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CENTRO BRASILEIRO DE PESQUISAS FÍSICAS

Notas de Física

CBPr-N'F-02 9/SÍ

POITS FI-RROMAINJ-T CORRECTION LRNPI'H JN UíPEW.VDJC

LATTICES: RIíXOíMALISATION-GRaip APPROACH

by

Eva 1 tio M.P. Curaclo find Paulo R. Mauser

DF: JAr . ' i : IR

NOTAS DE FÍSICA e uma pré-publicação de trabalhooriginal em Física

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í

ISSN 0029 - 3865

CBPF-NF-029/84

POTTS FERROMAGNET CORRELATION LENGTH IN HYPERCUBICLATTICES: RENORMALISATION-GROUP APPROACH

by

Evaldo M.F. Curado and Paulo R. Hauser*

; Centro B r a s i l e i r o de Pesqu isas F í s i c a s - CNPq/CBPF~, Rua Dr. X a v i e r S i g a u d , 150

22290 - Rio de J a n e i r o , RJ - B r a s i l

:< *0n leave of absence f romi Departamento de F í s i c a'"i Un ivers idade Federa l de Santa C a t a r i n a•'? Cidade U n i v e r s i t á r i aU SS.000 - F l o r i a n ó p o l i s , SC - B r a s i l

CBPF-NF-O29M

ABSTRACT

Through a r e a l space r eno rma l i s a t i on group approach, t h e

q - s t a t e P o t t s ferromagnet correlation length on h i e r a r c h i c a l l a t ~

tices is calculated. These hierarchical lattices are build

in order to simulate hypercubic lat t ices. The high-and-lov

temperature correlation length asymptotic behaviours tend

(in the Ising case) to the Bravais lattice^ correlation length

ones when the size of the hierarchical lat t ice^ cells tends

to infinity. We conjecture^that the asymptotic behaviours

for several values of q and d (dimensionality) so obtained are

correct. Numerical results are obtained for the full tempera-

ture range of the correlation length. '•"••""

Kcv-vords: Potts; Correlation Length.; Hiornrchicnl lattices.

CHl'F-NI-029/84

1 INTRODUCTION

The study of Bravais l a t t i c e s through the use of hierarchy

cal ones (which are not trans la tionally invariant) lias received a grow

incr attention in the last few years, especially in the area of Dhase transi-

tions (Reynolds et a l . 1977, Yeanans and Stinchcembe 1979, deílagalhães

e t a l . 1980, Levy e t a l . 1980, Curado e t a l . 1981).

The quest ion whether the l imi t of functions ca lcu-

lated on families of hierarchical lattices, with basic cells

(Melrose 1983) of increasing size b (see, for example, fig. 1),

converges to the respective functions on Bravais lattices is a

point that is not clear today. An argument that favours the

convergence is obtained if we adopt the Melrose (1983) defini^

tion of dimension (D) and connectivity (Q) of hierarchical lat

tices (en whose values depend the crit ical exponents). For

example, in fig. l , the limit b -*• °° leads to D = 2 and Q=lwhidi

coincides with the values of the square la t t ice . Other arguments

are given in several works which exhibit an apparent convergence

towards the corresponding results in Bravais lattices (Curado

et al. 1981, Martin and Tsallis 1981, Oliveira 1982, Hauser and

Tsallis 1984, Curado et al. 1984). However, some works (Tsallis

and Levy 1981, Kaufman and Griffiths 1983) have provided re-

sults which show that convergence is not a transparent point.

In this work we investigate this problem tiiroujh anoüier point",

ot view. We calculate, within the framework of real space re-

normalisation group (RG), the Potts model correlation length

for hierarchical lat t ices and investigate how both high and low.

temperature asymptotic behaviours converge to the true hyper_

CBPF-NF-029/84

-2-

cubic lattice correlation length as b + •,

2 POTTS MDDEL ON HIERARCHICAL LATTICES AND RG

The families of cells that we choose for simulate hypercu-

bic lattices are shown in fig. 1 for d = 2 (square lattice),

in fig. 2a for d = 3 (simple cubic), fig. 2b for d = 4 and so

on. The reasons for the word "simulate" are the following:

<L) the intrinsic dimension D and the connectivity Q of these

families of Cells lead to D = d and Q = d-1 as b •• «> (d is

the dimension of the Euclidean space where the cells are

embedded, see Melrose 1983). This leads to D = 1 + Q in that lira

it, which is typical for Bravais lattices.

ii) the number of sites and bonds, of a basic cell with size band

intrinsic dimension D (embedded in a Euclidean space of di

mension d) are

' N ... =b Q (b-1) +2=b d" 1(b-l) +2, (1)sites

N, . =b =b + (d-l)b (b^l)2 (2)

which yields

lim Nb ° n d s = d (3)

sites

as for a d-dimensional hypercubic lattice.

In order to build a model on these cells we associate with

each of their sites a Potts variable o^(=0,1...,q-l) and to

each bond of the cells a coupling constant J > 0. The interac

n a -

tions between the variables arc given by (Potts model)

. / a . a . (4)J > x j

where Ó i s the Kroenecker's delta and <i,j> means nearesto • o •1 J

neighbours sites.

Performing the partial trace over the "internal" sites (full

circles, see figures 1 and 2) of the chòósen cell we renormali.

ze it into a smaller cell (b1) with a coupling constant K1 .This

can be done in a simple way adapting the break-collapse method(Tsallis and Levy, 1981). Using the transmissivity

1 - e"qK 1

+ (q-l)e J AB

we obtain recursion relations in the form

R?,(f ) = R?(t) . (6)D O

where rf' ( t ) = N. (t)yD ' ( t ) b e i n g K. ( t ) and D f ( t ) p o l v n o m i a l s fvnc

t i o n s o f t .

For cxn-nplo t o b = 2, b 1 = 1 and d = 2 ( f i q . 1) we have

3 + 2 ( q - l ) t 3 + ( q - l ) t " + ( q - 1 ) ( q - 2 ) t 5 •

It is important to observe in the two dimensional case that

the cells are self-dual, like the square lattice, and yield the

correct critical tmoperaturc of the square lattice Potts model

:.'r ;!tiv v.il uc of b .

—nollapse jnethõd enables 'ns ±o T^il-culate ±he

qptrrtlc .behaviour (Curado e t .aL 19B1+ llauser and Tsal l is 1984,

Ttorado e t aJ- 1984) of ±íiese fractions J^(±) for -t-l(—— - D)

t —• 0 (Jí_T/J -+ « ) . !EfaB xesul-ts are

I b t b + 2 ( b - l ) 2 t b ' f l + . . . ( t - 0 ) (Ba)

( t -* l ) (Bb)

2 ( d - l ) b d " 2 ( b - l ) 2 t b + 1 -f . . . ( t - 0 ) (9a)

• + 2(d-l)

(t-*l) (9b)

vhere in eg. (9b) we have indicated only the f i rs t dcrainant

"terms which contribute to the asymptotic correlation length

i>eha"viour in the b •* «• limit. VJe remark that the tv.o-dimensionál

case i s different frcm a l l the d > 2 cases, because of i t s

peculiar topological features . ibr example if we break one

"".horizontal" ** * bond (any one) of figure lb(b = 3, d = 2)

-and collpse the rest of the horizontal ones, the

resulting graph is different from that obtained if we col-

-lapse a l l the horizontal bonds.-However if we break one hori-

-zoatal bond of figure 2 íb ~ 2 vr 3f d=3) and collapse the re-

d e f i n e a " h o r i z o n t a l " p l a n e as t h a t drrrrsjir.i-tl hy a l l hc

bonds i o c a t r d a t a sane " d i s t a n c e " t o an a r b i t r a r y

"vertex ( t e r m i n a l ) , where "disianrt;" means t h e a i n i s u n number

of h r n d s t h a t connect tht- bonds of t he plane to t h i s vert*-*.

In a c e l l of s i r r fc t h e r e art- b - 1 hpr i20ni .1 l plane-.,

1 and 2»

maining horizontal ones we obtain the same graph that is obtained

if a l l the horizontal bonds are collapsed.- This result is true

for al l d > 2.

In order to obtain the longitudinal correlation length (mea-

sured along one axis of a hypercubic lattice) as a function of

temperature, v/e must have another equation besides of (6). This

new equation arises from the well-known length scaling under

the FG operations. So, we obtain

b/b1

We observe that ecu (10) remains invariant when i t is multi-

plied by any factor c(q;d) independent of T but that may de-

pend of q and d. This factor cannot be obtained within the

present formalism.

3 TW.) DIMENSION'S: ASYMPTOTIC BEHAVIOUR AND NUMERICAL RESILTS OF

The asymptotic behaviours of the correlation length Ç, ,Si b

a r e o b t a i n e d t h r o u g h t h e use- of e q u a t i o n s (6) w i t h d = 2 , (8)

and (10) (Cuiv^lr ot. a l . 19*315. K? obtained t h e follov;incj expressions:

ai)

( 1 ? )

CtíPF-NF-029/84

-6-

lt is interesting to note that if in eq. (12) wo adopt for the

invariant factor the value c(q) =q we obtain qJ/k_T as a "na-

tural" variable of this equation suggesting that this can be

the correct value for this constant. The same is not possible

to do in eq. (11) where there is no factor that leads to a

"natural" variable. The exact longitudinal correlation length

of the square lattice Ising model (q=2) is known (Onsager 1944,

Fisher and Burford 1967, Baxter 1982)and is the following

XTT-T~T T > T (13a) |iKB i kB T c %

. _ ().k T k T 'J K BI j j.

ã " £n coth K7T T < Tc (14a)

^ j~^U--J~ exp(-2J/kBT)..J - j - -> 0 (14b)

Therefore, the asymptotic behaviours of eqs. (11) and (12)

[this with c(q) = 2] in the limit b-» °°, with q = 2, reproduce the

exact asymptotic behaviours given by expressions (13b) and (l-'b)

respectively. So, we conjecture th.it the asymptotic expres-

sions given by eqs. (11) and (12) are the correct ones fcr the

square lattice, in the limit b-*^, for all q.

Along the lines of Curado et ai. (19 81) w e

uso equations (6), (10), (11) nnd (12) to obtain lhe numerical

results for the correlation lcnoth as a function of the tem-

perature. They are shown in figures (3) and (4). In figure (j)

wo plot the ', , correspond iiui to b- 2,3 and •'. d'1 • I ) Í>T VW

CWF-NF-029/84

case q = 2 (v;herc the exact answer is known) and we note that,

the RG results tend to approach the exact answer for

increasing b. In figure (4) we plot the RG results for typical

values of q. We note that for a fixed T/T (q) the correlation

length decreases as q increases. This can be intuitively

understood as the higher probability that two variables be-

come uncorrelated if there are more states to choose.

To analyse the behaviour of the Ç, ,, near T ,J b 11> c

T - T (q)Yvb,b'<q)

we studied the amplitude A, , (q) (T > T (q)) for several valuesb,l i c

of q with b=2,3,4 (the critical exponent vfc b«(q) has been

studied in previous works, see for example Tsallis and Levy

1981). Some values of them are shown in Table I. The results for

T-< T (q) are obtained dividinq those of Table I by cia),'v c '

4 d. DIMENSIONS: ASYMPTOTIC BEHAVIOUR OF £ (d > 2)

In a similar way, with the use of equations (6), (9) and

(10) we obtain the following expressions for t, (in the unit

b •* «) for dimensions d > 2:

b'<b

/ kn T \V d> 2, Vq [•—- - «) (16)

/ kn T \> 2, Vq [•—- - «)

CBPF-NF-029/84

b'<b/k T

Vq,Vd>2 (-5-^01 (17)1

The two terms of expression (16) reproduce (with q = 2 and

d=3) the f i rs t two terras of the high-temperature correlation

length expansions (the only oossible canparison, in the best of our

knowledge) carried outby Fisher and Burford (1967). It i s worth-

while to note that in three dimensions the hierarchical l a t -

tices we have used are r.ot self-dual, which exhibits that this

property is not a necessary condition to obtain the correct re-

sults as might bethought if we look only for the two-di en-

sional case. For the low-temperature case, eq.(17), we do not

know of any result to compare with. Then we conjecture that the asym

ptotic behaviours of the correlation length of the adopted hi

erarchical l a t t i ces , given by equations (16) and (17), repro

dace the exact asymptotic behaviours of the corresponding hy-

percubic la t t ices .

CBFF-NF-029/84

5 CONCLUSION

We calculated, with IG techniques on the familyof hierarchical

lattices shown in figures (1) and (2), the asymptotic behav-

iour of the Potts model longitudinal correlation length. In

the limit b •*• » this correlation length reproduces (in the few

known cases where we can compare) the exact asymptotic behav-

iour of the longitudinal correlation length in the corresponding

hypercubic lattice. For the cases where there are no exact

results to compare, we believe that the asymptotic behaviours

obtained by us are the correct ones in the b •*•» limit. It is

interesting to note that in several works, with different func

tions (see, for example, the specific heat in the work of Mar

tin and Tsallis 1981, the surface tension in the works of Cu-

rado et ai. 1981, 1984, and Hauser and Tsallis 1984), theí

convergence of the asymptotic behaviours of these j

functions (constructed on hierarchical lattices) to the cor-

responding functions on hipercübic lattices as b + « was shown. So, we

believe that the asymptotic behaviours (for a large class of

functions) obtained for these families of hierarchical latti-

ces converge to the exact ones on the corresponding hypercu-

bic lattices as b •> ».

Therefore, these results strongly suggest that if any

discrepancy exist between the b -*• «limit of these families of

hierarchical lattices, and the corresponding hypercubic latticesthen it must be localized in the neighbourhood of T . To test this

c

suggestion, full analyt ic expressions for arbitrary b fcnuld be nc-

crasary. The numerical calculations of the amplitude of :. ncarTr scon to

CBPF-NF-029/84

-10-

converge to the exact values but are not conclusive. Finally

we remark that these IG 's, on hierarchical lattices, are a

good alternative way to high and low temperature series, at

least for the Potts model.

We want to thank especially Professor C. Tsallis for sev

eral discussions on this problem. We*are indebted to A.C.N.

deMagalhães, A.M. Mariz, L.R. da Silva and U.M.S. Costa for

many valuable discussions. Also we acknowledge I. Roditi.

CBPF-NF-029/84-11-

CAPTIONS (Figures)

Fig. 1. - Family of hierarchical lattices adopted to simulate

the square la t t ice . They are self-dual (Vb) . Hie full

circles are the "internal" sites and the open are

the "terminals".

Fig. 2. - Family of hierarchical lattices adopted to simulate (a) the

simple cubic lat t ice and (B) the hypercubic lattice

for d = 4.

Fig. 3 . - Comparison among the RG - Ç. . results for severalD i 1

sizes of cells (full lines) and the exact one (dashed

lines) for the cases q = 2 (Ising) , d = 2.

Fig. 4. - RG - £. . results for several values of q and b = 4.D t 1

CAPTION (Table I)

Table I. - The RG - Ç two dimensional amplitude A^ (q) for sev

eral values of b and q (T>T (q)). The only exact

value known is the q = 2 case where the amplitude

value is (2 ín(l+/2]"' 0.5673.

-12-CBPF-NF-029/84

õ

b=1

b = 4 b = 5

FI6.1

- 1 3 -CBPF-NF-029/84

CM

O

CBPF-NF-029/84

- 1 4 -

\

\

\\

\

-15-CBPF-NF-029/84

—: o

-16-

Table I

CBPF-NF-029/84

2

3

4

Extrap.

linear

8

6

4

0 . 1

.130

.998

.734

1

1

1

0 .5

.765

.596

.258

1 .0

1.144

1.049

1.00

0.857

2 . 0

0.80.5

0.746

0.715

0.62 6

3 .0

0.675

0.629

0.604

0.534

4 . 0

0.602

0.563

0.541

0.481

CBPF-HF-029/84

-17-

References

Baxter R.J. 1982 Exactly Solved Models in S ta t i s t i ca l Mechanics

(London: Academic Press)

Curado E.M.F., Tsallis C.,Levy S.V.F. and Oliveira M.J. 1981 Phys.

Rev. B 23! 1419-30

Curado E.M.F., Tsal l is C., Schwachheim G. and Levy S.V.F. 1984

To be published

Fisher M.E. and Burford RJ 1967 Phys. Rev. 15 6 583-622

Hauser P.R. and Tsallis C. 1984 Tb be published

Kaufman M. and Griffiths R.B. 1983 Phys. Rev. B 2_8 3864-5

Levy S.V.F., Tsallis C. and Curado E.M.F. 1980 Phys. Rev.3 £1

2991-8

de Magalhães A.C.N., Tsallis C. and Schwachhein: G. 1980 J.

Phys. C: Solid St. Phys. L3 321-30

Martin H.O. and Tsallis C. 1981 Z. Phys. B £4_ 325-31

Melrose J.R. 1983 J. Phys. A: Math. Gen. If 3077-33

Oliveira P.M.C. 1982 Phys. Rev. B 2_5 2034-35.

Onsager L. 1944 Phys. Rev. 65 117-49

Reynolds P.J., Klein W. and Stanley H.E. 1977 J. Phys. C: Solid

St. Phys. H) L 167-72

Tsallis C. and Levy S.V.F. 1981 Phys. Rev. Lett. AJ_ 950-3

Yeomans J.M. and Stinchcombe R.B. 1979 J. Phys. C: Solid St.

Phys. \2_ L 169-72