore, it seems tobe a promising scenario that these anomalousfeatures come from the highly nonlocal scat-tering of the charge carrier due to the orbitalcorrelation or short-range ordering such as(x2 2 y2) ordering (41–43). This situation isthe orbital analog of the heavy fermionswhere the local spin polarization is induced,but its quantum fluctuation eventually leads

to the singlet state caused by the Fermi de-generacy. Actually, the quantum mechanicsof TW shows low-dimensional dispersion sim-ilar to that discussed after Eq. 4, which en-hances the quantum fluctuation and enablesthe quantum-disordered orbital state to re-main stable down to zero temperature, i.e.,orbital liquid state.

However, a variety of scenarios have beenproposed so far for such an anomalously “badmetal” feature (44–47). The microscopicphase separation (45–47) is one of the mostimportant candidates. In any case, furtherstudies, both experimental and theoretical,are needed for this issue.

Other Materials and TheoriesIn the last section, we focused on manganeseoxides, but the orbital physics are universal intransition-metal oxides. We review heresome of the interesting features of orbitalphysics, both experimental and theoretical.

For the classic material V2O3, which hascorundum structure, two electrons occupy t2g

orbitals. In the conventional view, the orbitalordering in the Eg state of the original t2g

manifold has been assigned to the origin ofthe specific spin order in the AF ground state(48, 49). An effective Hamiltonian for spinand orbital has been derived on the basis ofthis picture, and the magnetic properties werediscussed. The important feature is that themagnetic exchange interaction depends onthe orbital occupancy as represented in Eq. 1,

i.e., even the sign could change. Therefore, itis possible that the magnetic correlation in thenormal state can be very different from that inthe ordered phase when the orbital order isaccompanied by the magnetic transition, asobserved experimentally (50). This strong in-terplay between the spin and orbital isthought to be the origin of a strong first-orderphase transition. However, a recent articleopposes this conventional picture and propos-es the spin triplet (S 5 1) formation at eachV site (51).

A more transparent example is the case ofperovskite type RTiO3 and RVO3 with 3d1

and 3d2 electron configuration, respectively,both retaining the orbital degree of freedomin the t2g state. For example, a Mott insulator,YTiO3, shows the ferromagnetic, not AF,ground state with ferromagnetic temperatureTC of ;30 K, a clear indication of someorbital order. According to the LDA calcula-tion on LaVO3 with the C-type or (p, p, 0)spin order and on YVO3 with the G-type or(p, p, p) spin order, the orbital order isconjectured to take the G type and C type,respectively, converse to the spin order (52).A temperature-induced magnetization-rever-sal phenomenon observed in YVO3 has beenattributed to the combined effect of the sin-gle-spin anisotropy, Dzyaloshinsky-Moriyainteraction, and the orbital transition (53).The actual orbital order pattern in these t2g

electron systems is not straightforwardly vis-ible from the crystal structure alone because

Mn3+

Mn4+

77 K 298 K

a

b

a

b

out

in

100 µm

T < TCO T > TCOB

A Fig. 3. (A) The orbital[(3x2 2 r2)/(3y2 2r2)] and charge orderof the CE type project-ed on the MnO2 sheet(ab plane). (B) Polar-ization microscope im-age with cross-polar-ization of incident andreflected light parallelto a and b axes for aLa0.5Sr1.5MnO4 (x 50.5) crystal. The leftbright image is for thecharge/orbital–orderedstate at 77 K. The orbit-al-disordered state at298 K shows the iso-tropic optical responseand hence gives thedark image.

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of the relatively weak JT distortion of the t2g

electron. This is in contrast to the case of aneg electron with strong JT distortion, but itprovides a more challenging problem. In thiscontext, the resonant x-ray scattering methodrecently developed by Murakami and co-workers (27) is a very powerful tool forprobing these orbital-ordering patterns.

An example of dynamical orbital corre-lation is seen in the spin-state transition inLaCoO3 with 3d6 configuration of Co.Around 100 K, the nonmagnetic and insu-lating state in LaCoO3 undergoes a gradualtransition from the low-spin state t2g

6 (S 50) to the intermediate-spin state t2g

5 eg1 (S 5

1). [The high-spin state t2g4 eg

2 (S 5 2) ispredicted to be located at a higher energylevel than the intermediate-spin state (54),as the latter gains a larger d-p hybridizationenergy in crystal lattice.] The intermediate-spin Co ion with one eg electron is a JT ion,whereas the low-spin state has no orbitaldegree of freedom. In fact, the correlatedlocal lattice distortion clearly shows up inthe infrared phonon spectra in accord withthe spin-state crossover (55), although theaverage lattice structure appears to be un-distorted from that of the ground state. Thisis another example of thermally induceddynamical JT effect due to short-range or-bital order, in addition to the case of theCMR manganese oxides. However, La-

CoO3 undergoes the insulator-to-metaltransition by warming above 500 K. Aspredicted by the LDA calculation (54 ), thisphenomenon may be interpreted as the lossof the orbital (short-range) order.

We have discussed only these limited ex-amples, but most of the bandwidth- and/orfilling (doping)–control Mott transition in thetransition-metal oxides are more or less af-fected by the orbital order-disorder transition.Therefore, the orbital correlation and the spincorrelation are expected to most often play animportant role in charge dynamics in themetallic state near the Mott transition, whichis termed “anomalous metal.”

Theoretically, it is a fascinating problemto look for exotic states realized only with theorbital degrees of freedom. Especially intenseinterests have been focused on the quantumliquid states of the spin and/or orbital. Toapproach this problem, many authors studythe model Hamiltonian (Eq. 1) with the rota-tional symmetry, i.e., SU(2) symmetry, alsoin the orbital space, where TW enters into theHamiltonian in the form of TW i z TW j as

H 5 Oij&

~ x 1 SW i z SW j!~ y 1 TW i z TW j! (3)

where ^ij& is the nearest neighbor pair. Thepoint x 5 y 5 1/4 is a special one where thesymmetry is enhanced from [SU(2)]spin 3[SU(2)]orbital to SU(4) (56). The former cor-

responds to the respective rotation in the spinand orbital space, and the latter also in-cludes the rotations between the spin andorbital space. At this SU(4)-symmetricpoint, the quantum fluctuations of both thespin and orbital are enhanced and the“SU(4) singlet” is more stable in compari-son with the usual “spin SU(2) singlet.”Therefore, the orbital degrees of freedomhelp the realization of the resonating va-lence bond spin liquid, which has beenlooked for with great interest but has notyet been found in quantum magnets withoutorbital degeneracy (e.g., La2CuO4). The ba-sic unit of the SU(4) singlet is the pla-quette, which might be realized in LiNiO2

(56 ). The 1D model of Eq. 3 has recentlybeen extensively studied (57– 62). It showsfive phases, including (i) the dimerizedstate of both SWi z SWi11 and TW i z TW i11 with thegap for the excitation spectra and (ii) thegapless state governed by the SU(4) sym-metry (62).

In real materials, there should be no rota-tional symmetry in the orbital pseudospinspace, and this anisotropy is expected to sta-bilize the ordered state because the quantumfluctuation is suppressed by the Ising-typeanisotropy. However, it is also possible thatthis anisotropy gives rise to another type ofquantum fluctuations, as will be discussedbelow. With doubly degenerate eg orbitals,

Fig. 4. The CE-type charge/orbital–ordering phase diagrams in the planeof magnetic field and temperature for various R0.5A0.5MnO3 crystals, Rand A being trivalent rare-earth and divalent alkaline-earth ions, respec-

tively. TC, TN, and TCO stand for the ferromagnetic, antiferromagnetic,and charge-ordering transition temperature, respectively.

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the dominant part of the effective Hamiltoni-an for 3D perovskite structure is (63)

H0 5 Oi

Oa 5 a,b,c

JF4~SW i z SW i1a!Stia 2

1

2D3 Sti1a

a 21

2D1 Stia 1

1

2DSti1aa 1

1

2DG (4)

where tia(b) 5 (1/2)(2 Ti

3z 6 =3Tix) and ti

c 5Ti

z. The classical ground state of H0 is found tobe highly degenerate; namely, an A-type 2D AFstate with (x2 2 y2) orbital, a G-type 3D AFstate with (3z2 2 r2) orbital, and two mixed-orbital phases are degenerate. Assuming that thefluctuation from the ordered state is small, the

dispersion of the Gaussian orbital mode is 2D,which gives rise to the logarithmic divergenceof the fluctuation. Therefore, it is possible thatthe ground state becomes quantum disorderedwhen the quantum fluctuation is treated beyondthe Gaussian approximation.

Another possibility is that the directionalordering of the orbital resolves the magneticfrustration by specifying the strong and weakbonds. In a triangular lattice (e.g., LiVO2), ithas been proposed that a pattern of orbitaloccupancy will lead to the isolated triangleswhose spin ground state is the singlet (wheneach spin is S 5 1) (64).

ConclusionWe have described some experimental and the-oretical aspects of orbital physics in transition-metal oxides. The orbital degree of freedom isthought to play important roles, not only in theprominent case of the CMR manganese oxidesbut also in anisotropic electronic and magneticproperties of many transition-metal oxides. Tak-ing the analogy to conventional materials phas-es, we may consider various states of orbital—not only solid (crystal) and liquid but even liquidcrystal and glass. In fact, the short-range orbitalorder as seen in the CMR state (Fig. 5) may beviewed as the nematic-like liquid-crystal state ofthe orbital. Likewise, the orbital glass state maybe realized in some frustrated or disorderedlattice of transition-metal oxides, although thestate has seldom been argued or unraveled ex-perimentally. In this context, another attempt isthe control of the orbital state in terms of exter-nal fields, not only with magnetic and stressfields as discussed above, but also throughan electric field and light or x-rays. Theelectric field may directly affect the direc-tional order of orbital, when the compoundis insulating enough, and may alter the mag-netic state. This is the orbital version of theliquid-crystal functionality. Additional vis-ible effects by irradiation of light or x-rayshave already been observed for the charge/orbital– ordered manganese oxide (Pr12xCax

MnO3) as the light-induced melting of thecharge/orbital order and resultant metalliza-tion (65, 66 ). Such an exotic and possiblyultrafast control of electronic and magneticphases may find some applications in thefuture. Theoretically, the spin-charge-orbit-al coupled systems in transition-metal ox-ides offer the most fascinating and challeng-ing arena to test many theoretical ideas,including quantum liquid, solid, and liquid-crystal states.

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X-r

ay D

iffus

e S

catte

ring

Inte

nsity

(a.

u.)

B

A

C

TC

0 100 200 300Temperature (K)

Res

istiv

ity (

ohm

•cm

) 0 T

7 Tcooling

warming

1.5

1.5 2 2.5

2

2.5

h

k

(1.7,2,0)

130 K

(2,2,0)

Sm1-x Srx MnO3 (x = 0.45)

0

10-4

10-3

10-2

10-1

100

Fig. 5. (A) The diffused scattering feature aroundthe (2, 2, 0) (in the orthorhombic setting) Braggpeak in the reciprocal space at 130 K, where thecompound shows a typical CMR behavior. Tem-perature dependence of (B) x-ray diffuse scatter-ing intensity (a.u., arbitrary units), reflecting theshort-range charge-orbital correlations (JT pol-aron correlation) and (C) magnetotransport prop-erty for a Sm12xSrx MnOI3 (x 5 0.45) crystal onthe verge of the charge/orbital–ordered state.

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Kawasaki, G. Khaliullin, D. Khomskii, S. Maekawa, R.Maezono, Y. Murakami, Y. Okimoto, T. Okuda, E. Saitoh,K. Terakura, H. Yoshizawa, J. Zaanen, and F. C. Zhang forvaluable discussions and collaborations. This work wassupported by the Center of Excellence and Priority AreasGrants from the Ministry of Education, Science, andCulture of Japan and by the New Energy and IndustrialTechnology Development Organization.

R E V I E W

Advances in the Physics ofHigh-Temperature Superconductivity

J. Orenstein1 and A. J. Millis2

The high-temperature copper oxide superconductors are of fundamentaland enduring interest. They not only manifest superconducting transitiontemperatures inconceivable 15 years ago, but also exhibit many otherproperties apparently incompatible with conventional metal physics. Thematerials expand our notions of what is possible, and compel us to developnew experimental techniques and theoretical concepts. This article pro-vides a perspective on recent developments and their implications for ourunderstanding of interacting electrons in metals.

In a paper published in Science very shortlyafter the 1986 discovery of high–critical tem-perature (Tc) superconductivity by Bednorzand Muller, Anderson identified three essen-tial features of the new superconductors (1).First, the materials are quasi–two-dimension-al (2D); the key structural unit is the CuO2

plane (Fig. 1), and the interplane coupling isvery weak. Second, high-Tc superconductiv-ity is created by doping (adding charge car-riers to) a “Mott” insulator. Third, and mostcrucially, Anderson proposed that the combi-nation of proximity to a Mott insulating phaseand low dimensionality would cause thedoped material to exhibit fundamentally newbehavior, not explicable in terms of conven-tional metal physics.

In the ensuing years this prediction of newphysics was confirmed, often in surprisingways. The challenge has become to charac-terize the new phenomena and to develop theconcepts required to understand them. Thepast 5 years have been particularly exciting.Advances in crystal chemistry and in exper-imental techniques have created a wealth ofinformation with remarkable implications forhigh-Tc and related materials. Here we focuson four areas where progress has been espe-cially rapid: spin and charge inhomogeneities(“stripes”); the low-temperature properties ofthe superconducting state; phase coherenceand the origin of the pseudogap; and the

Fermi surface and its anisotropies in the non-superconducting or normal state.

Mott Insulators, Superconductivity,and StripesHigh-Tc superconductivity is found in copperoxide–based compounds with a variety ofcrystal structures, an example of which isshown in Fig. 1A. The key element shared byall such structures is the CuO2 plane, depict-ed with an occupancy of one electron per unitcell in Fig. 1B. At this electron concentrationthe plane is a “Mott insulator,” the parentstate from which high-Tc superconductors arederived. A Mott insulator is a material inwhich the conductivity vanishes as tempera-ture tends to zero, even though band theorywould predict it to be metallic. Many exam-ples are known, including NiO, LaTiO3, andV2O3. [For recent reviews, see (2, 3).] How-ever, the high-Tc cuprates are the only Mottinsulators known to become superconductingwhen the electron concentration is changedfrom one per cell.

A Mott insulator is fundamentally differ-ent from a conventional (band) insulator. Inthe latter system, conductivity is blocked bythe Pauli exclusion principle. When the high-est occupied band contains two electrons perunit cell, electrons cannot move because allorbitals are filled. In a Mott insulator, chargeconduction is blocked instead by electron-electron repulsion. When the highest occu-pied band contains one electron per unit cell,electron motion requires creation of a doublyoccupied site. If the electron-electron repul-sion is strong enough, this motion is blocked.The amount of charge per cell becomes fixed,

leaving only the electron spin on each site tofluctuate. Doping restores electrical conduc-tivity by creating sites to which electrons canjump without incurring a cost in Coulombrepulsion energy.

Virtual charge fluctuations in a Mott in-sulator generate a “super-exchange” (4) in-teraction, which favors antiparallel alignmentof neighboring spins. In many materials, thisleads to long-range antiferromagnetic order,as shown in Fig. 1. Anderson proposed thatthe quantum fluctuations of a 2D spin 1⁄2system like the parent compound La2CuO4

might be sufficient to destroy long-range spinorder. The resulting “spin liquid” would con-tain electron pairs whose spins are locked inan antiparallel or “singlet” configuration.The motion of such singlet pairs is akinto the resonance of p bonds in benzene,thus the term “resonating valence bond”(RVB). Anderson pointed out that the valencebonds resemble the Cooper pairs of Bardeen-Cooper-Schrieffer (BCS) superconductivity.A compelling picture of a Mott insulator as asuppressed version of the BCS state emerged:electrons dressed up in pairs, but with noplace to go. Because the Mott insulator isnaturally paired, Anderson argued, it wouldbecome superconducting if the average occu-pancy is lowered from one.

Soon after the discovery of high-Tc super-conductivity, experiments revealed that thespin liquid state is not realized in the undopedcuprates. [It now seems likely that a spinliquid ground state exists for spin 1⁄2 particleson geometrically frustrated 2D lattices suchas the Kagome (5).] Instead, the spins orderin a commensurate antiferromagnetic patternat a rather high Neel temperature between250 and 400 K, depending on the material.The extent of the antiferromagnetic phase inthe temperature versus carrier concentrationplane of the high-Tc phase diagram is illus-trated in Fig. 2. The Neel temperature dropsvery rapidly as the average occupancy isreduced from 1 to 1 2 x, reaching zero ata critical doping xc of only 0.02 in the

1Department of Physics, University of California,Berkeley, CA 94720, USA, and Materials Science Di-vision, Lawrence Berkeley National Laboratory, Berke-ley, CA 94720, USA. 2Department of Physics andAstronomy, Rutgers University, Piscataway, NJ08854, USA.

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