Some aspects of singular interactions in condensed Fermi systems

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Fermi systems

P. Stamp

To cite this version:

P. Stamp. Some aspects of singular interactions in condensed Fermi systems. Journal de Physique I,

EDP Sciences, 1993, 3 (2), pp.625-678. �10.1051/jp1:1993153�. �jpa-00246747�

Classification

Physics

Abstracts

67.40D 67.50 71.25H

Some aspects ofsingular interactions in condensed Fermi systems

P.C.E.

Stamp

Physics Department, University

of British Columbia, 6224

Agricultural Road, Vancouver, B.C.,

Canada V6T 121

(Received

30

August

1992,

accepted

in final form 5

November1992)

Abstract. This article

gives

_a

fairly

detailed _survey of _some of the problems raised when the interaction _energy

If()

_{between 2 fermionic} quasiparticles

(in

2

dimensions)

is

singular

when (k

k'(

-

0. Before

dealing

with

singular

interactions, it is shown how _a

~

f(()

_{leads to}

a 2-dimensional Fermi

liquid

theory, which is

internally consistent,

at least _as far

ailts

_infrared

properties

_are concerned. The

quasiparticle properties

_are calculated in detail.

The

question

of whether

singular

interactions arise for the dilute Fermi gas, with

short~range repulsive

interactions, is

investigated perturbatively.

One finds _a weak

singularity

in

f(()

when the

dimensionality

D

= 2, but it does not destabilize the Fermi

liquid.

A _more sophisticated analysis ^is then

given,

to all orders in the interaction,

using

the Lippman-Schwinger equation

as well _{as a}

phase

shift

analysis

for _a finite box. The conclusion is that any breakdown of Fermi

liquid theory

must _come from

non-perturbative

effects. An examination is then made of _some of the consequences

arising

if

a

singular

interaction is introduced the form

proposed

by Anderson is used _{as an} example. A hierarchy ^of

singular

terms arise in all

quantities

this is shown for the

selLenergy,

and also the 3-point and

4-point scattering

functions. These _may be summed in _a perfectly consistent _manner. Most attention is

given

to the particle~hole channel, since it _appears to lead _to results different from those of Anderson. Nevertheless it _appears that it is

possible

to

derive _a sensible

theory starting

from _a

singular

effective Hamiltonian

although

Fermi

Liquid theory

breaks

down,

all fermionic

quantities

_may be calculated

consistently. Finally,

the effect of _a

magnetic

field

(which

cuts off the infrared

divergences)

is

investigated,

and the de Haas-van

Alphen amplitude calculated,

for such _a

singular

Fermionic system.

Prologue.

This article is written in honour of Rammal

Rammal,

for the

symposium

held in his _memory

on the 22

May, 1992,

in Grenoble. I _{am very}

glad

to have this

opportunity

to honour

him,

for not

only

is he now

generally recogn12ed

as

having

been the finest condensed matter theorist of his

generation

in

France;

he _,vas also _{a very} fine _man, whose loss _was difficult to absorb.

My personal experience

of Rammal

began

with _my first visit to Grenoble in late 1985.

Amongst

the _many of hundreds of _anonymous

physicists

in the various

labs,

Rammal _was

immediately

noticeable

by

his

hospitable

overtures to

conversation,

and I recall with

pleasure

several

wide-ranging

discussions of localisation

theory

at this time.

Later,

in Autumn

1987,

it

was his idea to

begin

_a

collaboration, looking

at RVB states on

triangular

lattices

(with

_{an eye}

on

~IIe films). Alas,

this work _was cut short

by

his terrible heart attack in

early

1988. He _was much _more

fragile thereafter,

but in the _summer and autumn

of1990,

_we were able to _resume

extended

discussions, particularly

_{on one} of his favourite

topics,

the "W.A.H."

problem.

I _was

looking

forward to

renewing

^the debate _{on my} return to Grenoble at the

beginning

of June last _year,

only

to learn _upon

landing

in France of his sudden and

tragic

death.

One is left with

striking

recollections. His love for the

jab, feint,

and thrust of

argument

and debate

(scientific

or

otherwise), punctuated by

blasts of irreverent and sometimes ribald humour. An incredible _sense of

humour, this,

which ceded to

nothing

and

nobody

I will

never

forget being

entertained for

nearly

6 hours _one

day

in his

office,

after he had

begun by

announcing

^"how much work he had to do". And

a

passionate

_nature, which when channeled

by

his

wide-ranging curiosity

and love of

enquiry,

led to _a constant flow of ideas. One wishes

now that he had been less driven

although, given

his character and

background,

would this have been

possible?

Above all _one is left in frank admiration of his

extraordinary

_courage and

tenacity,

and his relentless refusal to let

things

_as

they

_were. His _{was a}

spirit

too

strong

for his

body;

it is still

an

inspiration

_to those who where

lucky enough

to have known him.

The

following

contribution deals with _a set of themes which fascinated

Rammal,

and I

regret

very much not

being

able to discuss it with him. He would have

improved

it

considerably having first,

of _course, tried to

pull

it to

pieces!

1 Introduction.

The idea that

we may describe the

low-energy

excitations of _an

interacting

fermion

system

as fermionic

quasiparticles, interacting

via _an interaction energy function

f((',

_is

one of the

fundamental

lynchpins

of modern

physics.

Elaborated in the form of the

Ffiili liquid theory (FLT)

of Landau

[I],

^{this idea is}

generally

believed to describe

~He

and metals in 3 dimensions.

Various modifications which

incorporate

BCS

pairing _[2],

_or other kinds of order

parameter

built "on

top"

of the basic FLT

structure,

^{allow the idea} to be extended

throughout

3 di-

mensions,

and _a similar idea appears

implicitly

_or

explicitly

in nuclear

physics

and

quantum

field

theory.

One _can of _course also

develop

theories for bosonic

systems involving interacting

bosonic

quasiparticles,

in 2 and 3 dimensions.

Leaving

aside for the _moment the issues

surrounding

I-dimensional

systems,

it is

really quite

remarkable that until _very

recently nobody

had

thought

to ask what would

happen

if the

interaction function

f(('

were a

~

function of its

arguments (I.e.,

it

diverged,

_{or some}

derivative

diverged, etc.). Although

at first

sight

this

question

_seems to make _{no sense}

(in,

_eg.,

the context of

FLT),

_we shall _see in this paper that in fact such

singular

interaction functions

are

quite meaningful physically,

_even in the context of

FLT;

and _we shall _see how to deal with them

theoretically (see

also Ref.

[3]).

The idea that such

singular

interactions

might

exist

amongst

2-dimensional fermions _was

suggested by

Anderson

[4],

who discussed _one

particular singular form,

for which

fkk>

_~-

k'. (k k')/(k _k'(~.

We shall examine this form in this article.

Meanwhile,

the assertion of Anderson

suggests

another _more difficult

question, viz.,

under what

physical

circumstances

can such

singular

interactions arise? This

question

is

highly

controversial at the moment,

particularly

in 2 dimensions in this article I _can

give

_no definite _answer, but

only partial

results

(see

Sect.

3).

Thus most attention will be focussed here _on

describing

how to deal with

singular interactions,

_once

they arise,

and

deducing

_some of their

physical properties.

This will

be done

using

_a method first described in reference

[3],

^which deals

systematically

with the

hierarchy

of infra,red

singularities arising

from such

singular

terms in

fkk>,

as well _as

showing

that the

theory

is self-consistent

(by re-deriving fkk>,

after _a summation of

singular terms).

We shall find that the

theory

of

'~singular interacting

Fermi

liquids"

is _very different from the usual FLT. In

fact,

as

conjectured by

Anderson

[4],

^there are certain resemblances to the

properties

of I-dimensional

interacting

Fermi

systems [5],

^{in which the fermionic}

quasiparticle description

breaks down

coiupletely

in

fact,

_{as uJ -} 0 the fermionic

quasiparticles disappear completely.

However the detailed connection between the _two kinds of

systems

^is not at

present clear,

and _some of the results described here appear ^to

disagree

with those of Anderson.

Once _one has adiuitted that

singular

interactions _are

possible,

_a rather rich _new vein of

physical

ideas and

possibilities

is

opened

_up, both to the theorist and the

experimentalist.

The most obvious

applications

_are to known 2-dimensional fermionic

systems,

and attention has concentrated _on the _new

superconductors.

Ilowever there _are

probably

better

systems

available in which to test these ideas _one

example

which

springs

to mind is the 2-dimensional

~He

film

[6]. Particularly interesting

is the role of

applied magnetic

fields. This is because the

crucial effect of the

singular f[[)

_{is to create}

a

hierarchy

of IR

divergences

in the

low-energy physical properties

of the

system (~vhich eventually destroys

the fermionic

quasiparticles)

and

a

magnetic

field _can be used to at least

partially

_suppress these

divergences.

The net result is shown in

figure I,

in ,vhich the

quantity Z(uJ) (the

fermionic

quasiparticle

wave-function

renormalization

factor)

is shown for finite

H,

and H

= 0

(and temperature

^T =

0).

Thus _we

see the _very

interesting experiiuental possibility

of

passing

between the

singular interacting system,

and _an

ordinary spin-polar12ed

Fermi

liquid, just by switching

_{on a}

magnetic

field. In fact this sort of

suppression

of infi.ared

(IR)

effects

by

_a field has

already

been

suggested

[7] ^for 3-dimensional Fermi

liquids (with non-singular interactions),

but these IR effects _were rather weak

(they certainly

do not

destroy

the FLT

ground state!).

The modifications due to H _are much _more drastic for

singular systenis,

and _{so some} discussion is

given (in

Sect.

5)

of what

might

be _seen in the dHvA effect for _a

charged systeiu. Unfortunately, apart

from the dHvA

eject,

the effects _are not

quite

_so drainatic _{as one}

might hope,

and for

really strong

^{field effects}

it _seems that _one

ought

to go ^to

transport

iueasureiuents rather than

thermodynamic

_ones.

These _{are more} subtle to

calculate,

and at the time of

writing

I have _no

complete

results.

Other

interesting spin-offs suggested by

the

problem

at issue _concern the

possible

_crossover

between

singular

and

non-singular

behavior _{as a} function of

diiuensionality (at

T

= 0 this

should involve _a

phase transition).

A rather controversial

example

at the moment is the

~'coupled

chain

problem",

in which I-diiuensional

Luttinger liquid

chains _are

coupled by

a

perpendicular hopping

ii recent discussions of this _can be found in reference

[8].

More abstract

(but arguably

iuore

interesting

and

fundaiuental) spin-offs

from the

questions

addressed

here,

_{concern our}

understanding

of the different kinds of condensed matter

system

which _iuay

exist,

and how _we should

classify

and describe them. This

problem

of

"generic

states" _can be

approached

in ,~arious _,vays. It is often useful to start front _sortie sort of

general

"effective Haniiltoiiian" for the class of systems under

discussion,

and _{we are} interested in

something

like

Jiii

=

~ ~ li/~>bp~(q)bP~>(-q)

⁺

it bi~(q)

b

i~>(-q)I ⁽ⁱ⁾

kk' I

for _a systeiu

lacking spin-orbit coupling;

here b@;

(q)

describes

density

fluctuations in the _sys- tem, ^and

6R~ (q)

describes

spin fluctuations; f/~,

and

f£,

are the interaction _energy

functions,

which _may

jr liiay

not be

singular.

In the context of

FLT,

such Hamiltonians have been dis- cussed before

[9, 10],

^and the foriuulation of

FLT,

for normal and

superfluid _systems,

in terms

Z(tJ)

FLT

SFL: x#o

u

fi w,

Fig.

1. The fermionic

quasiparticle

wave-function renormalization

z(w),

shown for _a

typical

Fermi

liquid (FLT),

for the type of

singular

interaction dealt with in this _paper

(SFL),

and for the _same

singular

Fermi

liquid

_now in _an applied field

(SFL:

H #

0).

All of these _are at temperature T ₌ 0;

and /h is the Zeeman

splitting

when H

#

0.

of

"dynamic

molecular

fields", _[11]

is

closely

related to

this;

_very

recently

Haldane

[12]

^has discussed fermion systems ⁱⁿ 1,

2,

and 3 dimensions in terms of _a bosonized He~r _very similar to

(I.I) (notice

that 6@k

(q)

and

6R~ (q)

are bosonic

operators).

What _one would like to

understand,

both for

singular

and _non-

singular f([(,

_{is the}

_general

behavior _open to

7i[fl.

Such _a

question

is often formulated in the

language

of renormalization group

(RNG) theory.

It has been _common in _{recent years to} think of the FLT

ground

state

as a fixed

point,

described in its

vicinity by

_an effective Hamiltonian like

7i[fl.

Even for _non-

singular

interactions there _are certain difficulties in this

point

of view

[10, 13].

^For

example,

there exists in

principle

_an infinite set of "FLT

parameters" ([or

_an

isotropic

Fermi

liquid,

these _are the

Fi~

^and

Fi~,

derived from

N(0) f(((

on the Feriui surface

SF),

but if _we consider these _as "irrelevant

variables", describing

the

approach

toward the fixed

point,

then it is hard to understand the infinite set of "Pomeranchuk instabilities" which arise at finite values of the

Fi~(q

₌

S, A),

in fact when Fi~ -

-(21+ 1)

^{from above in 3-d.} Two

possible

_ways

_[10]

_one

might

^{resolve this}

problem

_are to either

(a)

insist that the real irrelevant variables _are

actually

the

Al _(w,here Al

=

Fi~ / _II

+ Fi~

/(21+ _1)] ),

so

instability

arises for infinite

Al

or

(b)

_suppose

that the

Fj~

are

marginally

relevant

[14].

^In any case, it is clear that the extension of

7i(fl

to include

singular

interactions lutist throw _some

light

_on this

question, by enoriuously enlarging

our space of

possible

effective IIaiuiltonians. In

fact,

_as will be

briefly

discussed later

(Sect. 6),

it _{seems as}

though

certain

singular

forms for

ff((

_{lead to} "non-normalizable" but

physically

meaningful theories, lying quite

outside the

usuiiRNG machinery.

Since the methods of RNG

theory

have _so far been

singularly unhelpful

in

understanding quantum liquids

like

~He

_which lack _an order

parameter,

this may well turn out to be rather useful.

We shall not _use RNG methods in the _paper in fact most of the discussion will

rely ultimately

_{on a}

technique

which has

hardly

been

exploited

in such

problems before,

that of the "eikonal

expansion".

This semi-classical

approximation

has been

previously exploited

in

high-energy

^field

theory

calculations

[15],

^{where it takes}

advantage

of the Lorentz construction of

wave-packets;

here it turns out to be useful for many

singular

interactions in the

low-energy

limit,

for

quite

different

physical

_reasons

(essentially

because forward

scattering

dominates _as

uJ -

0,

because of

phase

_space

restrictions).

Thus this method turns out to be the natural _way to set up a

description

of

singular interactions,

even those

having

_a non-normalisable character.

However in this _{paper we} will not follow this method. This is

partly

because there is _no space in what is

already

_{a very}

long

_paper, but also because the

diagrammatic methods,

which will be used

instead,

allow

a much clearer

understanding

of the

hierarchy

of IR terms which result from

singular

interactions.

Finally,

_{on a more}

speculative

_{note, one may} ask whether it is

possible

to set up some

kind of "Statistical

Quasiparticle theory"

for

singular

interactions. This _was done for _non-

singular

Fermi

liquids by

Balian and de Dominicis

[16],

^{and it is}

fairly

_{easy to}

generalize

to 2-dimensional Fermi

liquids (see

Sect.

2)

and to Fermi

liquids

in

magnetic

fields

[7, 17].

The obvious obstacle for

singular

interactions is the

large degeneracy

involved _as

uJ - 0

(almost

_as bad _as that

occurring

in the Fractional Hall

effect),

and the associated statistics transmutation. Nevertheless the

possibility

is _a

tantalizing

_one,

particularly

in view of the

possible

connections to _a semi-classical

description

in terms of the closed orbits of the

equivalent

classical Hamiltonian.

The

plan

of the _paper is _as follows. In section

2,

discuss the

theory

of the

low-energy properties

of _a 2-dimensional

system

of ferInions in which

f[[(

_is

non-singular.

This leads to _a Fermi

liquid theory having

a form somewhat different

from

_the 3-dimensional

theory.

Nevertheless,

at least _as far _as the

low-energy properties

_are

concerned,

it is

internally

consistent

(as usual,

_{one can say} little about the

high-energy behaviour,

which could

conceivably

_{cause a} breakdown of Fermi

liquid behaviour).

In section

3,

_a rather

thorough

discussion is

given

of the

perturbation theory

results for _a 2-d dilute Fermi gas with

short-range repulsive interactions;

this is based

largely

_on work done with G.

Beydaghyan

and N.V. Prokofev. _In sections 4 and

5, give

the results and calculational details of _a

partial investigation

of the

singular

interaction advocated

by Anderson, again

in 2 dimensions. In section 4 the

theory

is

developed

for _zero

magnetic

field the results _were

already reported

in reference

[3],

^{but with} an error which is

corrected here. In section

5, 1explore

the effect of the IR cut-oft in _a finite

magnetic field,

with

particular

attention

paid

to the dHvA effect. Section 6 concludes the paper. The

general

idea of this _paper is to survey the

important

themes

arising

in the discussion of

singular interactions,

whilst

giving enough

details of the calculations _so that interested readers _may follow them.

Solve of these details _are

relegated

to 3

Appendices, following

after section 6. I

emphasize

here that this article is

by

_{no means}

complete although

the discussions of FLT and the dilute Fermi gas in D ₌ 2 dimensions _at least _come to a

conclusion,

iuuch _more could be added to

them;

and _{as we} shall _see, the work

described,

in sections 4 and

5,

on

singular

interactions is

really only

_a

beginning

_on what

promises

to be _a very

interesting ^topic

^{of research.}

2.

Regular

interactions: FLT in D ₌ 2.

In this section _{we assume}

regular

interactions between the ferinionic

quasiparticles,

_so that

jaa' fad' _{(~ ~)}

k k' ^~ k k'

where

f(((

_{is bounded and has}

no

singular

^behavior _{as a} function of its

arguments,

_{or any}

derivatiii

_{of these}

_arguments.

_From

now on in this _paper the tilda "~-" will denote

quantities

derived from _or associated with such

non-singular

interactions.

We shall

investigate

the

~

of such _a

description

in 2 dimensions

(as

Landau and

many others did for 3

dimensions),

and derive _some of the main

properties

in terms of

f(((.

Some of the

results,

in their

general

form at

least,

have

already

been

given previously,

but almost

always

in the context of

perturbation theory.

It is of _course

possible

to show that if

perturbation theory

^is

^valid,

then _a 2-dimensional FLT will follow

(see

next

section);

but here _we wish to go ^to the next

stage,

^and see what FLT then leads to in 2 dimensions. The results will also be

important

when _{we come} to

non-singular

interactions. We concentrate _on 2 dimensions for _reasons which will become clear later in this _paper.

A

good place

to start is

by investigating consistency.

There _{are a} number of different ways of

doing

this here _we

just

_{use one} and _start with the

Bethe-Salpeter equation

for

fermionic

quasiparticle-quasiparticle scattering,

which is of _course central to the formulation of

microscopic

FLT

(note

that this is not

quite

the _{same as}

phenomenological

FLT

[13]).

At low energy transfer _v, and momentum transfer _q, with the "3-vector"

Q

=

(q, v),

we have _a

quasiparticle

T-matrix

I([(

for

non-singular

interactions which takes the form

(in

_zero

applied field)

ilk, _(Q)

=

tk

₊

~l

k>

tk

~.

il,'fir

+ i~

fir°' ii _{k(Q) (2.2)}

in which

Ail

=

fl(°) (ik")

is the

density

of states for

quasipartides

_{as a} function of the _non-

singular quasipartide

_energy

ik.

The function

I( _~,(Q)

is related to the usual

4-point

vertex

i~(P, _{P; Q) by}

I(~,(Q)

=

I(0)ik?k>i~ _(k, ik; k', _{ik'; Q)}

₊

I(0)il~,(Q) _(2.3)

around the Fermi surface

SFi

and the

equivalent equation

for the

4-point

vertex is

where the

frequency integral

dc has yet to be

done; integration

_over the factor

ji(It; _Q)

=

©(I(

₊

Q)©(I<) gives

the usual

singular

factor in

(2.2),

under the

assumption that,

at T ₌ 0 and low

Q (I.e.,

_q «

kF,

_v «

cF)

~~~~ ~~

_~'

~ q

i~~~

+ id ~~~

~~~

~~

~f~

~

~~~~ _~~'~~

where

<(It)

is

regular.

All of this is

standard,

of _course; the difference between 2 and 3 dimen- sions becomes

immediately _apparent

if _we

integrate ji(I(, _Q)

over

frequency

c and momentum k to find _a lowest-order

susceptibility,f$(Q)I

,vhen the

dimensionality

D

=

3,

((°~(Q)

~

13(0) (1- ^(/2

^In ₁₊

_I

⁺

^{ii(fl (I (()j (2.6)}

2

for low

Q (here (

=

v/qif)

but when D

=

2,

lg (-2 ij

g

(1 -2j

~~~~~

_"

~~~°~ li~- _il~

^~

_{J i~~}

~

~~ ~~

and _{we see} that Im

([(Q) actually diverges

_{as v -}

qif,

_a much

stronger singularity

than in the usual 3-dimensional Lindhard function in

(2.6) _[in

these

expressions, fi3(0)

and

k2(0)

are the usual renormalised densities of states;

k3(0)

=

p[/x~if

_and

12(0)

=

fli*/2x].

There is also _a much weaker behavior around the

point Q

_~-

(2kF,0), corresponding

to

low-energy

excitation of

quasiparticles right

_across the Fermi surface. In D

=

2,

for

example,

_one finds that

~IJl

I(

_(Q

2kF,

_{V "}

0)

_~

^~~~~~ ([(~ ^~/~F)

⁺

^W/2fFj

^fl

^{[(2 ~/~F)}

⁺ W

/2fFj

q

[(2

q

/kF)

_uJ

/2cF]

fl

[(2 q/kF)

_uJ

/2c~] (2.8)

This behavior _can be visualised

by plotting

Im

I((Q)

_in the

((q(, v) plane (see Fig. 2);

it will

play

_no ^role in what follows.

~l o

-0.2

etc.

-I i

V

a)

'

-i v

b)

Fig.

2. Plots of

Imi$ _{(qiv +16)}

_{in the}

_{(q, v) plane;}

_{these plots}

are

simply

contour maps. In

(a)

the 3-dimensional Lindhard function is shown, and in

(b),

the 2-dimensional version of this function

(similar _graphs

_can be found in Ref.

[18]).

We _{may now}

reasonably

ask whether the enhanced

singular

^behavior in

(2.7)

is

capable

of

leading

to _a breakdown of FLT in D ₌ 2. A

fairly convincing

_way of

seeing

that it will not lead

to _a

breakdown,

is

by calculating

the form of the fermionic

selLenergy

at low _{energy uJ.} For _a

general /[[(

_{to be}

a constant

I-e-, /[[(

-

It,

_for definiteness _we will _assume the

coupling

to be in the

spin antisymmetric channel,

and let

it

_%

/o.

A realistic

calculation, incorporating

all the different

spin

and

angular

_momentum channels

merely disguises

the essential results

(in

a paper

currently

under

preparation,

this kind of calculation is

done,

and the

general theory

of Fermi

Liquids

in 2 dimensions is

given

in full

detail).

Thus _we wish to determine the form of

fp(w

₊

_id)

as p - pF, uJ - 0

(with

uJ measured from the Fermi

energy).

This will be deduced

mainly

from _a calculation of Im

fp _(cp

₊

_{id) (I.e.,}

_the

"on-shell"

imaginary part

of

f~(uJ)).

We start

by considering

the lowest-order "reduced

graph"

contribution to Im

fp _(cp

₊

id),

sho,vn in

figure 3a,

both in _terms of the

4-point

vertex

I,

_{and also in terms of the}

_complete

dynamic susceptibility I(o(Q) _[which

we henceforth write _as

fl(Q),

in this model

calculation].

As shown in

Appendix A,

one can calculate Im

fp _(cp)

in these two different _ways, but _{as we} shall

presently

_see, it is _more convenient to calculate in _terms of

I(Q).

We then have in this

simple isotropic

model _a lowest-order

[in I(Q)]

contribution to Im

fp _(cp

₊

_ib)

_{of the form}

_[18]

im

[6(1)L~(w

⁺

id)]

⁼

S fl fl _jio(Q)j~

_n~,

(i _{n~_~)} (i _n~,+~)

6

(w

^{+ <~, <~_~}

<~,+~)

zp P' q

e

$ /~

_du

£ _{(I qp-q)} I(

_Im

_fl(q,

v

+16)b (v ^uJ cp-q)) ^(2.9)

zp _o

q

f f a)

^~

(

where qp ^e q

(cp)

is

again

the Fermi

function;

and

(in

this

isotropic approximation) iolo)

=

io/ Ii

+

ioi[(Q)) (2.io)

I(Q)

=

ill Ii

+

jai[(Q)) 12.ii)

In

(2.ll),

^the

"6(~)" signifies

^the contribution first-order in

((Q).

We notice that the

graph

in

figure

3b

gives

_no contribution to Iin

ip(uJ

+

id), although

it

obviously gives

_a contribution to Re

fp(uJ);

_we return to this

point

below.

Equation (2.12)

is

easily

evaluated and _we find _a standard result

[19]

ⁱⁿ 3 dimensions

(at

T =

0)

Im

(6l~Jfp(uJ

⁺

^6)j

^{~- uJ~}

(2.12)

However in D

= 2 _we find

(Appendix A)

(the

exact

expression

is

given

in the

appendix

_see

equation (A.15)).

There _are two main

differences between these two

expressions: first,

when D ₌ 2 the

leading

term in Im

(61~)f)

is

~-

uJ~lnuJ instead of the usual uJ~; ^and

second,

the

leading

IR correction to

this,

which in 3 dimensions is

~- (uJ(~, ^is now also

~-

uJ~ln

^" _{This IR term, in D}

=

3,

is

just

that

uJo

responsible

for the notorious

T~lnT

_terms

_{[9, 20]}

_in

Cv(T);

its IR nature is shown

by

the _way it is

suppressed by

_a

magnetic

^field

[7].

^{The 2} dimensional term is

stronger,

and will show the

same field

suppression

it leads to _a term in

Cv(T)

_~-

^T~.

At this

point

_one is

tempted

to

conclude, solely

from

equation (2.13),

that FLT is

clearly internally

consistent when D

=

2, simply

because the ratio Im

(6(~)f(uJ)j /Ref(uJ)

tends _to

zero with _uJ. However this conclusion would be

premature (although

it is

probably

correct in the

end!).

It is rather instructive to see what lacunae exist in the

argument,

^{and how}

they

_can

be dealt

with,

_up to _a

point.

There _are

essentially

three

points

that must be taken _care

of,

and

they

_are

(I)

We must

investigate

the effect of

higher

fluctuation

graphs.

(ii)

We should look at

3-particle, 4-particle,

etc.,

scattering

_processes.

(iii)

We should check the behavior of

ReLp(uJ) ^directly.

We shall deal with these in turn.

(I) Higher

fluctuations: The nature of these contributions _can be understood

intuitively by

_re-

ferring

back to the effective Hamiltonian in

equation (I.I).

This describes interactions between

bosonic

fluctuations,

and it is natural to ask whether the

higher-order

interactions between the bosons will

give significant

corrections to the result in

(2.12)

and

(2.13).

From _a formal

point

of

view,

in

calculating

the effect of these _on the fermionic

self-energy,

it is convenient to work in terms of the

"dynamic

molecular fields"

[11]

^which can be

generated

from

(I.I),

and which mediate the interactions between the fermions. These fields _are

again bosonic,

and in fact the relevant

graphs

_can be

generated

from the effective interaction in

(I.I) just by breaking

_up

JOURNAL DE PHYSIQUEI T 3. N'2. FEBRUARY ,993 22

Fig.

4.

Higher~order

fluctuation contributions to

fp(w +16)

: these _are discussed in the _text and also in

appendix

A.

one of the bosonic lines into its fermionic constituents

(plus

all

permutations

_over the

original

bosonic

graph).

This then

generates

the series shown in

figure 4,

for the fermionic

self-energy.

Now the

point

here is that if FLT is _to be _a consistent

theory,

then these

higher

contributions must not

seriously

affect the results

already

derived in

(2.12)

and

(2.13).

That this is true is

easily

seen

by noticing

that for

regular

interactions

I[[(, higher-order graphs

ⁱⁿ the fluctuation fields

I(Ql'lead

to terms

higher

order in _uJ, in the

low-energy expansion

of Im

f(uJ),

at least when D

= 2 _or 3. This

point

is shown in _more detail in

Appendix

A. Thus this

consistency

check is satisfied _we shall _see that for

singular

interactions it most

definitely

is not.

(it) Multi-particle scattering:

The next set

graphs

^that must be checked for

consistency

_go

completely

outside the framework of the effective Hamiltonian

(I.I),

since

they

involve 3-

particle

_or

higher multi-particle scattering.

Some

typical examples

_are shown in

figure

5. Now the usual Landau

expansion

of the energy of _a Fermionic

liquid

E ₌ ED +

~ _eka6qka

₊

~ ~ /(((bqkabqk,a,

_+.

(2.14)

ja

~ La j'a'

can of _course be

supplemented by higher

terms:

~~~~~

~ ~ ~ #~~~~~/~llkiai~llk~a~~lJ

_%a3

(2.15)

k~aik~a~k~a3

~~~~~

4 ~ ~ ~ ~

~~(~~~~~ ~flkiai

~flk~a~

$fl%a~ fink

a~

(2.16)

ki~i k~a~k~a3[a~

~

corresponding

to the

6-point

and

8-point

vertices in

figure 5a;

_{we assume a}

system

of unit

volume. The

j-

and

I-

_functions

can be related _to the

6-point

and

8-point

vertices in _a way similar to the relation between

f[[(

_and

)(Ii, Ii'; Q)

^{in the Landau}

theory [I]; thus,

for

example;

the

3-particle

T-matrix _can be defined in _{a way}

analogous

to

(2.3),

in terms of the

6-point

vertex

r6 (Pi P2P31QIQ?),

_as shown in

figure

5a:

~PIP~L ^~~l~~l2)

"

~PilP~lP3f ^P~

_<pi

_p~

_fp~

_~j,

_cp

^;Qi, Q2) (2.17)

a)

(

~

b)

Fig.

5. some

multi-particle

_processes that _can contribute to

lbp(w).

In

(a)

the

complete 6-point

and 8-point vertices _are shown these go into

higher

terms in the Landau

expansion.

In

(b),

the lowest-order contributions _to

Imibp(w), coming

from the

6-point

and

8-point

vertices, _are shown in reduced

graph

form.

and then for low _momentum transfers

Qi,

Q2> we have

j~~~~~~ =

(fi(0))~

^{lim lim liin lim}

ip

p p

(qi,

_{Vi q2, V2)}

(2.18)

Qi-oQ~-oqi/vi-oqa/v~-o -1-2~3

(the spin

indices in

(2.17)

and

(2.18)

_are

suppressed

for

simplicity).

If for the moment _we

ignore r8,

^and

higher vertices, equations (2.14) (2.16) imply

_a

quasiparticle

_energy functional

~ka (flka)

" fk _a +

~ _~~~~flk'a'

₊

~ ~ §~~~~~~~flk'a'~flk"a" (~.19)

k'a' k'a'k"a"

and _an effective interaction between

quasiparticles

of the form

~l' _(flka

" ~~~ +

~j j~~~~'~'dflk"a" (2.20)

k"a"

Now the staiidard and well-known _reason

why

we may

ignore

these

higher contributions,

in the context of

FLT,

is that at least at low _uJ and low

T,

the

expansions

in

(2.14), (2.19),

and

(2.20)

_are

expansions

in the "small" deviation

6qka

from the T

= 0 fermionic

quasiparticle

distribution. Of _course at

higher

T and _uJ, the

single expansion

in

(2.14)

will break

down,

and this will

happen roughly

when

)bq

~-

I. _{Notice, however,}

that this

argumentation

is

only

valid if the

)-

and

[-[unctions

are

-.

If not, then not

only

does the whole FLT framework

break

down,

but _so does the _more

general

effective Hamiltonian

(I,I) (in

which

I(((

_{is not}

necessarily singular).

A number of remarks _are in order at this

point. First,

_we notice form

(2.19)

and

(2.20) that, just

_as the

quasiparticle

_energy ⁱⁿ the usual FLT

depends non-trivially

_on

I(((,

_in the _same

way it

acquires

less than non-trivial

dependence

_on the

)-function

if

we go

it higher

orders

in,

_{say, energy uJ;} and likewise the

original f-function

_now also becomes _a functional of

bqka.

Thus when

calculating

terms

beyond (2.12)

and

(2.13)

in

Imf(uJ),

_we must not

only

calculate the

higher

fluctuation

terms,

but also terms

arising

from the _g-,

h-,

etc. functions.

A second

remark,

which is well known to anyone

working

^with the Fadeev

equations

for 3-

particle scattering,

is that the form of the

f-function

also feeds back into the

higher vertices;

all of those contributions _to

higher

vertices which _are

2-particle

irreducible will

bring

the

f-function

in. Thus these contributions to the

J-

and h-functions _can be calculated. What _we cannot calculate in the

original

FLT framework _are the interactions which _are

3-particle

irreducible and

higher.In

the _{same way}

as for

FLT,

it then follows that these irreducible functions will have to be introduced

phenomenologically. However, provided they

_are

regular,

it is

immediately

obvious that

they

will not

change

the results in either

(2.12)

_or

(2.13),

because the extra

integrations

_{over energy} in

figure

5b increase the _power of _uJ, in the usual _way.

However,

this

brings

_us to _our third

remark,

which is that there is _no obvious _reason to sup- pose

that,

_even if the

2-quasiparticle

interaction _energy

f(((

_is

_regular,

_that

a

higher

function like

g[(([I'

must also be

regular.

If for _{some reason} the contribution to the

g-function coming

from the irreducible

3-point

vertex

happens

to be

singular,

then there is _not

a lot _{we can} do

without

going

^outside the framework of _our

original

^effective Hamiltonian

(I.I).

This in fact takes _us

beyond

the _scope of this article.

Since there is not _a

great

^deal _{we can say} here about these

higher functions,

_we will _assume from _{now on} that

they

_are

benign (except

where

singularities

in

f((( _imply otherwise).

This

proviso applies

with

even more force to Re

ip(uJ).

(iii) ~ f(uJ)

:

Up

to _{now we} have been

discussing

^the

imaginary part

^of

f(uJ)

in order to show that

liin Iin

(fp(uJ

⁺

ib)) /Refp(uJ

+

ib)

₌ 0

(2.21)

w-o

If _{we assume} that

Ref(uJ)

~- uJ, then this

equation

is

certainly correct,

as we have _seen

above;

provided

that

I, _), I,

_{and all}

_higher

interaction fractions _are

non-singular,

then

Imfp(uJ+16)

~-

uJ~lnuJ,

and

(2.21)

is correct.

However _one must be careful

here,

because

our conclusions about the

ener behavior of

fp(uJ)

do not

necessarily

tell _us much about the

~ behavior,

and _{so we} cannot deduce

anything

about

Refp(uJ),

for low _w, from _our results for

Imfp

_(uJ

(except

for IR

singular terms,

such _as the (uJ(~ ^terra in 3

dimensions,

which leads to _an uJ~lnuJ _term in

Refp(uJ);

in 2

dimensions,

the

uJ2lnuJ

_{term in}

Iinfp(uJ)

leads to _a term _{~- uJ} in

llefp(uJ)).

Thus _we must

be _wary that _even

though

_our results for

Imfp(uJ)

_may be consistent with

FLT,

the form of

Refp(uJ)

_may not be.

It is not hard to find

examples

of such _a situation. The _most obvious is

provided by

the Pomeranchuk instabilities which arise in both 3 dimensions and 2 dimensions when

j/i~ -

-(21+ 1) (2.22)

from above. A

typical

such

instability, involving fit,

is the

ferromagnetic instability,

but in fact its _appearance ,vill be visible in

Imfp(uJ),

since the coefficients of the various terms in the low-w

expansion

of

Iinfp(uJ) depend

_on

2i(,

which

diverges

in the limit

(2.22).

Much _more

subtle is the

instability

which _{can occur} if I*

0,

when fii~ -

-3;

there is _no soft mode

accompanying

this Pomeranclmk

instability,

and _so the

approach

towards this

instability

is not obvious in

Imfp(uJ).

An

example

which is _nearer to the theme of this article _was

provided recently by

Khodel and

Shaginyan [21],

who

pointed

out that _any interaction

/ff'

_{which is}

_{reasonably strongly}

peaked

in the forward direction will

give

_an

instability

in the _very lowest-order contribution to

Refp(uJ),

shown in

figure

3b. The interaction does not have to be _a

singular

function of

)k k')

for this

instability, although

it is

clearly

easier to achieve

instability

if it is.

It is

actually

much _more difficult to check the behaviour of

Refp(uJ)

at low energy then it is

Imfp(uJ),

since the

higher

multi-fluctuation contributions to

fp(uJ)

of

figure

3a _can all

contribute to the

leading-order (in

_uJ) behavior of

Refp(uJ) (although,

_as

we have _seen,

they

contribute to

steadily higher-order

terms in

Imfp(uJ

₊

id)).

Thus it is rather obvious

that,

for

example,

the calculation of Khodel and

Shaginyan by

_{no means} establishes _a real

physical instability

in their model

(see

^{also NoziAres}

[21]);

to establish

this,

_one would have to _sum all the

higher-order

terms. However in fact the forward

scattering peak

in their model

actually

leaves it _open to treatment

by exactly

the _same

approximation

which _{we are}

going

to _use here for

genuinely singular interactions,

and _so this should allow at least _a

partial

solution.

Naturally,

in the framework of

perturbation theory

this will

correspond

to _{a sum} to infinite order of the multi-fluctuation contributions _we shall _see in section 4 how this is done.

This concludes _our

analysis

of the

self-consistency

of FLT in 2 dimensions. We _see that there is

nothing

in the behaviour of

Iinfp(uJ)

which leads _one to

suspect

a breakdown of

FLT,

and

provided

_{we can} be _sure that

Refp(uJ)

is

well-behaved,

then FLT is _a

perfectly good theory.

As such it has been

developed by

_a number of

authors, particularly Miyake

k Mullin

[22],

who noticed in

particular

that the

properties

of

a

spin-polarized

Fermi

liquid

would be rather

interesting

in 2 dimensions. The discussion in this section

(and

in

Appendix A)

confirms that calculations in _a D

= 2 FLT _are

perfectly consistent,

^{and that} criticisms of this

approach

_are

quite unjustified _[23], provided

_{we assume}

I[1' _~.

As

such,

FLT _can be

developed

in all of its usual

aspects, including transport theory _[22],

the

theory

of

non-equilibrium

and

high-frequency phenomena,

and

iven superfluid pairing.

One _tray also

develop

_a "Statistical

Quasiparticle" (SQP) theory

to describe its

equilibrium properties.

All of this is

interesting, particularly

for

spin-polarized

svstems, but is too far off the main theme of this

article,

_{so we} leave it for _now.

At this

point

two natural

questions arise, viz., (a)

is there _{any way}

we can

justify

_{a non-}

singular Ifl'

_from

more

microscopic

considerations

(at

least for _some class of

systems);

and

(b)

how do _we treat

singular Ifl'

_{forms? In} the next section _we look at

question (a),

where

we find that

perturbation theory

indicates that

I[p'

_must be

regular

in D

=

2,

at least for

isotropic

and

translationally

invariant

systems (provided,

^of _course, we

ignore

the _supercon-

ducting instability). However,

^{this does not exclude} a

non-perturbative

contribution to

ill'

which is

singular

_{moreover, we} shall _see that the

general

effect of unscreened interactions is to

give

rise to

singular

forms for

ifl',

_and

so it then becomes _necessary to examine

singular

functions _anyway; this is done in section 4.

3.

Microscopic origin

of

singular

interactions.

In the last section _a few consequences of the

assumption

that

f(((

was

regular

_were

studied;

in

particular,

^it _was found that for D ₌

2,

such _an

assumption

_was

perfectly

consistent with

an FLT.

However,

we would

really

like to known _a little bit _more about

f((( _apart

_from

anything else,

we would like to know if it is

regular

for _some class of

modefi and,

if

possible,