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Renormalization Group and Fermi Liquid Theory

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Renormalization Group and Fermi Liquid Theory Renormalization Group and Fermi Liquid Theory ACHewson Dept of Mathematics Imp erial College London SW BZ Abstract We give a Hamiltonian based interpretation of microscopic Fermi liquid theory within a renormalization group framework We identify the xed p oint Hamiltonian of Fermi liquid theory with the leading order corrections and show that this Hamiltonian in mean eld theory gives the Landau phenomeno logical theory A renormalized p erturbation theory is develop ed for calcula tions b eyond the Fermi liquid regime We also briey discuss the breakdown of Fermi liquid theory as it o ccurs in the Luttinger mo del and the innite dimensional Hubbard mo del at the Mott transition email address ahewsonicacuk Intro duction The recent theoretical attempts to understand the anomalous b ehaviour of the high temp erature sup erconductors have raised some fundamental questions as to the form of the low energy excitations in strongly correlated low dimensional Fermi sys tems Before the discovery of the high T materials the sup erconductivities in most c sup erconducting metals and comp ounds have b een explained within the BCS theory as instabilities within a Fermi liquid induced by an attractive retarded interaction due to coupling with the phonons Possible exceptions are the heavy fermion sup er conductors such as U P t where purely electronic mechanisms have b een prop osed for the eective interelectron attraction Even in these cases the sup erconducting instability is b elieved to b e one within a Fermi liquid However in their normal state high temp erature sup erconductors are not go o d metals and their b ehaviour app ears to dier from that of a conventional Fermi liquid for a review of the ex p eriments on these systems see This has led to conjectures that the normal state has no well dened quasiparticles at the Fermi level so that it can not b e a Fermi liquid there have b een conjectures that the b ehaviour is b etter describ ed by some form of marginal Fermi liquid or Luttinger liquid More extreme breakdowns of Fermi liquid theory have also b een prop osed where no Fermi surface remains in the usual sense b ecause the imaginary part of the selfenergy is always nite These are still a controversial issues However these theories all prop ose that the sup erconductivity in the high T materials is not an instability in a Fermi c liquid and that we are dealing with a novel situation As the characteristic feature of these materials is the C uO plane the basic question is whether the Fermi liquid breaks down in these two dimensional systems due to strong correlations induced b etween the d electrons at the C u sites In one dimension it is well established that no matter how weak the interelectronic interaction the Fermi liquid theory breaks down and the low energy excitations for short range repulsive interactions are well describ ed by a suitably parametrized Luttinger liquid Andersons theory for the high T materials is based on the assumption that a similar Luttinger liquid state c o ccurs in strongly correlated two dimensional mo dels If this is so then could such a breakdown o ccur for higher dimensional systems with strong correlation Is there a critical interaction strength for the breakdown of Fermi liquid theory for the two dimensional systems These are the sort of questions have led to a reexamination of Fermi liquid theory It is nearly forty years since Landau prop osed his phenomenological Fermi liquid theory and it was very so on after that the approach was veried for mi croscopic mo dels within the framework of manyb o dy p erturbation theory Since that time new techniques for tackling problems in condensed matter physics have b een devised such as the renormalization group approach and others such as b osonization have b een develop ed further These approaches can bring a fresh p ersp ective to the sub ject Whether any of these techniques can provide answers to the question of the b ehaviour of two dimensional Fermi systems cannot b e answered at this stage What is clear however is that they can give new insights which are of value in themselves and provide a new context for addressing these problems The b osonization approach in one dimension gives the essential physics for many interacting fermion mo dels in a very simple way This was shown in the pioneering work of Tomonaga Luttinger Mattis and Lieb then further develop ed by Luther Emery Haldane amongst others Haldane recognized uni versal features in these results and showed that low energy b ehaviour of a large class of one dimensional interacting fermion systems can b e derived from a suitably generalized Luttinger mo del He coined the term Luttinger liquid to describ e these in analogy with concept of a Fermi liquid for higher dimensional Fermi systems More recent b osonization work has b een concerned with showing how this approach can b e generalized to higher dimensions and showing how the Fermi liquid theory can emerge from b osons when they are constrained to the region of the Fermi surface In this article however we will fo cus our attention on the renormalization group approach b oth as develop ed by Wilson in the s to tackle problems of critical phenomena but also as originally develop ed in eld theory QED etc as a re organization of p erturbation theory Recently the Wilson renormalization approach to these fermion mo dels has b een the sub ject of a review by Shankar In this article we cover similar ground but with a dierent emphasis and dierent examples so that there is in detail little overlap b etween the two surveys The basic message however I think is the same that the renormalization group gives us new insights into Fermi liquid theory its stability and the circumstances which may cause it to breakdown We develop a renormalized p erturbation theory which can b e directly related to the Wilson typ e of calculations and also to the microscopic derivation of Fermi liquid theory Before embarking on the renormalization group approach we briey lo ok at the main features of Fermi liquid theory as originally intro duced by Landau The Landau phenomenological theory is based on the assumption that the single quasiparticle excitations at very low temp eratures of an interacting Fermi system are in onetoone corresp ondence with those of the noninteracting system In terms of the one electron states a total energy functional E is constructed of the form tot X X 0 0 0 E E n n n f tot gs 0 0 where E is the ground state energy n is the deviation in o ccupation numb er of gs the single particle state ji with an excitation energy from its ground state 0 value and f is the leading term due to the quasiparticle interactions A free en 0 ergy functional F is constructed from retaining only the rst two terms together with the FermiDirac form for the entropy of the quasiparticles Minimization of F with resp ect to n leads to asymptotically exact results for the thermo dynamic b ehaviour as T H for systems in a normal paramagnetic ground state The reason why the higher order terms in give negligible eects in this limit is b e cause the exp ectation value of n h n i as T and H The eective quasiparticle energy in the presence of other excitations is given by X 0 0 0 i h n f 0 0 From this result asymptotically exact results for the sp ecic susceptibility and other low temp erature prop erties can b e deduced By taking into account scattering of the quasiparticles the low temp erature b ehaviour of the transp ort co ecients can b e calculated and equations for the coherent excitation of quasiparticlequasihole pairs lead to a description of collective mo des such as zero sound Application and further discussion of the Landau phenomenological approach can b e found in ref erences This approach was veried within the framework of manyb o dy p erturbation theory in the early s and detailed derivations of the basic theory can b e found in the pap ers of Luttinger and the b o oks of Nozieres and Abrikosov Gorkov and Dzyaloshinskii Due to the mathematical complexity of the diagram matic p erturbation theory however some of the more intuitive ideas of Landau are lost Here we intend to show that the renormalization group approach can make a useful conceptual link b etween the two approaches For those not familiar with the renormalization group we very brief review of the philosophy of this approach as develop ed by Wilson for tackling the problems of critical phenomena and also for the Kondo problem The renormalization group is a mapping R of a Hamiltonian H K which is sp ecied by a set of interaction parameters or couplings K K K into another Hamiltonian of the same form with a new set of coupling parameters K This is expressed formally by K K RfH Kg H K or equivalently RK K where the transformation is in general nonlinear In applications to critical phe nomena the new Hamiltonian is obtained by removing short range uctuations to generate an eective Hamiltonian valid over larger length scales In the cases we consider here the transformations are generated by eliminating higher energy excited states to give a new eective Hamiltonian for the reduced energy scale for the lower lying states The transformation is characterized by a parameter say which sp ec ies the ratio of the new length or energy scale to the old one such that a sequence of transformations generates a sequence of p oints or where is a continuous variable a tra jectory in the parameter space
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