Proc. Natl. Acad. Sci. USAVol. 92, pp. 6668-6674, July 1995Colloquium Paper

This paper was presented at a coUoquium entitled "Physics; The Opening to Complexity, " organized by Philip W.Anderson, held June 26 and 27, 1994, at the National Academy of Sciences, in Irvine, CA.

New physics of metals: Fermi surfaces without Fermi liquidsP. W. ANDERSONJoseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544

ABSTRACT I relate the historic successes, and presentdifficulties, ofthe renormalized quasiparticle theory ofmetals("AGD" or Fermi liquid theory). I then describe the best-understood example of a non-Fermi liquid, the normal me-tallic state of the cuprate superconductors.

For some 40 years, almost all electronic phenomena in metalshave been interpreted in terms of a general theoretical frame-work, which one could variously call renormalized free particletheory, Fermi liquid theory, or "AGD" after the best-knownbook on the subject (1).

I came to the conclusion a few years ago that this theory is,in many of the most interesting cases, basically a failure. Forthe first 20 years of its history, until the mid-1970s, it served usvery well; but then as we began to focus on the most interesting(or the most anomalous) cases, more and more of the copiousliterature of our subject came to be engaged in fitting theproverbial square peg into a round hole. It is not that there areno instances that fit the framework but that, contrary to theclaims for universality which have been made for it, it seemsthat for systems with strong interactions, it often is completelymisguiding.To make my point I must first describe the nature of this

conventional theory. It arose in the 1950s, just after thesuccesses of the Schwinger-Feynman-Dyson theory in quan-tum electrodynamics, and it borrows the techniques that wereso successful in that theory. In quantum electrodynamics, thescheme was to map the properties of the real physical vacuumand the real physical particle excitations onto the correspond-ing entities of a supposed bare vacuum with bare particles bythe process of renormalization. One defines a propagator orGreen's function, G(r - r', t - t'), which is the amplitude forfinding a particle at point r and time t if it was inserted at pointr' and time t' into the real vacuum. The particle can encountervarious interactions with vacuum fluctuations on the way,which are sorted out into a series with Feynman diagrams. Ifthis series is well-behaved, its sum can be written in terms ofa self-energy, which merely renormalizes the unperturbedpropagator without changing its essential character.

In the condensed matter physics of metals there is novacuum, but there is a Fermi sea if the electrons are nonin-teracting. This is treated formally as a vacuum in which bothhole and particle excitations can propagate, in parallel to thetreatment in quantum electrodynamics of the Dirac sea ofnegative-energy electrons as a vacuum for positrons. There isa surface in p space of zero energy, the Fermi surface. Theunperturbed Green's function [Fourier transformed into mo-mentum (p) and energy (t) space] is

1G(o, p) = - (e -,

The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertisement" inaccordance with 18 U.S.C. §1734 solely to indicate this fact.

where Ep is the single-particle band energy, and ,u is thechemical potential EF. Positive t refers to electron-like prop-agators, negative (backwards-moving) co refers to holes. TheFeynman diagram series can, if convergent, be resummed interms of a self-energy, which is the sum of all self-energy partsand appears in the exact Green's functions' denominator:

1 1G G- - - (e, - IL) - M(,p),

The assumption is that I is sufficiently regular that the onlysingularities of G are poles at a modifiedp-dependent energyEp - ,u of strength 0 < Zp = 1/[i - (aX)/ato)] c 1

G = _(E= ) + incoherent part.

These poles are the renormalized quasiparticles.This theory was made useful and meaningful by a series of

theorems proved in the late 1950s, which depend on the ideathat quasiparticles at EF do not decay, because the exclusionprinciple blocks off all states into which they can decay, toorder t2 = (EP - EF)2.

Migdal: If Z is finite there is a jump at PF in nk ofmagnitude Z; there is a real, measurable Fermi surface.Landau: The dynamics can be completely described atlow energies by the quasiparticles, except for a smallfinite number of collective modes near q = 0 (the Fermiliquid theory).Luttinger: The Fermi surface contains a number of pstates exactly equal to the number of electrons.

Finally (Migdal again), phonons (lattice vibrations) can beadded in simply to the theory including only the lowest-orderdiagrams (the buzzword is "neglect vertex corrections") be-cause the ion's mass is much heavier than the electron's mass.The very elegant final form of the theory, although invented

by three groups simultaneously, is expressed in the "AGD"book (1). Its greatest achievement almost coincided with itsbirth: it turned out to require only a formally trivial (ifconceptually profound) redefinition of the vacuum and thetheory as revised by Schrieffer, Nambu, and Eliashberg ele-gantly encompassed Bardeen-Cooper-Schrieffer (BCS) su-perconductivity (2). By 1965 Schrieffer, I, and later W. L.McMillan, working with the beautiful experiments of Giaeverand Rowell, had made the theory quantitative, dealing with thereal complexities of real materials so efficiently that thesuperconducting Tc of metallic elements like Pb, Hg, and Almay be the best predicted of all condensed matter phasetransitions (3). Triumphs in such fields as "Fermiology," themeasurement of complex Fermi surfaces of real metals, led usto feel that the problem of the electron liquid in metals wasfinished in principle, with only quantitative or marginal prob-lems left, some of the simpler of which were solved in the late

Abbreviation: NFL, non-Fermi liquid.

6668

Proc. Natl. Acad. Sci. USA 92 (1995) 6669

1960s and early 1970s-like magnetic impurities in metals, theso-called "Anderson model," which led to the "Kondo effect,"which turned out to be the Fermi liquid in a new guise. Finally,our confidence was bolstered by understanding much aboutthe superfluidity in 3He, the original Fermi liquid referred toby Landau, as a consequence of Landau's theory supple-mented by the spin fluctuation theory of Schrieffer andDoniach, in 1973-1974 (these developments are well describedin ref. 4).Two more developments contributed to the general sense of

accomplishment of these years. First, there was the developmentof many useful and accurate experimental probes such as tun-neling spectroscopy, photoemission with spectacularly enhancedresolution, and other similar high-energy probes, etc. Second wasthe development of methods of electronic energy band andenergy level calculations that were extraordinarily successful andaccurate for semiconductors and ordinary metals, so that anelectronic structure even for a complex material could be calcu-lated, although often little attention was paid to its experimentalreality, if any.

It was, ironically, in the triumphant field of superconduc-tivity that this beautifully clear picture began to waver and losefocus. Superconductors were finding more and more techno-logical uses starting from the discovery of high-field super-conductivity. But the superconductors of practical value, withhigh critical fields and T, values between 15 and 25 K, were notsimple metals but outlandish intermetallic compounds oftransition metals with formulas like V3Si, Nb3Sn or Ge,Pb(Mo6S8) (this situation is discussed at length in ref. 5), etc.B. T. Matthias, the paladin of the field, taunted theorists withtheir inability to understand these more complex and inter-esting metals, which came to be called the "bad actor"superconductors (ref. 6; see also ref. 5). In the same period ofthe 1970s and 1980s, Matthias and his experimental friendsand collaborators in the world of exotic materials (for instance,T. H. Geballe) devised or brought under study a number ofmetals that tested the limits of the theory of metals in variousother ways: two-dimensional layer materials such as the "di-chalcogenides" NbSe2 and TaS2 (7, 8); quasi-one-dimensionalchain metals such as NbSe3 and the tungsten bronzes (9-11);"mixed valence" metals where electrons from the innerf shellsof the rare earths and actinides break out of their shells, at leastat low temperatures, and hybridize with Fermi sea electrons;and the "organic" superconductors or metals such as theTCNQ compounds, or the Bechgaard salts, where stacks ofaromatic molecules form metallic chains or layers (for review,see ref. 12). There had also been considerable interest inmetals with metal-insulator transitions, such as the metallicoxides of vanadium and titanium (13). The variety of nature isinexhaustible, but this list will do.

All of these materials represent, for one reason or another,cases in which the interactions between electrons in the metalare particularly strong, effective, or both. There came intoexistence a field of physics that specialized in these "stronglyinteracting electrons," of which I was a happy and activeparticipant throughout the 1970s and 1980s. Like all of mycolleagues in the field, I assumed that eventually some cleverreworking of the time-worn diagrammatic technique wouldsolve every problem; I was, as I have come to realize, "brain-washed by Feynman" into believing that these diagrammatic,perturbative, particle-based techniques were all of physics (notimplying with this slogan anything negative about Feynmanhimself; he was the most flexible of theorists). It was only withthe discovery of the "high-Tc" cuprate superconductors in1986-1987 (14, 15) that I began to realize that for almost 20years this type of theory had not had a single unequivocalsuccess and to speculate that the reason might lie not in ourlack of skill or in the complexity of the physics of these newmaterials but in a fundamental breakdown of the canonicaltheory; new ideas and concepts were needed. The one great

success of the past decade reinforces this point; the quantumHall effect is the case par excellence in which perturbationtechniques are not used at all and the entire system isdominated by impurities (in the integer effect) and interactions(in the fractional one). In the latter case, one finds elementaryexcitations completely unlike renormalized free electrons,having, for instance, fractional charge and statistics.

Let me describe a few of the anomalies exhibited by thesematerials, before settling on the cuprates as, actually, thesimplest and most unequivocal case of a non-Fermi liquid(NFL) metal. One may count no less than five classes ofsuperconductors that do not resemble the classic BCS, ele-mental metals. The characteristics of the BCS class are easilyunderstood in terms of the dynamic screening theory devel-oped in the early 1960s: (i) They are polyelectronic metals withlarge Fermi surfaces. Matthias (6) developed a set of empiricalcorrelations of free electron density with Tc that work very welland that make mechanistic sense. (ii) They are nonmagnetic;magnetism anticorrelates with Tc, and magnetic impurities aredeadly to Tc. This is easily understandable; magnetism usuallyresults from dominance by the repulsive Coulomb interactionsbetween electrons as opposed to the attraction caused byphonon-electron coupling. (iii) They are good conductors,well below the Mott limit of l/Ade Broglie = 1. (iv) They tend tohave stable, symmetrical structures. (v) Tc is limited to afraction of the lattice vibration energy Oi. Tc ' 1/3 - 1/40D.In no particular order, I list the new classes of superconductorsthat have been observed in the past decade or two.

(i) The organic superconductors BEDT, Bechgaard salts,etc. (12): These are layer- or chain-like arrays of stacked,charged aromatic molecules. (An early suggestion by Littlemotivated their discovery but has no predictive or explanatoryrelevance.) They violate several of the rules; superconductivityis closely associated with antiferromagnetic insulating phasesas well as with various other rather confusing magnetic phasetransitions, and the electron density is very low (<1 per largemolecule). No plausible suggestion as to a mechanism for theTC values, which range up to 12 K, has been advanced, but theresemblance to the cuprates in their association with magneticinsulators and in their low-dimensional, anisotropic structuressuggests that the mechanism may be the same.

(ii) The heavy-electron superconductors (16, 17): These aremixed valence metals such as UBe13, CeCu2Si2, UPt3, with lowTc values ('1 K) but very high electronic-specific heats so thatthe total entropy of condensation can be thousands of timesthat in conventional metals. The superconducting electronscome from anf band no more than 0.01 eV wide or less (1 eV= 1.602 x 10-19 J), which is magnetic in the room-temperaturestate. Most of these have magnetic spin density wave phasetransitions closely associated with superconductivity and af-fecting electrons from the same bands. No mechanism forsuperconductivity has been suggested, but it has been plausiblyproposed on experimental grounds that they are not isotropics waves and are therefore not phonon driven. Much investi-gation of all kinds of transport anomalies, magnetic phasetransitions, and other anomalies continues in this field.

(iii) The layer superconductors NbSe2, TaS2, etc (7, 8): Herethe anomaly is not only the low electron density and unusualstructures but, particularly, the association with charge densitywave distortions, which are not plausibly explained on the basisof nesting Fermi surfaces, which give anomalous responses atthe spanning vectors. Such nesting Fermi surfaces should causephase transitions at a temperature scale comparable with theFermi energy, 1-2 eV, not well below room temperature(<0.01 eV).

(iv) Cluster compounds, a vaguely defined category includ-ing C3- (18), "chevrels"-i.e., (X) Mo6S8 (19, 20), BaKBiO3(21), and similar materials-and among others perhaps theA15s: All have moderately high (15-40 K) Tc values. All ofthese seem plausibly motivated by phonons but have various

Colloquium Paper: Anderson

6670 Colloquium Paper: Anderson

puzzling anomalies indicating that straightforward theories donot apply. The chevrels, for instance, are almost immune tomagnetic constituents. The bismuthates have highest T, whennear a metal-insulator transition. The electron density of C3-is low, and its bands are very narrow; K4C6j0 is an insulator forno obvious reason. The A15s undergo mysterious low-temperature density wave transitions.

(v) Finally, there are by now some 2-3 dozen chemicallydistinct cuprates with Tc values ranging up to 150 K, which I willdiscuss shortly.

I have focused on superconductors mainly because that issuch a striking and easily measured electronic property, but inall the above cases there are other anomalies that are ofteneven farther from theoretical explanation. I am not claimingthat I know an explanation for all of these anomalies; rather,I am trying to express the sense of almost complete incapacityof what was supposed to be a complete and perfectly generaltheory to deal with any of the problems being posed by theexperimentalists. This does not mean that there is not amassive theoretical literature, but this seemed not to deal withthe real world of experiment but with artificial models too farfrom reality to be relevant. Another subculture seemed con-tent to calculate electronic energy bands without any exami-nation of whether they are relevant to the real materials-as,for instance, a full three-dimensional Fermi surface wasclaimed to have been obtained for cuprate materials that wereknown to have no metallic conduction along one direction inspace. Most disturbing was the experimentalists' claim to haveverified this Fermi surface experimentally, when a cursorylook at their data convinced me that the correlation betweenexperiment and theory was no better than with a randomlychosen band structure, possibly not even constrained to havethe right number of electrons. Basically, false confidence in thevalidity of renormalized quasiparticle theory is delayingprogress in this field, not enhancing it.

Let me now turn to a brief discussion of the cuprates as thebest example for the failure of the old theory and as a case inwhich the outlines of a new theory are clear.

First let me set up the basic phenomenology. Few will not befamiliar with the typical structures of these materials (Fig. 1),but not as many will appreciate the oversimplified, but correct,theorists' picture of them (Fig. 2). Mobile electrons live (for

° I]Conduction layer

Charge reservoir layer

] CuO2 planes

CuO, planes

* CopperO OxygenA Barium* Yttrium

b0_

FIG. 1. Crystal structure of a typical cuprate.

CuO2 plane + "stuff'

For example: YBa2Cu307 =

or (LaSr)2CuO4

/1--7 CU2

f/17 CuO2- BaO

------ BaO

/117 CUO2.____. etc.

/£17 C02------ LaO

LaO

/£17 C2CuO2 planes have all mobile electrons"stuff" has two functions:Doping Cu2+ > Cu25Transmitting (tunneling) electrons

FIG. 2. Schematic model of cuprate structures: CuO2 layers car-rying electrons, separated by insulating, charged layers of "stuff."

practical purposes) only in the CuO[2-(+x)] layers, betweenwhich are essentially inert layers of "stuff' that carries out twofunctions: (i) "doping": neutralizing the charge of theCu+(2+x)O(2-) layers-i.e., a charge reservoir function; (ii)providing a transmission medium, more or less effective, forquantum hopping of electrons between the CuO2 layers.The CuO2 layers are remarkably stable as compared to the

flexible structure and stoichiometry of the "stuff." They are inevery case of a square planar structure (Fig. 3), which may beslightly deformed but never in such a way as to modify the basicenergy level structure seriously. Clearly, the square planarbond of Cu to its four 0 neighbors is the strongest structuralelement in the problem.There is an almost universal generalized phase diagram

describing the materials, of which not all pieces have beenfound for all compounds, but no contradictory data exist. Thisis plotted in the temperature doping percentage plane (Fig. 4).Doping percentage 8 is the difference of the numerical Cuvalence from 2+. In general, 8 is positive, but for one or two

Cu 0 Cu

0 0 etc.

Cu - 0 Cu

FIG. 3. Square planar structure of the CuO2 layers. In the relevantband, there is one orbital per unit cell of this lattice, a hybrid of0 andCu functions but centered at Cu.

Proc. Natl. Acad. Sci. USA 92 (1995)

Proc. Natl. Acad. Sci. USA 92 (1995) 6671

Superconductivity involves at least one other parameter, thehopping integral between planes, which will have a complexand variable structure depending on the "stuff"; I refer to it interms of a hopping term

3D metal

O4-0< TC<250 K

20 8= % Cut-++ in planes

TN ind of "stuff"

Tc = function of "stuff"

STRIKING RESEMBLANCE TO 1D HUBBARD (EXACT)

FIG. 4. Generalized phase diagram for cuprates.

unique cases it is negative, although these are not as clearcutas the more common Cu2+8.The striking facts are (i) the narrow region of 8 = 0 + 1-2%

of stable, insulating (Mott) antiferromagnetism. J is large anddoes not vary by more than 20%. (ii) A transition regioncharacterized by defects and/or instabilities of 1% < 8 <-10%, where the material is a poor conductor with low (if any)T, or an insulator. (iii) A region of considerable stability 810-30% (the material often self-dopes-adjusts its stoichio-metry to a concentration within this range), which is optimumfor superconductivity, although Tc may vary from 10 to 150 KThere are characteristic anomalous behaviors in this state; Iam indebted to N. P. Ong (personal communication) for theconcept that the normal metal in this range is in some sensean "ideal" two-dimensional metallic system, from which de-viations are seen as one varies the doping from the ideal range.Finally, for 8> -30%, the material becomes a somewhat moreconventional metal, and T, drops rapidly to 0.The existence of the square planar structure and the narrow

region of the antiferromagnetic phase suggests, and muchadditional data confirm, that the basic electronic structure isdetermined by a very simple, one-band, two-dimensionalHubbard model. The orbital of the d shell of Cu, whichinteracts most strongly with the oxygen neighbors is the d.2-y2orbital, which will hybridize strongly with thep, orbitals on the02-. The bonding linear combination is deep (4-6 eV) belowthe Fermi level, while the antibonding linear combination ispushed up above the other d levels and forms the basis functionfor the single partially occupied band, which can be adequatelyrepresented with only two parameters, t and t', the nearest- andnext-nearest-neighbor hopping integrals.

Hlelectron = Eti,yicrcja.

To this energy the Hubbard model adds only one extraparameter, the repulsive energy U, which prevents doubleoccupancy of a site:

ae= E tijC4+Cfjt + UEnitfnim.i,J,a i

where n refers to the various planes, and Ck0(n) are the electronoperators in momentum space. But it is known that tI is anorder of magnitude smaller than t or t', and in fact it plays norole in the ideal two-dimensional region. Superconductivityalso brings in residual interactions in the Landau Fermi liquidsense, which affect T, slightly and the symmetry of the gap agreat deal; but this is not my concern here. In the true sense,these are "irrelevant" parameters in the normal state.The central qualitative fact about this ideal metallic phase is

that it has two independent energy scales: a region of energiesand temperatures >50-150 K (region A) and a region belowthis region, B, where superconductivity occurs-and occasion-ally other phenomena such as the spin gap. Region A ischaracterized by (i) power law transport and electromagneticproperties (scale-free-i.e., with only one scale); (ii) twodimensionality; (iii) rough quantitative universality for allcuprate planes. Region B has a widely variable scale-Te-andsuperconductivity is clearly three-dimensional, nor are thesuperconducting properties very universal.To show the contrast between the two scales, I borrow a

graph originated by Batlogg (14) and updated for me by N.-P.Ong. In Fig. 5, I plot a characteristic measurement on thetwo-dimensional metallic planes, the temperature coefficientof planar resistance (dpab)/(dT), against Tc, the superconduct-ing transition temperature. The two parameters seem to beabsolutely independent of each other. Many theorists persist inassuming that the properties of the planes determine Tc, butsuch theories (anyons, gauge theories, spin fluctuation theo-ries) have little relation to reality if they cannot explain thisindependence.The most striking power law of the higher energy region is

the conductivity itself. Fig. 6 shows how strikingly linear it isfor a pure single crystal of YBa2Cu307, while Fig. 7 shows thatover a very broad range of energies, from -100 cm-1 to nearly104 cm-1, the complex conductivity is proportional to(iW)-l+2a (22). This observation also has an implication aboutthe type of theory that is relevant. By many authors, the

1.4

1.2

i.0

C-

'a.

0.8

0.6

0.4 k

0.2

0.6

0 20 40 60 80 100 120 140

FIG. 5. Lack of correlation (for optimal doping) of T, and of a

typical planar electronic property, dPab/dT.

MESSy

NSULATING

pH

NA

L-

0._ci

0

CV0)

E

.

C

2D tra

ASE

Etflnn'(k)Ck'a(n)Ck.T(n'),n#n' k

0cu+

10

untwinned *Bi-2201 } Y YBCO7 Hg-1223

YBCOX, x-6.63 X

I_ Bi-2212LSCO, x=0.15

.-

I Al .I. ..

U) U

Colloquium Paper: Anderson

6672 Colloquium Paper: Anderson

YBa2CU307 90K

a a~~~~~~~~~~~~~~~~~~

800

*C -

600 I

I

a) 400F

200

C/ 25 10 15 20 25 20 2:irv

FIG. 6. Pab vs. T for a pure single crystal of YBa2Cu307.

Hubbard model is transformed, by a canonical transformationvalid at low energies, into the t - J model, which introduces anexchange parameter J t2/U of order 500-1000 cm-1. Thisartificial low-energy scale is indeed the correct one for the spindegrees of freedom of the insulating antiferromagnet, whereno particle motion is possible; but in the metal there is no signof it and the physics is uniform over a much wider range ofenergy. J is introduced artificially by projecting the kineticenergy term on that of an infinite U model and has the effectof correcting the resulting errors. The effective U in the metalis not as large as in the insulator and the transformation toinfinite U is a poor approximation except at very low energy.Unfortunately, most attempts at gauge theories have startedout from t - J rather than Hubbard physics and are notrelevant, as Fig. 7 demonstrates.The striking power law, which is even more universally

observed than the linear resistivity, is the T-2 power law of theHall angle OH = wrTH (Fig. 8) (for review, see ref. 15). Thisstrange and beautiful behavior shows unequivocally that elec-trical conduction is a composite process carried out by acomplex entity, not simple quasiparticles. That it is uniform

8000

6000

EC.

4000

20001

U'0

00 5 10 15

T2 (104 K2)20 25 30

FIG. 8. Hall angle vs. T for a single crystal of YBa2Cu307.

around the Fermi surface is shown by the T-4 dependence ofmagnetic resistance (Fig. 9), which is the variance of the Hallangle. TH is a qualitatively different quantity from T,A third power law, in a sense, is the nonexistence of metallic

conductivity along the c axis in the presence of large conduc-tivity in the ab plane. This is strikingly shown in infraredreflectivity measurements (Fig. 10) for c polarized radiation;the crystals of (LaSr)2CuO4 reflect like lossy insulators (R0.5) above T, but are good superconductors (R 1) below T,.The power I predict is c(o) X o92c (2a = 1/4) and somemeasurements (Fig. 11) suggest I am right. Note that of- 0 asc --> 0, at least at T = 0.What kind of theory can we use to understand this anom-

alous behavior? The quasiparticle theory fails generically in

T (K)

-2.0

0..

C0

m0.

0)

-3.0

-4.0

-5.0

-6.0

-7.0

2000 4000 6000 8000Energy, cm-1

100

1.8 1.9 2.0 2.1

200 300 400

2.2 2.3 2.4 2.5 2.6

log(T)FIG. 7. Infrared conductivity plotted as m*(co) and 1/T(w), with a

= (ne2)/[m*(iW + 1/X)], over a wide range of frequency and for avariety of single-crystalline films. Dashed line is a fit to o- = (Wp2-2a)/cW(i(O)l-2a] with 2a = 0.30. [Reproduced from ref. 30 with permission(copyright 1994, American Physical Society).]

FIG. 9. Magnetoresistance (MR) vs. T for a single crystal ofYBa2Cu307 (YBCO). Line shows the fit to T-4 MR is theoreticallyproportional to the variance of the Hall angle over the Fermi surface:(i2) (OH)2 = MR. [Reproduced from ref. 31 with permission(copyright 1992, American Physical Society).]

a

IS Bao36KO.6Bil.04O.EB YBa2Cu307 6* A YBa2CU307 0

o BYBa2cu307 G* DGdBa2Cu3O7O E Bi Sr2ca2cu30,o 082221A F Bi2Sr2cacu208 0

20

A~

00

0

QSA-o ~ o g

I M

Proc. Natl. Acad. Sci. USA 92 (1995)

.7.

.'7

t-..

p

Proc. Natl. Acad. Sci. USA 92 (1995) 6673

0.5

> ~~40Kj

0.5 ie-,; x=0.13&> 25Kx=01020K E11c

8K b

T= 40 K

30 K28 K

25.K x= 0.16300 K 20 K EIc

8K

.I

C

0 50 100 150 200 250 300 350

Frequency, cm-1

FIG. 10. Infrared reflectivity [measured by Uchida and colleagues(32)] in the c-axis direction for single crystals of (LaSr)2CuO4. Normalmetals have no plasma edge and hence no free electrons; one appearsnear 50 cm-' for the superconductors. [Reproduced from ref. 32 withpermission (copyright 1992, American Physical Society).]

one dimension, where in fact there exists an exact solution ofthe Hubbard model (among others) in one dimension, by Lieband Wu (23), as well as a considerable tool bag of techniquesfor one-dimensional electronic models. Initially, I simply be-gan to use these solutions as a template for the NFL case, butas time went on both experimental and theoretical reasoningled me to realize that the basic features reappear in twodimensions at least.

Let me describe the features of the Lieb-Wu solution. Thekey to this solution, as pointed out by Haldane (24), is that it

600

400E0

e 200

00

Normal A state: Spinons and holonsTemplate: Exact results on 1 D Hubbard model

2 kinds of excitations(a) holons and "antiholons" q = ±

a=0

El

2kF

+ek

(b) spinons q-0 c= 1/2spinon = antispinon: "real"

E

I I

Physicallyalwas IL pa rs

I kF k

electron - * antiholon (2kF) + spinon (-kF)(+ doud of pairs)

Not a stable excitation

FIG. 12. Holon and spinon dispersion curves (w vs. k) in theHubbard model at moderate doping.

is a Luttinger liquid, in the sense that it is not a Fermi liquidbut that it has a Fermi surface satisfying Luttinger's theorem,in that excitation energies go to 0 at a surface in momentumspace. But Migdal's construction does notwork: Z = 0, so thereare no quasiparticles and no jump in nk at the Fermi surface.

If there are no quasiparticles, what is there? There are twokinds of excitations, which in physical situations must becreated in pairs but are independent once made (Fig. 12).There are charge excitations called holons, which have chargee, no spin, carry momenta near 2kF, and have a velocity vc; theirdispersion curve looks like that of an electron, crossingthrough 0 at 2kF, and there are holons and antiholons.There are spin excitations called spinons, which have no

charge, S = 1/2, velocity vs, and that go to 0 at kF but do notextend through the 0; there are no antispinons; these behavelike Majorana Fermions. If one tries to make an antispinon, aspinon is created at a different momentum. Exactly the samethings were found in Bethe's 1931 solution of the Heisenbergmodel. F. D. M. Haldane and students (personal communica-tion) have exhaustively studied their properties.When an electron is added or taken away, at least one of

each must always be made. The electron decays very rapidlyinto a spectrum and is not a stable particle; this I take to be thedefinition of NFL. The spectrum of electron-like states nearthe Fermi surface at kF is shown in Fig. 13. The Green'sfunction for the electron is (25, 26) (that for free electrons isgiven for comparison)

Fermi liquid: G e ikFX _x - Vt

FIG. 11. High-frequency infrared conductivity in the c direction forYBa2Cu307. [Reproduced from ref. 33 with permission (copyright1993, American Physical Society).]

1NFL: G eikFX

(x -vst)l2(x -vct)l12(x2 - V2t2)cr/2'

Colloquium Paper: Anderson

c_-

6674 Colloquium Paper: Anderson

kF k

FIG. 13. Range of allowed composite electron-like states.

Fig. 13 shows that the spectrum's breadth is xw. The threeparts of G are (i) the spin moving at velocity v,; (ii) the chargeat velocity v,; (iii) a backflow due to the electron's modificationof all the other electrons' wave functions as it passes throughthe sea of opposite spins: a - 1/8; although small, it is this partthat causes the Fermi surface to smear.What we find theoretically and experimentally is that the

two-dimensional system, at least with strong interaction, is justa tomographic superposition of effective one-dimensionalmodels, one for each point on the Fermi surface. It wasdiscovered by Luther (27), and recently enlarged on by severalauthors, that this is the case for Fermi liquid theory, becauseof the exclusion principle's restriction to only forward (non-diffractive) scattering, which is exploited in Landau's theory.What I have discovered is that the same applies to NFL as well.As Haldane (28) points out, the Fermi surface can be

thought of as the order parameter of a critical point at T = 0,the excitations as fluctuations of this order parameter, and thepower laws as valid throughout the neighborhood of thiscritical point (whether they be Fermi liquid or NFL). Whathappens as this critical point is approached is that suddenly theinterlayer interactions grow to relevance and cause supercon-ductivity; but that part of the theory-region B-is not thesubject here.The power laws are straightforwardly explained within my

NFL theory: (i) the linear dependence of l/cond on T or cw.

This is the decay of the accelerated electron into spinon andholon. This is not a resistivity process unless something else isadded: impurity or phonon scattering of the holons (spinonsare scattered only by magnetic, time-reverse breaking, impu-rities). This is exactly analogous to phonon resistivity; under"phonon drag" conditions, phonon scattering is not a resistiv-ity process, but phonons are, normally, scattered and if they arethey control resistivity. So this is called the "holon nondrag"regime. Most Luttinger liquids are in the holon drag regime,which is quite different. The slight deviation of the power from1 is also predicted by the theory. (ii) Hall angle. A magneticfield rotates only the Fermi surface, so it does not affect theelectron-spinon-holon process and does not cause any addi-tional electron decay. Reciprocally, electron decay cannotaffect the Hall angle. Thus, the underlying spinon-spinonscattering, which, like electron-electron processes, is oc2, isthe Hall angle controlling process. The current is mostlycarried as a backflow by spinons, not by the holon chargecarriers, and in fact the holon current will not be colinear withthat of the spinons.

Finally, the absence of c-axis conductivity is an effect of Z= 0; the direct, coherent hopping is ineffectual, and in generalhops will take place incoherently to high-energy states (29).The proof is a bit subtle but the principle (and the fact) isobvious.There is no reason to suppose that the NFL phenomenon is

restricted to cuprates, and in fact infrared spectra resemblingFig. 7 are common to many of the weird materials mentionedearlier. Most of the anomalies have possible explanations; for

instance, the density wave responses of the Luttinger liquid aremuch more singular than those of the Fermi liquid. What isimportant is to have one case firmly tied down.

I should acknowledge many more students and coworkers than Ipossibly can. Some who contributed to specific things mentioned areG. Baskaran, S. Strong, D. G. Clarke, N. Bontemps, T. Timusk, D.Khveshchenko, A. Tsvelik, F. D. M. Haldane, N.-P. Ong, Z.Schlesinger, S. Chakravarty, Y. Ren, T. Hsu, B. Batlogg, and J.Wheatley. This work was supported by National Science FoundationGrant DMR-9104873.

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