Phase diagrams of antiferromagnetic Z(q) models

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Phase diagrams of antiferromagnetic Z(q) models

A. Benyoussef, L. Laanaït, M. Loulidi

To cite this version:

A. Benyoussef, L. Laanaït, M. Loulidi. Phase diagrams of antiferromagnetic Z(q) models. Journal de

Physique I, EDP Sciences, 1993, 3 (12), pp.2397-2409. �10.1051/jp1:1993252�. �jpa-00246875�

Classification Physics Abstracts

05.50 05.70F 75.10H

Phase diagrams of antiferromagnetic Z(q) models (*)

A.

Benyoussef (I),

L. Laanait

(~)

^and M. Loulidi

(1)

(')

Laboratoire de

Magn6tisme

et de

Physique

des Hautes

Energies,

Facult6 des Sciences de Rabat, B-P-1014, Morocco

(~) Ecole Normale

Supddeure,

B-P- 5118, Rabat, Morocco

(Received 25 May 1993,

accepted

in final

form

19

August1993)

Abstract. The

phase diagrams

of

general

q-state antiferromagnetic Z(q models _are studied for different values of q within _a real space

renormalization-group

scheme for _a d-dimensional

hypercubic

lattice. The model exhibits _a rich

variety

of phase

diagrams

for _even values _{of q. The} odd values of q give _no phase transition for bidimensio~lai lattices except at temperature T

= 0. Nevertheless, _an

antiferromagnetic

order appears ^at low temperature for d _m 3. Between

phase

transitions between

antiferromagnetic phases

and the disordered phase, ^{the model} undergoes

a phase transition between antiferromagnetic

phases

and

ferromagnetic phase

for q ~ 5.

1. Introduction.

It is well-known

that,

for the

antiferromagnetic

Potts

model,

it is

energetically

^{favourable for}

two

neighbouring spins

to be in distinct

spin

states. As _a consequence, the

ground

state of the q m 3 model _on

bipanite

lattices

(and

_q

=

2 model if the lattice is _not

bipartite)

has _{a non zero} entropy.

Then,

_{art argument can} be made _as in Wannier

[Ii

that _a transition to a

phase

with

long-range

order will not arise.

Nevenheless,

Berker and Kadanoff

[2]

have

argued

from _a

rescaling

argument that such systems can have a distinctive

low-temperature phase

in which correlations

decay algebrically.

For the q-state

antiferromagnetic

Potts

model,

this behavior is

permitted

if the

spatial dimensionality, d,

is

Sufficiently high

_or, for _a fixed

d,

if q is less than _a cutoff

value, qo(2)

₌ 2.618

[3].

The vector Potts model is _a

special

case of _{a more}

general

^discreet

planar

model, the

Z(q)

model. The number of

couplings

in this model increases _as

[q/2],

^the

integer

pan ^of

q/2.

In the present ^work we

study

the Z

(q )

models

using

_a

Migdal-Kadanoff

type renornJalization

grofip (MKRG) procedure [4]

and construct the

phase diagram

^{of the}

antiferromagnetic

Z

(q )

models for different values of _q. However it is necessary ^to present ^the

phase diagram

in

(*)

Supported

by the agreement of

cooperation

between C-N-R--Morocco and the D-F-G--Germany-

different

regions

of the parameter space then _{we can} show the bond between different

phase

transitions of the

ferromagnetic

and

antiferromagnetic Z(q )

models. We remind the reader that the MKRG method cannot

give

the first-order transition _{so we} will _not indicate the order of

transition _{at some} fixed

points.

The paper is

organized

_as follows _: in section

2,

_we define the

Z(q)

model and

give

_some

details _on the MKRG method. Section 3 contains _our results for different values of q, In section

4,

we discuss _our results and

give general

conclusions for the

antiferromagnetic Z(q)

^models.

2. Model and MKRG method.

Consider _a

simple

_square lattice whose lattice

points

_are

occupied by

classical _«

spins

» of unit

length pointing

in _one of the

equiartgular coplanar directions, given by

the

angles (2 ark/q ) (k

=

0,

1,

..., q I

)

the interaction between the

spins

^is assumed to

depend only

_on

the absolute value of the

angular

difference between two

nearest-neighbor spins.

Its _most

general

fornJ _can be written _as :

iq,21 ~ _~

flH

=

£ £ K~

_cos

m(«, «~ )) ⁽¹⁾

<u> _m i q

Here, _«,

= 0, 1,

...,

q -1, and

ii j)

denotes

pairs

of

nearest-neighbor spins.

The model defined

by (1)

is called the

general

discreet

plartar

model _or Z

(q )

model

[5]. Hereafter,

_we will denote

by

N the

integer part

of

q/2,

N

=

[q/2].

For

special

values of the

K~'s,

all

positive (all negative),

~the

Z(q)

model reduces to the

ferromagnetic (antiferromagnetic)

Potts model

[6]

defined _{as :}

flH

=

£ K(q8 («; «j) 1). (2)

<>J>

The standard

antiferromagnetic

Potts model does not order for _q

~ 2.618 _on square ^lattice

[3].

Moreover,

by using

the

Fonuin-Kasteleyn representation [7]

one can prove, ^{for d} m 2 that there exists _a

q~(d)

such that for _q

~

q~(d)

the free energy and the correlation function _are

analytic

for all values of the temperature

[8].

Therefore for _q

~

q~(d)

^the

arttiferromagnetic

Potts model is in its disordered

phase

for all temperatures.

If the

K~'s

_are all _zero for _m ~1, the model is the Clock _or vector Potts model.

The MKRG

method,

_{we use,} is the _one

technique already applied

in reference

[5].

It has

already

^been

explained

in many papers so it is not described here. The recursions

equations

_are

written in ternJs of the t~ variables which _are defined

by

_:

+ 2

£~

w~ cos

(m

~ "

r)

+ Y~w~ cos

(N

^{~ "}

r)

~~

q q

t~ =

~

(3)

+ 2

£

_{w~ + Y~w~}

m=1

r =

1, 2,

,

N.

Y~ ⁼

2 for odd values of q, and

Y~ ⁼ I for _even values of _q

w~ = exp

( K~ (cos ⁽

^{~ "}

mr) 1) 1. ⁽⁴⁾

~

q

For scale factor

b,

_on d-dimensional

hypercubic lattice,

the recursion relations write _{as :}

i

(

)b~ ^'

(5)

Wr ~ ~°r

with

+ 2

£ t$

_cos

(m

^{~ "}

r~

^{+ Y~t( cos}

(N

^{~ " r}

q q

&~ =

~

(6)

+ 2

£ ^t$

+ Y~t(

m =1

Hereafter,

_we shall consider d

= 2 and b

=

3 to remain in the

antiferromagnetic

state. We will also discuss the results for d

=

3.

3. Results.

A.q=2

and

q=3

q = 2 and _q

= 3

correspond respectively

to

Ising

and three-state Potts models. In the

ferromagnetic

_case, for d

=

2,

there is _a

Single

second-order

phase

transition with _a fixed

point

at K*

=

0.721 for _q

= 2 and _at K*

=

I.091

_for

q = 3. In the

antiferromagnetic

_case, the three-

state Potts model does _not order, while

Ising

model orders at K*

= 0.721.

For d

=

3,

the

antiferromagnetic

3 -state Potts model does not exhibit _a

phase

transition. The

minimum value of q, for which the model does not

order,

is

q~(3)

= 2.976. This result

disagrees

with that obtained _on the _one hand

by

Banavar _et al.

[9]

and _on the other hand

by

the infinitesimal MKRG method which

gives q~(3

= 3.31. This

disagreement

is due _{to a} small difference between the

q~(dl's

obtained

by

different

approximate

methods.

Using

a

cluster-flip

Monte Carlo simulation

algorithm Wang

et al.

[10]

have showed that the low temperature

phase

has

long-range

order with _a finite-size

dependence

similar to that of the XY model.

Within the renormalization group scheme, this

phase

in which correlations

decay algebraically

is observed

(for

^b

=

3 at

high dimension,

and for infinitesimal scale factor at d

=

3)

at finite temperature

B.q=4.

This is the

symmetric

Ashkin-Teller model

II ii.

The various fixed

points

of this model _are determined. Their coordinates and the

phase

transitions

they

characterize _are

given

ⁱⁿ table I.

The Z

(4

model _can be rewritten in ternJs of _two

coupled Ising

models

II 2]

with the two

spin coupling

_L~

=

)Kj

and the four

spin coupling

_L~

=

Ki

flH

=

£ L~(«,

_{«~ + r,}

r~)

+

£ L~(«,

_{«~ r, r~}

), (7)

<'J> <'J>

It is clear from this Hamiltonian that the model

Z(4

is

symmetric.

It has five different

phases

which _are

regions

of attraction for five

phase

sinks

ferromagnetic phase (F),

disordered

phase (D), ferromagnetic phase

with difference _two between

neighbouring spins (F~),

antifer-

romagnetic phase

with difference _one

(AFj ),

and

antiferromagnetic phase

with difference two

(AF~).

In the

region

I

(Kj ~0, K~ ~0) (Fig. I),

the model is

equivalent

to two

coupled ferromagnetic Ising

models with

ferromagnetic

four

spin coupling (L~

_~ 0

). II

in

figure 1,

is the

decoupling point

of these

Ising

models. The

phase

boundaries _are lines of

Ising-like

transitions

II 3].

The intersection of these lines is localized _at the four _state

ferromagnetic

Potts fixed

point P~

on the line

Kj

= 2

K~.

The

region

2

(Kj

_<

0, K~

~

0),

represents two

coupled antiferromagnetic Ising-models

with

ferromagnetic four-spin coupling (L~ ~0). 1)~

is the

JOURNAL DE PHYS>QUE -T 3, N'12 DECEMBER 1993 88

Table I. Coordinates and

classification of

the

fixed points of

the Ashkin-Teller model.

Fixed

points (Ki*, Kf)

coordinates Domain in

' d

=

2 d

=

4

(Ki, K~)

_space

F

= + co,

K?

_<

Kl) (K(

=

2

Kf,

+ co Surface F

D

0) (0, 0)

^{Surface D}

F2

+

co) (0,

₊

co)

Surface

F2

AFT co) (0, co)

Surface

AFT

AF2

= co,

Kf

_<

Ki* (Ki*

= co,

Kf

_<

Kf )

Surface

AF2

(«) (1.892, 0.946)

^Line

(«)

(«)~~ (- l.892, 0.946)

Line

(«)~~

Ii (0,

₊

0.721) (0,

₊

0.196)

Critical line

1(~ ^{(0, 0.721) (0, 0.196)}

^{Critical line}

1?~ (0.721,

₊

co)

+

co)

Critical line

If~~ (- 0.721,

+

co) 0.196,

₊

co)

Critical line

II (1.443, 0) (0.392, 0)

Critical line

1)~ ^{(- l.443,} 0) (-

0.392,

0)

Critical line

P4 (0.890, 0.445) 16, 0.158)

Tricritical

point

P( (- 0.890, 0.445)

0.316,

0.158)

Tricritical

point

P ~,~

0.362)

Tricritical

point

p(F

0.724,

0.362)

Tricritical

point

decoupling point

of these

Ising

models. This part of the

phase diagram

^is

symmetric

to the _one

represented

in the

region

I. The

phase

boundaries

P( I(, P( If~~

and

P(1)~

shown in

figure

I

are lines of

Ising-like

transitions. The

P( point

where these lines meet is _on the line

Ki

= 2

K~.

^The line

P(1)~

is extended to the

region

3

(Kj

_<

0, K~

_~

0),

where the model is

equivalent

to two

coupled antiferromagnetic Ising models,

with _an

antiferromagnetic

four

spin coupling (L~ ~0).

The unstable fixed

point1(~

_dominates the D

AFT

^line of

Ising-like

transition. The extension of the line

P~ I(

is

represented

in the

region

4

(Kj

~

0, K~

_<

0),

where the model is

equivalent

to two

coupled ferromagnetic Ising

models with

antiferromagnetic

four

spin coupling L~<0.

The line

Kj

=

2K~

in the

region

3

corresponds

to the four-state

antiferromagnetic

Potts model which is in the disordered

phase according

to a well-known result. The model is also not ordered _on the line

Kj

=

2

K~ (region 4).

It _was

argued

that the truncated correlation functions for the

antiferromagnetic

4-state Potts model _at

high

dimension _go from _an

exponential decay,

in the

high

temperature

regime

to an

algebraic

_power ^law

decay,

ⁱⁿ the low temperature

regime.

This result is based _on numerical

computations perfornJed

in reference

[2]. Using

the MKRG method _on the invariant

subspace Kj

=

2

K~ (Ki

_<

0, K~

_<

0),

_one finds that the

antiferromagnetic

4-state Potts model exhibits _a

phase

transition for d

= 4. From the

symmetry

^{of the AT}

model,

this _{one can} exhibit _a

phase

transition between _a disordered

phase

and _{a new}

phase

called

«(«)»

_on the line

Kj

=

2

K~ (Fig. lb).

This _new

phase

is located at finite temperature ^{and is} characterized

by

an

algebraic

_power law

decay

^for the correlation function.

Recently

the whole

phase diagram

for the AT model _was studied

analytically by using

the restretched ensembles method which is based _on the

Pirogov-Sinai theory

and its extension

II 4].

It _was

argued

that the model presents

a new

phase

on the line

Kj

₌ 2

K~

for d

~

d~.

^{For d} <

d~,

this _new

phase

does _{not occur} (d~ ^is a critical

dimension). Using

Monte Carlo

simulations,

_mean field

theory

and series

K2

6

F ,,%=2K>

', ,'

, ,,

,, ,/

A~ "~, ,,'~

,

%, ~' ,

', ^,'

j~

, ^'

a)

K~

G

I

F

,,

,"

$"2K2

,'

(o,o)

^~

~

' ,

lG

A6

Kim-2K~

b)

Fig,

I. a) Phase diagram of the Ashkin~Teller model for d

=

2. The _arrows indicate the stability of the

fixed points. b) ^Phase diagram of the Ashkin,Teller model for d

= 4. We notice that the phase

«

(« )

_,> is located at finite temperature. ^The arrows indicate the

stability

of the fixed

points.

The

region

Kj _~ 0 is obtained by symmetry.

expansion

Dizian _et al.

[15]

have shown that d~ ₌ ^3.

Figure

2 shows the

phase diagram

obtained

by

the MKRG method for d

=

4 in the

region (Ki

_~ 0

).

^{The other} part ^{of the}

phase diagram

is

symmetric

to this _one

c.q=5.

The

ferromagnetic Z(5)

model has been

investigated by

several authors. The

phase

structure

proposed by

Wu

[16]

and Alcaraz _et al.

II?]

is based _on the

duality

transformation and symmetry considerations.

They suggested

that the

phase diagram

_presents three

phases

: a

ferromagnetic phase (F),

_a disordered

phase (D)

and _a soft

phase. They argued

that the last

region

is _a

spin-wave phase. Rujan

et al.

[5],

Nishimori

II 8]

and

Roomany

et al.

[19]

obtained

only

the

ferromagnetic phase

and the disordered

phase. By exploring

finite-size

scaling

ideas and the conformal invariance of the critical infinite system, ^Alcaraz

[20]

showed that the

massless-spin

_wave

phase originates,

for the

Z(5)

model, at a bifurcation

point.

Bonnier et al.

[21]

have obtained the _same result.

Although

^{the MKRG method is unable} to show this soft

phase

for the

Z(5)

model

[5],

_{we present} in

figure

2a the

phase diagram

to show the

antiferromagnetic

pan ^{of the}

model,

in all

regions

of parameter space and the difference between d

=

2 and d

= 3. We obtain

only

two stable fixed

points

F and D. All fixed

points

of

Ki

,' Kin Ki

, -

,,

,

,, ,'

,

f

,,

,

,~

,

,.,

#

v ,

, '1

"'

' ' '

a)

Fig.

2. a) Phase

diagram

of the Z(5) model in dimension d

=

2. The _arrows indicate the

stability

of the

fixed

points.

b) Phase

diagram

of the Z(5 ) model in dimension d

= 3. The arrows indicate the

stability

of

the fixed

points.

K2

,' K =K

,

#Z$

, , , , '

t

'

,

,f /~

2 ,

~

' /

,'

l' j

Ji / Ji

b)

Fig,

² (continued).

this model and their coordinates _are indicated in table II. The

phase

boundaries _are lines described

by ii

and

j~ respectively. P~ ii

and its extension for

(Ki

_< 0,

K~

_~

0)

_represents the F- D

phase

transition

(Ki<K~, +co).

The symmetry ^{of this model} on the

permutation

Kj-K~

leads to the _same

phase

transition _on the line

P~j~

and its extension for

Kj

_~ 0,

K~

_~ ^{0. The} intersection of these lines,

Ki

=

K~,

is described

by P5

which

corresponds

to the 5-state

ferromagnetic

Potts model. All the

region (Kj

< 0,

K~

_<

0)

is disordered which

means that the

Z(5)

model does _not order

antiferromagnetically.

The

antiferromagnetic Z(5)

model _was studied

by

Den

Nijs [22]

in the SOS model limit which

corresponds

to

Kj

_- + co and

K~

_- co, or

Kj

- co and

K~

_- + co. He showed that the low

temperature

phase

is _a

floating

solid with

impurities

and

logarithmically

bound vortices with _a transition between this

phase

and _a disordered

phase

at finite temperature. ^{Within the MKRG method} this

phase

transition _occurs

only

at infinite

couplings.

For d

=

3,

the

ferromagnetic Z(5)

model has _a

qualitatively

similar

phase diagram,

but this model orders _at the stable fixed

point AF~

which characterizes _an

antiferromagnetic

order with difference _two between

neighbouring spins

and _at stable fixed

point AFT

which characterizes

an

antiferromagnetic

order with difference _one between

neighbouring spins.

The

phase

transition

AF2-D

is located at the line which is described

by

the unstable fixed

point j( (Fig. 2b).

The symmetry ^{of the} model leads _to another critical line described

by

the unstable

fixed

point it namely

the

AF2-D

transition is

replaced by AFT-D

transition. We mention also that the

antiferromagnetic

Potts model _on the line

Ki

=

K~

^does not order. The coordinates of these fixed

points

_are indicated in table II.

D.q=6.

Table II. Coordinates and

classification of

the

fixed points of

the

Z(5)

model.

Fixed

points (Ki*, Kf)

coordinates Domain in

d

=

2 d

=

3

(Kj, K~)

_space

F

(Kf

= + co,

Kf

_m

Kf) (Kf

= + co,

Kf

_~

Kf)

Surface F

and and

(Ki*

_m

Kf, Kf

= +

ml (Ki*

_m

Kf, Kf

= + co

)

D

(o, 0) (0, 0)

Surface D

AF2 (Ki*

₌ co,

Kf

m

Kl)

Surface

AF~

AFT (Ki*

_m

Kf, Kf

= co

)

Surface

AFj

ii (0.738, 0.103) (2.049, 0.168)

Critical line

j2 (- 0.103, 0.738) (- 0.168, 2.049)

Critical line

Ii _{(- 0.742, 0.132)}

_{Critical line}

ji (-

0.132,

0.742)

^{Critical line}

P5 (0.454, 0.454) (0.759, 0.759)

Critical

point

The parameter space is three-dimensional

(Kj, K~, K~ ).

For

good clarity,

we have

separated

the

phase diagram

into _two different

regions.

The various fixed

points

of this model _were determined. Their coordinates and the

phase

transitions

they

characterize _are

given

in table III.

We _use the _same nomenclature _as above. The model has six different

phases

:

D, F,

F~, F~, AFj

and

AF~.

In the

region

I

(Ki

m

0, K~

_m

0, K~

_m ⁰

) (Fig. 3a),

_we have

represented

the critical surfaces between different

phases.

The _two

Ising points ^1*~

and

I~

_describe

respectively F~-D

and

F~-F phase

transitions.

P~

^and

Pi

_are two 3-state Potts fixed

points they

describe

respectively

the

F~-F

and

D-F~ phase

transitions. The critical surface D-F is described

by

the fixed

point

S which defines the

universality

class of the vector Potts model. It becomes _a

region

of

stability

which grows if q increases. This

region

of

stability

characterizes _a massless

spin

_wave like

phase [5, 23].

The line _on which the critical surfaces

D-F, D-F~

and

F2-F

meet is the domain of attraction of the cubic fixed

point

C. Its dual C* describes the critical line

separating

the critical surfaces

F~-D,

F-D and

F-F~. ^D~

is the

decoupling point

of two

independent ferromagnetic Z(2)

and

Z(3)

models. It describes the line

separating

the four critical surfaces

F2-D, F3-F, F-F2

and

F~-D.

The intersection of all these lines is localized _on the 6-state

ferromagnetic

Potts fixed

point P~ along

the line

Ki

=

K~

₌ 2

K~.

In the

region 2(Ki

<

0, K~

_<

0, K~

_<

0) (Fig. 3b)

we show all

antiferromagnetic

transitions _:

AFj-D, AF~-F~, AF~-AFT, AFj-F~. They

_are localized _on critical surfaces which _are the

respective

domains of

repulsion

of five unstable fixed

points

: the _two

Ising

fixed

points (1*~~, I~~), K,

E and J.

D~~

_{is the}

decoupling

fixed

point

of the _two

independent models,

_an

antiferromagnetic Z(2)

and _a

ferromagnetic Z(3).

It describes the

separation

of

D-AFj

and

AF~-F~

critical surfaces. The unstable fixed

point 52

describes the coexistence line of

F3-D, AF~-D, F~-AF~, AFj-F~

and F-D critical surfaces.

The

plane

defined

by Ki

=

K~

+a =

2K~ (where

a is _a translation

factor)

is in the disordered

phase according

to the well-known results. A

special example

is the 6-state

antiferromagnetic

Potts model which is _on the invariant

subspace Ki

=

K2

_# ²

K3 (Ki

<

°,

K~

_< 0,

K~

_<

0).

The

phase diagram

^{for d} ₌ ^{3 is}

qualitatively

similar to the _one obtained for d

= 2.

E.q

₌ 7.

Table III. Coordinates and

classification of

the

fixed points of

the

Z(6)

model

for

d=2.

Fixed

points (Kl, Kf, Kf)

coordinates Domain in

(Kj, K~, K~)

space

F

(Kf

₌ + co,

Kf

_<

Kl, Ki

<

Kl)

Voiume F

and

(K(

_<

K?

_~

Kf, K?

₌ + co,

Ki

# +

°')

D

(0, 0, 0)

Volume D

F~ (0, 0,

₊

co)

Volume

F~

F~

(0,

_{+ co,}

0)

Volume

F~

AFj (0, 0, co)

Volume

AFj

AF~ (Kf

= co,

K?

_<

K(

,

Ki

_<

Kl )

Volume

AF~

( [Kj*

_<

[Kf _[, Kf ~~

co,

K?

₌ co

I*F

(0,

0,

0.721)

Critical surface

Pi (0,

09,

0)

Critical surface

IF

(0.481,

_{+ co,}

0.481)

Critical surface

P~

(0.545, 0,545,

+

co)

Critical surface

S

(2.814,

0.474,

0.l14)

Critical surface

I*~F (0, 0, 0.721)

Critical surface

I~F

(0,

+ co,

0.721)

Critical surface

E

(0, 1.09, co)

Critical surface

K

(- 2.814, 0.474, 0.l14)

Critical surface

J

(+

_co,

0.287, co)

Critical surface

C

(0.980, 0.245, 0.490)

Critical line

C*

(0.832, 0.832, 0.35)

Critical line

DF

(0, 09,

0.

721)

Critical line

D~F

(0, 1.09, 0.721)

Critical line

Si (- 0.980,

0.245,

0.49)

Critical line

S~

(- 0.832,

0.832,

0.35)

Critical line

P~ (0.665, 0.665, 0.665)

Tricritical

point

The

Z(7)

models _are

symmetric

under the

permutation Ki

-

K~

-

K~ (such

_symmetry is

always

present for _q

equal

to _a

prime number).

^The

phase diagram

is shown in

figure

4a.

MKRG method

gives

two trivial sinks

describing by

the stable fixed

points

F and D. The fixed

points

of this model and their coordinates _are indicated in table IV. The

phase

boundaries _are surfaces described

by sj, S~

and

S~.

These fixed

points

have _a

region

^of strong

stability

which indicates the presence of _an intermediate

phase [5, 23].

The

Si Point

describes the critical surface

F(Kj*

= + co,

Kf

<

K?, K?

_<

Ki*)-D.

The symmetry ^of the model leads _to the _same critical surfaces described

by

the unstable fixed

points 52

and

53 respectively.

The critical lines

separating

different surfaces _are described

by

the unstable fixed

points [, _J~

and

J~. ^The intersection of these lines is localized _on the unstable fixed

point _P~ (Kj

=

K~

₌

K~)

which

corresponds

to the ?-state

ferromagnetic

state Potts model. The antifer-

romagnetic Z(7)

model does _not order except at temperature ^T ₌ ^{0. In}

particular

_on the line

(with (Kj

_< 0,

K2

_< 0,

K3

_<

0))

it

corresponds

to the ?-state

antiferromagnetic

Potts model

according

to the well-known result.

K3

~3

F~

IF

o K2

,"'

j

F ,"'

," /OAF

."' _I'

F3

," j

s ,'

O ,'

,"

"'

o

Ki

a)

<j

Ii

,,~ K

=

K z2K

f

O,,

,"

3 ,'

,' ,' .'

b)

K~

/

S~l =K~= K~

, '

j

,~f

)

J~j ~

2

,/

⁰

K~

F

,/ ^J3

/ /

~ Sj

a)

,

K

,' ,

~ kQ

~

~ j~ -

/

0

/

~

/ '

/

'j

i

AF~

Fig.

4. a) Phase

diagram

of the Z(7) model in (Kj, K~, K~) space in dimension d

=

2. The _arrows

indicate the

stability

of the fixed points. b) Phase diagram of the Z(7) model in dimension d

= 3. We have shown only the critical surfaces AF~-D and

AF3-F.

The critical line described

by

the fixed point

J(

is indicated by a

heavy

drawn line. The _arrows indicate the

stability

of the fixed

points,

and the wavy lines denote _a smooth continuation of surfaces. The other critical surfaces _are obtained

by

permutations

Kj _~ K~ _~ K~.

4 Fig. 3.-a) Phase diagram of the Zj6) model in the region1

(Kj~0, K~~0,

K~m0) for

d

=

2. The _arrows indicate the

stability

of the fixed points and the wavy lines denote _a smooth

continuation of surfaces. The dotted lines (-.-.-.-.) indicate the boundaries of critical surfaces located behind the D-F and F3-F critical surfaces, b) Phase

diagram

of the Z(6) model in the region ² lKj ~ 0, K~ ~ 0, K~ _< 0) for d ₌ 2. The _arrows indicate the

stability

of the fixed

points

and the wavy lines denote _a smooth continuation of surfaces.

Table IV. Coordinates and

classification of

the

fixed points of

the

Z(7)

model.

Fixed points (Kj*,K~*, K?) coordinates Domain in

d 2 d ₌ 3 (Kj, K2, K3) _space

F (K~* = + cc, Kf

< Kj*, K?

< Kj*) (Kj* _{= +} cc, Kf _{< K~*, K~*} < Kj*) Volume F

and and

(Kj* _~ Kf, Kf + cc, K?

< Kf) (Kj* < Kl, Ki + cc, Ki

< K~)

and and

(Kj* _< K?, Kl

< K?, K?

= + cc ) (Kj*

< Ki, K?

< K3*, Ki + CC)

D (0, 0, 0) (0, 0, 0) Volume D

AF~ (K(

= cc, K?

~ Ki, Ki _< Kj*) Volume AF3

AF~ (Kj* < Kf, K? ₌ cc, Ki

~ Kj*) Volume AF~

AFj (K~* < K?, Kf

~ Kf, K?

= cc ) ^Volume AFj

Sj (0.740, 0.018, 0.000277) (3.442, 0.613, 0.124) Critical surface

S~ (0.000277, o.740, 0.018) (0.124, 3.442, 0.613) Critical surface

53 (- 0.018, 0.000277, 0.740) (- 0.613, 0.124, 3.442) Critical surface

S( (- 0.740, a-a119. 0.00029) Critical surface

S( (- 0.00029, 0.740, a-a119) Critical surface

Sj (- a-a119, 0.00029, 0.740) Critical surface

Lj (- cc, + cc, + cc) Critical surface

L~ (+ _{cc, cc, +} cc) Critical surface

L~ (+ _{cc, + cc,} cc) Critical surface

j, ^{(0.476, 0.588,} 0.018) (0.744, 1.022, 0.093) Critical line

J~ (o.018, 0.476, 0.588) (0.093, 0.744, 1.022) Critical line

J~ (0.588, 0.018, 0.476) (1.022, 0.093, 0.744) Critical line

J( (- 0.495, 0.699, 0.0563) Critical line

Ji (0.0563, 0.495, 0.699) Critical line

Ji (0.699, 0.0563, 0.495) Critical line

P~ (0.376, o.376, 0.376) (0.594, 0.594) 0.594) Critical point

As for q = 5 Den

Nijs,

in the SOS

40del limit,

showed that the low

temperature phase

for the

antiferromagnetic Z(7)

model is _a

floating

solid and that the F and AF

floating phases

are

directly

connected.

The

phase diagrams

for the

ferromagnetic

model for d

= 2 and d

= 3 _are very similar. The fixed

points

for d

=

3 _are indicated in table IV.

Nevenheless,

the

antiferromagnetic Z(7)

model for d

= 3 possess an order _at the stable fixed

points

_:

AF~, AF~

and

AFj.

The critical surface

AF3~D

is described

by

the unstable fixed

point S( (Fig. 4b).

The symmetry ^{of the} model

gives

other identical critical surfaces

AF~~D

and

AFT-D. They

_are described

by

the

unstable fixed

points S(

and

Sj respectively.

The

antiferromagnetic Z(7)

model for

d

= 3 _possesses like

Z(6) model,

_a transitions between

ferromagnetic

order and antifer-

romagnetic

order

namely

for the critical surfaces

AF~-F, AF~-F

and

AFT-F

which _are

described

by

the unstable fixed

points Li,

_L~ and

L~.

^{The unstable fixed}

points J(,

ii

and

ii

describe the critical lines which separate ^these surfaces. The

antiferromagnetic

Potts model which is _on the invariant

subspace Ki

=

K~

₌

K~

^does not order.

4. Conclusion.

We have studied the

antiferromagnetic Z(q)

model within the MKRG method. We find _a rich

variety

of

phase diagrams

for _even values of q. Odd values of q

give

_no

phase

transitions _on square lattice

except

at temperature ^T =

0.

Indeed,

for even values of _q the

Z(q)

_group has

Z(2)

_as

subgroup.

It is easy to see, for the

antiferromagnetic

Clock

model,

that there exist q non translation invariant

antiferromagnetic

phases

at low temperature ^{whenever the dimension d} m 2.

Thus,

the

antiferromagnetic

^Clock

models for _even values of _{q are} in the universal class of the

antiferromagnetic Ising

model. The

antiferromagnetic

Z

(2

_q

models, however, present

^several

type

^of transitions between AF and D

phases

_on the _one hand and between AF and F

phases

_on the other hand. A _new

phase

called

«

(«)

_{» appears} at finite

temperature

^for q = 4 and d

= 4. Since the

antiferromagnetic

Potts

model orders for

large

values of _q and

d,

the _same

phase

_{can appear} for every value of

q m4 at

large

d.

For odd values of q, the

antiferromagnetic

Z

(q model,

within the MKRG

method,

does not exhibit _a

phase

transition _at finite

couplings K8

at d

= 2. In the SOS model limit

[22] (infinite K~),

it _was shown that the low temperature

phase

is _a

floating

solid and that the model exhibits

a

string melting

into _an mq fluid _{at non-zero} temperature. ^More

study using

_more

sophisticated

methods is needed in order _to

chop

_up construct the

phase diagrams

for these models.

In three dimension the

antiferromagnetic Z(q)

model for odd values of q exhibits _a

phase

transition _on the _one hand between disordered

phase

and

antiferromagnetic phases

and at q m 5 _on the other hand between

ferromagnetic phases

and

antiferromagnetic phases.

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