Variational approach for uniformly frustrated 2D XY spin systems. I. Phase transitions in modulated arrays

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Variational approach for uniformly frustrated 2D XY

spin systems. I. Phase transitions in modulated arrays

D. Ariosa, A. Vallat, H. Beck

To cite this version:

Variational

_approach

for

_uniformly

frustrated 2D XY

_spin

systems.

I. Phase transitions

in

modulated

_arrays

D.

_Ariosa,

A. Vallat and H. Beck

Institut de

physique,

Université de Neuchâtel, Rue A.-L.

Breguet 1,

CH-2000 Neuchâtel,

Switzerland

(Reçu

le 20

_juillet

1989, révisé le 13

février

1990,

accepté

le I S mars

₁₉₉₀₎

Résumé. 2014 Pour clarifier la nature de la transition de

phase

du modèle XY-2D

complètement

frustré,

nous étudions le cas d’un réseau avec des liaisons

modulées,

où la transition se dédouble en deux transitions distinctes. La méthode utilisée est une

_{approximation harmonique}

autoconsis-tante

_(«

SCHA

_{») qui}

nous fournit un

système d’équations

_pour les

_couplages

effectifs

Kij

et les moyennes

thermiques 03B8ij>

des différences

angulaires

entre

_{plus proches}

voisins. Nous

prédisons

l’occurrence d’une transition de

_phase

du _type

_Ising,

suivie à

_plus

haute

température

d’une transition du _type BKT. Dans le

voisinage

de la 1re _transition, les

_équations

SCHA ont été modifiées en accord avec les exposants

critiques

connus du modèle

Ising-2D

en évitant ainsi la

divergence

des fluctuations inhérente à la méthode. Dans le cas non modulé, les deux transitions

sont confondues en une seule _ayant un caractère mixte. La validité de « SCHA », reliée aux fluctuations de

phase,

est discutée. Nous comparons nos résultats à ceux obtenus avec une

approche

de _type

Champ Moyen

sur un modèle à deux couches, à des simulations Monte-Carlo et

à des résultats

_{expérimentaux.}

Abstract. 2014 In

order to

_clarify

the nature of the

phase

transition

taking place

in the

fully

frustrated XY _model, we examine the case of the array with modulated bonds, where one observes a

_splitting

of this transition in two distinct ones. The method used is a Self Consistent Harmonic

_{Approximation (SCHA)}

that

_provide

us with a set of self consistent

_equations

for effective

_couplings

_Ki,j

and for thermal averages of

angular

differences

_{03B8i-03B8j>}

between

nearest

_neighbours.

We

predict

the occurrence of an

_Ising

like

_phase

transition followed

_by

one of the Berezinski-Kosterlitz-Thouless

_(BKT)

_type at

_higher

_{temperatures.} In the

_vicinity

of the 1st

_transition,

the SCHA

equations

were modified in accordance with the known critical

exponents of the

_2D-Ising

model in order to avoid the

_divergence

of

_phase

fluctuations inherent to

the method. For the unmodulated case both transitions merge in a

_single

one of mixed character. The range of

validity

of the SCHA, related with the

phase

fluctuations, is discussed. The results are

_compared

with those of the mean-field solution of a

_two-layer

model, Monte-Carlo simulations and

_experimental

data.

Classification

Physics

Abstracts 75.10H - 74.50

1. Introduction.

The 2D X Y

_spin

model describes the

_{thermodynamic properties}

of

_planar

arrays of

proximity

junctions.

In recent _years there has been a _{great deal} of theoretical and

experimental

interest

_[1]

in the

_properties

of such arrays. When a transverse

_magnetic

field is

_applied,

the

system

behaves as a frustrated XY model. In

_particular,

a _square lattice of such

_junctions

in a

transverse

_magnetic

field with half of an

_elementary

flux _quantum per

plaquette,

corresponds

to a

_fully

frustrated XY model

_[2].

In this case, frustration introduces a discrete

_symmetry

(related

to the double

_degeneracy

of the

_{ground state)}

on

_top

of the continuous

_symmetry

(global

rotation of the

_{spins) already}

_present

in the unfrustrated case. These two distinct

symmetries

are related to two kinds of

_phase

transitions : the

_Ising

transition

_{(driven by}

domain wall

_excitations)

and the BKT one

_(driven

_by

vortex

_{excitations).}

Both

exper-iments

_[3]

and

_computer

simulations

_{[4, 7]}

show a

_single

transition. However the occurrence

of this

_single

transition of mixed character is not

_yet

well

_understood,

since different

approaches disagree

about its nature.

The work

_{by Berge et}

al.

_[5]

was, to our

_knowledge,

the first tentative to

_split

these transitions

_{by modifying}

the frustration

_{through varying}

the

_stength

of

_negative

bonds of Villain’s

_fully

frustrated model

_{[6]. They performed}

a Monte-Carlo

_simulation,

_finding

the

Ising

transition

_preceding

the BKT one in

_temperature.

_By

_looking

at _{the cusp of the}

_specific

heat

_they

conclude that for the transition in the non-modulated case the

_Ising

character was

dominant.

The same model was

_{investigated recently by}

H. Eikmans et al.

_{[7] ;}

this contribution contains two different and

_{complementary}

_approaches

of the

_{problem :}

a derivation of the

Coulomb gas

picture

of the modulated

_system

and a Monte-Carlo simulation. The first leads the authors to the conclusion

_that,

in the modulated

_system,

existence of

_dipoles

of fixed

length

centered on the weak bonds can inhibit the formation of free fractional

_charges

(responsible

for the

_screening

of the Coulomb

_interaction)

associated with the

_growth

of domains at the

_Ising

transition. The

_helicity

modulus

_(or

inverse dielectric

_constant)

will then contain an _{« energy like}

_{anomaly »}

at

_TIS

and will go to 0 at a

_higher

_temperature

T BKT.

Both

_predictions

were confirmed

_by

the MC simulation. For further

_comparison

with

our variational

_approach,

we want to focus on two

_particular

results in reference

_{[7] :}

the

density

of domain walls vs.

_temperature

that

_supports

the

dipole picture,

and the

shape

of

helicity

modulus that contains both the

_anomaly

at

_Tjs

and the

_jump

at

_TBKT·

The purpose of our _{paper is} to

_investigate

the frustrated and modulated

_(F-M)

model,

introduced

_by

_{Berge [5],}

within the Self Consistent Harmonic

Approximation

(SCHA).

In

section 2 we recall the definition and the

_principal

features of the

_{model ;}

in section 3 we

derive the set of

_equations

for effective

_couplings

_K;j

_{and average}

_angular

differences

( () i -

_Oj>.

In section

_4,

we

_present

an iterative method for

_solving

the

_{coupled equations}

of

section 3. In section 5 we

_present

the results for the

_helicity

modulus,

domain wall densities and the

_{phase diagram}

for the double

_{transition ;}

_comparisons

are made with other theoretical

approaches

[5,

7,

8]

and

_experimental

work

[9].

We summarize our results and draw conclusions in section 6. Some

_precautions

must be taken in

_evaluating

_{thermal averages} near

the

_Ising

critical

_point

to avoid

_divergence

of the fluctuations inherent to the SCHA. This

point

and other technical difficulties are discussed in the

_appendix.

2. The model.

Our

_{starting point}

is the usual

_fully

frustrated 2-D XY model. This model

_applies

to an _array

of proximity

junctions

in a transverse

_magnetic

field of flux

Ba2

_{= ~ 0/2}

(~o = hc

₀

_being

the

2e e g

magnetic

flux

_quantum

and a the lattice

_constant).

We thus consider the Hamiltonian :

In _{the Landau gauge}

_(i.e.

A =

(0,

Bx,

0 )),

the

quantities

Aij

are 0 for all horizontal bonds

and nx

7r for vertical _ones, located at horizontal distances _{nx a} from the

_{origin. Consequently}

Hamiltonian

(2.1)

can be written as follows :

with

_.l;j

= J for horizontal bonds and for bonds on _{every second vertical} row, and

- J on the other half of vertical bonds.

The

_{generalization}

introduced

_by

_{Berge et}

al.

[5],

consists in

_multiplying

the

_negative

bonds

by

a factor a. A

_picture

of the modulated _array is shown in

_figure

_1,

where the dotted line

encloses the unit cell of the array we shall consider in the next

_section,

in accordance with the

symmetries

of the well known

_{doubly degenerate ground}

state

_[2].

Fig.

1. - Modulated

array. Double vertical lines stand for

antiferromagnetic (J

0) bonds. The dotted line encloses the unit cell of the

_ground

state.

3. The variational

_equations.

We consider a trial Hamiltonian

_{HTR, quadratic}

in the deviations from thermal

equilibrium

of

angular

differences between nearest

_{neighbours :}

The variational free energy is

given

as usual

by :

where the averages with the o TR »

subscript

have to be

_computed

with the trial

_density

matrix :

with

The

_quantities

_Xij

are

_calculated,

as

_usual,

_by

the

_eigenvalues

of the harmonic Hamiltonian

HTR (see

appendix

for some

_details).

If we _{define the average} current on the

_{(1, j )}

bond

by :

the minimization of F with

respect

to the

_{parameters (o i)}

_{yields the, current}

conservation on

every node of the array.

On the other

hand,

the minimization of F with

respect

to the

_parameters

_Xij

_provides

us

with a set of self consistent

_equations

for effective

_{couplings :}

Taking

into account the

_symmetries

of the

_system,

we can define three non

_equivalent

kinds

of bonds : the horizontal bonds

_(subscript

« h

_»),

the unmodulated vertical bonds

_(subscript

« v

_»)

and the vertical modulated bonds

_(subscript

« «

_»).

If we define in

addition,

the bond

variables :

(where

sites

_{1, 2,}

3 and 4 are defined on a

typical plaquette

in

Fig.

2),

we can express the

current conservation as follows :

Fig.

2. -

Labelling

of the sites for a

_{typical plaquette (see Eq. (3.7)).}

(3.8)

can be written in a more useful _{way :}

with

and

Using

the same

_notations,

the set of

_{equations (3.6)}

reduces to three self consistent

equations :

As a

_check,

the

_ground

state for a

given alpha

can be evaluated from

_(3.9)

and

_{(3.10) by}

making Xh

=

0,

Xv

= 0 and

X a

= 0

( T

=

0) :

At a =

1,

we recover the well known result for the

_fully

frustrated XY model. Notice that

equations (3.12)

admit non trivial solutions

_{(Oh :0 0) only}

if a

> 1 .

i

*

In

_figure

3 we

_display

the evolution of the

_ground

state

_{angular configuration}

when

varying

a.

Fig.

3. - Evolution of the

two

_{degenerate ground}

state

_{configurations}

when

varying

the modulation a.

4. Iterative solution of the SCHA

_equations.

The

_starting

_point

is the

_ground

state

_given

_by

_{equations (3.9)-(3.11)}

at T = 0

_(i.e.

Xh

=

X v

=

X a

=

0).

The iteration from a

_temperature

T to a

_temperature

(T + 8 T )

is

performed

in the

following

way :

a)

we first

change

T

by ( T + 8 T )

in the

density

matrix of

equation (3.3) ;

b)

we then

update

the values of

_{Xh, X,}

_{and Xa using}

the

_{updated density}

matrix and the last calculated values for

_Kh,

_{K, and Ka ;}

c)

we introduce the news X’s in

_{equations (3.10)}

in order to

_update

the Os.

d)

the

_updated

X’s and Os are then used to

_compute

the K’s

_{through équations (3.11).}

Steps b)

to

_d)

have to be

_repeated

up to obtain

_stability

of the results

_{(in practice,}

for

8 T = J

_{10- 2,}

the

_procedure

_{converges in less} than four

_cycles).

kB

All

_operations

described above are

_{straightforward}

_except

the thermal

_{averaging using}

Eq.

(3.3)

in

_step

_b).

This calculation is detailed in the

_appendix.

At this

_point,

we should discuss the

validity

of SCHA on our

_specific

_system.

_Indeed,

the

original periodic potential

in cos

_{(03B8i -}

_03B8j)

is

_{replaced by}

an effective one in

(p i - P j)2 ;

_thus,

for weak

_{enou h}

_couplings

_Kij,

_phase

fluctuations

_{(characterized by}

their

_quadratic

mean

( P i - P j )2) T¡J

can _{carry the average}

_phase

difference outside its definition interval

[- 7T,

7T ]. Furthermore,

as

_pointed

out

_by

Fishman

[10],

even if the

_phase

difference remains in its allowed

_interval,

the

approximation

becomes doubtful for

_phase

fluctuations of the

order 1 .

_Fortunately

in our case we can

_{check, a}

_posteriori,

that _average

_phase

fluctuations

4

are

_always

well

below 1

_(the

value where the cosine curvature

_changes

its

_sign)

at

2

equilibrium

values of

_K;j

found

_{self-consistently.}

The eventual existence of another free _energy minimum at lower values of K is not a

problem

for us

_since,

_starting

from the known

_ground

_state, we follow the solution

continuously

in

_temperature.

_{(This continuity}

_argument

must be reconsidered in the case of

quantum

fluctuations induced

_{by charging}

effects

_[11].)

5. Results.

The

_procedure

described in section 4 was used for different values of the modulation

parameter

a.

Figure

4 shows a

_typical

result of our self consistent

_{calculation ;}

the X and Y

_components

Kh

of the

_{anisotropic helicity}

modulus r

_(within

our

_{approximation r x}

= h

and

J

Kv+K03B1

e

kB T

T Y

=

2 J )

are

plotted

versus the reduced

_temperature

t

= 4 .

J

(we

use the SCHA

critical

temperature

of the unfrustrated and unmodulated

system

Tco

= 4 - J

as our

e

_kB

temperature

unit [13]).

The dotted line

_represents

the

_Ising

like order

_parameter

S = sin

(Oh),

that

goes

rapidly

to 0 at the

_Ising

transition. Notice the

_singularity

in the first derivative of T

_appearing

at the same

_temperature

where S goes to 0.

_Although

this is

obviously

not

_apparent

in the numerical curves, the cusp at

TIS corresponds

to the

_logarithmic

derivative

_singularity

in the internal energy or the nearest

neighbour

correlation function of

the 2D

_Ising

model. This can be seen as follows : the renormalized force constants

Kh, K,

and Ka

depend, according

to

_{expressions (3.11)}

on the nearest

_neighbour

fluctuations

Xh, X,

and X,,

which are

_{given by expressions}

like

_(A7).

In a mode

decomposition

the latter

Fig.

4.

Fig.

5.

Fig.

4. - Two

components of the

_helicity

tensor

_{r a{3}

_{(full line)}

and

Ising

like order

_{parameter S}

(dotted

line)

vs. reduced _{temperature t}

= 4 . e

kB Tfor a

= 0.5.

Fig.

5. -

Helicity

modulus T vs. reduced temperature t, for different values of the modulation

parameter

(a

= - 1 ; 0.4 ; 0.7 and

_{1). (The}

dotted line is

plotted

to exhibit the

universality of the jump).

belongs

to the

_eigenvector

_v3

_(see

_appendix)

the form of

which,

at q

=

0,

is

with This mode

_{corresponds precisely}

to the

pattern

of the

_ground

state of our

_system,

in other words : rotations of the

_spins

on the 4 sites

of

_figure

3

_by

_{angles corresponding}

to the

_components

of vector _V3,

_starting

from a

ferromagnetic configuration, precisely

generates

the

_ground

state

_pattern.

One can thus

interpret

the

_{field 0 3}

as

_{representing,}

in the

_spirit

of a o continuous

spin» l/J 4-model,

the

Ising

spin

variable,

and

_Xh,

_X,

and

_Xa

_corresponds

to the

_{Ising spin}

nearest

_neighbour

fluctuations (Si - Si +

1)2>

= 2 -

_{2 (Si.}

_{Si +}

_{1 > .}

The correlation function

_{(Si. Si + 1)}

have the same critical behaviour

_of

_{T - Tc}

_{long ( 1 T - Tc 1)}

as the internal energy.

The same feature is observed for all values of a

above 3

i.e.

an

_Ising

like

_phase

transition

3

followed

_by

one of BKT

_type

at

_higher

_temperature.

The

_{rapid drop}

of S reflects the fact

that,

above

_TIS,

the current conservation

equations

(3.10)

admit

_only

the trivial solution

_{0 h =}

_{Oy = ° a}

= 0. The

_jump

of T at

TBKT

reflects the fact

that,

for

_strong

_enough

fluctuations,

the self consistent

_{equations (3.11)}

admit

_only

the trivial solution

_Kh

=

K,

=

Ka

= 0.

(In

fact,

the free

energy minimum

disappears

at this

_temperature,

and constitutes the

_signature

of the BKT transition in the

harmonic

_{approximation.)}

The case a = 1

(fully

frustrated without

modulation)

is

specially interesting

_to discuss : for

~2

~2

a =

1,

S =

2

at T = 0 and all the Ks are

2

J. In

increasing

the

temperature,

the mean

fluctuations on different bonds will evolve

_identically

since

_they

have the same

_starting

strength. Thus,

the

_quantities

a and b of

_equations

_(3.10)

will remain

_equal

to

_unity

and the average

angular configuration

will be constant _up to the

_temperature

above which the non

Bellow a

== 3 ,

S = 0 at all

_{temperatures,}

thus the self consistent

equations

(3.11)

evolve

3

independently

up to the

_jump

at

_TBKT.

In

_figure

_5,

we show for

_comparison,

the

_helicity

modulus versus reduced

_temperature

for four different

_{systems :}

the

simple

unfrustrated _array

( a = -1 ),

the

fully

frustrated array without modulation

( a

=

1 )

and two frustrated and

modulated arrays

(a

= 0.4 and

0.7).

As in reference

[7]

we take into account the

_anisotropy

by plotting

the

_geometric

mean T =

Tx T y.

At this

_point

we want to stress that the

present

calculation is

_{mainly aiming}

at

_studying

the various

_aspects

of the

Ising

transition. The B-K-T transition itself is

_probably

no _very well described

by

SCHA

_{[14] ; although}

it does take thermal fluctuations into account, which renormalise the

_helicity

_modulus,

as it is done

_by

calculations based

_explicitly

on the existence of vortices

_[15].

Nevertheless it is

_interesting

to note the «

_{universality »}

of the

_jump

of F at the

_higher

_temperature

transition.

The

_{phase diagram (a, t c)}

in

_figure

6 resumes the situation described above.

_Comparison

of

our results with the

_{phase diagram}

of

_{Berge et}

al.

_[5]

and with the curves for

_helicity

modulus of Eikmans et al.

_[7]

_(both

obtained

_by

Monte-Carlo

_simulation)

are

_highly

satisfactory.

In addition our results _agree with those obtained earlier

_by

us in a mean field

approach

on a

two-layer

model

_[8].

_Finally,

recent measurements of the

_helicity

modulus in square arrays of

Josephson

Junctions

_by

Théron et al.

_[9]

seem to show a weak

_singularity

in

agreement

with our

predictions.

Fig.

6. - Phase

_{diagram (a, te).}

Notice the absence of

_Ising

transition below a =

1/3.

z

The

_analysis

of the

_helicity

modulus

_only

is,

of course, not conclusive as to the nature of the transitions. In order to _compare our results with the

_{dipole picture}

of reference

_[7]

we should

evaluate,

within our

_approach,

the

_density

of different

_types

of domain walls at each

temperature.

To do

_it,

we shall use a « zero current criterion » to detect domain boundaries in the

_system.

This

_{approximation,}

that

_clearly

overestimates domain walls

_densities,

is

_justified

at low

_temperatures

_by

_{the fact that minimal energy} domain walls are centered on bonds

through

which there is zero current. In

_addition,

within our variational

_approach

we deal

naturally

with effective

_couplings

and currents

_{(see Eq. (3.5))}

that are

_provided

without additional numerical cost

_by

the iterative

_procedure.

Let us write the

_probability

for a certain bond

(i, j )

to be

_sitting

on a domain wall as

A truncated cumulant

expansion,

followed

_by

a Gaussian

integration gives :

The

_{quantities 8) 2 and 82) c’}

for S = sin

(0

l - e J - A jj),

can be evaluated in the SCHA :

and

Curves for

_Dij

vs. t on

_h,

v and a-bonds are shown in

_figure

7 for three different values of the modulation

_parameter

a. One can see from the

_plot

_that,

_excepting

the

_isotropic

case,

the domain wall

_density

on a-bonds is _very small

_compared

with those on unmodified vertical

and horizontal bonds and it becomes

_{important just}

close to the

_{Ising point.}

The same

qualitative

behaviour was observed in reference

_[7],

where the authors evaluate the densities

of domain walls

_using

nearest

_{neighbour charge-charge}

correlation functions on the dual

lattice.

Fig.

7. -

6. Conclusions.

In this

work,

we have used a variational

_technique

to deal with the

_problem

of

_{Ising phase}

transitions in frustrated 2D XY models. Our results

_{(in good}

_agreement

with those obtained

by

numerical simulations

_{[5, 7],}

earlier

_{analytical approaches [8]}

and recent

_experimental

work

_[9])

shows that the

_Ising

character associated with the discrete

_symmetry

is

_present

all

over the range of modulation above «

= 1;

it causes a double

_phase

transition for

3

a # 1 and renormalize the critical

_temperature

at a = 1. For values of « in the interval

1,1

]

the

_picture

that emerges

is,

as

_proposed

in reference

_[7],

those of an _array of

_dipoles

3

flipping

around the center of a-bonds that

_undergoes

an

_Ising

transition at low

_{temperatures.}

At a

_higher

_temperature,

_unbinding

of thermal induced vortex-antivortex

_{pairs (integer}

charges),

causes a BKT transition. At a =

1,

there _{are no more} fixed

dipoles,

and

isotropic

domains,

by

liberating

fractional

_charges

when

_{growing, precipitates}

the BKT transition. The fact that the

_screening

of Coulomb interaction is due to fractional

_charges

does not seem to affect the value of the

_jump

in

_helicity

modulus at a = 1. On another

hand,

the observed

singularity

of the

_specific

heat

[5]

shows that the

_Ising

character of the transition is

_always

present

in the

_isotropic

case.

Topological

order is observed to coexist with

_Ising

disorder but not the

_contrary.

We believe that this last

_property

is a

_{particularity}

of the model and not an universal feature

_[16].

In other

_words,

we can

_imagine

2D

_systems

in which the

_coupling

between the

_Ising

and the

BKT variables allows

_topological

disorder in the interior of

_Ising

ordered domains. However in our

_system,

the

Ising

order

_parameter

in a sort of

_chirality

measure of

_topological

excitations and looses its sense when

topological quasi-long-range

order

disappears.

The variational

_technique

used in this paper is able to account not

_only

for the double transition but also for the

_{microscopic picture}

of the transition mechanism. In view of the

preceding

remark,

we can trust our

phase diagram

of

figure [7]

and conclude that fractional

charges

remain confined at the

_{Ising point}

for a

_{approaching unity.}

Both transitions are

always splitted

and

they

merge

only

for a = 1. This

fact,

proposed by

H. Eikmans et al.

[7]

based on domain wall energy

_{considerations,}

_{was, up to now,} never contradicted

_by

_any

experiment

or numerical simulation.

Acknowledgments.

We would like to thank one of the referee of this _paper for his clear and constructive

suggestions.

This work has been

_{supported by}

the Swiss National Science Foundation.

Appendix

Calculation

_{of ( c/J j -}

_{c/J j )2> TR .}

In order to use the

_density

matrix _{p TR} of

_{equation (3.3)}

to _{evaluate statistical averages,} we

A four site basis is introduced on the _array, in the

following

_{way :}

where

a

being

the lattice constant of the array, x

and ÿ

the unit vectors on the x

and y

axes, nx and

ny

the

integer

coordinates of the center of the unit cell in the new _array

_(see

_Fig.

_8).

Fig.

8. -

Parametrization of the lattice with basis introduced in

_{equations (A2).}

Expanding

all functions of p in Fourier

series, HTR

can be rewritten as follows :

the

_dynamical

matrix

_{M(q )}

_{being given by :}

Its

_eigenvalues

and

eigenvectors

are listed in the

_following

table :

Eigenvectors :

with the notations :

Thus,

for each site

_{(p ; s ; t )}

we can write :

where

_{au (q ; s ; t )}

is the

_(s,

_{t )}

_component of the

_eigenvector

_Vu defined above.

_HTR

can then be

_expressed

as follows :

and the thermal

_{averages (1 p u (q) 12) TR}

with the Gaussian

_{density matrix (3.3)}

take the

value :

We have now all the tools we need to

_perform

_{the averages}

Xij

entering

in

_equations

_(3.10)

and (3.11 ).

Approximating

the

_{q-dependent quantities by}

their

_{small q}

limit,

and

_replacing

the sum

by

an

_integral

d2q

on a circular Brillouin zone of area

7T 2 we

find :

In a similar _way one finds :

Expressions (A8)

can be used

directly

in the iterative

procedures

of section 4.

However,

a

_problem

arises with the

_procedures

when we reach

_TIS

from below : the X’s

diverge

because the denominator

_{D = G (G + 2 ) - F2 in}

_{equations (A8)}

goes to 0 at the

transition.

Indeed,

at

_TIS

the

_Ising

like order

_parameter

S is 0.

_Thus,

from

_{equations (3.10)}

we

get

4 a Zab -

(a - a )2 = 0,

G =

_{(ab)1/2-a(b/a)1/2}

and F =

(ab)1/2+a(b/a)1/2.

_Simple

algebraic

manipulations

yields

G ( G

+

_{2 ) - F 2}

=

0,

or in an other form :

This

_divergence

is related to the

softening

of the third

_optical

mode at the

Ising

transition,

since

_{for q -->}

0 one can write :

where 03BC ~ 0 at

_TIS.

The

_softening

of an

_optical

mode is an usual

_signature

of

_Ising

transition in

SCHA,

and

causes

_divergencies

in two

_dimensions,

since the

_dispersion

relation _goes as

q2.

However,

we known from RG

_{analysis [12]}

that

_{the q}

_dependence

for 2D

_Ising

models near

the critical

_point

is

_actually

_{q (2 - 1/}

4)0

In order to avoid the

_difficulty

we have chosen to

_replace

the third

_eigenvalue

in the

_{small q approximation}

and near

_TIS

as follows :

since this correction to the mass of the third

_optical

mode will induce a correction of the

denominator D near

_{TIs :}

The above

_replacement

is conceived in order to _{suppress the}

_divergence

at its

origin.

Quantitatively

Au

is chosen in order to simulate the effect of the actual _power

_{of q}

in the

dispersion

relation,

when

_computing

_{statistical averages :}

The actual

_dispersion

relation of the third

_optical

mode near the

Ising

critical

point

should

be of the form :

It will be

_changed

in :

1

will be

_changed

in

Since we are interested in

using thé correction

AD

just

close to the critical

point,

we can assume the relation G + 1 =

F2 +

1 in

evaluating

it.

Finaly,

the

expression

of D

(that

comes from the

_{product U 3 U 4)}

will be

_given,

near

_{TIS by :}

as we wish in

expression (A9).

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