Phase transitions in an extended hubbard model

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Phase transitions in an extended hubbard model

F. Brouers, C.M. Chaves, A.A. Gomes, A. Troper

To cite this version:

F. Brouers, C.M. Chaves, A.A. Gomes, A. Troper. Phase transitions in an extended hubbard model. Journal de Physique, 1977, 38 (10), pp.1257-1263. �10.1051/jphys:0197700380100125700�.

�jpa-00208695�

PHASE TRANSITIONS IN AN EXTENDED HUBBARD MODEL

F. BROUERS

Institut für Theoretische

Physik,

^Freie Universität D1000 Berlin 33 C. M.

CHAVES (*)

Laboratoire de

Physique

^des

Solides,

^Bâtiment

510,

⁹¹⁴⁰⁵

Orsay,

^France

A. A. GOMES and A. TROPER

Centro Brasileiro de

Pesquisas Fisicas,

^{Rio de}

Janeiro,

^Brazil

(Reçu

^{le 28 mars} 1977,

accepté

^le

28 juin 1977)

Résumé. ^- Nous établissons l’expression de la fonctionnelle de l’énergie libre d’un modèle de Hubbard avec interactions interatomiques. ^{On est} amené à résoudre un problème ^de quatre

champs

couplés, ^deux champs vectoriels de

spin

^{et deux} champs ^de charge scalaires. Deux de ces

champs

peuvent ^{induire une} transition (un ^de

spin

^{et un de} charge). ^{Nous montrons} qu’en ^ce

qui

^concerne

la transition ferromagnétique, l’introduction d’une interaction à courte portée ^entre voisins conduit à la même fonctionnelle que dans le modèle de Hubbard

simple

à condition d’éliminer les termes de

couplage ^non pertinents. ^{Ceci est conforme au}

principe

d’universalité. Pour la transition vers un état ordonné de

charge

^on utilise des arguments de la théorie du _groupe de renormalisation pour ^montrer que le couplage

spin-charge

^{est non} pertinent ^et que le couplage ^{entre les deux} champs ^de charge peut rendre la transition du premier ^ou du second ordre. La nature de la transition

dépend

^du détail de la structure de bande du métal et de la valeur des interactions intra- et interatomiques.

Abstract. ²⁰¹⁴ We derive the free _energy functional of an extended Hubbard model Hamiltonian with interatomic interactions. One has to solve a four coupled ^fields (two ^vectorial spin fields and two scalar charge fields)

problem.

^{Two are}

potentially

^soft (one

spin

^{and one} charge field). ^{We show that}

as far as the

ferromagnetic

transition is concerned, universality holds and that introducing ^{a short}

range interaction between neighbours yields ^{the same} free energy functional ^as for the

simple

^Hubbard

model if irrelevant

coupling

^{terms are} discarded. For the charge ordered transition we show using

renormalization _group theory arguments ^{that the}

spin-charge coupling

is irrelevant and that the

coupling

between the two charge ^{fields can} give ^{rise to a first or} second order transition depending

on the details of the metallic band structure and the value of the interaction energy parameters.

Classification

Physics ^Abstracts

75.10

1. Introduction.

- Recently [1] phase

transitions of the

isotropic non-degenerate

Hubbard Hamiltonian have been studied within the renormalization

group

°

formalism

using

^{the s = 4 - d}

expansion

^{to first}

order in E. The _purpose of these _papers was to consider the effects of the

spin-charge coupling

^on the critical

properties

of the electron _{gas. In} the usual non-

degenerate

Hubbard model the

charge

^{field cannot}

become soft when the intra-atomic electron-electron interaction is

positive.

^The

question

which arises is to know what

happens

^{to a} second order

magnetic

transition when the order parameter ^is

coupled

^to

this non-soft

degree

of freedom. One of the conclu-

sions of _papers

[1]

^{is that}

spin-charge coupling

may inhibit a first order transition. The effects of

simple

,

constraints

imposed

^{on the}

charge

^{field was consi-}

dered as well as the

possibility

^{of Fisher renorma-}

lization of the critical

exponents.

The

present

paper deals with the extended Hubbard

model,

^{an extension of the} conventional Hubbard model which consists of

taking

^account both into the intra-atomic Coulomb

repulsion

and the Coulomb interaction between electrons at different sites. This model was considered _among others

by Ropke et

^al.

[2]

using

functional

integral

^{methods. It} is much richer than the

original

^one. Instead of one interaction

para-

meter, ^{one can} consider three _energy

parameters.

^In

the molecular field

approximations

^a

variety

^{of solu-}

tions :

paramagnetic, spin ferromagnetic

^{and anti-}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197700380100125700

1258

ferromagnetic, charge-ordered

^states

(see

^{also ref.}

[3])

are obtained

depending

^on the value and

sign

^{of these} parameters. Moreover the extended Hubbard model

was used

by

^Bari

[4]

ⁱⁿ the atomic limit to

investigate

the influence of

electron-phonon

interactions on the metal-insulator transition. This model was also consi- dered

by

^{Ihle and Lorenz}

[5]

^{to discuss the}

phase diagram

^{of the ordered}

charge

transition which can be first or second order

depending

^on the value of the inter- and intra-atomic electron-electron interaction.

In the

present work,

^{we derive the}

free-energy

functional of the extended Hubbard Hamiltonian

using

^the Hubbard-Stratonovich transformation

[6]

and

taking

^{account of the} vector nature of the

spin-

field. If one considers that all

neighbours

^are

equi- valent,

the number of fields is

four;

^{two vectorial}

spin

fields and two scalar

charge

^{fields. Two are soft}

(one spin

^{and one}

charge field)

^{and the two others are non-}

soft. One has to solve the

problem

^{of four}

coupled

fields.

We show that as far as the

ferromagnetic

^transition

is concerned

universality

holds and that

introducing

^a

short-range

interaction between

neighbours yields

^the

same functional as for the

simple

Hubbard model considered in 1.

However here we shall

mainly

^{focus on the}

charge-

ordered transition. The

spin-charge coupling

^{is irrele-}

vant

(see

^section

3)

^{in the} renormalization _group

theory

sense, and we

investigate

^the influence of the

coupling

^{of the two}

charge

^{fields on} the transition

by considering

^{the relevance} of the various

coupling

terms in the free _energy functional

expressions.

^For

the three-dimensional model one of the

couplings

^is

marginal

^and

requires

^a separate

analysis.

^{We show}

however that the effect of the interaction between the two

charge

^{fields can result in a shift of the}

quadratic

and

quartic coupling

parameters ^{in the Landau-}

Ginsburg-Wilson functional,

^{r and} u, and conse-

quently

^{as in the}

ferromagnetic

transition of the

simple

^Hubbard

^model,

^{one has the}

possibilities

^{of a}

tricritical

point

and first and second order

transitions, depending

^on the details of the metallic band structure and the value of the interaction _energy parameters.

2. The free energy functional for the extended Hubbard model. ^- We start with the hamiltonian

where as usual

Tij

^{is the}

hopping integral

^{between i}

and j, a£

^and aia ^are the creation and destruction

operators

^for electrons on site i and with

spin

^(I,

Viji,j,

represents the Coulomb and

exchange

interactions between electrons

on

sites i, i’, j and j’.

We shall consider

only

^{one site}

(i)

^{and two site terms}

(i # j),

^{and we define}

Viiii

⁼

U, viji j = Ki j

^and

Vijji

⁼

I i j.

The one site terms read

using

^{the relation}

already

discussed in reference

[1].

The two site terms can be written

Defining

one can show that

and

This allows us to write the

rotationally

invariant extended Hubbard hamiltonian we consider in this _paper in the form

with

and

In this _way we have taken into account the vector nature of the

spin.

^{In reference}

[2],

^the

spin flip

^{terms are}

neglected

^and

Now we use the Hubbard-Stratonovich

identity

^{for a}

product

^{of two}

operators

^{in its vector} form for the last term of

(8)

and its scalar version for the second term of

(8).

In that way ^we obtain

One has the symmetry ^relation

and

using

^{the same} symmetry arguments, ^{one has for} the vectorial part

The

partition

^function

where

To

^{is the}

time-ordering

operator ^{needed to} preserve the

non-commutativity

^{of the}

operators,

^{which are}

now

(imaginary)

^time

dependent,

^{can be}

^written, making

^{use of}

(14)

^and

(16)

with

(cf.

^Ref.

] 1, 6[

^and

]7[

for details about

the.

method used to derive _eq.

( 1 9))Zo andgo

^{are the}

partition

function and the Green’s function

respectively

for the non

interacting

system ^{when the}

potential

Vhas matrix elements

given by

1260

If one assumes that all

neighbours

^are

equivalent,

^and

using

^the symmetry ^relations

(15)

and similar relations

for wand

cp, ^{one can} reduce the number of

auxiliary

^fields

by

^the

following unitary

transformations

70 and _vo are the

diagonal

^and yl and _Vl the first

neighbour of’-diagonal

interactions of hamiltonian

(8).

^{One can}

see

using

^the symmetry

properties of t7, A,

(p

and ij*

fields that the fictitious fields _{xl, x2, Yl} and _y2

correspond

^to

the

following

^real

fields,

^chemical

potential, staggered

electrical

field,

uniform and

staggered magnetic

^{fields so}

that the

potential (20)

^{can be written :}

The

correspondence

^{with the notations} of reference

[2] being

^{A H}

y2, B H

Yl,

C H x2

^{and D H} xl

where a and b

correspond

^{to two} sublattices. These fields _express the fact that intra- and inter-atomic electron- electron interactions can induce

ferromagnetic, antiferromagnetic

^and

charge

^{ordered states.}

We

develop

^Tr

Log (1 - VGO)

up ^to fourth order in V

(second

^{order in the} interaction

energies) (cf.

^ref.

[7]) using

the Fourier transform

representation with qi

⁼

(q;, Mj

^to include both the momentum qi and the Matsubara fermion

frequencies

Wi.

We first consider the second order terms. The free energy is then :

where a

corresponds

^{to the vector} components. ^{The function}

x°(q)

^{is the} non-enhanced

susceptibility

^and

the other

coupling

^{constants are}

given

^{later in}

appropriate

^units.

The

coupling xl x2

^is

generated by

^the

expansion

^{of Tr}

log.

^{It is intrinsic to} the formalism that _every field that appears in V will generate

couplings.

^{What is}

specific

of the extended Hubbard hamiltonian is the existence of two scalar

charge

fields and so the

possibility

^of

having couplings

in the second order term of the

expansion.

In the limit

Ul, U 2 -+ ^0,

one recovers the results of reference

[1]

^{i.e. two fields. From}

(24),

^{one can see that}

two

fields ;

^one

charge (x2)

^{and one}

spin (yl)

^{can become soft.}

3.

Charge

^{ordered state} transition. ^- Now we are interested in

studying

^the

system

^{close to a}

charge

^insta-

bility eventually

^{at low} temperature ^but not very close to T ⁼ 0. This means that it is

enough

^{to consider the}

dependence

^{of constants} and fields at m ⁼ 0 with respect ^to small fluctuations of _q around the wave-vector

ko

corresponding

^{to the} maximum value

of x°(q).

^{Here we} shall consider the case when the

charge-wave density

^is

periodic

^with

ko

⁼

g/2

⁼

n/a (1, 0, 0),

^the half lattice wave-vector. We therefore

expand

^the

susceptibility

around its maximum

and

with

In the same way ^one has

with

If we

put

^{q -}

ko

⁼

Q

^{and if we} consider the

expansion

^near

Q = ^0,

^{we must consider the} fluctuations

near

Q =

⁰ for all fields. We get

The field _xl is centered at q ^{= 0 but cannot} be soft within this model even if we consider fluctuations around

q=0.

We have

already neglected the Q 2-dependence

of the Gaussian term of the non-soft field which will turn out to be irrelevant.

All third order terms will have the form

where j

^{stands for} any of the fields _{xl, x2, yl, Y 2.}

Transforming

^{to the new}

variables qi

⁼

Q

⁺

ko

^{one can see}

that one cannot have momentum conservation for

Qi N

⁰ and all third order terms vanish.

We now consider the fourth order terms and

investigate

the relevance of the various

coupling

^terms

[8].

^We

first

integrate

^{out all the variables}

xl(q), X2(q), yi(q), Y2(q)

^{with momenta} greater ^than

Ilb

^{where b} >

1 ;

^{we then} rescale the momentum variables

by b

and the fields _xl, X2, yl, y2

by C1, C2, C3

^and

C4 respectively.

^{The choice of}

the

Ci

^is made such that the coefficient of the

q2-term

of the soft mode

x2 in

^the transformed hamiltonian remains

equal

^to

unity

^{and that the}

quadratic coupling

^constants of the non-soft fields are left

unchanged.

^Therefore

The terms

x2.xl, x2.y2

transform like

b-2+£,

^the terms x2.

xi, x2.xi.y2

^like

^b-3+£

^{and the terms}

xf, yf, x i . y2

^like

^{b - 4 + E}

^{and are for that reason} irrelevant after _many iterations. The same argument

applies

^{to the}

Q 2-term

^{in the} non-soft field as stated before.

We have considered the

charge

ordered transition. If we use the same discussion for the

ferromagnetic

transition the

free-energy

functional has the structure

with

appropriate coupling

^{constants and}

integrations.

^{If one} introduces the non-soft field x ⁼ _X, + x2 ^{and if}

one

neglects

Y2, which is not

coupled

^to any other

field,

^the functional for the soft field _{Y 1} reduces to that obtained in reference

[1]

^{and this is} consistant with the

universality principle.

^This

point

is discussed in reference

[9].

For the

charge

^ordered

transition,

^{if the}

dimensionality d

^{= 3}

(s

⁼

1)

^we

keep only

^{the terms}

x" and x2.xl,

1262

which transform like b’

and b-’

^+E

respectively.

^So

putting Q

= q, the

Landau-Ginsburg-Wilson

^free energy functional reads

In contrast to what

happens

^{for the}

magnetic

transition in the

simple

^Hubbard

model, spin

^and

charge

^fields

are not

coupled ;

^{all the}

coupling

^{terms are now} irrelevant and the terms

containing spin

^{fields can be}

dropped

from JC if we consider

only

^the

charge instability.

Therefore one is left with the free _energy functional

where p

⁼

,u 12

^and y ⁼ y 12. The

coupling

^{constants are}

given by

where N" is the fourth order electron

loop (cf.

^ref.

[7]).

The

coupling

^term

x2 . xl

^is

marginal (for d

⁼

3)

in the renormalization _group

theory

^{sense. Let us note}

however that if we define

and

integrate

^{over this new variable we are left with}

where

which is a

simple

^Wilson type functional with

coupling

constants u’ and r’ modified

by

^the

coupling

^{of the two}

charge

fields. With

(36)

^one obtains the usual

Ising

^and

Gaussian fixed

points.

^The

coupling

^{between the two}

charge

^{fields can affect the nature of the transition}

since u’ can become

negative inducing

^a first order transition. The

quantities

^u, /1, y,

fl depend

^{on the band}

structure of the metallic band and in

particular

^{on the}

shape

^{of the} Fermi surface

nesting.

^For

instance, using

^{the results of the} present paper ^one finds a

condition for a first order transition

depending

^{on the}

sign

^{of N".}

Therefore one can conclude that an

investigation

^of

the condition for the occurrence of a first or second order transition

using

^the functional

(36) requires

^a

detailed

analysis

^{of the}

susceptibility

function and of

the fourth-order bare fermion

loop

^{in the}

neighbour-

hood of the wave-vector

ko corresponding

^{to the}

charge density

transition.

Acknowledgments :

^{F. B. is}

grateful

^to Pr. Benne-

mann for

inviting

^him in Berlin. C. M.

C.,

^{A. A. G.}

and A. T. express their

acknowledgment

for the kind

hospitality

ⁱⁿ

Orsay

^{and F. B. wants to thank}

Dr.

Ropke

and A. Richter for useful discussions.

A. T. would like to thank the IAEA for a grant

during

^his stay ^at

Orsay.

References

[1] GOMES, A. A. and LEDERER, P., ^J. Physique ³⁸ (1977) ^231.

CHAVES, C. M., LEDERER, ^P. and GOMES, A. A., To be published.

[2] RÖPKE, G., ALBANI, ^B. and SCHILLER, W., Phys. Stat. Sol. 69

(1975) ^{45 and 407.}

[3] IONOV, S. P. et al., Phys. Stat. Sol. 72 (1975) ^{515 and 77} (1977) ^403.

[4] BARI, R., Phys. ^{Rev. B 3} (1971) ^2662.

[5] IHLE, ^D. and LORENZ, B., Phys. ^{Stat. Sol. 60} (1973) ^319.

[6] STRATONOVICH, ^R. L., ^Sov. Phys. ^{Dokl. 2} (1958) ^416.

HUBBARD, J., Phys. Rev. Lett. 3 (1959) ^77.

[7] HERTZ, ^{J. A. and} KLENIN, ^M. A., Phys. ^{Rev. B 10} (1974) ^1084.

[8] ^{FISHER, M.} E., Rev. Mod. Phys. ⁴⁶ (1974) ^597.

[9] BROUERS, F. and CHAVES, ^C. M., J. Phys. F Lett. 7 (1977)

L-233.