Modern solid-state physics explains the physical prop-erties of numerous materials, such as simple metals

and some semiconductors and insulators. But materialswith open d and f electron shells, where electrons occupynarrow orbitals, have properties that are harder to ex-plain. In transition metals, such as vanadium, iron, andtheir oxides, for example, electrons experience strongCoulombic repulsion because of their spatial confinementin those orbitals. Such strongly interacting or “correlated”electrons cannot be described as embedded in a staticmean field generated by the other electrons.1 The influenceof an electron on the others is simply too pronounced foreach to be treated independently.

The effect of correlations on materials properties isoften profound. The interplay of the d and f electrons’ in-ternal degrees of freedom—spin, charge, and orbital mo-ment—can exhibit a whole zoo of exotic ordering phenom-ena at low temperatures. That interplay makes stronglycorrelated electron systems extremely sensitive to smallchanges in external parameters, such as temperature,pressure, or doping.

The dramatic effects can range from huge changes in theresistivity across the metal–insulator transition in vanadiumoxide and considerable volume changes across phase transi-tions in actinides and lanthanides, to exceptionally high tran-sition temperatures (above liquid-nitrogen temperatures) insuperconductors with copper–oxygen planes. In materialscalled heavy fermion systems, mobile electrons at low tem-perature behave as if their masses were a thousand times themass of a free electron in a simple metal. Some strongly cor-related materials display a very large thermoelectric re-sponse; others, a great sensitivity to changes in an appliedmagnetic field—an effect dubbed colossal magnetoresistance.Such properties make the prospects for developing applica-tions from correlated-electron materials exciting. But therichness of the phenomena, and the marked sensitivity to mi-croscopic details, makes their experimental and analyticalstudy all the more difficult.

The theoretical challengeTo understand materials made up ofweakly correlated electrons—siliconor aluminum, for example—band the-ory, which imagines electrons behav-ing like extended plane waves, is agood starting point. That theory helpscapture the delocalized nature of elec-trons in metals. Fermi liquid theorydescribes the transport of conduction

electrons in momentum space and provides a simple butrigorous conceptual picture of the spectrum of excitationsin a solid. In that description, excited states consist of in-dependent quasiparticles that exist in a one-to-one corre-spondence to states in a reference system of noninteract-ing Fermi particles plus some additional collective modes.

To calculate the various microscopic properties of suchsolids, we have accurate quantitative techniques at ourdisposal. Density functional theory (DFT),2 for example, al-lows us to compute the total energy of some materials withremarkable accuracy, starting merely from the atomic po-sitions and charges of the atoms. See box 1 for a more de-tailed prescription of DFT.

However, the independent-electron model and theDFT method are not accurate enough when applied tostrongly correlated materials. The failure of band theorywas first noticed in insulators such as nickel oxide andmanganese oxide, which have relatively low magnetic-or-dering temperatures but large insulating gaps. Band the-ory incorrectly predicts them to be metallic when magneticlong-range order is absent.

Neville Mott showed that those insulators are betterunderstood from a simple, real-space picture of the solidas a collection of localized electrons bound to atoms withopen shells. Adding and removing electrons from an atomleaves it in an excited configuration. Because the internaldegrees of freedom (like orbital angular momentum andspin) in the remaining atoms scatter the excited configu-rations, these states propagate through a crystal incoher-ently and broaden to form bands, called the lower and theupper Hubbard bands.

The modeling problem becomes more complicated asone moves away from these two well-understood, extremelimits and works with materials made up of electrons thatare neither fully itinerant (propagating as Bloch waves inthe crystal) nor fully localized on their atomic sites. Thedual particle–wave character of the electron forces theadoption of components of the real-space and momentum-space pictures.

Systems with strongly correlated electrons fall withinthat middle ground. Traditionally, such materials havebeen described using the model Hamiltonian approach.That is, the Hamiltonian is simplified to take into accountonly a few relevant degrees of freedom—typically, the va-lence electron orbitals near the Fermi level. Reducing the

© 2004 American Institute of Physics, S-0031-9228-0403-030-3 March 2004 Physics Today 53

Gabriel Kotliar is a professor of physics at the Center for Materi-als Theory at Rutgers University in Piscataway, New Jersey.Dieter Vollhardt is a professor at the Center for Electronic Cor-relations and Magnetism at the Institute for Physics, University ofAugsburg in Germany.

Materials with correlated electrons exhibit some of themost intriguing phenomena in condensed matter physics.A new theoretical framework is now allowing theorists tocalculate the electronic structure of these materials, whichcan exist in a rich variety of phases.

Gabriel Kotliar and Dieter Vollhardt

Strongly Correlated Materials: Insights From Dynamical Mean-Field Theory

full many-body Hamiltonian to a simpler, effective modelretains the essence of the physical phenomena one wantsto understand, but is itself a complicated problem.

One of the simplest models of correlated electrons isthe Hubbard Hamiltonian, defined in equation 2b of box 2.This Hamiltonian describes electrons with spin directionss ⊂ R or A moving between localized states at lattice sitesi and j. The electrons interact only when they meet on thesame lattice site i. (The Pauli principle requires them tohave opposite spin.) The kinetic energy and the interac-tion energy are characterized by the hopping term tij andthe local Coulomb repulsion U, respectively. These two

terms compete because the kinetic part favors the elec-trons’ being as mobile as possible, while the interaction en-ergy is minimal when electrons stay apart from eachother—that is, localized on atomic different sites. Thiscompetition is at the very heart of the electronic many-body problem. The parameters that determine the proper-ties described by the Hubbard model are the ratio of theCoulomb interaction U and the bandwidth W (W is deter-mined by the hopping, tij), the temperature T, and the dop-ing or number of electrons.

The lattice structure and hopping terms influence theability of the electrons to order magnetically, especially in

54 March 2004 Physics Today http://www.physicstoday.org

+

–

+ ¬0 +R¬ +RA¬

Vn Vn

e– e–

Electron reservoir

DMFT

R A

Time

Figure 1. Dynamical mean-field theory (DMFT) of correlated-electron solids replaces the full lattice of atoms and electronswith a single impurity atom imagined to exist in a bath of electrons. The approximation captures the dynamics of electronson a central atom (in orange) as it fluctuates among different atomic configurations, shown here as snapshots in time. In thesimplest case of an s orbital occupying an atom, fluctuations could vary among +0¬, +R¬, +A¬, or +RA¬, which refer to an unoc-cupied state, a state with a single electron of spin-up, one with spin-down, and a doubly occupied state with opposite spins.In this illustration of one possible sequence involving two transitions, an atom in an empty state absorbs an electron from thesurrounding reservoir in each transition. The hybridization Vn is the quantum mechanical amplitude that specifies how likelya state flips between two different configurations.

In DFT, the basic quantity is the local electronic charge den-sity of the solid, r(r). The total energy of the full, many-body

problem of interacting quantum mechanical particles is ex-pressed as a functional of this density:

(1a)

The equation contains three parts: the kinetic energy of anoninteracting system T[r]; the potential energy of the crys-tal, Vext(r), plus the Hartree contribution to the Coulomb in-teraction between the charges; and the rest, denoted as theexchange and correlation energy term Exc. Minimizing thefunctional results in the Kohn–Sham equations

(1b)

which have the form of one-particle Schrödinger equationswith a potential VKS(r). This Kohn–Sham potential representsa static mean field of the electrons and has to be determined

from the self-consistency condition

(1c)

The Kohn–Sham equations serve as a reference system forDFT because they yield the correct ground-state density via

(1d)

where f (ei) is the Fermi function. That is, although theKohn–Sham equations describe a noninteracting single-particle system, they give the correct density of the many-body inter-acting system. Practical implementations require explicit, al-beit approximate, expressions for Exc (for example, the localdensity approximation obtained from the uniform electrongas). Although the eigenvalues and eigenvectors of theseequations cannot be identified rigorously with the excitationsof the solid, if electrons are weakly correlated the energies eiare often a very good starting point for computing the true ex-citation spectra by perturbation theory in the screenedCoulomb interaction.

Box 1. Density Functional Theory

G r r r

r r_ _

r r r r r

r rr r

r

( ) = ( ) + ( ) ( )

+ ( ) ′( )− ′

∫T Vext3

3 3

d

12

d d ′′ + ( ) ∫ r rExc .r

− + ( ) =∇• /2 KSm V i i ir c e c ,

V VE

KS ext3 xcd .r

r_ _

d r

drr r

rr r

rr

r( ) = ( ) +

′( )− ′

′ +( )

( )∫

r e _ c _r r( ) = ( ) ( )∑f i ii

2,

the insulating state. If the magnetic ordering is impeded—crucial for observing subtle transitions between nonmag-netic phases—the system is said to be “frustrated.”

Despite using model Hamiltonians to simplify theproblem, even simple properties such as the phase dia-gram of a strongly correlated electron system are difficultto calculate exactly. Given the rich variety of physical prop-erties in such systems, though, it is important to developtechniques that are tractable and yet remain flexibleenough to allow theorists to incorporate material-specificdetails into the calculations. Dynamical mean-field theory(DMFT) has been allowing researchers to make strides inthat direction.

Dynamical mean-field theoryA wide variety of numerical techniques and analyticalmethods have been used to treat strongly correlated elec-tron systems. In 1989, Walter Metzner and one of us (Voll-hardt) introduced a new limit to the correlated electronproblem, that of infinite lattice coordination: Each latticesite is imagined to have infinitely many neighbors.3 Thatapproach retained the competition between kinetic energyand Coulomb interaction of electrons while simplifying thecomputation. And it triggered a multitude of investiga-tions of the Hubbard and related models taken in thislimit. Consequently, a better understanding of various ap-proximation schemes emerged along with an exact solu-tion of some simpler models.4

A second advance came when Antoine Georges and oneof us (Kotliar) mapped the Hubbard model (a lattice model)onto a self-consistent quantum impurity model—a set oflocal quantum mechanical degrees of freedom that inter-acts with a bath or continuum of noninteracting excita-tions.5 (For a subsequent significant derivation, see MarkJarrell’s article in ref. 6.) That construction provides thebasis of the dynamical mean-field theory of correlated elec-trons.7 It allowed many-body theorists to formulate andsolve a variety of model Hamiltonians on the lattice usinganalytic5 and numerical techniques such as quantumMonte Carlo,6 previously developed to study impurity mod-els.7 The DMFT solutions become exact as the number ofneighbors increases.

In essence, a mean-field theory reduces (or maps) amany-body lattice problem to a single-site problem with ef-fective parameters. Consider the classical theory of mag-netism as an analogy: Spin is the relevant degree of free-dom at a single site and the medium is represented by aneffective magnetic field (the classical mean field). In thefermionic case, the degrees of freedom at a single site arethe quantum states of the atom inside a selected centralunit cell of the crystal; the rest of the crystal is describedas a reservoir of noninteracting electrons that can be emit-ted or absorbed in the atom. Figure 1 depicts that emis-sion or absorption as mediated by a quantum mechanicalamplitude Vn, and box 2 describes it mathematically. Theeffect of the environment on the site is to allow the atomto make transitions between different configurations. Un-like the classical case, in which a number—the effectivemagnetic field—describes the effect of the medium on thecentral site, the quantum case requires a hybridizationfunction D(w) to capture the ability of an electron to enteror leave an atom on a time scale 1/w.

The local description of a correlated solid in terms ofan atom embedded in a medium of noninteracting elec-trons corresponds to the celebrated Anderson impuritymodel, but now with an additional self-consistency condi-tion. The hybridization function plays the role of a meanfield and describes the ability of electrons to hop in andout of a given atomic site. When the hybridization is very

http://www.physicstoday.org March 2004 Physics Today 55

To treat strongly correlated electrons, one has to introducea frequency resolution for the electron occupancy at a

particular lattice site. A Green function that specifies theprobability amplitude required to create an electron withspin s (R or A) at a site i at time t� and destroy it at the samesite at a later time t will do the job:

(2a)

The Green function contains information about the localone-electron photoemission spectrum. The dynamical mean-field theory (DMFT) can be used to investigate the full many-body problem of interacting quantum mechanical particlesor effective treatments such as the Hubbard model1,7

(2b)

which is the simplest model of interacting electrons on a lat-tice. The equation contains a matrix element tij that de-scribes hopping of electrons with spin s between orbitals atsites i and j, and a local Coulomb interaction U between twoelectrons occupying the same site i; nis ⊂ cis

† cis is the densityof electrons at site i with spin s.

The Anderson impurity model

(2c)

(in which h.c. is the Hermitian conjugate) serves as a refer-ence system for the Hubbard model because it yields theexact local Green function in DMFT when the Vn fulfills aself-consistency condition. It provides the mathematical de-scription of the physics in figure 1: Starting from a generalHamiltonian, one separates a lattice site’s atomic degrees offreedom, described by Hatom, from the remaining degrees offreedom, treated as a bath of electrons with energy levels en

bath. Electrons may hop in and out of that site via the hy-bridization Vn between the atomic (c0,s) and the bath elec-trons (an,s

bath). The parameters enbath and Vn appear in a simple

combination in the hybridization function

(2d)

which here plays the role of a mean field.5 Its frequency de-pendence makes it a dynamic mean field. Because the bathdescribes the same electrons as those on the local site, D(w)has to be determined from the self-consistency condition

(2e)

where the self-energy term S[D(w)] [ D(w) ⊗ 1/G[D(w)] ⊕ wtakes on the meaning of a frequency-dependent potential,and tk is the Fourier transform of the hopping matrix ele-ments ti j of the solid.

By analogy with density functional theory, an exact func-tional of both the charge density and the local Green func-tion of the correlated orbital is introduced as

(2f)

The functional9 has a similar decomposition as in DFT.However, the kinetic energy is no longer that of a free elec-tron system because T[r, G] is the kinetic energy of a systemwith given density r(r) and local Green function G. DMFTprovides an explicit approximation for Exc[r(r),G].

Box 2. Dynamical Mean-Field Theory

G c ci i is s st t t t- ′( ) ≡ − ( ) ′( )† .

H t c c U n nij i jij,

i ii

= +∑ ∑ ↑ ↓s ss

† ,

H H n V c a h.c.AIM atombath

,bath

,0, ,

bath

,

+= + + ( )∑ ∑en n sn s

n s n sn s

†

D w_ _

w en

nn

( ) =−∑ V 2

bath ,

G tD w w S D w( ) = − ( ) −{ }−

∑ kk

1,

G r r r

r r_ _

r r r r r

r rr r

( ) = ( ) + ( ) ( )

+ ( ) ′( )− ′

∫, , d

12

d

ext3G T G V

33 3xcd , .r r r′ + ( ) ∫ E Gr

small, the electron is almost entirely localized at a latticesite and moves only virtually, at short durations compati-ble with the Heisenberg uncertainty principle. On theother hand, when it is large, the electron can movethroughout the crystal.

We thus obtain a simple local picture for the competi-tion between itinerant and localized tendencies underly-ing the rich phenomena that correlated materials exhibit.The mapping of the lattice model onto an impurity model—the basis of the dynamical mean-field theory—simplifiesthe spatial dependence of the correlations among electronsand yet accounts fully for their dynamics—that is, the localquantum fluctuations missed in static mean-field treat-ments like the Hartree–Fock approximation.

Besides its conceptual value of providing a quantumanalog of the classical mean field, the mapping of a latticemodel onto the Anderson impurity model has had greatpractical impact. Applications of DMFT have led to a lot ofprogress in solving many of the problems inherent tostrongly correlated electron systems, such as the Mottmetal–insulator transition, doping of the Mott insulator,phase separation, and the competition of spin, charge, andorbital order.

The potential for system-specific modeling of materi-als using DMFT was recognized early on.7 But actual im-plementation required combining ideas from band theoryand many-body theory.8 Vladimir Anisimov’s group in Eka-terinburg, Russia, and one of us (Kotliar) first demon-strated the feasibility of this approach using a simplifiedmodel of a doped three-dimensional Mott insulator, La1⊗xSrxTiO3. The electrons in that material are dividedinto two sets: weakly correlated electrons, well describedby a local-density approximation (LDA) that models the ki-netic energy of electron hopping, and strongly correlated(or more localized) electrons—the titanium d orbitals—well described using DMFT. The one-body part of theHamiltonian is derived from a so-called Kohn–ShamHamiltonian. The on-site Coulomb interaction U is thenadded onto the heavy d and f orbitals to obtain a modelHamiltonian. DMFT (or more precisely, “LDA + DMFT”) isthen used to solve that Hamiltonian.

Just as the Kohn–Sham equations serve as a referencesystem from which one can compute the exact density of a

solid, the Anderson impurity model serves as a referencesystem from which one can extract the density of states ofthe strongly correlated electrons. The parallel betweenDFT and DMFT is best seen in the functional approach toDMFT,9 outlined in box 2. It constructs the free energy ofthe system as a function of the total density and the localGreen function. Finding the extrema of the functionalleads to a self-consistent determination of the total energyand the spectra. These and related efforts to combinemany-body theory with band-structure methods are an ac-tive area of research.10,11

The metal–insulator transitionHow does an electron change from itinerant to localized be-havior within a solid when a control parameter such aspressure is varied? The question gets at the heart of theMott transition problem, which occurs in materials as var-ied as vanadium oxide, nickel selenium sulfide, and layeredorganic compounds.1,11 Those materials share similar high-temperature phase diagrams, despite having completelydifferent crystal structures, orbital degeneracies, and bandstructures: In each, a first-order phase transition separatesa high-conductivity phase from a high-resistivity phase. Atlow temperatures, in contrast, the materials exhibit verydifferent ordered phases: Organics are superconducting,vanadium oxide forms a metallic and an antiferromagneticinsulating phase, and nickel selenium sulfide forms a broadregion of itinerant antiferromagnetism.

The Mott transition is central to the problem of mod-eling strongly correlated electrons because it addresses di-rectly the competition between kinetic energy and corre-lation energy—that is, the wavelike and particlelikecharacter of electrons in the solid. Indeed, systems nearthe Mott transition display anomalous properties such asmetallic conductivities smaller than the minimum pre-dicted within a band picture, unconventional optical con-

56 March 2004 Physics Today http://www.physicstoday.org

1

0.5

0

1

0.5

0

DE

NS

ITY

OF

ST

AT

ES

a

b

c

1

0.5

0

U = 0

U W/ = 0.5

U W/ = 1.2

EF

ENERGY+EF

U2

W

⊗EFU2

1

0.5

0

U W/ = 2U

d

Figure 2. The density of states (DOS) of electrons in a material varies as a function of the local Coulomb interac-tions between them. This series illustrates how the spectral

features—as measured by photoemission or tunneling experiments, say—evolve in the dynamical mean-field

solution of the Hubbard model at zero temperature andhalf filling (the same number of electrons as lattice sites).

U is the interaction energy and W is the bandwidth ofnoninteracting electrons. (a) For the case in which elec-

trons are entirely independent, the DOS is assumed tohave the form of a half ellipse with the Fermi level EF, lo-cated in the middle of the band, characteristic of a metal.

(b) In the weakly correlated regime (small U), electronscan be described as quasiparticles whose DOS still resem-

bles free electrons. The Fermi liquid model accounts forthe narrowing of the peak. (c) In strongly correlated met-

als, the spectrum exhibits a characteristic three-peak struc-ture: the Hubbard bands, which originate from local

“atomic” excitations and are broadened by the hopping ofelectrons away from the atom, and the quasiparticle peaknear the Fermi level. (d) The Mott metal–insulator transi-

tion occurs when the electron interactions are sufficientlystrong to cause the quasiparticle peak to vanish as the

spectral weight of that low-frequency peak is transferred tothe high-frequency Hubbard bands.

ductivities, and spectral functions outside of what bandtheory can describe.1 A crucial question is how the localdensity of states—or more generally, the spectral func-tion—varies as the ratio of the correlation strength U tobandwidth W increases (see figure 2). The extremes areclear enough. When the electron is delocalized, its spectralfunction closely resembles the local density of states ofband theory; when the electron is localized, the density ofstates peaks at the ionization energy and the electronaffinity of the atom. Those atomic-like states compose theHubbard bands, illustrated in figure 2d.

But how does the density of states evolve betweenthese well-characterized limits? In DMFT, the spectralfunction of the electron contains both quasiparticle fea-tures and Hubbard bands in the intermediate correlationregion.5 A three-peak structure appears naturally withinthe Anderson impurity model, but was unexpected for alattice model that describes the Mott transition. In fact,part of what makes DMFT an advance over previous tech-niques is that it allows researchers to obtain and equallytreat the Hubbard bands and the quasiparticle peakwithin this three-peak structure. (DFT treats the quasi-particles alone, atomic theory treats the Hubbard bandalone, but DMFT treats both.) Within DMFT, the Motttransition appears as the result of the transfer of spectralweight from the quasiparticle peak to the Hubbard bandsof the correlated metallic state.12 The transfer of spectralweight can be driven by pressure, doping, or temperature,and is responsible for the anomalous behavior near theMott transition.

To appreciate the DMFT description of the Mott tran-sition, consider the qualitative features of the phase dia-gram shown in figure 3. The figure essentially summarizesthe features of a partially frustrated Hubbard model in dif-ferent temperature regimes. The disappearance of metalliccoherence and the closing of the high-energy Mott–Hubbardgap are distinct phenomena that occur in different regionsof the phase diagram—at low temperatures in the form oflines at energies UC1 and UC2 that frame a hysteretic re-gion, and at high temperature (above a second-order crit-

ical point) in the form of two distinct crossover lines thatgradually separate metallic from insulating phases.7

The Coulomb interactions and the matrix elementsthat describe electron hopping from site to site are thebasic ingredients necessary to calculate the experimentalphase diagram of materials near a Mott transition. Thequestion is, Can the phase diagram be obtained from anelectronic model alone, or is the coupling of electrons to thelattice required to obtain a first-order phase transition be-tween a paramagnetic insulator and a paramagneticmetal? According to DMFT studies, an electronic modelalone will do the job. Lattice changes across the phasetransition are therefore the consequence, rather than theorigin, of the discontinuous metal–insulator transition.

Now widely accepted, that conclusion was not reachedwithout controversy. An analytic approach to the Motttransition was first required,13 which deepened the anal-ogy between the local spectral function and a statistical-mechanics order parameter. The electronic properties ofthe system—for example, the DC conductivity—behavelike the order parameter of a liquid–gas transition or likethe magnetization of the Ising spin model in statistical mechanics.

Experimental confirmationNumerous efforts have been made to square the surprisingaspects of the theory of the Mott transition with experimen-tal observations. For instance, a collaboration of groups fromBell Labs and Rutgers University observed that pure vana-dium oxide (V2O3) is located in the vicinity of the crossoverline that continues the UC2 line (where the metallic phasedisappears in figure 3) above the endpoint of the first-ordertransition line.14 Using DMFT, this collaboration predictedthat, with increasing temperature, resistivity would stronglyincrease and the weight of the quasiparticle peak wouldstrongly decrease. And, indeed, optical conductivity meas-urements bore that prediction out. A group led by GordonThomas detected a transfer of optical intensity from low tohigh frequency with increasing temperature, a finding con-sistent with a temperature-dependent quasiparticle peak.14

Only photoemission can measure the density of statesof the correlated electrons directly. Z. X. Shen’s group atStanford University observed a quasiparticle peak, welldefined and separated from the Hubbard bands in nickelselenium sulfide and depending strongly on temperatureand proximity to the Mott transition. More recently, an in-ternational collaboration of experimentalists and theorists

http://www.physicstoday.org March 2004 Physics Today 57

Figure 3. Schematic phase diagram of a materialundergoing a Mott metal–insulator transition.

Temperature and the strength of Coulombic repul-sion U are plotted for a correlated electron material

that is described using a Hubbard model. At lowtemperature, the system has long-range (for exam-

ple, magnetic or orbital) order (red) according to solutions of the dynamical mean-field theory; the

type of ordering is material- and model-dependent.In the orange region, the model has two distinct

paramagnetic solutions bounded by the lines UC1(where the insulator disappears) and UC2 (where the

metal disappears). In equilibrium, a dotted first-order phase transition boundary indicates the posi-tion at which the free energy of these two solutionscrosses. The first-order line terminates at a second-

order critical point. At higher temperatures, thephase diagram is more universal. Systems as diverse

as vanadium oxide alloys, nickel selenium sulfide,and organic materials exhibit the same qualitative

behavior. Two crossover regimes indicate thechange in materials properties as the electron inter-

actions increase from left to right. The first illustratesthe crossover from a Fermi liquid to a bad metal

(blue), in which the resistivity is anomalously large.In the second crossover, materials take on the

properties of a bad insulator (green), in which theresistivity decreases as temperature increases.

kT

WB

0.05

0.025

0U W/

2

Fermi liquid

Critical point

Paramagneticinsulator

UC1 UC2Orderedphase

1

Hysteresis

Bad m

etal Badin

sula

tor

observed well-defined and separated peaks in the metallicphase of vanadium oxide after first overcoming problemsrelated to surface sensitivity that prevented clear observa-tion in earlier studies (see figure 4).15

In the Sr1⊗xCaxVO3 system, researchers observed a sim-ilar quasiparticle peak well separated from the Hubbardband. The quasiparticle signal decreased in intensity withincreasing calcium doping, x. Attempts to interpret thespectra within DMFT as enhanced electron correlationwith increasing x, however, did not square with other ex-perimental results; the specific heat, for instance, wasmeasured as essentially independent of x. A group led byD. D. Sarma and Isao Inoue and one by Shigemasa Sugahelped resolve the puzzle by performing bulk-sensitive photo-emission experiments.16 Earlier photoemission experi-ments had given the false impression that the electronicstructure was evolving toward a more correlated system asdoping increased, when, in fact, only the surface stateswere doing so. On the theoretical side, first-principles cal-culations demonstrated that the strength of the correla-tions do not significantly depend on the concentration x.

This kind of successful blending of DMFT and exper-iment is also occurring in studies of electronic transport.In materials such as vanadium oxide, nickel selenium sul-fide, and k-organics, researchers have been able to maptheoretically the various regimes of the phase diagrams insome detail.11,1 After many unsuccessful attempts, even thecritical behavior predicted by the theory has been observedin transport measurements around the critical endpoint ofvanadium oxide and organic materials.17

Other materialsThe application of DMFT concepts and techniques to abroad range of materials and a wide variety of strong-

correlation problems is now an activeresearch frontier. The materials andtopics under study include mangan-ites, ruthenates, vanadates, actinides,lanthanides, fullerenes, quantum crit-icality in heavy fermion systems, mag-netic semiconductors, and Bechgaardsalts (quasi-1D organic materials). Adeeper understanding of the metal–insulator transition is useful not justfor predicting the properties of that listof materials, but provides insights intodiverse electronic structure problems.Plutonium and cerium, for instance,exhibit itinerant magnetism and suf-fer a large change in the crystal vol-ume at the Mott transition. The fol-lowing brief case studies outline therelevance of DMFT to these elements.� Plutonium, in its stable d phase, isnot well described within LDA. Thatapproximation underestimates thephase volume by nearly 30% and pre-dicts a nonexistent unstable phase.Using DMFT, researchers realized

that the plutonium f electrons stabilize the material neara Mott transition, and they were able to interpret calcu-lated photoemission spectra in terms of the quasiparticlepeak and Hubbard bands described previously. More re-cently, a collaboration among Rutgers University, NewJersey Institute of Technology, and Los Alamos NationalLaboratory predicted the theoretical spectrum of vibra-tions of d-Pu using DMFT. Figure 5 illustrates the re-markable agreement between the theoretical vibrationalspectrum and recent experimental measurements.18

� Cerium displays an isostructural transition10 and alsoexhibits a large volume change (the so-called volume col-lapse) between two of its phases, a and g. Jim Allen andRichard Martin, then at Xerox Corp, interpreted the ther-modynamics and spectral changes across the a–g transi-tion in terms of the Anderson impurity model more than20 years ago. Recently, Andy McMahan, Karsten Held, andRichard Scalettar calculated the spectra, total energies,and thermodynamics of cerium using DMFT10 and foundclose accord with the experimental spectra. The calcula-tion indicated a dramatic reduction of the 4 f quasiparticleweight at the transition, responsible for the energychanges and the volume collapse at the transition.� Iron and nickel exhibit metallic ferromagnetism atlow temperatures at which band theory applies. And acombination of DFT and LDA bears that out. However, athigh temperatures, those elements behave more like a col-lection of atoms, described by a Curie susceptibility and alocal magnetic moment reduced below the atomic value.DMFT provides a natural framework to describe both thehigh- and low-temperature regimes and the crossover be-tween them. Indeed, in the case of nickel, Alexander Licht-enstein and collaborators found that the theoretical photo-emission spectrum contains an experimentally well known

58 March 2004 Physics Today http://www.physicstoday.org

3 2.5 2 1.5 1 0.5 0ENERGY (eV)

INT

EN

SIT

Y (

arbi

trar

y u

nit

s)

DMFT, T = 300 K= 700 eV= 60 eV

hh

nn

EF

Figure 4. Photoemission spectrum of metallic vanadium oxide (V2O3) near themetal–insulator transition. The dynamical mean-field theory calculation (solidcurve) mimics the qualitative features of the experimental spectra. The theoryresolves the sharp quasiparticle band adjacent to the Fermi level and the occu-pied Hubbard band, which accounts for the effect of localized d electrons in the lattice. Higher-energy photons (used to create the blue spectrum) are less surface sensitive and can better resolve the quasiparticle peak. (Adaptedfrom Mo et al., ref. 15.)

satellite at energies well below the Fermi level. That fea-ture, like the Hubbard band, is not reproduced in band the-ory. DMFT also predicts the correct magnetic propertiesabove and below the Curie temperature.

Future directionsJudging by how reliably the theory reproduces complicatedphysical properties in a variety of materials, dynamicalmean-field methods clearly represent a new advance inmany-body physics. The strongly correlated electronregime of transition-metal oxides, for instance—in whichelectrons are neither fully itinerant nor localized—hassimply not been accessible to other techniques.

Generalizations of the theory to an even wider varietyof materials systems is an active area of research. By ex-tending DMFT from single sites to clusters, for example,researchers can capture the effects of short-range correla-tions. And combining DMFT with advanced electronic-structure methods should make it possible to evaluate theHamiltonian and the frequency-dependent screenedCoulomb interaction from first principles without havingto first construct a local density functional.11 Similarly,combining realistic DMFT total energies (calculated as afunction of atomic positions) with molecular dynamics totreat the motion of ions and electrons simultaneously isanother great challenge.

As the number of DMFT implementations and stud-ies of new materials increases, more detailed comparisonswith experiments are becoming possible. Such compar-isons should help separate the effects that can be under-stood simply from local physics (captured by DMFT) fromthe effects that require long-wavelength modes not cap-tured by the DMFT approach.

The DMFT equations have also been recast in a formuseful for modeling strongly inhomogeneous materials,such as disordered alloys or interfaces—two new areas inwhich to explore strongly correlated phenomena. They arealso areas of great technological importance. As growingcomputer power, novel algorithms, and new concepts bol-

ster the ability of theorists to better model complicatedsystems, one can foresee a time when investigating corre-lated electron phenomena at nanoscales and in biologicalmolecules is possible.

We are grateful to our research groups, collaborators, andcolleagues for fruitful interactions; to NSF’s division of ma-terials theory, the US Department of Energy, the Office ofNaval Research, and the German Research Foundation(DFG) for financial support; and to the Aspen Center forPhysics for hospitality.

References1. For a recent review of the electronic correlation problem, see

M. Imada, A. Fujimori, Y. Tokura, Rev. Mod. Phys. 70, 1039(1998); for an early review of fermionic correlations, see D.Vollhardt, Rev. Mod. Phys. 56, 99 (1984).

2. R. O. Jones, O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989).3. W. Metzner, D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989).4. E. Müller-Hartmann, Z. Phys. B: Condens. Matter 74, 507

(1989); U. Brandt, C. Mielsch, Z. Phys. B: Condens. Matter75, 365 (1989); V. Janiš, Z. Phys. B: Condens. Matter 83, 227(1991).

5. A. Georges, G. Kotliar, Phys. Rev. B 45, 6479 (1992).6. M. Jarrell, Phys. Rev. Lett. 69, 168 (1992); M. Rozenberg,

X. Y. Zhang, G. Kotliar, Phys. Rev. Lett. 69, 1236 (1992); A.Georges, W. Krauth, Phys. Rev. Lett. 69, 1240 (1992).

7. For a general review of dynamical mean-field theory, see A.Georges, G. Kotliar, W. Krauth, M. Rozenberg, Rev. Mod.Phys. 68, 13 (1996); for a review of applications to the Hub-bard model away from half filling, see Th. Pruschke, M. Jar-rell, J. K. Freericks, Adv. Phys. 44, 187 (1995).

8. V. I. Anisimov et al., J. Phys.: Condens. Matter 9, 7359 (1997);A. I. Lichtenstein, M. I. Katsnelson, Phys. Rev. B 57, 6884(1998).

9. R. Chitra, G. Kotliar, Phys. Rev. B 62, 12715 (2000); G.Kotliar, S. Savrasov, in New Theoretical Approaches toStrongly Correlated Systems, A. M. Tsvelik, ed., Kluwer Aca-demic, New York (2001), available at http://arXiv.org/abs/cond-mat/0208241.

10. K. Held et al., Psi-k Newsletter, April 2003, p. 65, available athttp://psi-k.dl.ac.uk/newsletters/News_56/Highlight_56.pdf;A. I. Lichtenstein, M. I. Katsnelson, G. Kotliar, Electron Cor-relations and Materials Properties, 2nd ed., A. Gonis, N.Kioussis, M. Ciftan, ed. Kluwer Academic/Plenum, New York(2003), available at http://arXiv.org/abs/cond-mat/0211076.

11. Recent developments in the field are documented on thehomepage of the Program on Realistic Theories of CorrelatedElectron Materials (2002) at the Kavli Institute for Theoreti-cal Physics, Santa Barbara, CA. See http://online.kitp.ucsb.edu/online/cem02.

12. A. Fujimori et al., Phys. Rev. Lett. 69, 1796 (1992); X. Y.Zhang, M. Rozenberg, G. Kotliar, Phys. Rev. Lett. 70, 1666(1993); A. Georges, W. Krauth, Phys. Rev. B 48, 7167 (1993).

13. G. Kotliar, Eur. J. Phys. B 11, 27 (1999); G. Kotliar, E. Lange,M. Rozenberg, Phys. Rev. Lett. 84, 5180 (2000); C. Castellaniet al., Phys. Rev. Lett. 43, 1957 (1979).

14. M. Rozenberg et al., Phys. Rev. Lett. 75, 105 (1995).15. A. Matsuura et al., Phys. Rev. B 58, 3690 (1998); S.-K. Mo et

al., Phys. Rev. Lett. 90, 186403 (2003).16. K. Maiti et al., Europhys. Lett. 55, 246 (2001); A. Sekiyama

et al., http://arXiv.org/abs/cond-mat/0312429.17. P. Limelette et al., Science 302, 89 (2003); F. Kagawa et al.,

http://arXiv.org/abs/cond-mat/0307304.18. X. Dai et al., Science 300, 953 (2003);. J. Wong et al., Science

301, 1078 (2003). �

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Figure 5. Predicted theoretical phonon spectrum of plutonium (red), calculated using dynamical mean-fieldtheory, compares well with the experimental values (solidsquares), measured by inelastic x-ray scattering. The dis-persion curves plotted here map the vibrational frequen-cies of the atoms in momentum space. The frequenciesvary with the wave vectors and are evident in thebranches as one moves in a particular Brillouin zone di-rection. (Adapted from ref. 18.)