Remarks on coupled spin and charge fields in the Hubbard hamiltonian

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Remarks on coupled spin and charge fields in the Hubbard hamiltonian

A.A. Gomes, P. Lederer

To cite this version:

A.A. Gomes, P. Lederer. Remarks on coupled spin and charge fields in the Hubbard hamiltonian.

Journal de Physique, 1977, 38 (2), pp.231-239. �10.1051/jphys:01977003802023100�. �jpa-00208582�

REMARKS ON COUPLED SPIN

AND CHARGE FIELDS IN THE HUBBARD HAMILTONIAN

A. A. GOMES

(*)

^{and P. LEDERER}

Laboratoire de

Physique

^des

Solides,

Université

Paris-Sud,

⁹¹⁴⁰⁵

Orsay,

^France

(Reçu

^le

9 juillet 1976,

^{révisé le 12 octobre}

1976, accepté

^{le 21 octobre}

1976)

Résumé. ²⁰¹⁴ Cet article traite des effets conjoints des fluctuations de spin ^{et de} charge ^sur les pro-

priétés

thermodynamiques

^{d’un gaz} d’électron paramagnétique représenté ^par l’Hamiltonien de Hubbard. Nous discutons d’abord la formulation intégrale fonctionnelle de la fonction de partition.

D’après nous, ^une formulation correcte doit préserver l’invariance de l’Hamiltonien _par rotation du spin. ^En conséquence la fonctionnelle d’énergie libre à la

Landau-Ginzburg-Wilson

^{contient à}

la fois les champs ^de spin ^{et de}

charge,

^{avec des termes de} couplage ^du type

03C1M2, 03C12 M2,

^{etc... où} p est un

champ

^scalaire (de charge) ^{et M est un}

champ

^vectoriel (de spin) ^{à n = 3} composantes. ^Le couplage pertinent ^entre champs ^de

spin

^{et de}

charge

^{est $$ où} n(03B5) ^{est la densité} d’états, 03B5F ^est le niveau de Fermi, ^{et u est} l’intégrale de Coulomb intra-atomique. ^{Il est}

imaginaire

pur. Ensuite, ^nous exploitons l’analogie ^entre la forme de

l’énergie

libre obtenue et celle d’autres

systèmes présentant ^des

champs

couplés ^{comme les} métamagnétiques, ^{ou encore} celle de théories

phénoménologiques. ^Nous discutons le rôle de certaines contraintes, ^comme la neutralité de charge,

pour les propriétés critiques, par la méthode du groupe de renormalisation.

Abstract. ²⁰¹⁴ This paper deals with the combined effect of spin ^and charge fluctuations on the

thermodynamic properties ^{of an} itinerant electron paramagnet ^described by the Hubbard Hamilto- nian. Firstly, ^we discuss the functional integral formulation of the partition function. We argue that ^a correct formulation must preserve the spin rotational invariance of the Hamiltonian. As a conse- quence, the

Landau-Ginzburg-Wilson

free energy functional contains both spin ^and charge ^fluctua-

tion fields, ^with coupling ^terms of the form

03C1M2, 03C12 M2,

^{etc... where} p is a

charge

(scalar) field,

and M is a vector spin field with 3 components. The relevant coupling ^between spin ^and charge ^{fields is}

$$ ^where n(03B5) ^{is the} density ^{of states, 03B5F} is the Fermi level, ^{and u is} the intra-atomic Coulomb integral. ^{It is} pure imaginary. ^{Next, we}

exploit

^the similarity between the free energy which we obtain and various other cases of coupled fields, such as metamagnets, ^{or other}

pheno-

menological functionals. The role of constraints such as charge neutrality ^is discussed, ^with respect

to critical properties, using the renormalization group method.

Classification

Physics ^Abstracts

8.510

1. Introduction. ^- The

non-degenerate

^Hub-

bard hamiltonian

[1] has

been studied

using

^functional

integral techniques [2, 3] by

several authors in recent years

[4, 5].

This formulation relies on the Hubbard- Stratonovitch transformation

[6]

which enables the transformation of the

original many-body

^hamilto-

nian into an effective free _energy

density,

^{from which}

the

thermodynamic quantities

may be evaluated.

Some exact

results,

e.g.

showing

^the

equivalence

between this fermion

problem

^{and a} boson-like system,

(*) Permanent address : Centre Brasileiro de Pesquisas Fisicas,

Av. Wenceslau Braz 71, Rio de Janeiro, ^Brasil.

have been obtained

using

^this transformation

[7].

The main characteristic of this formulation as

applied

to the Hubbard model is to show the existence of two

fluctuating

effective fields

coupled respectively

to the

spin

^and

charge

densities.

Charge

fluctuations have

usually

^been

disregarded (except

^{in the exact}

results above

[7]).

Considering only spin fluctuations,

it has been shown

[4]

that the free _energy

density

^is

given by

^an

infinite series of

interacting spin fields,

^{the nth}

order

vertex

being

the nth order bare fermion

loop.

^We

recall that as a consequence of the

non-commutativity

of the kinetic and interaction terms in the Hubbard

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003802023100

232

(or

^Anderson

[2, 3]) ^models,

^the

fluctuating spin

fields

depend

^{on the}

imaginary time,

^{and conse-}

quently

the Matsubara boson

frequencies

^are present.

The functional

integral

formulation has been shown

[5]

to

provide

^a way to construct a

generalized

^Landau-

Ginzburg-Wilson (L.G.W.)

^free energy suitable for

studying

quantum systems ^{at low} temperatures.

Renormalization _group

techniques [8, 9]

^{have been}

applied

^{in the}

study

^{of this}

spin-field

^free energy

functional;

^{in this} way the critical dimension separat-

ing

classical and non-classical critical behaviour has been obtained.

In this

work, charge

fluctuation effects are taken into account in the derivation of the free _energy

density.

^The

resulting

functional shows

spin-charge coupling

^{terms and we} compare this two-field

problem

to others studied in the literature

[13, 16].

2.

Ambiguity

^{of the} Hubbard-Stratonovich ^trans- formation. ^- In this

paragraph,

^{we would like to}

point

^{out some delicate}

points concerning

^{the cons-}

truction of the

generalized

^L.G.W. for the Hubbard model.

Although specific

characteristics of the Hub- bard model are considered

here,

^a similar discussion

can be

applied

^{to the} Anderson hamiltonian.

First of

all,

^we recall that

spin

rotational invariance is a fundamental property of the interaction term in the Hubbard hamiltonian.

Consequently,

ⁱⁿ the construction of the L.G.W.

adequate

^to describe the

properties

of the Hubbard

model,

^this

requirement

should be taken into account.

In the literature

[2, 3,10,11],

^{two kinds of}

procedures

have been

proposed

^to transform the Coulomb term.

The first one

[2, 3, 10]

^uses the idea of

constructing squared

operators ^{in order to use} the Hubbard- Stratonovich transformation in its classical form :

a

being

^{a constant and A a bounded} operator. ^Since this

approach provides

^a

variety

^of

possibilities

^we

discuss it in a

quite

detailed form.

2.1 SCHRIEFFER’S TRANSFORMATION

(Ising type) [2].

- One uses the

identity nta

= n« ^to write :

The term

U12(niT

⁺

nil)

^{is then}

incorporated

^{in the}

one-electron term of the hamiltonian and the

squared

operator

(nit

^-

nil)2

^is transformed

using (1).

^The

representation (2),

^{as discussed}

by

^Keiter

[10],

^intro-

duces a

spurious

interaction

(see below)

among

equal spins.

^A second alternative is

provided by

^Hamman’s

representation.

Furthermore the

original

rotational invariance is lost in the

remaining

^{term :}

(nit

^-

ni 1) 2 = SiZ2.

2.2 HAMMAN’S TRANSFORMATION

[3].

^{- Hamman}

suggested using

^the

following identity

instead of

(2) :

It should be

emphasized

^{that the}

proof

of the iden-

tity (3)

^{does not use the}

identity ni26

^{= ni6. In fact}

the

squared

^{terms cancel out} in the difference between the

charge (nit

⁺

nil)2

^{and the}

spin

^term

(nit - nil )2.

The

representation (3)

substituted into the Hubbard- Stratonovitch transformation

gives

^{rise to}

coupling

between

charge

^and

spin

^fields

(see below).

^An

impor-

tant remark concerns the dimension of these fields.

From

(1)

^and

(3)

^the

fluctuating

fields have dimension

n ⁼ 1 which reflects the

Ising

^{character of the trans-}

formation.

Although

rotational invariance is ensured for the sum of the two terms in

(3)

^the

spin

^{term alone}

is not

rotationally invariant,

^thus

showing

^{the one-}

dimensional character of the fields. As a final comment about Hamman’s

representation

^{let us} recall the

identity :

which is

clearly

^a consequence of

n2i6

⁼ ni6.

Eq. (4)

combined with Schrieffer

representation (2)

repro- duces Hamman’s

proposal.

^The

repeated

^{use of the}

identity ni6

^{= nia} regenerates the result

(3)

^which

is free from the difficulties associated with the _spu-

riously-introduced equal-spin

interaction

[10].

2.3 HEISENBERG-LIKE TRANSFORMATION. ^- We

begin by deriving

^a

general expression

^for

Si.Si

in terms of the

occupation

^numbers ni,,; this _expres- sion is obtained without

using

^the

identity n3

^{= nia.}

From

and the usual definitions :

one gets :

We

emphasize

^{that no use of}

n2ia

⁼ ni6 has been made in the derivation of

(5c) ;

^from

(5c) by completing

the _square one gets :

From this

general

^{result one} gets :

2.3.1

Schrieffer’s first transformation.

^{- Use the}

identity nfa

^{= nia to} transform :

Combining (7a)

^and

(6)

^{one has}

or Schrieffer’s three-dimensional

representation [2] :

The result

(8) clearly implies

^the

spin

rotational inva- riance of the last term

(the

nit

+ ni,

^term is absorbed in the one-electron

part). However,

it relies on the

use of

ni6

= niq and

again (compare

^with

(2)) ^spin- charge coupling

is lost due to the use of this

identity.

2.3.2 Second version

of Schrieffer’s transforma-

tion. ^- From the

general

^result

(6) squared charge

operators ^{can be}

regenerated, again using n2i6

^{= ni,,,;}

one

just

^{needs to transform in}

(6) :

to get :

or

In this

expression,

rotational invariance is present ⁱⁿ both terms and at the same time

spin-charge

^inter-

action is restored.

However, again

this result

depends

on the use of

ni26

^{= ni6}

^[26].

2.3.3 Alternative version. ^- From the

general

result

(6),

^one gets ^the

corresponding

three-dimen- sional order parameter Hamman-like

representation

At this

point

^{let us make some comments} about these

representations.

Schrieffer’s transformation

(2)

^has

been discussed in detail

by

^Keiter

[10]

^{in the context}

of the functional

integral

formulation of the Anderson hamiltonian. Keiter noted that the

representation (2)

introduces a

spurious many-body

interaction among

equal-spin

electrons.

Using

^a

diagrammatic analysis,

he shows that the initial interaction

Uni1 n¡¡

^and

(2) give

^{the same} result for the

partition

^function

provided

a careful summation of the

diagrams involving equal- spin

interaction is

performed.

^In other words in order to get rid of the

spuriously-introduced

interaction an

infinite summation has to be done over these

equal-

spin

^{terms. If}

approximations

^{are made in the treat-}

ment of the functional

integral, spurious

^{results are}

then obtained.

Conversely,

in Hamman’s

approach,

the

equal-spin

interactions in the

charge

^and

spin

terms of

(3)

^{cancel out and} contrary ^to Schrieffer’s

expression (2)

^no

spurious

results associated with this

equal-spin coupling

^are

generated.

There remains however the

question

of rotational invariance or in other words the

question

^{of the}

dimensionality

^{of the}

order parameters ^or

fluctuating

fields. Hamman’s

representation implies (cf.

^eq.

(1))

^a one-dimensional

fluctuating field,

^{and this contrasts} with the Hei-

senberg

three-dimensional fields of

representation (11).

Clearly

^Hamman’s

representation

^should

give

^iden-

tical results to

Unit

nil which is

rotationally

^invariant.

However this _supposes that no

approximation

^is

performed

^{in the} functional

integral

formulation.

By

^{this we mean that the sums over}

frequencies

^and

wave vectors

together

^{with the}

expansion

^{of trace}

log (1 -

^V’

G) (cf. [4])

^{should be}

performed exactly

in order to recover rotational invariance.

Clearly

this is associated with the fact that the

spin

^{term of}

(3)

is not

rotationally

invariant.

Then,

if rotational invariance is to be

preserved,

^both

charge

^and

spin

terms should be conserved in other words

spin- charge

interaction is a clear consequence of

spin

rotational

invariance,

^{and the}

approximation

^of

replacing

^the operators ^{ni, in the}

charge

^term

by

^an

averaged

^value

(thus suppressing spin-charge

^inter-

action)

violates rotational invariance. The three- dimensional

representation (11)

^{conserves rotational}

invariance in both terms

U/8 p2

^and

U/2 Si. Si.

Clearly

^the

price

^{to be}

paid

^for this is the occurrence

of the

coupling

^terms

(to charge

^and

spin respecti- vely) U/8

^and

U/2. However,

^{since we intend to use} renormalization _group

techniques

^{where the}

coupl- ing

^{terms are} renormalized

by

^the

fluctuations,

^we expect that these non-usual

coupling

^{terms do not}

introduce

larger

troubles than different

starting

^free

energy densities. Before

briefly discussing

^{the trans-}

formation

procedure suggested by

Amit and Kei-

ter

[11]

^{let us}

investigate

^the

possibility

^{of a} parame- trization of the Coulomb interaction term,

involving

as a fundamental step ^{the use of the}

identity ni62

⁼ ni6.

Start from :

where

a, B

^{are constants and I is an}

effective

^Coulomb

interaction. The constants a,

fl

and I are related to the

original

Coulomb interaction

U through :

234

this relation

implying

^the

representation :

Two remarks are in order :

firstly

^the represen- tation

(13)

^{reduces to Hamman’s}

(3)

^{if a =}

fl.

Secondly

^{a can be chosen to be}

negative,

^thus

implying

a real

charge

^field.

By taking

^a

negative

^{one could be}

tempted

^{to use}

(13)

^{in order to simulate a situation}

where simultaneous

charge

^and

spin

^mode

softening

could occur. In this case an

expansion

of the free energy functional _up to say, fourth order in the fields

(charge

^and

spin)

^could

provide

^a simulation of

coupled

^and

simultaneously

soft fields. However it should be noted

that, similarly

^to Schrieffer’s trans- formation

(2), spurious many-body

interactions _among

equal spins

^are introduced in

(13) by

^{the use of}

ni6

^{= nia. In}

^fact,

^for

a #B (which

is essential to get ^a real

charge field)

^the

ni 2

^and

ni 2

^{terms do not cancel out}

and

consequently

the results obtained from the fourth-order

expansion

may include artifacts due to the

spurious

interaction.

Clearly

these interactions

disappear

^{when a}

= fl

^{since the}

squared occupation

numbers cancel out

exactly

^as in Hamman’s case. The second alternative

representation

of the Hubbard term, which does ^{not start} from the classical result

(1),

was

suggested by

Amit and Keiter

[ 11 ].

^This represen- tation avoids

(like

^Hamman’s

approach (3)

^{and the}

result

(11))

^{the use of the}

identity ni6

⁼ nia. The central idea is to start with the usual

representation

^of

the

partition

^{function :}

and to use the

identity :

Introducing

the Fourier

(frequency) representation

^of

the 6-function

the interaction term can be rewritten as :

where

Now one makes use of the

generalized

^Hubbard-

Stratonovitch transformation

(for

^two

commuting

operators ^{A and}

B) :

Combining (14), (13c), (13d)

^and

(13e)

^{one sees}

that the Amit-Keiter transformation leads to the Hamman’s construction of the

generalized

^L.G.W.

for the Hubbard model.

To conclude this section we

emphasize

^{that :}

1)

^{the correct} derivation of the L.G.W. functional from the Hubbard hamiltonian contains

coupled spin

and

charge

^{fields and}

2)

the dimension of the

spin

^order parameter ^{is not}

given adequately by

^a derivation which does not

explicitly

take into account the

spin

^rotational

invariance of the

original

hamiltonian in both

spin

and

charge

operators. Therefore the Hubbard model is not

equivalent

^{to an}

Ising-like

model since the

spin

field is a

three-component

^{vector. We}

emphasize again

^{that the}

Ising-like representation

^derived

by

Hamman is exact

provided

^no

approximation

^is

made in the _energy

density functional;

^{since one needs}

a finite

expansion

in powers of the order parameter,

one has to

keep

^a formulation where both

charge

^and

spin

^terms retain rotational invariance in all orders.

This is not the case in Hamman’s

representation.

In the

following

^{we shall} concentrate on the conse-

quences of

spin-charge coupling

^on the critical beha- viour of an itinerant electron

magnetic

system, ^as

compared

^to situations where no such

coupling

^has

been taken into account.

3.

Coupled spin

^and

density

fluctuations. The free energy functional. ^-

Following exactly

^{the same} pro- cedure as used in

(4)

^and

(5), namely

^the

expansion

of Y

^trace

^log (1

^-

V’(T)),

^but

conserving

^{both the}

C1

charge

^and

spin

^{terms of the}

representation [11]

^one

obtains a free _energy

density

which exhibits

coupled

^order parameters. ^The

one-body potential

^{is now a}

complex potential,

and the free _energy

density

^{reads :}

As was discussed in part ^{2 of this} paper, the relevant value of n for the Hubbard model is n ⁼ 3.

However,

for

simplicity,

^{in the rest of this} paper ^we will restrict the discussion to the case n ⁼ 1. The

general

^{case for}

any n is treated in reference

[18].

In the above

expression

^{we have}

adopted

^{the nota-}

tion qi (qi, cvi)

^to include both the momentum and

frequency dependences.

^{The function}

xo(q

is the usual non-enhanced

susceptibility

^{and the term 1 +}

Uxo(

in the

gaussian charge

^{term of}

(15)

^{is a} characteristic feature of the initial Hubbard hamiltonian. We have denoted

by

U4,

V3

^and

V4

^the

coupling

^constants

among

spin

^and

charge

^fields

respectively.

^The

couplings Ils’

^and

12"

^are likewise the

spin

^and

charge couplings, given by

^{a linear} combination of bare fermion

loops.

^{The free} energy

0(,u, v)

^describes

coupled spin

^and

charge

^fields _Jlq ^and vq. ^{It is clear} that the

low q

^{and low} temperature

expansion

^{of the}

1 +

Uxo(q)

coefficient

of vq 2

^{shows no}

possibility

of a

charge instability,

^as

expected

for the Hubbard model. A simulation of a

charge instability

may be introduced if one

adopts

^the

ex, P representation

discussed in 2 with a 0

(real charge fields).

^{If in}

expression (15)

^one considers finite temperatures

(only

c ⁼ 0

frequencies

^are

retained)

^and

neglects

the _qi

dependence

of the interaction vertices.

It can be shown that the

q-dependent

^{terms of their}

Taylor expansion

^are irrelevant

parameters (at

^least

in the

isotropic

^{case, which}

corresponds

^{to the}

Hubbard

model).

The first two terms

give precisely

^a

Wilson-type

^free

energy for one-dimensional

fields;

the nth-order Fermion

loop

^{in this}

approximation

is discussed in reference

[4].

^In

particular

the third-order Fermion

loop corresponding

^to the lowest order

spin-charge coupling

^{term is}

where

n(c)

^{is the}

density

^{of states. The} coefficient of the

p2

^{M2 term is}

as is the coefficient _u4 of the

M4

^term. Note that at T ⁼ 0

K,

^the

spin-charge coupling

vanishes when the Fermi level sits at an extremum of the

density

^{of states.}

We have put

everywhere kT,,

^{= 1 in the} expres- sions

The terms

involving only

the fields vq ^{describe a} system

(charge

fluctuation

system)

^{with a}

non-ordering

parameter. ^The

interesting

feature of

(15)

is the exis-

tence of

coupling

^terms between the

magnetic

modes

(j,lq) ^(which

^{may become}

^soft

^near the transition

temperature)

^{and the}

charge

modes which are

always non-soft.

^{As a final} comment note that the

spin

^inde-

pendence

^{of the}

charge

^{term in va}

gives

^{rise to the}

third-order

coupling V(qi , q2).

This

coupling

^is

purely imaginary,

^{as is the}

coupling 7;

^{this is} necessary ^{to ensure} the convergence of the functional

integral

^{over fields} _vq.

Clearly

^{we have}

expanded

^trace

log (1 - Va G) only

^to fourth order in

V ; higher

^{order terms are}

expected

^{to be}

irrelevant, although

ⁱⁿ

special

^situa-

tions

(cf. below)

^a sixth-order term has to be intro- duced.

In the Hubbard

model,

interaction forces are

short-range.

One may wonder if our results would hold for a real system ^and

specially

^{for the case of}

a

charged

^Coulomb gas. In

particular

the relevance of the

spin-charge coupling

^term may

depend (and

indeed does

[18])

^{on the range} of the interaction. Let us

consider the case of transition

metals ;

^{in this case} the broad s-band contributes a

screening

^mechanism

for the interaction between d-electrons

[19].

^This

mechanism results in an effective

short-range

^inter-

action. Thus we do expect ^the

spin-charge coupling

to be relevant at finite

temperatures

in transition- like metals. The situation

might

be different in a

metal like

Cr02

^where

only

d-character states are

thought

^{to be} present ^near the Fermi level. In the

cas of

liquid He3,

the interaction term is due to hard

core

repulsion

^{and is}

effectively short-ranged

eq.

(15)

should therefore describe

reasonnably

well the inter-

236

actions between

spin

fluctuations and

density

^fluc-

tuations in that system.

In order to obtain a

comparison

with other

problems dealing

^{with a} L.G.W. free _energy functional describ-

ing coupled fields,

it is convenient to

approximate

the true Brillouin zone

by

^a

spherical zone I q I 7r/a

with some suitable mean lattice

spacing

^{a. With this}

procedure,

the coefficients

U4, V3, V4, 7

^and

I2s’

are

slightly

^{altered. We shall}

neglect

^{this in the remain-}

ing

parts of this work.

4.

Analogies

with other

coupled-fields problems.

^-

Starting

^{from a}

completely

^different

microscopic hamiltonian,

^{suitable for}

describing

^a metamagnet in the _presence of an external

magnetic field,

^Nelson

and Fisher

[13]

^{derived a free} energy

density

^which

shows

exactly

^{the same} characteristics as

(15)

except that

li

is real. The _presence of two order

parameters

in the

metamagnet

^free energy reflects the existence of alternate

planes

^of

spins, coupled ferromagneti- cally

^{within a}

plane

^and

antiferromagnetically

^bet-

ween

adjacent planes.

^In the absence of external

fields, antiferromagnetic Ising-like

behaviour is observed with a second-order

phase

transition. This

behaviour,

observed for small external

fields, changes

to a first-order transition for strong

enough magnetic

fields

(note

the existence of a tricritical

point).

^The

cross terms

present

in the free _energy derived in

[13]

are shown to be associated with the external

magnetic

field. A renormalization _group

procedure [13]

^shows

the irrelevance of the fourth-order cross term

(cor- responding

^to

I2c

ⁱⁿ eq.

(15))

and of the third- and fourth-order

coupling

^terms between fluctuations in the

non-ordering parameter (V3

^and

^V4

^{in eq.}

(15)).

We recall that the renormalization _group

operations performed

ⁱⁿ

[13]

^are

quite special

^ones

(besides

momentum

integration), namely :

choice of the

scaling

of field variables such that the non-critical field has a constant propagator ^{and at each}

appli-

cation of these

operations

^a

secondary

shift is intro- duced in the field variables. This shift ensures that the zero-momentum linear term which is _spon-

taneously generated (through

the third-order cou-

pling

^term between soft and non-soft

fields)

^{is eli-}

minated. In the free _energy

(15)

^discussed

above,

this linear zero-momentum term would

correspond

to an electric

field

which is eliminated at each _opera- tion. Direct

comparison

with the free _energy functional eq.

(15)

shows that the

spin-charge coupling

^{is relevant}

for _space dimension d 4

(see

eq.

(4.19)

^{in ref.}

[13]).

The main results of Nelson and Fisher

[13]

^show

the existence for the

metamagnetic

system ^{of two}

gaussian-like

^{and two}

Ising-like

^fixed

points.

^The

tricritical behaviour is shown to be controlled

by

^a

gaussian-like

^fixed

point.

^{Once the} renormalization group

operations

^are

performed,

^a

partial

^trace opera- tion over the non-soft fields reduces the

problem

^{to a}

free energy which shows close

similarity

^{to the}

coupled spin-phonon

systems

[14].

^The

possibility

^{of a}

negative four-spin

interaction vertex is then

clearly

^shown.

Using

^eq.

(4. 30)

^{of reference}

[ 13]

^{we find} the follow-

ing

conditions for the occurrence of a first order transition

(1) :

Tricritical behaviour is obtained when the

equality

holds.

One should remember that this criterion is

approxi-

mate because of the corrections to the coefficients at finite temperature, ^{which are of order 0}

(kTIEF),

^and

because of the

rounding-off

of the Brillouin zone

mentioned at the end of part ^{4. Thus we find that the}

spin-charge coupling

term turns more difficult the appearance of a first order transition

which,

^{in a}

spin-only theory

^{would occur for 0}

d2n(E) ^de

fp^{. In}

liquid He3, for instance,

^since

n"(eF)

^{0 for a free}

fermion

dispersion relation,

^{one should never}

expect

a first order

ferromagnetic

transition. In transition metals or transition metal

compounds,

eq.

(16)

indicates that the occurrence of first order transitions is

quite

^{less common than a}

spin only theory predicts.

In the metamagnet

problem

^a first-order transition is obtained

by varying

^{the external}

magnetic

^field

strength.

^{In our} case, what matters is the detail of the band structure at the Fermi level. In the derivation of the criterion

(16)

^the temperature

plays

^{no role}

since we assumed that the various coefficients of the L.G.W.

expression (15)

take their T = 0 K value.

However in a more detailed

calculation,

^the tempera-

ture

dependence

of these coefficients can be

included,

and the transition temperature ^{enters as an additional} parameter ⁱⁿ

(16).

^It should be mentioned

(cf. below)

that no constraint is

imposed

^{in this treatment of}

coupled

^fields.

Remark. ^- In the quantum ^limit

(T

^{= 0}

K)

^{in the}

absence of

spin-charge coupling,

Hertz showed that the anomalous dimension of the

quantic spin-spin coupling

^{term is}

(d

⁼

1),

^so that the latter is irrelevant for d > 1. It can be verified that the anomalous dimen- sion of the

spin-charge coupling

^term

Isc

^is

(d - 1)/2

in the quantum ^limit.

Thus,

^as

expected,

^the

spin- charge coupling

^is also irrelevant for d > 1. However there remains the

possibility

^{of a} first-order transi- tion if criterion

(16)

is satisfied

since,

^as in eq.

(4.30)

of reference

[13], U4

= U4 -

(Isc)2 /2(l

⁺

Ux°(0))

^can

be

negative,

^if U4 is

sufficiently negative.

^Remember

Isc1’

is pure

imaginary.

5. Role of constraints. ^- The results of this _para-

graph

^are

strictly

^valid

only

^{when the}

spin-field

ⁱⁿ

eq.

(15)

^is

Ising-like (i.e. n

⁼

1).

^{The case of a vector}

spin

field will be

briefly

^{discussed at} the end of the

paragraph.

First recall the

general study

^made

by

Fisher

[15]

^of the role of constraints in the observed

(1) ^{We are} grateful ^{to M. T.} Beal Monod for pointing ^{out to}

us a numerical error in eq. (16) ^{in the} manuscript.

critical behaviour. In

particular,

^we mention the role of hidden variables

[15] (which

ⁱⁿ the Hubbard model may be associated with the

requirement

^of

charge neutrality

^or

charge conservation).

^Before

going

^into

recent work on constrained systems ^{let us}

emphasize

the

importance

^of

imposing

constraints in the free energy

density (15) describing

the correlated electron gas. The

general

arguments ^{of Fisher}

[15] suggest

^the

possibility

^of

obtaining

renormalized critical _expo- nents

through

^the

physical requirement

^of say,

charge

conservation. In a recent work

[16]

Achiam and

Imry

^{discussed a free} energy functional which

again

shows close

similarity

^to that derived in

equation (15).

It must be noted that in

[16]

^{a term}

describing long-

range energy-energy

density coupling of

^{the form}

j

,Uq /u- q U-’q’

9 - 4’

is included for ^reasons of _gene-

qq

rality

^{in the free} energy

adopted.

^We recall that such terms have been derived

previously by

^Sak

[17]

ⁱⁿ

connection with

compressible

magnets ^and

by Imry

et al.

[18]

^{in the context of}

imposing

constraints

(in

the

thermodynamic sense)

^to the usual L.G.W. free energy

density.

With their more

general

free energy

density

^and

consequently larger

parameter space, Achiam and

Imry [16]

^{obtained a} much richer chart of fixed

points (four

groups of four different

points)

and four different critical behaviours. These _groups of fixed

points correspond respectively

^to unconstrained and renormalized

Ising behaviour,

Gaussian-like and

spherical-like

^critical exponents. These different groups of two

possibilities (Ising

and renormalized

Ising

^for

instance)

arise from the _presence or other- wise of constraints

imposed

^{in the} system. This is the central

point

^{of our}

comparison

of the free _energy

corresponding

^to the Hubbard model

(15)

^{and the}

quite general

work of these authors. To

specify

^more

clearly

^our

point,

^consider

Ising-like

behaviour which is closest to our

adopted representation

of the Hubbard model.

Following

Achiam and

Imry,

Fisher-renor- malized critical behaviour

[15]

is obtained

by imposing

constraints in the

following

^two ways. One

possibility

is to use the constraint Vq=o ^{= constant and the} alternative choice is to

impose

^a 6-function constraint

6(yo - 0)

in the functional

integral defining

^the

parti-

tion function. In the second

possibility,

^the

integral representation

of the 6-function

(through

^the exponen- tial

function)

^{is to} be used. For both choices of the constraints one gets ^{the same critical}

behaviour,

^but

different free _energy densities and fixed

points.

^Another

interesting point

^{concerns the} energy-energy

density coupling

mentioned above : if one takes the coefficient of this

coupling equal

^{to zero in the} initial free _energy, the existence of

coupled

soft and non-soft modes

toge-

ther with a 6-function constraint

regenerates

^{this term}

in the constrained free _energy. The critical behaviour however remains

unchanged.

These rather

general

remarks of Achiam and

Imry [16]

support ^our sugges-

tion of the relevance of

charge

constraints in

dealing

with the Hubbard

model,

^at least when the

spin

^field

is

Ising-like.

Remember that all recent works

[8, 9]

on the renormalization _group

study

^of the Hubbard model deal with that case.

They

^{do not} consider the

possibility

^of

changes

ⁱⁿ the critical behaviour

through

the

physical

constraints on the

charge

fluctuations.

We have mentioned that this is correct at 0 K

[18],

when quantum ^{effects are} present. ^One may wonder what the effect of constraints is on the

charge

^fluctua-

tion field when the

spin

^field

is,

^{as we}

argued

^{it should}

be,

^a

three-component

^{vector. Fisher}

[15] points

^out

that renormalization of critical indices

by

^hidden

variables occurs when the index a is

positive.

^Such

is the case for the

Ising

model and for

Ising-like

Hamiltonians. However a number of theoretical

[20]

and

experimental

^results

[21]

^{seem to}

point

^{out that}

for a

Heisenberg

system

(i.e. n

⁼

3),

^{a should be}

negative

of the order of - 0.1. In that case no renor-

malization of critical indices

by

hidden variables should occur. In other

words,

^we expect ^the map of fixed

points

of the renormalization _group for the Hubbard model to be different from the _map obtained with an

Ising-like spin

^field.

This

remark,

among

others,

shows that it is

impor-

tant to take into account the

spin

rotational invariance of the Hubbard hamiltonian in the L.G.W.

expression

for the free _energy. The

problem

will be discussed elsewhere in more details

[18].

^However it is clear that the

possibility

of first-order transitions and of tricritical

points

^{should not}

depend

^on the dimension of the

spin

^field.

6.

Experimental

discussion. ^- Our discussion of the effect of

spin-charge coupling

^{on the}

thermodyna-

mic

properties

of metallic magnets ^has

neglected

^a

series of effects such as orbital

degeneracy, hybridiza-

tion between

bands,

interatomic

interactions,

^etc...

so that

quantitative comparison

^with

experiments

must be handled with care. Note that the few _pure usual

magnetic

^{metals do not} exhibit first-order tran-

sitions,

_except ^{Cr. The}

theory

^for the latter must

presumably

^{take into account the}

specific

^{nature of}

the incommensurate

spin density

^wave which orders at

7e [22] ;

^{this has not been done in’ the} present

treatment.

The

problem

^{is to have an}

experimental

^{handle on}

the criterion

(16). Any technique

which allows a

change

^{of the}

density

^{of states}

(and

^its

derivatives)

could be useful in this respect.

It is therefore

interesting

to comment on the results obtained

by

^{Bloch et al.}

[23]

^{on A}

CO2 compounds

where A is the series of Rare Earths. While

TbC02

and

GdC02

exhibit second-order

phase transitions,

first-order transitions are observed in

ErC02, HoC02

and

DyC02.

^{In reference}

[23]

this behaviour is

explain-

ed on the basis of a Landau

theory

^{for the}

magnetic

free _{energy in} the Co

d-band, suitably

^{corrected to}

include the interaction with the Rare Earth moments.