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

This paper was presented at a coUoquium entitled "; 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 : Fermi surfaces without Fermi P. W. ANDERSON Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544

ABSTRACT I relate the historic successes, and present where Ep is the single-particle band , and ,u is the difficulties, ofthe renormalized theory ofmetals EF. Positive t refers to -like prop- ("AGD" or Fermi theory). I then describe the best- agators, negative (backwards-moving) co refers to holes. The understood example of a non-Fermi liquid, the normal me- series can, if convergent, be resummed in tallic state of the cuprate superconductors. terms of a self-energy, which is the sum of all self-energy parts and appears in the exact Green's functions' denominator: For some 40 years, almost all electronic phenomena in metals have been interpreted in terms of a general theoretical frame- 1 1 work, which one could variously call renormalized free particle G G- - - (e, - IL) - M(,p), theory, , or "AGD" after the best-known book on the subject (1). The assumption is that I is sufficiently regular that the only I came to the conclusion a few years ago that this theory is, singularities of G are poles at a modifiedp-dependent energy in many of the most interesting cases, basically a failure. For Ep - ,u of strength 0 < Zp = 1/[i - (aX)/ato)] c 1 the first 20 years of its history, until the mid-1970s, it served us very well; but then as we began to focus on the most interesting G = _ + incoherent (or the most anomalous) cases, more and more of the copious (E= ) part. literature of our subject came to be engaged in fitting the proverbial square peg into a round hole. It is not that there are These poles are the renormalized . no instances that fit the framework but that, contrary to the This theory was made useful and meaningful by a series of claims for universality which have been made for it, it seems theorems proved in the late 1950s, which depend on the idea that for systems with strong interactions, it often is completely that quasiparticles at EF do not decay, because the exclusion misguiding. principle blocks off all states into which they can decay, to To make my point I must first describe the of this order t2 = (EP - EF)2. conventional theory. It arose in the 1950s, just after the Migdal: If Z is finite there is a jump at PF in nk of successes of the Schwinger-Feynman-Dyson theory in quan- magnitude Z; there is a real, measurable . tum electrodynamics, and it borrows the techniques that were Landau: The dynamics can be completely described at so successful in that theory. In , the low by the quasiparticles, except for a small scheme was to map the properties of the real physical vacuum finite number of collective modes near q = 0 (the Fermi and the real physical particle excitations onto the correspond- liquid theory). ing entities of a supposed bare vacuum with bare particles by Luttinger: The Fermi surface contains a number of p the process of . One defines a propagator or states exactly equal to the number of . Green's function, G(r - r', t - t'), which is the amplitude for Finally (Migdal again), (lattice vibrations) can be finding a particle at point r and time t if it was inserted at point added in simply to the theory including only the lowest-order r' and time t' into the real vacuum. The particle can encounter diagrams (the buzzword is "neglect vertex corrections") be- various interactions with vacuum fluctuations on the way, cause the ion's is much heavier than the electron's mass. which are sorted out into a series with Feynman diagrams. If The very elegant final form of the theory, although invented this series is well-behaved, its sum can be written in terms of by three groups simultaneously, is expressed in the "AGD" a self-energy, which merely renormalizes the unperturbed book (1). Its greatest achievement almost coincided with its propagator without changing its essential character. birth: it turned out to require only a formally trivial (if In the condensed physics of metals there is no conceptually profound) redefinition of the vacuum and the vacuum, but there is a Fermi sea if the electrons are nonin- theory as revised by Schrieffer, Nambu, and Eliashberg ele- teracting. This is treated formally as a vacuum in which both gantly encompassed Bardeen-Cooper-Schrieffer (BCS) su- hole and particle excitations can propagate, in parallel to the perconductivity (2). By 1965 Schrieffer, I, and later W. L. treatment in quantum electrodynamics of the Dirac sea of McMillan, working with the beautiful experiments of Giaever negative-energy electrons as a vacuum for . There is and Rowell, had made the theory quantitative, dealing with the a surface in p space of zero energy, the Fermi surface. The real complexities of real materials so efficiently that the unperturbed Green's function [Fourier transformed into mo- superconducting Tc of metallic elements like Pb, Hg, and Al mentum (p) and energy (t) space] is may be the best predicted of all condensed matter transitions (3). Triumphs in such fields as "Fermiology," the 1 measurement of complex Fermi surfaces of real metals, led us G(o,p) = - (e -, to feel that the problem of the electron liquid in metals was finished in principle, with only quantitative or marginal prob- The publication costs of this article were defrayed in part by page charge lems left, some of the simpler of which were solved in the late payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact. Abbreviation: NFL, non-Fermi liquid. 6668 Downloaded by guest on September 27, 2021 Colloquium Paper: Anderson Proc. Natl. Acad. Sci. USA 92 (1995) 6669 1960s and early 1970s-like magnetic impurities in metals, the success of the past decade reinforces this point; the quantum so-called "Anderson model," which led to the "," is the case par excellence in which perturbation which turned out to be the Fermi liquid in a new guise. Finally, techniques are not used at all and the entire system is our confidence was bolstered by understanding much about dominated by impurities (in the integer effect) and interactions the in 3He, the original Fermi liquid referred to (in the fractional one). In the latter case, one finds elementary by Landau, as a consequence of Landau's theory supple- excitations completely unlike renormalized free electrons, mented by the fluctuation theory of Schrieffer and having, for instance, fractional charge and statistics. Doniach, in 1973-1974 (these developments are well described Let me describe a few of the anomalies exhibited by these in ref. 4). materials, before settling on the cuprates as, actually, the Two more developments contributed to the general sense of simplest and most unequivocal case of a non-Fermi liquid accomplishment of these years. First, there was the development (NFL) . One may count no less than five classes of of many useful and accurate experimental probes such as tun- superconductors that do not resemble the classic BCS, ele- neling , photoemission with spectacularly enhanced mental metals. The characteristics of the BCS class are easily resolution, and other similar high-energy probes, etc. Second was understood in terms of the dynamic screening theory devel- the development of methods of electronic energy band and oped in the early 1960s: (i) They are polyelectronic metals with energy level calculations that were extraordinarily successful and large Fermi surfaces. Matthias (6) developed a set of empirical accurate for and ordinary metals, so that an correlations of free electron densitywith Tc that work very well electronic structure even for a complex material could be calcu- and that make mechanistic sense. (ii) They are nonmagnetic; lated, although often little attention was paid to its experimental anticorrelates with Tc, and magnetic impurities are reality, if any. deadly to Tc. This is easily understandable; magnetism usually It was, ironically, in the triumphant field of superconduc- results from dominance by the repulsive Coulomb interactions tivity that this beautifully clear picture began to waver and lose between electrons as opposed to the attraction caused by focus. Superconductors were finding more and more techno- -electron coupling. (iii) They are good conductors, logical uses starting from the discovery of high-field super- well below the Mott limit of l/Ade Broglie = 1. (iv) They tend to conductivity. But the superconductors of practical value, with have stable, symmetrical structures. (v) Tc is limited to a high critical fields and T, values between 15 and 25 K, were not fraction of the lattice vibration energy Oi. Tc ' 1/3 - 1/40D. simple metals but outlandish intermetallic compounds of In no particular order, I list the new classes of superconductors transition metals with formulas like V3Si, Nb3Sn or Ge, that have been observed in the past decade or two. Pb(Mo6S8) (this situation is discussed at length in ref. 5), etc. (i) The organic superconductors BEDT, Bechgaard salts, B. T. Matthias, the paladin of the field, taunted theorists with etc. (12): These are layer- or chain-like arrays of stacked, their inability to understand these more complex and inter- charged aromatic . (An early suggestion by Little esting metals, which came to be called the "bad actor" motivated their discovery but has no predictive or explanatory superconductors (ref. 6; see also ref. 5). In the same period of relevance.) Theyviolate several of the rules; the 1970s and 1980s, Matthias and his experimental friends is closely associated with antiferromagnetic insulating phases and collaborators in the world of exotic materials (for instance, as well as with various other rather confusing magnetic phase T. H. Geballe) devised or brought under study a number of transitions, and the electron density is very low (<1 per large metals that tested the limits of the theory of metals in various ). No plausible suggestion as to a mechanism for the other ways: two-dimensional layer materials such as the "di- TC values, which range up to 12 K, has been advanced, but the chalcogenides" NbSe2 and TaS2 (7, 8); quasi-one-dimensional resemblance to the cuprates in their association with magnetic chain metals such as NbSe3 and the tungsten bronzes (9-11); insulators and in their low-dimensional, anisotropic structures "mixed valence" metals where electrons from the innerfshells suggests that the mechanism may be the same. of the rare earths and actinides break out of their shells, at least (ii) The heavy-electron superconductors (16, 17): These are at low , and hybridize with Fermi sea electrons; mixed valence metals such as UBe13, CeCu2Si2, UPt3, with low and the "organic" superconductors or metals such as the Tc values ('1 K) but very high electronic-specific heats so that TCNQ compounds, or the Bechgaard salts, where stacks of the total of can be thousands of times aromatic molecules form metallic chains or layers (for review, that in conventional metals. The superconducting electrons see ref. 12). There had also been considerable interest in come from anfband no more than 0.01 eV wide or less (1 eV metals with metal- transitions, such as the metallic = 1.602 x 10-19 J), which is magnetic in the room- oxides of vanadium and titanium (13). The variety of nature is state. Most of these have magnetic phase inexhaustible, but this list will do. transitions closely associated with superconductivity and af- All of these materials represent, for one reason or another, fecting electrons from the same bands. No mechanism for cases in which the interactions between electrons in the metal superconductivity has been suggested, but it has been plausibly are particularly strong, effective, or both. There came into proposed on experimental grounds that they are not isotropic existence a field of physics that specialized in these "strongly s waves and are therefore not phonon driven. Much investi- interacting electrons," of which I was a happy and active gation of all kinds of transport anomalies, magnetic phase participant throughout the 1970s and 1980s. Like all of my transitions, and other anomalies continues in this field. colleagues in the field, I assumed that eventually some clever (iii) The layer superconductors NbSe2, TaS2, etc (7, 8): Here reworking of the time-worn diagrammatic technique would the anomaly is not only the low electron density and unusual solve every problem; I was, as I have come to realize, "brain- structures but, particularly, the association with charge density washed by Feynman" into believing that these diagrammatic, wave distortions, which are not plausibly explained on the basis perturbative, particle-based techniques were all of physics (not of nesting Fermi surfaces, which give anomalous responses at implying with this slogan anything negative about Feynman the spanning vectors. Such nesting Fermi surfaces should cause himself; he was the most flexible of theorists). It was only with phase transitions at a temperature scale comparable with the the discovery of the "high-Tc" cuprate superconductors in , 1-2 eV, not well below room temperature 1986-1987 (14, 15) that I began to realize that for almost 20 (<0.01 eV). years this type of theory had not had a single unequivocal (iv) Cluster compounds, a vaguely defined category includ- success and to speculate that the reason might lie not in our ing C3- (18), "chevrels"-i.e., (X) Mo6S8 (19, 20), BaKBiO3 lack of skill or in the complexity of the physics of these new (21), and similar materials-and among others perhaps the materials but in a fundamental breakdown of the canonical A15s: All have moderately high (15-40 K) Tcvalues. All of theory; new ideas and concepts were needed. The one great these seem plausibly motivated by phonons but have various Downloaded by guest on September 27, 2021 6670 Colloquium Paper: Anderson Proc. Natl. Acad. Sci. USA 92 (1995)

puzzling anomalies indicating that straightforward theories do CuO2 plane + "stuff' not apply. The chevrels, for instance, are almost immune to magnetic constituents. The bismuthates have highest T, when For example: YBa2Cu307 = /1--7 CU2 near a metal-insulator transition. The electron density of C3- is low, and its bands are very narrow; K4C6j0 is an insulator for no obvious reason. The A15s undergo mysterious low- temperature density wave transitions. f/17 CuO2 (v) Finally, there are by now some 2-3 dozen chemically distinct cuprates with Tcvalues ranging up to 150 K, which I will - BaO discuss shortly. I have focused on superconductors mainly because that is such a striking and easily measured electronic property, but in ------BaO all the above cases there are other anomalies that are often even farther from theoretical explanation. I am not claiming /117 CUO2 that I know an explanation for all of these anomalies; rather, .____. etc. I am trying to express the sense of almost complete incapacity of what was supposed to be a complete and perfectly general theory to deal with any of the problems being posed by the experimentalists. This does not mean that there is not a or (LaSr)2CuO4 /£17 C02 massive theoretical literature, but this seemed not to deal with the real world of experiment but with artificial models too far ------LaO from reality to be relevant. Another subculture seemed con- LaO tent to calculate electronic energy bands without any exami- nation of whether they are relevant to the real materials-as, /£17 C2 for instance, a full three-dimensional Fermi surface was claimed to have been obtained for cuprate materials that were CuO2 planes have all mobile electrons known to have no metallic conduction along one direction in "stuff" has two functions: space. Most disturbing was the experimentalists' claim to have Doping Cu2+ > Cu25 verified this Fermi surface experimentally, when a cursory Transmitting (tunneling) electrons look at their data convinced me that the correlation between experiment and theory was no better than with a randomly FIG. 2. Schematic model of cuprate structures: CuO2 layers car- chosen band structure, possibly not even constrained to have rying electrons, separated by insulating, charged layers of "stuff." the right number of electrons. Basically, false confidence in the validity of renormalized quasiparticle theory is delaying practical purposes) only in the CuO[2-(+x)] layers, between progress in this field, not enhancing it. which are essentially inert layers of "stuff' that carries out two Let me now turn to a brief discussion of the cuprates as the functions: (i) "doping": neutralizing the charge of the best example for the failure of the old theory and as a case in Cu+(2+x)O(2-) layers-i.e., a charge reservoir function; (ii) which the outlines of a new theory are clear. providing a transmission medium, more or less effective, for First let me set up the basic . Few will not be quantum hopping of electrons between the CuO2 layers. familiar with the typical structures of these materials (Fig. 1), The CuO2 layers are remarkably stable as compared to the but not as many will appreciate the oversimplified, but correct, flexible structure and stoichiometry of the "stuff." They are in theorists' picture of them (Fig. 2). Mobile electrons live (for every case of a square planar structure (Fig. 3), which may be slightly deformed but never in such a way as to modify the basic energy level structure seriously. Clearly, the square planar bond of Cu to its four 0 neighbors is the strongest structural element in the problem. ° I]Conduction layer There is an almost universal generalized phase diagram describing the materials, of which not all pieces have been found for all compounds, but no contradictory data exist. This is plotted in the temperature doping percentage plane (Fig. 4). Charge reservoir layer Doping percentage 8 is the difference of the numerical Cu valence from 2+. In general, 8 is positive, but for one or two Cu 0 Cu ] CuO2 planes

CuO, planes 0 0 etc.

b0_ * Copper O Cu - 0 Cu A Barium * Yttrium FIG. 3. Square planar structure of the CuO2 layers. In the relevant band, there is one orbital per unit cell of this lattice, a hybrid of 0 and FIG. 1. structure of a typical cuprate. Cu functions but centered at Cu. Downloaded by guest on September 27, 2021 Colloquium Paper: Anderson Proc. Natl. Acad. Sci. USA 92 (1995) 6671 Superconductivity involves at least one other parameter, the hopping integral between planes, which will have a complex N A M and variable structure depending on the "stuff"; I refer to it in E S terms of a hopping term S y L- Etflnn'(k)Ck'a(n)Ck.T(n'), N 2D tra n#n' k 0 S ._ci U 3D metal L where n refers to the various planes, and Ck0(n) are the electron 0 A T operators in space. But it is known that tI is an 0) I order of magnitude smaller than t or t', and in fact it plays no N CV G role in the ideal two-dimensional region. E Superconductivity p also brings in residual interactions in the Landau Fermi liquid H sense, which affect T, slightly and the of the gap a . A O4-0< TC<250 K C S great deal; but this is not my concern here. In the true sense, E 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 .- energies and temperatures >50-150 K (region A) and a region below 0 10 20 8= % Cut-++ in planes + this region, B, where superconductivity occurs-and occasion- cu ally other phenomena such as the spin gap. Region A is TN ind of "stuff" characterized by (i) power law transport and electromagnetic Tc = function of "stuff" properties (scale-free-i.e., with only one scale); (ii) two STRIKING dimensionality; (iii) rough quantitative universality for all RESEMBLANCE TO 1D HUBBARD (EXACT) cuprate planes. Region B has a widely variable scale-Te-and FIG. 4. Generalized phase diagram for cuprates. superconductivity is clearly three-dimensional, nor are the superconducting properties very universal. unique cases it is negative, although these are not as clearcut To show the contrast between the two scales, I borrow a as the more common Cu2+8. graph originated by Batlogg (14) and updated for me by N.-P. The striking facts are (i) the narrow region of 8 = 0 + 1-2% Ong. In Fig. 5, I plot a characteristic measurement on the of stable, insulating (Mott) . J is large and two-dimensional metallic planes, the temperature coefficient does not more A of planar resistance (dpab)/(dT), against Tc, the superconduct- vary by than 20%. (ii) transition region ing transition temperature. The two parameters seem to be characterized by defects and/or instabilities of 1% < 8 < absolutely independent ofeach other. Many theorists persist in -10%, where the material is a poor conductorwith low (if any) assuming that the properties of the planes determine Tc, but T, or an insulator. (iii) A region of considerable stability 8 such theories (, gauge theories, spin fluctuation theo- 10-30% (the material often self-dopes-adjusts its stoichio- ries) have little relation to reality if they cannot explain this metry to a concentration within this range), which is optimum independence. for superconductivity, although Tc may vary from 10 to 150 K The most striking power law of the higher energy region is There are characteristic anomalous behaviors in this state; I the conductivity itself. Fig. 6 shows how strikingly linear it is am indebted to N. P. Ong (personal communication) for the for a pure of YBa2Cu307, while Fig. 7 shows that concept that the normal metal in this range is in some sense over a very broad range of energies, from -100 cm-1 to nearly an "ideal" two-dimensional metallic system, from which de- 104 cm-1, the complex conductivity is proportional to viations are seen as one varies the doping from the ideal range. (iW)-l+2a (22). This observation also has an implication about Finally, for 8 > -30%, the material becomes a somewhat more the type of theory that is relevant. By many authors, the conventional metal, and T, drops rapidly to 0. The existence of the square planar structure and the narrow I Al .I. .. region of the antiferromagnetic phase suggests, and much additional data confirm, that the basic electronic structure is 1.4 untwinned * determined by a very simple, one-band, two-dimensional Bi-2201 } Y YBCO7 Hg-1223 . The orbital of the d shell of Cu, which 1.2 interacts most strongly with the oxygen neighbors is the d.2-y2 YBCOX, x-6.63 X orbital, which will hybridize strongly with thep, orbitals on the i.0 02-. The bonding linear combination is deep (4-6 eV) below I_ Bi-2212 the Fermi while level, the antibonding linear combination is 0.8 LSCO, x=0.15 pushed up above the other d levels and forms the basis function C- for the single partially occupied band, which can be adequately 'a. represented with only two parameters, t and t', the nearest- and 0.6 next-nearest-neighbor hopping integrals. 0.4 k

Hlelectron = Eti,yicrcja. 0.2 To this energy the Hubbard model adds only one extra U)0.6U parameter, the repulsive energy U, which prevents double 0 20 40 60 80 100 120 140 occupancy of a site:

ae = E tijC4+Cfjt + UEnitfnim. FIG. 5. Lack of correlation (for optimal doping) of T, and of a i,J,a i typical planar electronic property, dPab/dT. Downloaded by guest on September 27, 2021 6672 Colloquium Paper: Anderson Proc. Natl. Acad. Sci. USA 92 (1995) YBa2CU307 90K

a a~~~~~~~~~~~~~~~~~~ 800

*C - 600 I .7. .'7 .. t- I a) 400F

C/ 25 10 15 20 25 20 2: irv 200

a

0 0 5 10 15 20 25 30 FIG. 6. Pab vs. T for a pure single crystal of YBa2Cu307. T2 (104 K2) Hubbard model is transformed, by a canonical transformation FIG. 8. Hall angle vs. T for a single crystal of YBa2Cu307. valid at low energies, into the t - J model, which introduces an exchange parameter J t2/U of order 500-1000 cm-1. This around the Fermi surface is shown by the T-4 dependence of artificial low-energy scale is indeed the correct one for the spin magnetic resistance (Fig. 9), which is the variance of the Hall degrees of freedom of the insulating antiferromagnet, where angle. TH is a qualitatively different quantity from T, no particle motion is possible; but in the metal there is no sign A third power law, in a sense, is the nonexistence of metallic of it and the physics is uniform over a much wider range of conductivity along the c axis in the presence of large conduc- energy. J is introduced artificially by projecting the kinetic tivity in the ab plane. This is strikingly shown in infrared energy term on that of an infinite U model and has the effect reflectivity measurements (Fig. 10) for c polarized radiation; of correcting the resulting errors. The effective U in the metal the of (LaSr)2CuO4 reflect like lossy insulators (R is not as large as in the insulator and the transformation to 0.5) above T, but are good superconductors (R 1) below T,. infinite U is a poor approximation except at very low energy. The power I predict is c(o) X o92c (2a = 1/4) and some Unfortunately, most attempts at gauge theories have started measurements (Fig. 11) suggest I am right. Note that of- 0 as out from t - J rather than Hubbard physics and are not c --> 0, at least at T = 0. relevant, as Fig. 7 demonstrates. What kind of theory can we use to understand this anom- The striking power law, which is even more universally alous behavior? The quasiparticle theory fails generically in observed than the linear resistivity, is the T-2 power law of the Hall angle OH = wrTH (Fig. 8) (for review, see ref. 15). This T (K) strange and beautiful behavior shows unequivocally that elec- 100 200 300 400 trical conduction is a composite process carried out by a complex entity, not simple quasiparticles. That it is uniform -2.0 8000 IS Bao36KO.6Bil.04O. p EB YBa2Cu307 6 * A YBa2CU307 0 -3.0 o BYBa2cu307 G 6000 * DGdBa2Cu3O7 0.. O E Bi2221Sr2ca2cu30,o 8 0 A F Bi2Sr2cacu208 0 C0 -4.0

0. E 2 A~ m C. 0 4000 0 0) -5.0 0 0

20001 -o ~ QSAo g -6.0

M I -7.0 U' 0 2000 4000 6000 8000 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 Energy, cm-1 log(T)

FIG. 7. Infrared conductivity plotted as m*(co) and 1/T(w), with a FIG. 9. Magnetoresistance (MR) vs. T for a single crystal of = (ne2)/[m*(iW + 1/X)], over a wide range of frequency and for a YBa2Cu307 (YBCO). Line shows the fit to T-4 MR is theoretically variety of single-crystalline films. Dashed line is a fit to o- = (Wp2-2a)/ proportional to the variance of the Hall angle over the Fermi surface: cW(i(O)l-2a] with 2a = 0.30. [Reproduced from ref. 30 with permission (i2) (OH)2 = MR. [Reproduced from ref. 31 with permission (copyright 1994, American Physical Society).] (copyright 1992, American Physical Society).] Downloaded by guest on September 27, 2021 Colloquium Paper: Anderson Proc. Natl. Acad. Sci. USA 92 (1995) 6673 Normal A state: and holons Template: Exact results on 1 D Hubbard model 2 kinds of excitations (a) holons and "antiholons" q = ± 0.5 a=0

El

2kF > ~~40Kj k +e 0.5 &>ie-,;25 Kx=01x=0.13 020K E11c 8K b (b) spinons q-0 c= 1/2 = antispinon: "real" T= 40 K E 30 K Physically 28 K alwas IL pa rs

I I 25.K x= 0.16 I kF k 300 K 20 K EIc

8 K C I . electron - * antiholon (2kF) + spinon (-kF) 0 50 100 150 200 250 300 350 (+ doud of pairs) Frequency, cm-1 Not a stable excitation FIG. 10. Infrared reflectivity [measured by Uchida and colleagues (32)] in the c-axis direction for single crystals of (LaSr)2CuO4. Normal FIG. 12. and spinon dispersion curves (w vs. k) in the metals have no edge and hence no free electrons; one appears Hubbard model at moderate doping. near 50 cm-' for the superconductors. [Reproduced from ref. 32 with permission (copyright 1992, American Physical Society).] is a , in the sense that it is not a Fermi liquid but that it has a Fermi surface satisfying Luttinger's theorem, one dimension, where in fact there exists an exact solution of in that excitation energies go to 0 at a surface in momentum the Hubbard model (among others) in one dimension, by Lieb space. But Migdal's construction does notwork: Z = 0, so there and Wu (23), as well as a considerable tool bag of techniques are no quasiparticles and no jump in nk at the Fermi surface. for one-dimensional electronic models. Initially, I simply be- If there are no quasiparticles, what is there? There are two gan to use these solutions as a template for the NFL case, but kinds of excitations, which in physical situations must be as time went on both experimental and theoretical reasoning created in pairs but are independent once made (Fig. 12). led me to realize that the basic features reappear in two There are charge excitations called holons, which have charge dimensions at least. e, no spin, carry momenta near 2kF, and have a velocity vc; their Let me describe the features of the Lieb-Wu solution. The dispersion curve looks like that of an electron, crossing key to this solution, as pointed out by Haldane (24), is that it through 0 at 2kF, and there are holons and antiholons. There are spin excitations called spinons, which have no 600 charge, S = 1/2, velocity vs, and that go to 0 at kF but do not extend through the 0; there are no antispinons; these behave like Majorana . If one tries to make an antispinon, a spinon is created at a different momentum. Exactly the same things were found in Bethe's 1931 solution of the Heisenberg 400 model. F. D. M. Haldane and E students (personal communica- 0 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 rapidly e 200 into a spectrum and is not a stable particle; this I take to be the definition of NFL. The spectrum of electron-like states near the Fermi surface at kF is shown in Fig. 13. The Green's function for the electron is (25, 26) (that for free electrons is given for comparison) 0 0 Fermi liquid: G e ikFX _ x - Vt FIG. 11. High-frequency infrared conductivity in the c direction for 1 YBa2Cu307. [Reproduced from ref. 33 with permission (copyright NFL: G eikFXc_- 1993, American Physical Society).] (x - vst)l2(x -vct)l12(x2 - V2t2)cr/2' Downloaded by guest on September 27, 2021 6674 Colloquium Paper: Anderson Proc. Natl. Acad. Sci. USA 92 (1995) instance, the density wave responses of the Luttinger liquid are much more singular than those of the Fermi liquid. What is important is to have one case firmly tied down. I should acknowledge many more students and coworkers than I possibly can. Some who contributed to specific things mentioned are G. 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 Foundation Grant DMR-9104873. kF k 1. Abrikosov, A. A., Gor'kov, L. P. & Dzialoshinskii, I. (1963) FIG. 13. Range of allowed composite electron-like states. Methods of in (Prentice Hall, New York). Fig. 13 shows that the spectrum's breadth is xw. The three 2. Schrieffer, J. R. (1963) Superconductivity (Benjamin, New York). parts of G are (i) the spin moving at velocity v,; (ii) the charge 3. McMi!lan, W. L. & Rowell, J. M. (1969) Superconductivity, ed. at velocity v,; (iii) a backflow due to the electron's modification Parks, R. D. (Dekker, New York), Vol. 1, p. 561. of all the other electrons' wave functions as it passes through 4. Brinkman, W. F. & Anderson, P. W. (1978) in Physics ofLiquid - the sea of opposite spins: a 1/8; although small, it is this part and , eds. Bennemann, K. H. & Ketterson, J. B. that causes the Fermi surface to smear. (Wiley, New York). What we find theoretically and experimentally is that the 5. Anderson, P. W. & Yu, C. C. (1985) in Highlights of Condensed two-dimensional at least with is Matter Theory, eds. Fumi, F., Bassani, F. & Tosi, M. (North- system, strong interaction, just Holland, New York), pp. 767-797. a tomographic superposition of effective one-dimensional 6. Matthias, B. T. & Anderson, P. W. (1964) Science 144, 133-141. models, one for each point on the Fermi surface. It was 7. DiSalvo, F. J. & Rice, T. M. (1969) Phys. Today 34, 32. discovered by Luther (27), and recently enlarged on by several 8. DiSalvo, F. J. & Rice, T. M. (1969) Phys. Rev. B 20, 4883. authors, that this is the case for Fermi liquid theory, because 9. Monceau, P. & Ong, N.-P. (1976) Phys. Rev. Lea. 37, 6902. of the exclusion principle's restriction to only forward (non- 10. Gruner, G. (1988) Rev. Mod. Phys. 60, 1129. diffractive) , which is exploited in Landau's theory. 11. Ong, N.-P. & Monceau, P. (1977) Phys. Rev. B 16, 3443-3455. What I have discovered is that the same applies to NFL as well. 12. Jerome, D. et al. (1989) Phys. Scripta T27, 130. As Haldane (28) points out, the Fermi surface can be 13. McWhan, D. B. et al. (1973) Phys. Rev. B 7, 1920. 14. Batlogg, B. (1990) in High Temperature Superconductivity, eds. thought of as the order parameter of a critical point at T = 0, Bedell, K. S., Coffey, J. M., Meltzer, D. E., Pines, D. & Schrief- the excitations as fluctuations of this order parameter, and the fer, J. R. (Addison-Wesley, New York), pp. 37-80. power laws as valid throughout the neighborhood of this 15. Ong, N.-P., Yan, Y. F. & Harris, J. M. (1995) CCASTSymposium critical point (whether they be Fermi liquid or NFL). What on High Temperature Superconductivity and C60, (Gordon & happens as this critical point is approached is that suddenly the Breach, New York), in press. interlayer interactions grow to relevance and cause supercon- 16. Fisk, Z., Sarrao, J. L., Smith, J. L. & Thompson, J. D. (1995) ductivity; but that part of the theory-region B-is not the Proc. Natl. Acad. Sci. USA 92, 6663-6667. subject here. 17. Rice, T. M. (1988) in Frontiers and Borderlines ofMany-Particle The laws are within Physics, eds. Broglia, R. H. & Schrieffer, J. R. (North-Holland, power straightforwardly explained my New York), p. 172. NFL theory: (i) the linear dependence of l/cond on T or cw. 18. Hebard, A. F., Rosseinsky, M. J., Haddon, R. C., Murphy, D. W., This is the decay of the accelerated electron into spinon and Glarum, S. H., Palstra, T. T. M., Ramirez, A. P. & Kortan, A. R. holon. This is not a resistivity process unless something else is (1991) Nature (London) 350, 600-601. added: impurity or phonon scattering of the holons (spinons 19. Fischer, 0. (1981) in Ternary Superconductors, eds. Shenoy, are scattered only by magnetic, time-reverse breaking, impu- G. K, Dunlap, B. D. & Fradin, F. Y. (North-Holland, New rities). This is exactly analogous to phonon resistivity; under York), pp. 303-307. "phonon drag" conditions, phonon scattering is not a resistiv- 20. Anderson, P. W. (1981) in Ternary Superconductors, eds. Shenoy, ity process, but phonons are, normally, scattered and if they are G. K., Dunlap, B. D. & Fradin, F. Y. (North-Holland, New York), pp. 309-311. they control resistivity. So this is called the "holon nondrag" 21. Matthiess, L. R., Gyorgi, E. M. & Johnson, D. W. (1988) Phys. regime. Most Luttinger liquids are in the holon drag regime, Rev. B 37, 3745. which is quite different. The slight deviation of the power from 22. El Azrak, A., Bontemps, N., etal. (1993)1. Alloys Compounds 195, 1 is also predicted by the theory. (ii) Hall angle. A magnetic 663. field rotates only the Fermi surface, so it does not affect the 23. Lieb, E. & Wu, F. Y. (1968) Phys. Rev. Lett. 20, 1445-1448. electron-spinon-holon process and does not cause any addi- 24. Haldane, F. D. M. (1994) in Perspectives in Many-, tional electron decay. Reciprocally, electron decay cannot eds. Broglia, R. A., Schrieffer, J. R. & Bortignan, P. (North- affect the Hall angle. Thus, the underlying spinon-spinon Holland, N.Y. 1994), p. 5. like electron-electron is 25. Dzialoshinsky, I. & Larkin, A. (1973) Sov. Phys. JETP 38, scattering, which, processes, oc2, is 202-208. the Hall angle controlling process. The current is mostly 26. Menyhard, N. & Solyom, J. (1973) J. Low Temp. Phys. 12, carried as a backflow by spinons, not by the holon charge 529-545. carriers, and in fact the holon current will not be colinear with 27. Luther, A. (1973) Phys. Rev. B 19, 320. that of the spinons. 28. Haldane, F. D. M. (1981) J. Phys. C 14, 2585. Finally, the absence of c-axis conductivity is an effect of Z 29. Clarke, D. G., Strong, S. & Anderson, P. W. (1994) Phys. Rev. = 0; the direct, coherent hopping is ineffectual, and in general Lett. 72, 3218-3221. hops will take place incoherently to high-energy states (29). 30. El Azrak, A. & Bontemps, N. (1994) Phys. Rev. B 49, 9846. The proof is a bit subtle but the principle (and the fact) is 31. Harris, J. M., Yan, Y. F. & Ong, N. P. (1992) Phys. Rev. B 46, obvious. 14293. 32. Tanasaki, K., Nakamura, Y. & Uchida, S. (1992) Phys. Rev. Lett. There is no reason to suppose that the NFL phenomenon is 69, 1455. restricted to cuprates, and in fact infrared spectra resembling 33. Cooper, S. L., Nyhus, P., Reznick, D., Klein, M. V., Lee, W. C., Fig. 7 are common to many of the weird materials mentioned Ginsberg, D. M., Veal, B. W., Paulikas, A. P. & Dabrowski, B. earlier. Most of the anomalies have possible explanations; for (1993) Phys. Rev. Lett. 70, 1533. Downloaded by guest on September 27, 2021