Renormalization group for an Ising system with competing interactions

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Renormalization group for an Ising system with competing interactions

P.M.C. de Oliveira

To cite this version:

P.M.C. de Oliveira. Renormalization group for an Ising system with competing interactions. Journal

de Physique, 1986, 47 (7), pp.1107-1113. �10.1051/jphys:019860047070110700�. �jpa-00210296�

1107

Renormalization _group for an Ising ^{system with} competing interactions (*)

P. M. C. de Oliveira

Instituto de Física, Universidade Federal Fluminense, ^Outeiro de São João Batista s/n Niterói, ^RJ 24210, ^Brasil (Reçu ^le 29 juin 1985, révisé le 14 octobre 1985, accepté ^{le 19 mars} 1986)

Résumé.

Nous appliquons ^une approche de groupe de renormalisation dans l’espace ^{direct à un} hamiltonien

d’lsing ^sur réseau carré avec des interactions de proche (J1) et ^second (J2) voisins. Nous obtenons le diagramme

de phase complet. Celui-ci exhibe trois phases ordonnées distinctes : ferromagnétique (F), antiferromagnétique (AF)

et super-antiferromagnétique (SAF). ^{Dans cette dernière} phase, les interactions concurrentes J1 > 0 et J2 ⁰ jouent ^un rôle crucial entraînant une dégénérescence d’ordre deux des états fondamentaux. Ce phénomène ^se

traduit dans notre approche par ^un attracteur avec bifurcation d’ordre _{deux pour} cette phase. ^De plus la forme des

trajectoires de groupe de renormalisation près ^{de la} ligne critique suggèrent que la transition SAF - P n’est pas universelle.

Abstract

We apply ^{a Position} Space Renormalization Group approach ^{to the} Ising Hamiltonian in a square lattice with first (J1) ^{and second} (J2) neighbour exchange interactions. The full phase diagram ^{is obtained with}

three distinct ordered phases : ^ferro (F), ^antiferro (AF) ^and superantiferromagnetic (SAF). ^In particular, in the last one, the competing interactions J1 ^and J2 0, play a crucial role yielding ^a two-fold space degeneracy ^{of the} ground ^states. This behaviour is reproduced ^{in the} present approach by ^a bifurcated two-fold attractor for this

phase. Moreover, the non-universal character of the SAF - P transition is inferred from the Renormalization Group trajectories ^near the critical line.

LE JOURNAL DE PHYSIQUE

J. Physique ⁴⁷ ( 1986) ^{1107-1113 JUILLET} 1986,

Classification

Physics ^Abstracts

05.70J

05.50

75.10H

1. Introduction.

The magnetic properties ^{of the} square lattice with first and second neighbour coupling ^{have been} investigated extensively [1-5] by ^Position Space Renormalization

Group (PSRG) techniques [1-4] ^{and series} expansions [5]. ^The magnetic ^models adopted ^are Ising [2, 5],

Potts [1], ^bond percolation [3] ^{and site} percolation [4].

Various interesting features of these systems ^{have been} studied : i) ^the possibility ^of extracting ^the magnetic

critical exponents ^without introducing any external

field, ^{from the crossover behaviour at} the tricritical

point localized between the ferro (F), ^antiferro (AF)

and paramagnetic (P) phases ( 1-4] (see Fig. ⁹ =below) ; ii) the universal behaviour at the F - P and AF - P critical frontiers [1-5]; iii) the existence of a super-

(*) Supported by CNPq ^{of Brazil.}

antiferromagnetic phase (SAF) ^for sufficiently large negative values of the second neighbour Ising coupling

constant JZ [2, 5] ; iv) ^the impossibility ^of sustaining

the AF ordered phase for finite temperatures ^without sufficiently large positive ^values of J2, for Potts models with 3 or more spin ^states [1, 6] ; v) the non-universal behaviour at the SAF - P critical frontiers [2, 5].

Another interesting feature is the two-fold _space

degeneracy of the SAF ground ^{states for the} Ising ^case

(see ^{section 4 of Ref.} [1]).

The purpose of the present paper is to study ^the

SAF strange behaviour, using PSRG, by scaling ^finite

lattices with the appropriate symmetries ^{of the SAF} ground ^states.

Section 2 presents the method for calculating ^the

PSRG transformation, including ^{a review of recent}

progresses obtained by preserving geometrical sym- metries. Section 3 presents ^the results, ^and the conclu- sions about the SAF phase ^are in section 4.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047070110700

1108

2. The PSRG transformations.

In a cell-to-cell PSRG calculation, ^{one obtains} quali-

tative features and quantitative approximations ^{for an}

infinite lattice, ^from physical quantities exactly ^calcu-

lated for finite cells of the lattice (unlike the alternative PSRG procedure ^of decimating ^{sites of an infinite} lattice, the cell-to-cell PSRG does not generate higher

order interactions). First, the values of a particular physical quantity ^are exactly ^{obtained for two different}

sized cells, ^{the smaller one at} temperature ^{T’ and the} other at T. A scaling transformation T’(T) ^{is then}

obtained by imposing ^{that this} physical quantity ^be

invariant under scaling. ^The length scale value is known a priori ^as being the ratio between the length

sizes of the cells. The Renormalization Group Theory [7] ^{is then} applied ^{to the} scaling transformation T’(7)

to obtain critical temperatures ^{and other critical} properties ^{of the} system. ^The reliability of the results

depends ^{on various} points : i) ^the arbitrary ^{choice of a}

convenient invariant physical quantity ^to be calculated for both cells; ii) ^the geometrical properties ^{of the}

cells that must reproduce those of the entire lattice, ^and

those of the ground ^state configurations ^{of the ordered} phases ^under study; iii) the size of the cells. In general,

the invariant physical quantity depends ^on parameters others than the temperature. ^{A crucial} point ^{is to}

include in the PSRG transformation all the parameters being relevant for the physical system. ^{The number} of those parameters defines the number of different invariant physical quantities ^to be calculated for both cells. In this _case, the PSRG transformation is no more a simple ^function T’(T), ^{but it is a} map in a muti- dimensional _space (the space of parameters).

In the first attempt ^{to use PSRG} [8], good approxi-

mations for the exactly known critical values for planar Ising ^{models were} obtained. The physical quantity

chosen to be invariant under scaling ^{was the} sign ^{of the} majority ^{of the} spins ^{in the cell} (Ising spins ^{can be}

either _up or down corresponding ^to dynamical

variable values + 1 or - 1).

The critical temperature ^{of the} square lattice Ising ferromagnet ^{was first obtained} [9] exactly ^{from the}

self duality property ^{of the} Ising ^{model on} this lattice.

An improvement ^{over the} early ^PSRG calculation can

be obtained by choosing ^{as invariant} physical quantity

the transmissivity (defined below) which besides the

free-energy preserves the self duality property ^of the family of cells shown in figure ^{1. In this manner, the}

exact critical temperature of the square lattice Ising ferromagnet ^{has been} reproduced [10] ^as follows. The

Hamiltonian of the Ising model with only ^first neigh- bour exchange interaction J1 ^is

where a; = ^{± 1 is the dynamical variable at site i,}

and (i, j) ^{is a} pair ^{of nearest} neighbour ^{sites. The} transmissivity ^{of a} cell like the one shown in figure ^{1 b}

Fig. 1.

Family of self-dual cells appropriate for the square lattice Ising ferromagnet. ^The length sizes of the cells are

b

I (a), b

² (b), b

³ (c), ^{and b}

⁴ (d) in units of the lattice parameter. The full circles are sites inside the cell,

and the open circles ^are sites of the neighbouring ^{cells. The}

number of sites in each cell is N

b2.

for example, is defined by taking ^the following ^boun- dary conditions : i) ^{the two sites at} the bottom of the cell are taken as the same spin; ii) ^{the two} sites above the cell (open circles) ^are also taken as the same

(other) spin; iii) ^{these two} collapsed ^{sites are the}

terminals. Figure ^{2 shows the} graphs corresponding

to the cells of figure ^{1 after} applying ^these boundary

conditions. Each graph ^{has a} pair ^of terminal sites

(open squares) ^{that can be in two distinct states :} paired (both up ^or both down) ^or unpaired (one up, the other down). ^The transmissivity of the cell is defined

as the difference between the probability ^{of the termi-}

nal sites being paired ^{and the} probability ^{of their} being unpaired. ^This quantity ^{can be} calculated from the partition function of the system, obtained ^from the _energy (1) ^{of each} possible spin configuration. ^The

number of configurations ^{to be} computed ^is 2 (b2 -b+l)

for a b-sized cell; this number increases exponentially

with the number N = b2 of sites in the cell. For the

simplest ^b

^{1 case} (Figs. ^{1 a and} 2a) ^{there are no}

internal sites in the graph, ^{and one} obtains the trans-

missivity tl ^{of a} single ^bond

For the b

2 case (Figs. Ib and 2b), ^the transmissivity tl ^is given by equation (3), ^{as a} function of tl.

Fig. ^2.

Family ^of graphs corresponding ^to the cells of

figure ^{1 after} applying ^the boundary conditions described in the text Open squares correspond ^{to terminal} sites,

while full circles are internal sites. The horizontal bonds

from the cell to its right neighbour ^{cell are omitted, because}

they ^{have no influence on the vertical} transmissivity.

Equation (3) ^{is the} required ^PSRG transformation

T’(7) ^{for a} length scaling ^factor b/b’

^{2. In the} physical region 0 tl 1, ^it presents ^{two distinct} behaviours under repeated iterations, depending ^{on the}

initial value of ti : the successive iterated values of tl

will _converge to the low temperature attractor 11

1,

or, alternatively, ^{to the} high temperature ^attractor tl

0, if the initial value of 11 ^{is above or below the}

critical value tt = j2 - ^1, respectively. This critical value corresponds ^{to the} exactly known critical temperature [9], because the graphs ^of figure 2 pre-

serve the self-duality ^{of the} square lattice Ising ferromagnet, ^for any value of b. The flow diagram ^of equation (3), ^{shown in} figure 3, corresponds ^{to the}

exact phase diagram ^{of the} square lattice Ising ferromagnet. Moreover, good approximations [10]

can be obtained for the correlation length ^critical exponent ^v (Table I). ^{Note that the} approximation ^is improved ^for increasing cell size. The value of the critical exponent ^{measures the rate} of decrease of the correlation length ^near the critical point, and for this

reason its approximated ^{value is} obtained from the rate of iterative deviation from the critical point T*,

i.e. the derivative of the PSRG transformation, ^as

shown in equation (4).

For the study of the critical properties ^{of the} Ising antiferromagnet, where the first neighbour interaction

J1 ^becomes negative ⁱⁿ equation (1), the cells of

figure ¹ (or ^the graphs ^of Fig. 2) ^{are no} longer appro-

priate. ^{The reason} for this failure is easily ^understood by analysing figure ^{4. The} boundary ^condition of taking

all the bottom line of the cell in the same spin ^state

is correct for the ferromagnetic ground ^state (Fig. 4a),

but it does not work for the antiferromagnetic ^case (Fig. 4b). Consequently, equation (3) ^{does not} present

Fig. ^3.

^Flow diagram ^of equation (3). Full squares ^are the attractors, and the open circle is the critical point separating

two distinct phases : ferromagnetic (F), ^{where the} points

are attracted to tl ^{= 1} by repeated iterations of equa- tion (3) ; ^and paramagnetic ^{(P), where the} points ^{are attracted} to ti ^=0.

Table I.

Approximations [10] for the correlation

length ^critical exponent ^v obtained from ^{PSRG trans-} formation. ^{A cell} of ^{size b}

2, 3, ⁴ (Fig. 1) ^{is trans-} formed by scaling ^{into a cell} of ^{size b’}

^1.

Fig. ^4.

^{Ground state} configurations of the three possible

ordered phases for the first and second neighbour square lattice Ising ^model. Ferromagnetic ^F (a), Antiferromagnetic

AF (b) ^and Super-antiferromagnetic ^SAF (c) (c’) phases.

Decoupling of the lattice into two sublattices in the J1

⁰

limit (d). ^{The SAF} phase ^{has two} degenerate ground ^states (c) and (c’).

the ordered phase ^attractor at t1

1, neither the correct critical point tf

¹

J2. ^{The same failure}

due to the same geometrical constraint is also observed in the _presence of second neighbour interactions

[1, 3, 4], ^{as well as in the case of} anisotropic (different

horizontal and vertical interactions) systems [11].

The anisotropic square lattice [12, 13], ^{as well as the}

inclusion of second neighbour interactions [1, 3, 4],

besides the simple antiferromagnetic ^{case can be}

treated by using ^the family ^{of cells shown in} figure 5, and the corresponding graphs ^of figure ^{6. The boun-}

Fig. ^5.

Family of self-dual cells appropriate ^{for the}

square lattice Ising antiferromagnet ^with length ^sizes

b = (a), b

² (b), b

³ (c) ^{and b}

⁴ (d). ^{The number}

of sites in each cell is N

2 b2.

Fig. ^6.

Family ^of graphs corresponding ^to the cells of

figure ^{5 after} applying ^the boundary conditions. For sim-

plicity, only ^{the b}

¹ (a) ^{and b}

² (b) ^{cases are shown.}

1110

dary conditions for the simple antiferromagnetic ^case (J1 0) ^{are :} i) ^{the sites at} the bottom line of the cell

are taken as the same spin ^and ii) ^{the sites at the} upper line inside the cell (full circles) ^are also taken as the same

(other) spin. Comparing figures ^{4 and 5} (the lattice has suffered ^{a 450} rotation), ^{we note} that these boundary conditions ^{are now} appropriate ^{also to the AF} ground

state. The PSRG transformation for the scaling ^{of the}

b

2 cell into the b

1 single ^{bond is} presented ⁱⁿ equation (5).

The flow diagram ^of equation (5) is shown in

figure ^{7. In the} physical interval - 1 K ti 5 ^{1 there}

are now three distinct attractors, ^{each one} correspond- ing ^{to one of the three} possible phases ^{P, F and AF.}

The critical values tt

^± (VF2 - ¹⁾ correspond ^to

the exact critical temperature ^{of both the F - P or} AF - P phase transitions. The Ising antiferromagnet

can be mapped ^{into the} corresponding ferromagnet, by interchanging ^the paired ^{and the} unpaired ^energy

values J and - J of each interacting pair ^of spins.

As the number of the paired possible configurations (both up ^or both down) and the number of the unpaired

ones (one up the other down) ^{are the same, the} map-

ping ^is possible. ^In particular, the square lattice Ising antiferromagnet has also the self-duality property, and this symmetry ^is preserved by ^the graphs ^of figure 6, ^{also for} any value of b. The approximations

obtained for the correlation length ^critical exponent, using equation (4), ^{are shown in} table ^II.

The same cells can be used for the case of including

second neighbour interactions (J2 > 0) in the Hamil- tonian (1). ^{In this case, the} space of parameters ^{is two-} dimensional, ^{with two} independent bond transmissi-

vities ti and t2 for first and second neighbour pairs ^of sites, respectively. Two distinct invariant quantities

are now necessary for obtaining ^{the PSRG two}

Fig. ^7.

^Flow diagram ^of equation (5). By ^successive iterations, ^an initial value t1 > J2 - ^{1 (F phase) will}

converge ^to the attractor tl

^{1. Another initial value} tl ¹ - J2 ^{(AF phase)} will converge ^to other attractor

t1

1. Intermediate initial values 1

J2 ^tl J2 -.1 ^I

(P phase) will converge ^to the central attractor ti 1

0.

Table II.

Approximations for the correlation length

critical exponent v, using ^{the cells} of figure ^{5. The value} for ^b

^{2 and 3 coincides} with those of reference [13].,

dimensional transformation. The first is the same

transmissivity ^defined above where both terminals

correspond ^{to sites} inside the cell; the second is obtained [1, 3, 4] by taking ^the upper terminal ^as the bottom line of the _upper neighbouring ^cell (open

circles in Fig. 5). ^{The b}

^{2 PSRG} transformation obtained [1] ^{in the} (tl, t2) ^{space reduces to} equation (5)

in the t2 ^{- 0} limit, and also reduces to equation (3)

in the tl ⁰ limit. The same reductions occur for _any value of b. In particular, the tl ^{--+ 0 limit} corresponds

to the decoupling ^{of the whole lattice into two} square sublattices [1-5] (see Fig. 4d), ^{and this crossover}

behaviour allows the evaluation of the magnetic ^cri-

tical exponent ^without introducing ^a magnetic ^field

in equation (1) [1-4J. ^{As can be seen in} figure 4, ^{the AF} ground ^state corresponds ^{to two} interpenetrating

sublattices (a and fl ⁱⁿ Fig. 4d). Each sublattice _separa-

tely ^{has a} ferromagnetic (F) configuration (Fig. 1 a).

The cells of figure ^{5 are obtained} by superimposing ^two interpenetrating equal ^{cells of} figure 1, ^{one for each} sublattice (details ^{in Ref. [1], section} IV). ^{In this}

manner, the cells of figure ^{5 and the} corresponding boundary conditions preserve the geometrical sym- metries of the AF ground ^{state as} the cells of figure ¹

have preserved early ^the symmetries ^{of the F} ground

state.

The symmetry of the SAF ground ^states (Figs. ^4c

and c’) ^{is not} preserved by ^{the cells of} figures ^{1 or 5 and}

their corresponding boundary conditions, ^{and it is}

necessary to construct new cells to treat the J2 ⁰ problem. ^{Each one of the} degenerate ^SAF ground

states corresponds ^{to two} interpenetrating sublattices

a and p, ^{each one in an AF} configuration. Using ^the

same previous reasoning, the cells of figure ^{8 are}

constructed [1] by superimposing ^two equal ^{cells of} figure ^{5. The} boundary conditions to be adopted ^are taking ^{the sites A as one} terminal and the sites B as the other terminal in each cell : this gives the first invariant

quantity. Taking ^{A and C as} terminals, ^we get ^{the other} invariant quantity needed for mapping (tl, t2) ^into (t’, t2).

Fig. ^8.

Family ^{of cells} appropriate ^{for the} study ^{of the}

SAF phase. They ^are constructed by superimposing ^two equal ^{cells of} figure (5), ^one for each sublattice a and (see Fig. 4d). ^The length sizes are b === 1 (a) ^{and b}

² (b).

The continuous (dashed) ^bonds correspond ^{to first} (second)

neighbour interactions. The number of sites in each cell is

N

4 b2.

The important ^{role of} preserving ^the symmetries ^of

the various ground ^{states is} conveniently emphasized

in references [14].

An important ^{remark concerns the} degeneracy ^of

the ground ^{states for} F, ^{AF and SAF} phases (Figs. 4a, b,

c and c’). ^{There is a trivial} spin degeneracy ^common

to those four ground-states, determined by ^the opera- tion of reversing ^all spins : ^{this is a} global symmetry transformation performed ^{in the} space of spins. ^This kind of degeneracy ^cannot be reflected in the flow of a

PSRG transformation performed ^{in the real} space.

Besides this trivial _one, however, ^{the SAF} phase presents another non-trivial degeneracy (Figs. ^4c

and c’) ^{that cannot} be associated to a global symmetry in the _space of spins, ^{and must be related to a} geome- trical symmetry transformation performed ^{in real}

space (a 90° rotation around an axis perpendicular ^to

the lattice). This kind of _space degeneracy ^{must be}

reflected in the flow of a conveniently constructed PSRG transformation. Because any RG attractor

represents ^the ground-state(s) ^{of the} corresponding phase, space degeneracies ^must be reflected in the behaviour of these attractors.

3. Results.

The flow diagram of the PSRG transformation, ^ob-

tained by scaling figure ^{8b into 8a,} is shown in figure ^9.

There are four distinct attractors (full squares) P, F, AF and (SAFI, SAF2), the latter formed by ^a pair

of points, ^one being ^the image of the other by ^the

PSRG transformation. Each attractor has its own

basin of attraction (the region of initial points ^that

converge ^to it by successive iterations of the PSRG

transformation), corresponding ^{to each one of the}

Fig. ^9.

^Flow diagram of the PSRG transformation

(tl, t2) ^--+ (t,, t2). The cell of figure ^{8b is scaled} into the cell of figure ^8a.

four possible phases ^{of the} physical system. ^{The four} basins of attraction (four phases) ^are separated by ^the

critical frontiers F - P, ^{AF -} P, ^{F - AF and SAF -}

P. Hereafter, ^we present separately ^the analysis ^{of the}

critical behaviour on each frontier.

3.1 F - P LINE. 2013 Points located exactly ^{on this line}

are attracted by ^Al. The critical behaviour of the F - P phase transition, however, ^is determined by

how an initial point ^{near the line turns away from}

it by successive iterations of the PSRG transformation.

The flow trajectory ^{of such a} point is directed towards

point Al, nearly parallel ^{to the line.} Only ^after approaching point Al, ^the trajectory deviates from the line, regardless ^of where is its beginning (near ^the line). This kind of reasoning ^{is the RG} explanation ^of universality : ^the critical behaviour is the ^same

(determined by point Al) along ^{all the line. The} approximated ^{value v}

0.718 is obtained at point ^A1, using ^the b/b’

² scaling ^factor. Certainly, ^{this value}

would be improved ^{if the PSRG} transformation could be calculated for greater ^cells.

Point T is a critical point separating three distinct

phases F, ^{AF and} P, and is located at the exact value

t2 = /2 - ^{1. In the RG language, it presents two}

distinct critical behaviours : i) ^a strong ^{rate of devia-} tion from it along the horizontal direction; ^and ii) ^a

weak rate of deviation along the t2 axis where the a

and fl sublattices (Fig. 4d) ^are decoupled ^In this latter

case (ti

^{0) each cell of} figure ^{8 reduces to two} equal

cells of figure 5, ^one for each sublattice a and fl. ^{In this}

manner, the critical exponents ^{obtained are those of} table II. Moreover, ^{it can be shown} [1-4] ^{that the other}

critical behaviour at point ^T (along ^the horizontal) corresponds ^{to another} independent ^critical exponent : the inverse fractal dimension 1 /df ^{of the} incipient

infinite island of correlated sites along the lattice.

The magnetic susceptibility ^critical exponent, ^for example, ^{can be then obtained from the} scaling ^rela-

tion _y

df ^{v. The} approximated values for df ^are

shown in table III.

3.2 AF - P LINE.

This line is the mirror image ^of

the preceding ^{F - P line,} and has the same proper- ties. This symmetry ^{is also} present in the real physical

system.

3. 3 F - AF LINE. 2013 This line also displays ^universal behaviour, ^because point ^{B attracts all} points exactly

located on the line and determines the critical beha- viour for points ^{near it. The} corresponding ^critical

exponent lld can be calculated along the t2

^{1 axis}

Table III.

Approximations for ^the fractal ^dimension

df of ^the incipient infinite ^islands of correlated sites,

formed ^near the tricritical point ^T (Fig. 9), using ^the

family of ^cells of figure ^8.

1112

for _any value of b, ^as given by

This exponent ^{measures the} coupling ^{of two infinite}

islands of ferromagnetically correlated spins, ^{one in}

the a sublattice and the other in {3. ^The positive ^second neighbour exchange interaction J2 ^is strong enough

to allow the formation of both islands, ^{and a} very small first-neighbour exchange interaction J, couples

the islands together. ^The coupling ^{can be ferro or} antiferromagnetic ^for positive ^or negative ^values of J1, respectively. ^{The exact value d}

^{2 can} be obtained from the thermodynamic ^limit (b ---> oo) ^{taken in} equation (6). This value shows that the infinite islands

are two dimensional objects (like the lattice itself) along ^{all the F - AF} line, except ^at point ^{T where} they ^{become infinite with a} fractal dimension df ^{= 1.75.}

3.4 SAF - P LINE.

It is well known [2, 5] ^{that the}

critical behaviour is non-universal along ^{SAF - P line.}

The flow direction along this line in figure 9, ^however indicates that the critical behaviour would be the ^same

along ^{all the line,} determined by point ^C (the ^same

values presented ^{in Table} II). ^{In order to} reproduce ^the

non universal behaviour, point ^{C could} not attract the

points exactly ^{located on the line, as in} figure ^{9. To}

further investigate ^this point, ^we have determined the flow direction near point ^C along ^{the SAF - P} line, for greater ^cells. Namely, the horizontal derivatives of the PSRG transformation have been calculated for

increasing cell sizes (Table IV). ^{For b}

4, ^{this value} becomes negative (which merely ^{means that the}

flow becomes oscillatory ^around C), ^and greater ^than unity in absolute value, ^{which means that} point ^C

becomes fully repulsive.

4. Discussion of the SAF phase.

By preserving the fundamental geometrical ^features

of both the lattice and the four possible ground-states

of the system, ^the present ^PRSG gives ^{a SAF} phase

characterized by ^a bifurcated attractor (SAFI ^and

SAF2 in Fig. 9). The line denoted by ^numbers 1, 2, 3,

etc. in figure ^{9 is a} typical ^PSRG trajectory ^{for an}

initial SAF point. ^The oscillatory repulsive ^behaviour

is already present ^{in the} neighbourhood ^of point D, for this small b

2 transformation. The two-fold _sym- metry of the SAF attractor reproduces the two-fold space degeneracy of the SAF ground ^{states. The oscil-} latory repulsive behaviour is observed also in the neigh-

bourhood of point ^{C for b}

⁴ transformation

(see ^Table IV). Unfortunately, ^{we are not able to} compute the full b

4 phase diagram (there ^are

Table IV.

Derivative A of ^{the PSRG} transformation along ^the horizontal, ^at point ^C (Fig. 9).

257 ~ 1017 configurations ^to compute), ^{but we can}

infer its qualitative features from two facts : i) ^the

attractor is still the pair (SAFI, SAF2) ^{located at the}

same place (the ends of the critical line); ii) point ^C

becomes repulsive along the critical line. We then expect ^{that an initial} point exactly ^{located on the}

critical line will be directly ^{attracted to} (SAF1, SAF2) :

this is possible because the attractor is unusually

localized on the critical line (no ^{other RG treatment} presents ^{such a} behaviour, ^{to the best of our} knowledge). The critical behaviour along ^{the line}

would not be dominated by any special point ^{on it :}

initial points ^near the line will be repelled ^{from it} directly, without the usual approaching ^{to a} special point ^{on the} line, ^{and the rate} of deviation will become

a local property ^{of the} initial localization near the line.

In this _manner, the non universal character of the SAF - P phase transition is explained by ^{the present}

PSRG transformation.

Two final comments concern the successful _repro- duction of the two-fold _space degeneracy ^{of the SAF} ground ^states. First, ^the correspondence between the attractor (SAFI, SAF2) ^{and the two} ground ^states

cannot be splitted, ^SAFI corresponding ^to figure ^4c

and SAF2 to figure ^{4c’ for} instance. The correspon- dence is necessarily between both pairs ^{each one}

considered as a whole entity. Second, the existence of

competing interactions when J2 ^{0 is} by ^itself responsible for the two-fold _space degeneracy ^{of the}

SAF ground-states. ^If disorder is included in this system, ^one expects ^the degeneracy ^to be enhanced for

increasing ^{values of some} disorder control parameter.

In the RG language, ^one expects ^a cascade of Feigen-

baum bifurcations of the SAF attractor. For suffi-

ciently high ^{disorder, the} degeneracy ^becomes infinite,

the attractor becomes continuous, ^{and the} system becomes a spin glass [15]. ^We think the present approach ^{can be} applied ^{to the} study ^of systems ^with these features.

Acknowledgments.

The author acknowledges ^{Dr. F.} A. de Oliveira for a

critical reading ^{of the} manuscript, ^{and Drs. S. L. A. de}

Queiroz ^{and R.} R. dos Santos for comments before its

final form.

References

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