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Plasma Oscillations in Heavy-Fermion Materials A Plasma oscillations in heavy-fermion materials A. J. Millis, M. Lavagna, P.A. Lee To cite this version: A. J. Millis, M. Lavagna, P.A. Lee. Plasma oscillations in heavy-fermion materials. Physical Review B: Condensed Matter (1978-1997), American Physical Society, 1987, 36 (1), pp.864-867. 10.1103/Phys- RevB.36.864. hal-01896336 HAL Id: hal-01896336 https://hal.archives-ouvertes.fr/hal-01896336 Submitted on 16 Oct 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW B VOLUME 36, NUMBER 1 1 JULY 1987 Plasma oscillations in heavy-fermion materials A. J. Millis AT& T Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974 M. Lavagna Laboratoire Louis Neel, Centre National de la Recherche Scientifique, Boite Postale No. 166x, 38042 Grenoble Cedex France P. A. Lee Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received 2 February 1987) We calculate the dielectric function of the lattice Anderson model via an auxiliary-boson large-N method suitably generalized to include the eA'ects of the long-range part of the Coulomb interaction. We show that the model exhibits a low-lying plasma oscillation at a frequency m* on the order of the Kondo temperature of the model, in addition to the usual high-frequency plasma oscillation. We also analyze the Anderson model without the long-range Coulomb interaction, computing the Landau parameter Fo„and showing that the model has a zero-sound mode whose velocity we compute. We derive the version of the f sum rule applicable to our model and show that our results satisfy it. Heavy-electron metals are a class of compounds involv- The Anderson model as conventionally formulated does ing rare-earth or actinide elements which have recently not include the (physically necessary) long-range part of been much studied' because at low temperatures their the Coulomb interaction. We therefore add to it a term of properties are, crudely speaking, those of a Fermi liquid of the form mass m* —10 m (m is the usual electron mass) and Fer- H 4' mi temperature T*—10-100 K. Here we report results Coul g 2 of a theoretical study of the small-q limit of the dielectric q&q, function e(q, co). Our principle result is that heavy- where nq is the Fourier transform of the density electron metals should have a plasmon mode at the renor- n; =n„+nf;. We note that the short-range part of the malized plasma frequency co*=T*. The plasmon has Coulomb interaction between the f electrons is already in- low spectral weight, making an order m/m* contribution cluded in the Anderson model, where it produces the to the f sum rule, and is likely to be damped, but perhaps "infinite-U" repulsion leading to the constraint nI; ~ 1. ought to be observable in reflectivity or other measure- However, the long-range part of this interaction is not in- ments. We also relate our results to a previous calcula- cluded, as can be seen from a Gedankenexperiment in tion of the optical conductivity cT(to), and comment on which a few f electrons are moved from one end of the lat- the implication of our results for the physical interpreta- tice to the other, with everything else held fixed. The sum oA' tion of the heavy-fermion compounds. on q in Hc,„~ must be cut at q =q, &&kF to avoid dou- As a model for the low-temperature behavior of heavy- ble counting. In this Rapid Communication we are con- electron materials we use the auxiliary-boson (or "slave- cerned only with asymptotically-long-wavelength proper- boson" ) large-N version of the lattice Anderson model. ties, and will never have to specify q, . The Anderson model is believed to represent the essential The inequality constraint on nf makes the model physics of heavy-fermion materials. The slave-boson, dificult to attack by conventional methods. In the slave- large-% version was devised for the single-impurity An- boson method one introduces a new boson field b; derson model and has been also used to study various representing an empty set of f orbitals on site i, rewrites aspects of the lattice problem. ' For a comparison of the hybridization (Vctf bt+H. c.) and enforces the con- this with other methods, see Ref. 2. straint nf;+ng; =1 via a Lagrange multiplier field A, . To The Anderson model describes a structureless band of use the 1/N expansion one assumes both c and f electrons conduction electrons (operator ck, energy et, ) hybridizing are characterized by a N-fold-degenerate "spin" quan- via a hybridization matrix element V (conventionally as- tum number m, conserved in hybridization and e-electron sumed to be structureless), with a dispersionless band of f propagation, and one rewrites the constraint as nf; electrons at an energy Eo, and subject to the constraint +nt,; =qpN, where qp is regarded formally as independent that the number of f electrons on site i, nI; ~ 1. We mea- of N. (To recover the original Anderson model one sets sure all energies with respect to the chemical potential qp~ 1/N at the end of any calculation. 3) Next, one splits (taken to be zero) and we are interested in the Kondo lim- the boson operators into static and Auctuating parts. Re- it in which —Fp/ppV »1. pp is the c-electron density of taining only the static parts leads to a mean-field theory of states evaluated at sk =0. We assume T =0 throughout. electrons moving in a renormalized band structure in 864 1987 The American Physical Society PLASMA OSCILLATIONS IN HEAVY-FERMION MATERIALS 865 which a dispersionless band of f quasiparticles3 at energy rise to interband transitions. T* & 0 hybridizes via a renormalized hybridization The factor of N in Lc,„~ arises because we consider a a'p && V with the c-electron band. Corrections to the model with —1 electron per spin channel. mean-field theory come from Coulomb interactions and The expressions above are correct only up to terms of interactions between the electrons and the fluctuating relative order (q/kF) . The terms in L,«„ which we have parts of the boson fields. To study these corrections we not explicitly written, include a three-boson interaction use the radial gauge formulation, but we write the elec- (which does not contribute to the order to which we work) trons in terms of the operators d; which diagonalize the with the terms which fix the mean-field parameters, and mean-field theory, and we take the small-q limit in the in- the terms involving band-2 operators which do not con- teraction terms. The model is then specified by the La- tribute at T=O. grangian L LF+Lg+Lq+Lc«~+L„,t, where From Eqs. (1)-(4) one may easily compute the = density-density correlation function X(q, co) by standard LF g d;I [8,+c;(k)]d;k (la) diagrammatic techniques. Details of a similar calculation i,k, m are given in Ref. 10. To leading order in 1/N the only dia- N a(qT — —q' grams which contribute are shown in Fig. 1. They have Lc 2 g Ep) aq+ 2iapoqk 2V q the familiar random-phase-approximation (RPA) form of polarization bubbles connected by interaction lines, which 2Q'p in this case may be either boson propagators [from LI g d t k+qmdl mkaq+~~q Eq. k, q, m Ek (lb)] or the Coulomb interaction. The polarization bub- bles may be of inter- or intra-band type. Note that the 8k — 10'p Gf density operator (and thus the Coulomb interaction) cou- + (d jk~d i k+q ~+ H.c.) crq-' Ek Ek ples to an interband bubble via a vector vertex, but the bo- (lc) son propagator couples to an interband bubble via a scalar vertex. Summing the diagrams we find at small q, Lcoul 4' nqn — g 2 q (id) q(&q, ) Nq Il(q, co) +IIi2(q, co) L co q, 2 2 p 1+(4' /q N)[II(q, ro)+IIP co)] where the d~(2&k and ci&2lk(k) are the operators and en- (q, ergies for the lower and upper bands of the renormalized Here Hi2 is the interband density polarization bubble, band structure; one has II~2(q, pi) —N (pk q) ' g ci(2)(k) =-, [(ck+T ) (+) Ek] k(( kF) and 1 + 1 co —Ek +i6 —co —Ek —I 8' E.-i(ck T*)'+4al . — (4) We have assumed the Fermi level lies in the lower band. and II(q, co) is the intraband density polarization bubble, The dk; are related to the c and operators via f which includes the effects of electron-slave-boson interac- dk; -cos8k(ck; + sin8k; fk;, tions. One finds where II(q, co) -IIp(q, pi)/I +I &(cp)Iip(q, rp), (s) tan8k; -[c;(k) —ck]/ap crp/[c;(k) —T*] . where IIp is the bare intraband polarization bubble; The mean-field parameters and T*, to leading order f(., (k)) -f(., (k+q)) ap in, Ii,(q, ~) - —N . (6) 1/N, and are given T g qp— by Dexp(Ep/ppV ), a'p =qpV (1 nf), and nf =(1+T*/ppV ) '.
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