Proc. Natl. Acad. Sci. USA Vol. 92, pp. 6668-6674, July 1995 Colloquium 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 liquids 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 energy, and ,u is the difficulties, ofthe renormalized quasiparticle theory ofmetals chemical potential EF. Positive t refers to electron-like prop- ("AGD" or Fermi liquid 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- Feynman diagram 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, Fermi liquid 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 quasiparticles. 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 nature 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 Fermi surface. tum electrodynamics, and it borrows the techniques that were Landau: The dynamics can be completely described at so successful in that theory. In quantum electrodynamics, the low energies 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 renormalization. One defines a propagator or states exactly equal to the number of electrons. Green's function, G(r - r', t - t'), which is the amplitude for Finally (Migdal again), phonons (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 mass 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 matter 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 positrons. 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 phase 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 "Kondo effect," Hall effect 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 superfluidity 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 spin 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) metal. 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 spectroscopy, 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 semiconductors 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 magnetism 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- phonon-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.