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 K The transformation is constructed so that it

satises

0 0

K fR Kg R R

The key concept of the renormalization group is that of a xed p oint a p oint K

which is invariant under the transformation

R K K

The tra jectories generated by the rep eated application of the renormalization group

tend to b e drawn towards or exp elled from the xed p oints The b ehaviour of

the tra jectories near a xed p oint can usually b e determined by linearizing the

transformation in the neighb ourho o d of the xed p oint If in the neighb ourho o d of

a particular xed p oint K K K then expanding R K in p owers of K

R K K K L K O K

where L is a linear transformation If the eigenvectors and eigenvalues of L are

O and and if these are complete then they can b e used as a basis for a

n n

representation of the vector K

X

K K O

n

n

n

where K are the comp onents How the tra jectories move in the region of a par

n

ticular xed p oint dep ends on the eigenvalues if we act m times on a p oint K

n

in the neighb ourho o d of a xed p oint K then from we nd

X

m m

R K K K K O

n

n n

n

provided all the p oints generated by the transformation are in the vicinity of the xed

p oint so that the linear approximation remains valid Eigenvalues with

n

are termed relevant and the corresp onding comp onents of K in increase with

n

m while those with eigenvalues termed irrelevant get smaller with m Those

n

to linear order do not vary with m and are marginal The eigenvalues with

n

of the linearized equation lead to a classication of the xed p oints stable xed

p oints have only irrelevant eigenvalues so K unstable xed p oints have one

or more relevant eigenvalues and the tra jectories are eventually driven away from

the xed p oint A marginal xed p oint has no relevant eigenvalues and at least one

marginal one Whether a tra jectory is ultimately driven towards or away from the

xed p oint in this case dep ends on the nonlinear corrections

The application of the renormalization group to critical phenomena is based

on the fact that at a second order phase transition the correlation length b ecomes

innite so that at the critical temp erature the system app ears the same on all

length scales As a result at the critical p oint the parameter changes on applying the

renormalization group transformation at the critical p oint are small and tend to zero

as the length scale b ecomes large compared to the lattice spacing This b ehaviour

corresp onds to the asymptotic approach of the tra jectory to a xed p oint of the

renormalization group transformation For small deviations away from the critical

temp erature this scale invariance no longer holds and under the renormalization

group transformation the system tra jectory is eventually driven away from the xed

p oint For this to o ccur the xed p oint must b e an unstable one For critical

b ehaviour of the form jT T j the critical exp onent can b e calculated from the

relevant eigenvalues of the linearized renormalization group equations ab out this

xed p oint for further details see for example The universality of the critical

exp onents follows when a whole class of systems scale to the same xed p oint and

consequently all have the same critical b ehaviour

In the situations we consider here the renormalization group transformations

are asso ciated with the elimination of higher energy states to generate an eective

Hamiltonian for the lower lying levels If the system do es not undergo a phase

transition or develop an energy gap then we exp ect the physics on the lowest

energy scales not to change signicantly so that we nd a limiting form for the

eective Hamiltonian as the energy scale is reduced to zero This Hamiltonian

must corresp ond to a stable or marginal xed p oint of the renormalization group

transformation It will b e this xed p oint Hamiltonian and the leading correction

terms which determines the lowest lying excitations of the system and hence the

thermo dynamic b ehaviour as T

As one universal form of low energy b ehaviour is that of a Fermi liquid it seems

a natural question to ask whether we can reformulate Fermi liquid theory within the

renormalization group framework The idea of a Fermi liquid xed p oint has b een

used frequently in condensed matter theory see for example but very little work

has b een done in clarifying the concept and relating it to existing phenomenological

and microscopic theory Renormalization group calculations for fermion systems

in which higher energy excitations are eliminated and an eective Hamiltonian ob

tained for the lowest energy scales are dicult to carry out There have b een a some

calculations for translationally invariant systems and lattice mo dels see and

references therein but the most successful calculations of this typ e have b een those

for magnetic impurity mo dels the Kondo and Anderson mo dels for a mag

netic d transition or f rare earth ion in a metallic host These are relatively

simple systems yet with nontrivial physics with a lo cal moment regime at higher

temp eratures and a crossover to Fermi liquid b ehaviour at low temp eratures and

eventually to a spin comp ensated ground state Apart from the extensive renormal

ization group calculations there are p erturbational results as well as an exact

solutions for the thermo dynamics for these mo dels It will b e instructive to lo ok

at the renormalization group results for these systems for which we have a rather

complete quantitative description from a range of theoretical approaches The in

sights gained from these sp ecic examples will enable us to make conjectures and

devise metho ds to apply to a very general class of systems

Fermi liquid theory of impurity mo dels

We base our discussion primarily on the Anderson mo del as it gives a more

general description of a magnetic impurity than the Kondo mo del b ecause it includes

the p ossibility of charge uctuations at the impurity site In its simplest form the

mo del has an impurity d f level taken to b e nondegenerate which is hybridized

d

with the host conduction electrons via a matrix element V When the interaction

k

term U b etween the electrons in the lo cal d state is included the Hamiltonian has

the form

X X X

y y y y

c c H V c c V c c c c U n n

d d d

k k

k d k k d d d

k k

k k

P

where jV j is the function which controls the width of the

k k

k

virtual b ound state resonance at in the noninteracting mo del U In

d

the limit of a wide conduction band with a at density of states b ecomes

indep endent of and can b e taken as a constant For U where

d F d F

is the Fermi level and j j j U j the mo del is equivalent to the

F d F d F

Kondo mo del with an antiferromagnetic interaction J U j jj U j

d F d F

where is the conduction electron density of states

Figure A linear chain form of the Anderson impurity mo del with the impurity at

one end lled square coupled by the hybridization to a tightbinding chain of conduction

states lled circles Iterative diagonalization pro ceeds by diagonalizing a nite chain

adding a further conduction electron site to the chain and rep eating the pro cess

We lo ok rst of all at the renormalization group calculations of Wilson who was

the rst to obtain a precise description of the low temp erature b ehaviour of the mo del

in the Kondo regime where the spin uctuation eects dominate We need not go

into the technical details of the numerical renormalization group metho d he devised

but it will help to have a brief overview to see what was involved The mo del was

cast in the form with the conduction electron states in the form of a tightbinding

chain with the impurity at one end coupled via the hybridization see gure By

iterative diagonalization of chains of increasing length in which a nite numb er of

the low lying states were retained at each stage the lowest excitations on a decreasing

energy scale were obtained the details can b e found in the original pap ers of Wilson

and Krishnamurthy Wilkins and Wilson The transformation that maps

the mo del on one energy scale to one on a reduced energy scale when suitably

scaled is a renormalization group transformation The higher energy excitations

which are eliminated at each stage mo dify the couplings of the mo del In general

further interaction terms not present in the original mo del are intro duced It can

b e shown that the eective Hamiltonian for the lowest energy scales found by this

technique can b e expressed in the same form as the original Anderson mo del but

P

jV j U and with mo died renormalized parameters

k k

k

corresp onding to the eective Hamiltonian

d

X X X

y y y y

c U n n c V c c c H c c V c

d d d e

k k

d d d k k d k

k k

k k

where the energy level is measured with resp ect to the Fermi level

d

For the lo cal moment regime of the mo del there is only one energy scale involved

which is the Kondo temp erature T The imp ortant p oint which makes the renor

K

malized eective Hamiltonian tractable at low temp eratures is that it describ es

excitations from the exact ground state so that the interaction term U b etween ex

citations only comes into play when there is more than one excitation from the

ground state From the renormalization group p ersp ective the interaction terms

are the leading irrelevant corrections to the free fermion xed p oint They tend to

zero under the rescaling of the renormalization group as the low energy xed p oint

is approached T If a single particle excitation or single hole excitation is

created this interaction term plays no role and can b e omitted and the one electron

Hamiltonian remaining can b e diagonalized and written in the form

X

y

H U c c

e

l l l

l

y

and c the corresp onding creation where are the oneelectron energies and c

l l l

and annihilation op erators This is the xed p oint Hamiltonian as T and corre

sp onds to free or noninteracting fermions Comparing this with the Landau theory

we can identify the excitation energy as the energy of the noninteracting quasi

l

particles in the free energy functional Rather than diagonalize explicitly

for the excitation energies we can solve for the Greens function of the quasiparticles

by the equation of motion technique which gives a closed set of equations for U

This gives

G

d

i

d

the quasiparticle energies corresp ond to the p oles of this Greens function The cor

resp onding sp ectral density or quasiparticle density of states can b e straight

d

forwardly b e deduced and is given by

d

d

corresp onding to a Lorentzian resonance the socalled Kondo resonance

For U can b e written using the single particle eigenstates of as a

basis

X X X

y y y

000 0

c c c c c U H c

00 000 0 00

l l e l

l l l l l l l

l

00 000 0

l

l l ll

where

X

y y

c c

l d l

l

As it is the excitations of the interacting system from its ground state that

are describ ed by the eective Hamiltonian it is appropriate to transform this

Hamiltonian to op erators which describ e the single particle excitations

y y

c p p c

l l F

l l l

y y

c h c h

l l F

l l l

where the ground state is such that p ji and h ji The interaction

l l

term has to b e normal ordered in terms of these op erators so that H ji and

e

the Hamiltonian describ es interactions only between excitations from the ground

state In the mean eld approximation the exp ectation value of op erators in the

y y

interaction terms such as p p p p are approximated by

0 00 000

l l

l l

y y y y

00 000 0

hp p p p i hp p ihp p i

0 00 000 00 00

l l ll

l l l l

l l l l

The quasiparticle energy of the Landau theory in the presence of other excitations

can b e identied as the eective one particle energy in this approximation

X

0 0

j h n i j U j j

l l l l

l

0

l

y y

0 0 0 0

as j j and n hh h i for where h n i hp p i for

0 0 0 0

F l F l l l l

l l

l l

is prop ortional to N where N is the numb er of sites b ecause the scattering

s s

p otential is due to a single impurity the energy shift in is of the order N

s

y

p i as T the quasiparticle interaction do es not contribute to As hp

l l

the linear term in the sp ecic heat which can b e calculated from the noninteracting

quasiparticle Hamiltonian Hence the impurity sp ecic heat co ecient is

imp

given by

k

B

imp d F

where is the impurity quasiparticle density of states dened in equation

d

in calculating the susceptibility the quasiparticle interaction has to b e taken into

account Using the mean eld approximation for this interaction given ab ove the

impurity susceptibility can b e deduced and expressed in the form

g

b

U

d F d F imp

for a at wide conduction band The total charge susceptibility at T

c

dn d where n is the exp ectation value of the total numb er op erator for the

F

electrons can b e calculated following precisely the same argument The result for

the impurity contribution is

U

cimp d F d F

eliminating the term in U b etween and and the term in using

d F

gives the well known Fermi liquid relation for the impurity mo del relating

imp imp

and

cimp

imp imp

cimp

g k

B

B

The interesting physics of this mo del o ccurs in the strong correlation regime when

U is large and is well b elow the Fermi level so that the d electron at the impurity

d

is lo calized and has an o ccupation numb er n This is the lo cal moment regime

d

in which the mo del maps into a Kondo mo del At high temp eratures T T

K

where T is the Kondo temp erature there is a Curie law susceptibility asso ciated

K

with the lo cal moment but with logarithmic corrections lnT T which reduce the

K

eective moment In the low temp erature regime is valid for T T when the

K

impurity moment is fully screened giving a nite susceptibility at T The charge

susceptibility on the other hand must tend to zero in the strong correlation limit as

the d electron at the impurity is lo calized The condition is achieved in the

cimp

quasiparticle picture by the interaction term U which selfconsistently constrains

the manyb o dy resonance at the Fermilevel so that the impurity o ccupation is

maintained with n This was rst p ointed out by Nozieres This condition

d

can b e used to deduce U In the case of particlehole symmetry and the

d

resonance in the quasiparticle density of states is p eaked at the Fermi level and

We then nd on substituting into equations and

d F

U T

K

where the Kondo temp erature which is the only relevant energy scale for the mo del

in the lo cal moment regime is dened by g T

imp B K

4T /π∆ 1.0 K

∆∼/∆

U~/π∆

0.0 0.0 1.0 2.0 3.0 4.0

U/π∆

Figure A plot of the renormalized parameters U and for the symmetric Anderson

mo del in terms of the bare parameters U and In the comparison of these parameters

with T for U the value of T is given by

K K

These parameters give for the or Wilson ratio R

k

imp imp

B

R

g

B imp imp

and so is enhanced over the free electron value of unity where and are the

susceptibilities and sp ecic heat co ecients for the conduction electrons alone The

argument used here is essentially a reformulation of the one originally given by

Nozieres The result R was rst derived by Wilson from his numerical

renormalization group results

As the Anderson mo del is integrable and exact solutions exist for the thermo dy

namic b ehaviour it is p ossible to deduce the renormalized parameters exactly over

the full parameter regime For the symmetric Anderson mo del the parameters U

and have b een calculated using the exact Bethe ansatz results from equations

and These are shown in gure for the symmetric mo del n plotted

d

as a function of U At weak coupling U U as one would exp ect and U and

are indep endent energy scales At strong coupling there is only the single energy

scale T given by

K

U U

e T U

K

U

with U and given by Using these equations the form of the noninteracting

quasiparticle density of states given by normalized by for vari

d

ous values of U is shown in gure illustrating the exp onential narrowing of the

resonance at the Fermi level in the Kondo regime U as U is increased

0.40

0.30 ω)

( 0.20 d ∼ ρ

0.10

0.00 -5.0 -3.0 -1.0 1.0 3.0 5.0

ω

Figure The quasiparticle density of states normalized by as a function

d

of U for U in order of decreasing width and for the particle

hole symmetric Anderson mo del

This approach can b e used for other magnetic impurity mo dels such as the N

fold degenerate Anderson mo del U and the n S multichannel Kondo

mo del Though explicit renormalization group calculations do not exist for

most of these mo dels one can conjecture the form of the quasiparticle Hamiltonian

to describ e the Fermi liquid regime and deduce the Fermi liquid relations These

relations can b e checked with the exact results known from either the Bethe ansatz

or microscopic Fermi liquid theory For these mo dels there are more renormalized

parameters to take into account but in the strong correlation regime they can all

b e expressed in terms of the Kondo temp erature We give an example in gure

where we plot the quasiparticle density of states in the Kondo regime for the N fold

degenerate mo del for dierent values of N The case N corresp onds to

the Kondo resonance at the Fermi level of the large U Anderson mo del as shown in

gure With increasing N this resonance narrows and moves towards T ab ove

K

the Fermi level b ecoming a delta function at T in the large N limit Much of

K

the low energy and low temp erature physics of this mo del can b e deduced from this

asymmetric form for the quasiparticle density of states

0.8

0.6 8

6 0.4 ρ(ω)

4

3 0.2 N=2

0.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

ω/ΤΚ

Figure The quasiparticle density of states for the Nfold degenerate Anderson

mo del U in the Kondo regime for N as indicated as a function of

T where T is the Kondo temp erature

K K

What general conclusions ab out Fermi liquid theory might we b e tempted to

draw from these explicit renormalization group calculations First of all there

is strong parallel with the original phenomenological approach of Landau The

renormalization group calculations have given us an explicit Hamiltonian to describ e

the low energy excitations The mean eld exp ectation value of this Hamiltonian

corresp onds to the exp ectation value of the Landau total energy functional The

fact that this energy functional had to b e expressed in terms of the deviations in

the o ccupations of the states from their ground state values has its parallel in the

renormalization group xed p oint Hamiltonian the Hamiltonian had to b e normal

ordered b ecause it only describ es the excitations from the ground state The fact

that the mean eld theory is asymptotically exact as T is b ecause the deviation

in the o ccupation of these states tends to zero in this limit this corresp onds to a

low density of excitations For the mean eld treatment of the Hamiltonian to b e

valid there must b e no singular scattering b etween the quasiparticles at low energies

Such singular scattering o ccurs in one dimensional mo dels and in higher dimensions

with attractive interactions We will discuss this more fully later

A microscopic derivation of these Fermi liquid relations for the symmetric An

derson mo del has b een given in a series of pap ers by Yamada and Yosida Their

calculations were based on an expansion in p owers of U for the symmetric mo del

Though it is not p ossible to sum explicitly the dominant terms in the strong correla

tion limit in this mo del exact relations can b e deduced corresp onding to Fermi liquid

theory in terms of the lo cal selfenergy evaluated at the Fermi level

its derivative and the irreducible four p oint vertex Setting up a corresp ondence

with the ab ove we can identify the renormalized parameters and U in terms

d

of these With z the wavefunction renormalization factor given by

z

where prime denotes a derivative with resp ect to evaluated at the Fermi level

the renormalized parameters are given by

z z U z

d d

0

is the irreducible four p oint vertex function The selfenergy has where

two arguments set to zero to indicate that it is evaluated not only for but

also at zero temp erature and in zero eld

We have succeeded in showing that from the Wilson normalization group ap

proach for the impurity mo del a quasiparticle Hamiltonian can b e derived corre

sp onding to a renormalized Anderson mo del and that this in the mean eld ap

proximation gives the Landau total energy functional Both the Landau and the

renormalization group approaches give exact results as T If we try to extend

the renormalization group calculations b eyond the Fermi liquid regime by gener

ating an eective Hamiltonian over a higher energy range then further and more

complicated interactions will come into play as the renormalization group tra jecto

ries are no longer under the exclusive inuence of the low energy xed p oint On

higher energy scales the eective Hamiltonian approach only b ecomes useful when

the tra jectories once again b ecome dominated by a particular xed p oint For the

crossover regime the eective Hamiltonian approach is not feasible and we have to

rely on explicit numerical renormalization group calculations of the energy levels

for predictions The sp ectral density of the d electron Greens function over

d

the full energy range for the symmetric Anderson mo del as calculated by the nu

merical renormalization group is shown in gure The quasiparticle density

of states only describ es the small region near the Fermilevel the extra p eaks at

U are asso ciated with the broadened atomic p eaks The magnetic impu

rity problem is rather a sp ecial case in that we have a numerical renormalization

results over all the relevant energy scales As such calculations are dicult to carry

out in general for other interacting fermion systems it is worth while to lo ok at

another renormalization pro cedure in which we make a p erturbation expansion in

terms of the fully dressed quasiparticles This corresp onds to a reorganization

of p erturbation theory in close analogy with that develop ed originally for quantum

electro dynamics where the expansion is in terms of the particles with their observed

masses and charges interaction strengths ie fully dressed In the eld theory case

the reorganization of the p erturbation theory was a necessary one to eliminate the

ultraviolet divergences and obtain nite results Due to the nite upp er cutos in

condensed matter physics problems such a reorganization of p erturbation theory is

not a necessary on one in order to get nite results nevertheless it makes a lot of

sense to work with the fully dressed quasiparticles particularly for systems where

the renormalization eects are very large such as in heavy fermion systems What

is more it integrates the microscopic p erturbation theory derivation of Fermi liquid

theory with the more intuitive Landau approach

1.5 U/π∆ = 0 U/π∆ = 1.5 U/π∆ = 2 U/π∆ = 4 U/π∆ = 6 1.0 ω) ( d π∆ρ

0.5

0.0 -20.0 -10.0 0.0 10.0 20.0

ω/∆

Figure The sp ectral density for the d electron Greens function of the

d

symmetric Anderson mo del for values of U as indicated see reference

A renormalized p erturbation approach

We continue with the Anderson mo del and consider the retarded one particle double

time Greens function for the delectron expressed in the form

G

d

i

d

P

where was dened earlier and P jV j We continue to

k k

k

work in the wide band limit where is indep endent of and The

function is the prop er selfenergy within a p erturbation expansion in p owers

of the lo cal interaction U We will need the corresp onding irreducible four p oint

0

which is a sp ecial case of the more general irreducible vertex function

0 00 000

with four p oint vertex function

and Our aim is to reorganise this p erturbation expansion into a more

convenient form to consider the strong correlation regime which corresp onds to a

mo del with U large at low temp eratures and with weak magnetic elds

Our rst step is to write the selfenergy in the form

rem

rem

which is simply a denition for the remainder selfenergy Using this ex

pression the Greens function given in equation in the wide band limit can b e

written in the form

z

G

d

i

d

We assume that the general theorem of Luttinger that the imaginary part of

vanishes as at so that z is a real quantity The renormalized

quantities are dened by and the renormalized selfenergy by

rem

z

The next step is to intro duce rescaled creation and annihilation op erators for the

delectron via

p p

y y

c z c z c c

d d d d

We can now rewrite the Hamiltonian in the form

H H H

qp c

where H will b e referred to as the quasiparticle Hamiltonian which can b e written

qp

I

as H H The Hamiltonian H describ es noninteracting particles and is given

qp qp qp

by

X X X

y y y y

c c c c V c V c c H c

d

k k

k d d k k d d qp

k k

k k

I

and H is the interaction term

qp

I

H U n n

d d

qp

with U given by The second term in equation will b e known as the counter

term and takes the form

X

y

H c c n n

c d d

d d

where and are given by

z z U

Equations are simply a rewriting of the original Hamiltonian so

equation is an identity

We note that by construction the renormalized selfenergy is such that

so that O for small on the assumption that it is analytic at

As we also have

0 0

U

To develop a theory appropriate for the low temp erature regime we follow the

renormalization pro cedure as used in quantum eld theory so that we can make a

p erturbation expansion in terms of our fully dressed quasiparticles see for instance

reference We take our renormalized parameters and U as known and

d

reorganise the p erturbation expansion in p owers of the renormalized coupling U

I

The full interaction Hamiltonian is H H The terms and z are formally

c

qp

expressed as series in p owers of U

X X X

n n

n n n n

z U U z U

n n n

n n

n

The co ecients and z are determined by the requirement that condi

tions and are satised to each order in the expansion The p erturbation

expansion is ab out the free quasiparticle Hamiltonian given in equation so that

the noninteracting propagator in the expansion for the thermal Greens function is

G i

n

d

i isgn

n d n

where n and k T The retarded function is deduced by

n B

analytic continuation

We prop ose to make a direct contact with b oth the Landau phenomenological

and the microscopic formulations of Fermi liquid theory The o ccupation numb er of

the dlevel can b e deduced from the Friedel sum rule which takes the form n

d

at T and has the form

H

d

n H tan

d

in terms of the selfenergy in a nite magnetic eld H for the expansion

in p owers of U for the bare Hamiltonian It takes the same form for the

renormalized expansion

H

d

n H tan

d

This follows directly from by writing it in terms of and from equation

d

as the common factor of z in the argument of cancels The quasiparticle

interaction plays no role as T and H as and in this limit

n corresp onds to the noninteracting quasiparticle numb er As the eects of the

d

quasiparticle interactions go to zero as T due to the cancellation with the

counter term giving and the sp ecic heat co ecient of

the impurity is due to the noninteracting quasiparticles so we obtain the free

imp

electron result

Other thermo dynamic results for the low temp erature regime can b e obtained

from the lowest order term in the renormalized p erturbation for the tadp ole

diagram shown in gure a which gives

H T U n H T n

d d

There is no wavefunction renormalization to this order so z and also to this

order U and U n The complete cancellation by

d

the counter term only o ccurs for H and T The impurity spin susceptibility

at T of the impurity to rst order in U can b e calculated from g n n

B d d

by substituting the selfenergy from into equation and then dierentiat

ing with resp ect to H to give the earlier result and similarly for the charge

susceptibility It is not obvious that the results to rst order in U for these

quantities should b e exact However there are renormalized Ward identities

which can b e derived from charge and spin conservation

U

d

h

where h g H The spin and charge susceptibilities can b e derived from the

B

exact relations

g g

B B

h U

imp d d d

and

U

cimp d d d

using and conrming that the earlier results These relations show

that there are no higher order contributions in the renormalized expansion in U for

these quantities b eyond rst order

(a) (b)

Figure The two lowest order diagrams in the renormalized p erturbation theory

Exact results for the impurity Greens function to order follow from the

calculation of the second order diagram for shown in gure b There is no

second order counter term contribution to this diagram from as There

is a contribution to z which is required to eliminate the contribution from the linear

term in to this order and this is given by

z

Calculation of this diagram to order gives

U

Im O

The sp ectral density of the noninteracting renormalized dGreens function de

scrib es the Kondo resonance and the terms in give the low frequency

corrections to this picture

There is a temp erature dep endent contribution to T to rst order in U

given by For the particlehole symmetric mo del and more generally in the

Kondo regime n this vanishes as n T n and the leading

d d d

order temp erature dep endence arises from the second order diagram b This gives

the result

U k T

B

Im T OT

These results can b e used to calculate the leading order T contribution to the

impurity conductivity T which is given by

imp

k T

B

T R OT

imp

where R is the Wilson ratio given by

R U

This conductivity result was rst derived by Nozieres for the Kondo regime and

for the more general case by Yamada The formula for the Wilson ratio is a

more general one than that given earlier and go es over to R in the Kondo

regime on using the relation b etween the renormalized parameters Hence all

the basic Fermi liquid results can b e obtained within the renormalized expansion

up to second order in U The wavefunction renormalization factor z is required to

relate the bare sp ectral density to the quasiparticle density of states

d d

but do es not enter explicitly the calculation of the thermo dynamics which can b e

expressed entirely in terms of the renormalized parameters and U

d

We see that the eects of the counter terms play no really signicant role in

the Fermi liquid regime The renormalized p erturbation theory with the counter

terms however go es b eyond the Fermi liquid regime In this p erturbation theory

nothing has b een omitted so that in principle calculations can b e p erformed at high

temp eratures and high elds allowing the bare particles to b e seen If the primary

interest is in these regimes then the more appropriate starting p oint is the mo del

in terms of the bare parameters rather than the renormalized ones It should b e

p ossible to estimate the renormalized parameters from the bare ones by a separate

variational calculation for the single and two particle excitations from the ground

state In principle it is p ossible from the results of the renormalized p erturbation

theory to estimate the bare parameters from the renormalized ones by inverting

and

z z U U z

d d

where and z are implicit functions of and U In the very weak coupling

d

limit U z and and U U as we can seen in

d d

the results for the symmetric mo del in gure The most useful regime for the

renormalized approach however should b e in calculating corrections to Fermi liquid

theory The calculation with counter terms gives a systematic pro cedure for taking

such corrections into account

Renormalized Perturbation Theory for Trans

lationally Invariant Systems

The renormalized p erturbation approach describ ed in the previous section gives us

a very clear picture of the physics of the low temp erature regime for a magnetic

impurity as describ ed by the Anderson mo del It combines elements of the renor

malization group microscopic and phenomenological Fermi liquid approaches as

well retaining many of the more intuitive elements of the original Landau theory In

this section we investigate whether it is p ossible to generalize the approach to a sys

tem which has translational invariance and deal with the complications which arise

when the selfenergy dep ends on the momentum vector k as well as the frequency

We consider a quite general mo del of fermions in states classied by momenta k

and spin describ ed by a Hamiltonian

X X

y y y

c c H V qc c c c

k

k k k

k k q k q

k k k

where k m with m the mass and an interaction V q which is a function of

k

the momentum transfer q It will b e useful to express the Hamiltonian in a more

general form

X X

y y y

c c k k k k c c c c H

k

k k k

k k k

k ks s

where the interaction is antisymmetric under the exchanges k k and

k k For translational invariance and spin conservation we have

k k k k

For a spin indep endent interaction dep ending only on the momentum transfer q as

in with V q V q then the antisymmetrized form is

V k k V k k k k k k

We can then write as

X

y

c c H

k

k

k

k

X

y y

c c c k k k q k qc

k k

k q k q

k k q s

The interacting one electron Greens function can b e expressed in the usual form

G k

k

k

where k is the corresp onding selfenergy To follow our previous prescription

we expand the selfenergy in a Taylors series to rst order ab out the Fermi surface

and keep the remainder term

k k

F F F F

rem

k k k k k

F F F F

k

where we have assumed jk j We can rewrite this as

k

z

G k

k

k

with z the wavefunction renormalization factor on the Fermi surface given by

z

k

F F

and with the renormalized energies given by

k

F F

z k k k

k k F F F F

k

Similarly to we dene a renormalized selfenergy by

rem

k z k

with measured relative to the Fermi level We intro duce rescaled op erators

p p

y y

z c z c c c

k k

k k

and dene a quasiparticle Greens function

G k

k

k

The full irreducible renormalized four vertex is dened by

k k k q k q

z k k k q k q

So far every step has b een an obvious generalization of the impurity case When it

comes to dening the renormalized interaction we have to b e a little careful Due

to rep eated particlehole scattering the vertex has a singularity such that there is

no unique way of taking the limits and q We leave this problem for

the moment and assume that we can nd a suitable antisymmetrized renormalized

k k k q k q and hence construct a quasi quasiparticle interaction V

particle Hamiltonian

X

y

c c H

qp

k

k

k

k

X

y y

V k k k q k qc c c c

k k

k q k q

k k q s

As earlier the ground state is dened as the vacuum state with no excitations and

the Hamiltonian has to b e normal ordered and expressed in terms of quasiparticles

and quasiholes so that the interaction only comes into play when more than one

excitation is created from the interacting ground state Following the steps we

used in the previous section we write our original Hamiltonian in the form

H H H

qp c

where H is the Hamiltonian for the counter terms and takes the form

c

X

y

a b

c k k c H

F c

k

k

k

X

y y

c

k k k q k qc c c c

k k

k q k q

k k q s

where

k

F F

b a

z k z

F F F

k

and

c

k k k q k q

k k k q k q k k k q k q z V

We can dene a renormalized p erturbation expansion as earlier in p owers of V with

the co ecients of the counter terms and the z factor chosen to satisfy the conditions

k

F F

k k

F F F F

k

for zero eld and at zero temp erature These conditions reect the fact that the

quasiparticle energies and Fermi level used in the quasiparticle Hamiltonian are al

k k k q k q ready fully renormalized We now consider how to cho ose V

c

k k k q k q In the renormalized and how to x the counter term

k k k q k q is p erturbation approach the renormalized interaction V

a matter of choice b ecause it is added and subtracted but there should b e an op

timum choice for calculations of the low temp erature b ehaviour as there was in the

impurity case An obvious choice is

k k k q k q k k k q k q V

which is the quasiparticle scattering function Its antisymmetry follows from its

denition When we come to evaluate the lowest order Hartree diagram of the

renormalized theory this has to b e evaluated in the zero momentum exchange limit

q For a more general retarded dep endent interaction this diagram has to

b e evaluated in the limit q followed by there is a small imaginary

contribution to asso ciated with the time ordering for this diagram Rep eated

quasiparticlequasihole scattering gives a singular contribution to the retarded in

teraction and it is imp ortant to take this into account b efore we take the limits

so that we can take them in the correct order Similarly the Fo ck term which is

evaluated for q k k followed by also has a singular contribution from

quasiparticlequasihole scattering

kk -qk k -q k k +q 1 2 1 2 1 1 +

k +q kk +q kk -q k 1 2 1 2 2 2

k k -q 1 2

k +q k

1 2

Figure Particlehole diagrams corresp onding to the integral equation The

unlled square vertex corresp onds to U k k k q k q and the square lled

ph

vertex to k k k q k q

If we denote the renormalized vertex corresp onding to the sum the quasiparticle

ph

k k k q k q then we obtain quasihole series shown in gure by

the integral equation

ph

k k k q k q k k k q k q U

Z

X

00

ph

k q k k k q dk k k k q k qF k q U

0 00 0

0 00

Z

X

00

U k k k q k k k qF k k k q

0

0 00

ph

k k k q k k q k dk

00 0

where

k k k q k q U

c

k k k q k q k k k q k q V

and the propagator for the quasiparticlequasihole pair F k q is given by

f 00 f 00

k k q

F k q

00 00

V i

k k q

where V is the volume If we take the limit in then as do es not

vanish in this limit it is clear that the counter terms must b e nonzero as we have

implicitly already taken some of these contributions to k k k q k q

into account We have to cho ose the counter term so that at T we satisfy the

condition expressed in equation This implies

V k k k q k q U k k k q k q

Z

X

0 00

k k k q k qF k q V k q k k k qdk U

00 0

0 00

Z

X

00

U k k k q k k k qF k k k q

0

0 00

00

k k k q k k q k dk V

0

For a given V k k k q k q let us denote the solution of for

U k k k q k q at T by U k k k q k q The antisymme

k k k q k q with resp ect to the transformation k q try of U

k q or equivalently k k follows from equation

k k k q k q can b e We can now show that in the limit q U

k k which is the identied as the Landau quasiparticle interaction function f

more general form than that intro duced in which takes account of the p ossibility

of spin dep endent interactions

The contribution to the direct term in the integral equation has a p ole

arising from F k q for small q and At T and as q we have on

continuing to real frequencies i

n q v

F

F k q jk j k

F

n q v

F

where v k m and n k jk j

F F

We can follow the standard treatment given in reference chapter and

ph

k q near the k q k k k q k q by represent

p ole If we keep only the dominant terms for small q and we can cancel the

k q and then write common factor

n q

n k q

n q v

F

where n is a unit vector in the direction of k so that for q we nd

Z

k

F

n q v n n q k k k k n d U

F

where the integration is over the solid angle Equation corresp onds to the

Landau equation for the zero sound collective mo des see so it enables us to make

the identication

f k k U k k k k

Figure The HartreeFo ck diagram in the renormalized p erturbation theory The

vertex corresp onds to U k k k q k q and the double line represents the

dressed propagator There is also a counter term arising from the rst term in

In the renormalized p erturbation expansion we should include the more general

vertex with the rep eated quasiparticlequasihole scattering given by in evaluat

ing the Hartree Fo ck diagram gure so that we take account of all the singular

contributions in the limit We then get the correct results when we take the

limits q q k k followed by Evaluation of the HartreeFo ck

diagram is equivalent to taking the mean eld approximation on the quasiparticle

Hamiltonian

X

y

H c c

qp

k

k

k

k

X

y y

k k k q k qc U c c c

k k

k q k q

k k q s

This gives the total excitation energy in the form

X X

n n k k k k n U E E

tot gs

k k k k

k k k

when we have spin indep endent interactions This equation can b e identied with

k k k k k k U the Landau equation as f

We can cho ose to work with the quasiparticle Hamiltonian with the interac

k k k q k q This has the advantage that the mean eld theory tion U

corresp onds to the Landau total energy functional as in the impurity case and it

has no explicit interaction counter term when the low energy collective quasiparticle

quasihole excitations are taken into account It satises the conditions for the opti

mum choice of the quasiparticle Hamiltonian

k k k k k k q k q and U The two scattering functions V

k k k q k q are related by the integral equation for T with U

k k k q k q It follows from the denition that in q k q U

k k q k k q is the forward scattering function for the q limit V

k k two quasiparticles in the Landau theory which we denote by A

k k k k k k V A

If we take the limit q in equation however the equation do es not reduce to

the usual one see relating A k k and f k k due to the presence

of the exchange term In this resp ect this derivation diers from the standard

derivations of the microscopic Landau theory However equation would app ear

to b e more satisfactory b ecause it is applicable for general q and includes the singular

contribution at q k k arising form the p ole in the exchange term in at

n k k qv

F

To summarize the situation so far

i We identify the quasiparticle Hamiltonian with the interaction U k k k

q k q as the Hamiltonian which determines the low energy excitations of a Fermi

liquid We conclude that the mean eld approximation to this mo del corresp onds to

the Landau total energy functional This Hamiltonian has nonforward scattering

terms q which do not enter the Landau theory but these do not contribute

at the mean eld level For low energy quasiparticlequasihole pairs in the region of

the Fermi surface there is only a rather restricted phase space for such scattering In

three dimensions with a spherical Fermi surface and states k k close to the Fermi

surface b eing scattering into k k then k and k can only lie on or near the circle

on the Fermi surface where the plane p erp endicular to k k and which passes

through k and k intersects the Fermi sphere When the scattering is restricted

k k k k can b e expressed as U to states on the Fermi surface U

where is the angle b etween the wavevectors k and k and the angle b etwen

the plane containing the vectors k k and that containing k k There have

b een attempts to write down an eective Hamiltonian for interacting quasiparticles

with q dep endent scattering and parameters chosen so as to repro duce the Fermi

liquid theory for forward scattering see for instance However we can see from

the theory develop ed here the limitations and inconsistencies of any such approach

which do es not take account of the full Hamiltonian and include the counter terms

ii We conclude also that H given by is the Hamiltonian for the Fermi

qp

liquid xed p oint plus leading corrections that we would obtain if we could carry

out fully the sequence of renormalization group transformations to remove the higher

energy excitations This is in analogy with the impurity case where we could show

this directly The mean eld approximation for this Hamiltonian like that of the

impurity xed p oint Hamiltonian gives the Landau Fermi liquid theory The molec

ular eld nature of the Landau phenomenological theory has b een emphasized in the

review of Leggett here we can give it a more precise interpretation within the

framework of the renormalization group

This form for the xed p oint Hamiltonian is in general agreement with the calcu

lations of Shankar based on a p erturbative renormalization of diagrams up to one

lo op level The leading corrections to the free particle Hamiltonian for translation

ally invariant systems were found by Shankar to b e marginal whereas the leading

corrections in the Wilson impurity calculation discussed in section were the

leading order irrelevant ones This is just a reection of the fact that for an im

purity problem we are interested in the impurities eects which after summation

scale indep endently of the size of the system and which therefore b ehave as V

where V is the volume of the system

iii The leading order thermo dynamics as T can b e calculated either

by normal ordering the quasiparticle Hamiltonian ignoring the counter terms and

taking the mean eld theory or by selfconsistently taking into account the rst order

Hartree Fo ck diagram together with the counter term As in the impurity case

the counter terms play no role in the Fermi liquid regime other than subtract o the

ground state exp ectation values For calculations b eyond the very low temp erature

or low frequency regime the full Hamiltonian with the counter terms must

b e used This way of reorganising the p erturbation expansion closely parallels the

renormalized prescription used in p erturbative eld theory Here the approach is

rather more similar to that for the eld theory than quantum electro dynamics

b ecause we do not have the problems asso ciated with gauge elds In the eld

theory case there is a choice ab out which p oint to make the renormalization and

this freedom is exploited in setting up the CallenSymanzik renormalization group

equations This freedom do es not seem to extend to the situation considered here

b ecause if we expand ab out any p oint other than one on the Fermi surface the z

factor will in general b ecome complex this would seem to preclude the setting up

CallenSymanzik typ e of renormalization group equations in this context

This reformulation of Fermi liquid theory as a renormalized p erturbation the

ory links the renormalization group microscopic Fermi liquid theory and the phe

nomenological theory of Landau within a single theoretical framework There should

b e scop e for its further development and application to a range of physical systems

Breakdown of Fermi Liquid Theory

Most metallic systems are not Fermi liquids at very low temp eratures b ecause at

some critical temp erature they undergo a phase transition to a magnetic sup ercon

ducting or some other form of ordered state Nevertheless if this transition o ccurs

at very low temp eratures it can usually b e analyzed as an instability within a Fermi

liquid such as in the BCS theory of sup erconductivity In the case of sup ercon

ductivity the quasiparticle scattering b ecomes singular at the critical temp erature

T signalling the onset of the sup erconductivity if there is an attractive interaction

c

b etween the quasiparticles in one of the particleparticle scattering channels no

matter how small More commonly as for most magnetic transitions there is some

critical value of the interaction strength b etween the quasiparticles which has to

b e exceeded if the scattering b etween the quasiparticles b ecomes singular If quasi

particles are assumed to exist in one dimension the scattering is singular whatever

the magnitude nonzero or sign of the interactions and there is no nite critical

temp erature indicating the breakdown of the quasiparticle concept in this case

As mentioned earlier Haldane has shown that for one dimensional systems with re

pulsive interactions the b ehaviour at low temp eratures can b e describ ed within the

framework of a Luttinger mo del This would seem to preclude any description

in terms of the renormalized p erturbation theory that we intro duced in the earlier

sections as we assumed the selfenergy to b e analytic at to give a nite wave

function renormalization factor However we show here that this problem can b e

circumvented at least for the spinless Luttinger mo del by making our expansion

at nite and combining it with a form of p o or mans renormalization The cor

rect form for the Greens function is obtained with the exp onents of the singular

terms evaluated p erturbationally A form of renormalized p erturbation theory with

counter terms can then b e set up for the regular contributions to the selfenergy

We give an outline of the approach

In the Luttinger mo del there are two branches of the conduction states a and

b with Fermi p oints at k and k and energies relative to the Fermi energy of

F F

v k k and v k k resp ectively The Hamiltonian of the mo del is

F F F F

X X X

y y y y

c c v k k c H v k k c c c c v q c

0 0

F F F F ij

bk bk ak ak ik jk ik q jk q

k k k q ij ab

where v q is the interaction which is symmetric with resp ect to i and j In this

ij

version of the mo del the numb er of electrons of each typ e are conserved We take

a cuto of D on the momentum transfer q jq j D and a cuto on the bands

of W so that jk k j W for the a band and jk k j W for the b band with

F F

W D The Luttinger mo del is an approximation for a one dimensional system

of interacting fermions in which the disp ersion of k is linearized ab out the two

p oints of the Fermi surface

The corresp onding Greens functions can b e written in the form

G k

i i

v k k

F i i i

where k k k and k k k

a F b F

We now reduce the cuto D by D and separate the selfenergy into two parts

rem D D

k k k where k is the contribution from

i i i i i

i i i

rem

terms in which there is any scattering from the band edge region and k

i

i

D

is the remaining part For the selfenergy k we expand in p owers of

i

i

k D and keep the leading terms This should b e sucient for calculating the

i

D

b ehaviour in the region k D If k a D b D k

i i i i i

i

O k then for this region we can write

i

z D

i

G k

i i

v k k

F i i i

rem

where z D b D k z D k and a D is ab

i i i i i i i

i

sorb ed a a renormalization of k and the Fermi velo city v Initially we calculate

F F

D

k to second order in p erturbation theory and justify this later We do not

i

i

attempt a complete solution at this stage but just try to pick up the most imp ortant

contributions For the system with a reduced cuto we can set up a new p ertur

bation expansion for the renormalized selfenergy k using the renormalized

i i

parameters This can b e continued as long as all the steps in the calculation are valid

so generating a set of renormalization group equations The rst order p erturbation

terms only renormalize the chemical p otential so we concentrate on the second order

terms for the a branch which corresp onds to gure b taking v q so that

ii

the a propagator scatters only with a particlehole pair in the b branch This gives

Z Z

v q dq

dk f k q f k

ab

b b

k nq f k q

n a a

a

v i q k

F n a

where we have now scaled the k s to absorb the Fermi velo city v and nq is the

F

Bose distribution function We integrate over k for T to obtain

Z

q k q q k q

a a

v q q dq k

n a

ab a

v i q k

F n a

We take account of the momentum exchange terms that fall within the band edge

It helps at this stage if we distinguish the upp er and lower band edges taking the

running values to b e D and D The second order contribution due to scattering

in the band edges is given by

k D k D

a a

D D

k D D k

a a

D v and we have sp ecialized to the case k and con where v

i F a i

ab

tinued to real frequencies To eliminate a step function we have also displaced D

by the transformation to D k D We can set up scaling equations by ex

a

panding to order D D the most imp ortant one b eing that for the wavefunction

renormalization factor z which is given by

a

Z Z

D D

dD dD

D D lnz

a

j k j j k j

D D

a a

where is a suitable constant sub ject only to the condition Due to the

dierent regimes of validity of the expansions used the lower limits dier in the two

cases

There is also the p ossibility of a renormalization of the running eective inter

action to consider There are two second order corrections for the a part of the

i

interaction vertex for the simple mo del with scattering b etween the two branches

only one arising from the diagram in gure and the other from the z factor which

a

in the renormalized expansion is absorb ed into the vertex and similarly for the b part

of the vertex It is straightforward to show that these two corrections cancel and

v so if we take v q indep endent of q then is a constant v

F ab i i

ab

This result can also b e derived from a Ward identity based on the conservation of

the two sp ecies of fermions by the Hamiltonian a a b

a a

b

Figure The second order contribution to the renormalized a vertex for the Luttinger

mo del with the momentum exchange q to the b particlehole pair in the interval D D

jq j D

As is constant we can integrate which gives the result

j k j j k j

a a

z

a

D

k for remaining the range of q values in the The renormalized selfenergy

a

a

range j k j q j k j is calculated This is not singular at k

a a a

b ecause of the dep endent upp er and lower cutos The nal result is

j k j j k j

a a

G k

a a

D k

a

where D is indep endent of for to leading order in The renormalizations

of k have b een absorb ed into a renormalized Fermi velo city This result corresp onds

a

to the exact form for the spinless Luttinger mo del with the exp onent given to

leading order There is no quasiparticle excitation at k as z go es to zero

a a

at this p oint In this derivation we do not follow the usual scaling pro cedure of

changing the cutos on all the momenta We reduce the cuto on the momentum

transfer q only In progressively reducing this cuto we take into account the short

range parts of the interaction at each renormalization step

The second order terms neglected in setting up the scaling equations can b e

reg

taken in to account by writing the selfenergy as the sum k k

a a a

a

sing sing

k where is the singular k is the singular contribution we have

a a

a a

reg

just calculated and k is the remaining regular part Then we can apply

a

a

reg

standard p erturbation theory for k and deduce k exactly to

a a n a

a

second order from

ln k reg D

a

k k k k k e

a a a a a

a

sing

The terms arising from k cancel the nonanalytic terms in k to

n a a a

a

second order v and so regularize the expansion to this order There are dier

ab

ent p ossible pro cedures for calculating higher order corrections dep ending on the

range of of primary interest If the main interest is in the region near k

a

then by intro ducing an eective couplingv byv v and comp ensating

ab

ab F

counter terms one can x the exp onent and calculate the regular terms in p owers

of The counter terms are then determined by the condition that they eliminate

any further the singular terms in order by order in We can thus obtain

the essential features for the Greens function for the spinless Luttinger mo del from

this simple renormalization group approach The Greens function for the Luttinger

mo del can b e obtained exactly from a complete summation of diagrams or

via b osonization Here however we are concerned with nding approximate

techniques which can capture the essential physics of the mo del that can b e gener

alized to tackle some of the interesting physical mo dels which are not tractable by

the exact approaches An example in this context where this approach might b e

applicable is the mo del of two directly coupled Luttinger liquids which has b een

put forward to explain the enhanced sup erconductivity of the high T comp ounds

c

The two dimensional Hubbard mo del is the most basic mo del which has b een

put forward to describ e the electrons in the C uO planes of the high T materials

c

This mo del has the form

X X

y

H n n c U t c

i i ij

i j

i

hij i

where t is the hopping matrix element for electrons in the hybridized orbitals built

ij

up from the copp er dx y and the oxygen p state with U as a short range

repulsive interaction Bethe ansatz and conformal invariance metho ds

provide a comprehensive picture for the one dimensional version of this mo del but

so far there has b een limited progress in understanding the b ehaviour of the two

dimensional mo del despite much active work in this eld The low energy excitations

of the one dimensional mo del can b e analyzed in terms of the holons and spinons

of the general Luttinger mo del with spin The low energy features of the one

electron Greens function can b e obtained from conformal invariance metho ds

The nondop ed high T materials corresp ond to the insulating half lled Hubbard

c

mo del For an insulating state describ ed by the Hubbard mo del there must b e a

MottHubbard gap in the one electron sp ectrum In one dimension a gap exists no

matter how small the interaction U In mo dels of higher dimensionality this

is only likely to develop at some critical value of U U The sup erconductivity

c

arises with relatively small doping from this insulating state so it is of some interest

to understand the transition from a Fermi liquid to a MottHubbard insulator as

U is increased to the critical value U and also to determine the nature of the low

c

energy excitations away from halflling for U U where the electrons are the most

c

constrained In this parameter regime b ecause of the limited phase space available

for scattering the correlation eects are likely to b e at their strongest and nonFermi

liquid states might o ccur or transitions to magnetic order at half lling and nite

but large U the mo del is equivalent to an antiferromagnetic Heisenb erg mo del This

dicult regime app ears to b e the most tractable in the innite dimension version of

this mo del where it has b een shown that the selfenergy is purely lo cal indep endent

of the wavevector k and the mo del can b e mapp ed into an eective impurity mo del

with an additional selfconsistency condition By formally integrating out all

the states except those of a single site in a functional integral representation of the

partition function the mo del can b e mapp ed on to an impurity problem with an

eective hybridization matrix element The hybridization can b e absorb ed into the

resonance width function of the Anderson mo del The Greens function

for the noninteracting version of this eective Anderson mo del can b e put in the

form

R

G

d

d

d

which is equivalent to The selfenergy of the impurity mo del can b e

formally expressed in the form

F U G

d

where F is a functional of G and corresp onds to the complete sum of the

d

diagrammatic p erturbation series This sum is completely sp ecied given G

d

and U and the functional F is universal in the d limit The lo cal or onsite

Greens function for the innite dimensional mo del has the form

Z

X

c

d G

l

k

k

where is the density of states of the noninteracting mo del for a tightbinding

c

hyp ercubic mo del this has a Gaussian form The selfconsistency condition arises

from the fact that this Greens function corresp onds to that of the equivalent impu

rity and hence has to satisfy the condition

G G

l d

The resulting equations are those for the Anderson impurity mo del but instead of

the resonance width function b eing sp ecied as an initial condition it has

to b e determined selfconsistently As one is now dealing with an impurity mo del

there is a go o d chance that a reasonably accurate solution can b e obtained The

dicult step is to calculate the selfenergy given U for any given form of G

d

Many metho ds have b een devised for calculating the selfenergy and the sp ectral

density for the Anderson mo del such a p erturbation theory Monte Carlo

the noncrossing approximation and the renormalization group These all

have their advantages and disadvantages The easiest to carry out in practice is the

second order p erturbation theory which has b een found to work very well for the

standard impurity mo del with particlehole symmetry up to values of U into the

strong correlation regime where there are three well dened p eaks in the sp ectral

density corresp onding to that shown in gure Second order p erturbational results

for the lo cal density of states of the innite dimensional mo del on a Bethe lattice

are shown in gure for a value of U greater than then conduction band width but

less than the critical value U If this is compared with the corresp onding density

c

of states for an impurity hybridized with a wide structureless conduction band as

shown in gure one sees the tendency of a gap to op en up and the central p eak

asso ciated with the quasiparticle density of states to b ecome isolated The lo cal

density of states for the noninteracting version of this mo del has the single p eaked

semielliptical form In the results for the real part of the selfenergy shown in gure

for the same mo del one can see a large negative gradient developing at the Fermi

level resulting in a small value for the wavefunction renormalization factor z As

U U the selfenergy b ecomes singular at so z and the quasiparticle

c

p eak of the Fermi liquid disapp ears This basic picture of the breakdown of the

Fermi liquid state is conrmed in the Monte Carlo noncrossing approximation and

renormalization group approaches There are some complications arising from the

fact that there is a another solution corresp onding to an insulating state which

p ersists b elow U but this solution app ears to b e a unstable one for T

c

With several groups actively working on this problem we might exp ect to get so on

a detailed denitive theory of the breakdown of Fermi liquid theory at the Mott

transition for this mo del

0.8

0.6 ) ω

( 0.4 ρ

0.2

0.0 -4.0 -2.0 0.0 2.0 4.0

ω

Figure The lo cal density of states for the innite dimensional Hubbard mo del at

halflling for a Bethe lattice total bandwidth D for the noninteracting mo del

with U as calculated from second order p erturbation theory for the impurity mo del

with the selfconsistency condition 8.0

6.0

4.0

2.0 ) ω

( 0.0 Σ

-2.0

-4.0

-6.0

-8.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

ω

Figure The real part of the selfenergy for the innite dimensional Hubbard

mo del with the same parameters as given in gure The dashed curve corresp onds to

the selfenergy in the atomic limit

Breakdowns of Fermi liquid have also b een prop osed for some strong correlation

mo dels away from halflling and with dimensionality greater than one In

this regime the breakdown of Fermi liquid theory is more unusual as the resulting

state can b e a metal but lacking a Fermi surface in the usual sense as the imaginary

part of the selfenergy do es not vanish as in the recent theory of Herz and Edwards

There is an impurity mo del which shows nonFermi liquid b ehaviour the n

channel Kondo impurity mo del where an impurity spin S is coupled equally to the

spins of conduction electrons in each of the n channels The thermo dynamics of

this mo del can b e calculated exactly by the Bethe ansatz and nonFermi liquid

b ehaviour is found in the overcomp ensated regime n S The low frequency

b ehaviour of the lo cal Greens function can b e determined from conformal eld

theory This mo del has b een put forward to explain the anomalous b ehaviour

of some uranium heavy fermion materials The relevance of these various non

Fermi liquid theories to strong correlation systems and in particular to the high

temp erature sup erconductors is likely to remain a sub ject of lively debate in the

eld for sometime The nonFermi liquid theories of strong correlation systems is a

suitable topic for a future survey

Acknowledgement

I am grateful to the SERC for the supp ort of a research grant

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