Volume 96A, number 5 PHYSICS LETTERS 4 July 1983

CROSSOVER SCALING BEHAVIOUR OF THE QUASI-ONE-DIMENSIONAL n.VECTOR MODELS

Surjit SINGH 1 Solid State Physics Division, Tata Institute of Fundamental Research, Bombay 400005, India

Received 24 February 1983 Revised manuscript received 30 April 1983

Crossovers in the quasi-one-dimensional n-vector models are discussed by making new scaling hypotheses. Predictions re- gaxding critical point shifts and amplitudes are made for one- to two- and three-dimensional crossovers for 1 < n ~ oo. Many new scaling functions are obtained exactly and studied by series-expanding methods.

Crossovers in the general d-dimensional, classical n- vector exchange hamiltonian have been discussed in the last ten years or so by various techniques * 1. The predictions of the usual [3] and the extended [4] scaling hypothesis have been verified and the scaling functions obtained in some cases , 2 . Examples are the (3, 3) + (3, 1), (3, 3) + (3, 2) and (2, 1)-+ (3, 1) cases [5,6]. [Here the obvious notat ion (d, n ) ~ (d', n ' ) has been used to denote a crossover from the primary behaviour at a point (d, n) to the secondary behaviour at a point (d' , n ' ) in the (d, n) space.] In this paper, new crossover scaling hypotheses for the following quasi-one-dimensional cases are proposed: (i) (1, 1) + ( 2 , 1)and (3, 1), ( i i)(1, n ) + (3 ,n) , 2~<n ~<~. These hypotheses are analyzed to obtain many new predictions for the critical point shifts and the ampli- tudes. Scaling functions for the spherical-model case n = ** and for the case (1, 1) + (2, 1) are obtained ex- actly. For the case (1, 1) -* (3, 1), the behaviour nea r the point (1 ,1 ) is analyzed by means of a series expan- sions (ratio and Pad6) to obtain a sixth-order expan- sion of the scaling function. In this paper, we will main- ly concentrate on the susceptibility X, similar results may be obtained for other quantities.

i On leave from North-Eastern Hill University, Shillong 793003, lndia.

,1 For a review of the RG work see ref. [ 1 ], for series work see ref. [21 .

.2 For the RG work see e.g. ref. [5], for the series work see e.g. ref. 161-

Case (i) (1, 1) -+ (2, 1) and (3, 1): In this case the critical point for the primary system is at K = oo and X = exp(2K) where K = J/kBT in the standard nota- tion. The proposed scaling hypothesis for the cross- over to two or higher dimensions is

x(g, K) ~ e2K X(BgKe2K) , (1)

where g is the "anisot ropy" parameter which takes one from (at, n) to (d' , n ' ) in general. Naturally, X(0) = 1 and to reproduce the g ~ 0 behaviour, we assume the form [4]

x(x)~.k(1-x/;)-4, x+k, (2)

where "~ is the secondary exponent [relevant to the case (d ' , n ' ) ] defined by

× ( g , K ) ~ A ( g ) [ - 4 , ~ = [Kc(g ) - K ] / K , (3)

and Kc(g ) is clearly the shifted critical point. The hy- pothesis (1) entails the predictions,

1 * Kc(g ) - i In [~¢/BgKe(g)] , (4)

.~(g) = ( .~ /Bg)2- '~ [Kc(g)] - (1 +-~) (5)

Asymptotically, as g + O, Kc(g ) -+ ~*, eq. (4) agrees with the Onsager result [7] and the exact bounds by Weng et al. [8] which have been found to represent quite accurately the recent Monte Carlo data of Graim and Landau [9]. For the (1, 1) + (2, 1) case, exact re- suits for the correlation length ~U [7], magnetization M [10], and the specific)aeat C [7,11], can be put in

0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland 225

Volume 96A, number 5

the scaling form similar to (1). We get, trivially,

~ I I ~ ' A l e 2 K ( l - x ) - 1 , M ~ A 2 ( x -2 )1 /8 ,

with x = BgKe 2K. For the specific heat the calculation is somewhat lengthy. Because of the logarithmic di- vergence for g :/: 0, the scaling is found to be valid in the form ,a (for the linear chain to the sq case)

C ~ (8/Tr)K2e - 2K (c K (x)

+ ( K -1 + ¼K-2)[CK(x) - ¢ (x ) ]} ,

where cK (x) and g (x) are the complete elliptic inte- grals o f the first and second kind, respectively.

For the (1, 1) to (3, 1) case, we have studied the susceptibility series [ 12] for the linear chain to the sc case. The prediction

(amx[agrn)o = ~ an(m)K m n

~, Cm Km exp(2mK + 2 K ) ,

which follows from (1), is verified by plotting nan(m)~ an - 1 (m) versus n - 1. We find that these extrapolate nicely to 2m + 2, as expected. The amplitudes C m and the corresponding universal amplitudes are found by evaluating the diagonal PAs to the appropriate series at K = ~ . Using these results, we obtain the low-x ex- pansion of X(x):

X(x) = 1 +x + 0.8265 x 2 + 0 .6754x 3 + 0.5306 x 4

+ 0.4166 x 5 + 0.3232 x 6 + ....

This yields the preliminary estimates ~ ~ 1.34,3( 1.08 based on a Pad6 analysis o f the series using 3,

= 1.25. Further work on this model is in progress [ 13].

Case (ii): (1, n) -~ (3, n), 2 ~< n ~< ~ : In this case, X - A K , as K + ~ in the primary system [14]. The pro- posed hypothesis is

x(g, K) .~ AKX(BgK2) . (6)

This predicts that

Kc(g ) -~ ( x / B ) l / 2 g -1/2 , (7)

,a The author thanks Professor M.E. Fisher for explaining how to deal with the logarithmic singularity, although in a some- what different context.

PHYSICS LETTERS

,~(g) ~ A,~2-~; (~ /B) l /2g- 1/2 .

4 July 1983

(8)

These are new predictions for the quasi-one-dimen- sional n-vector models for n >1 2. For n = ~, these agree with the exact calculations o f Joyce [ 15]. In this case, however, the scaling function for × can be calculated in closed form. We find that

rrxl/2 = i dy W 2 [y2 + 4/xX(x)] , 0

where Wd(z) is the Watson integral [16]. For the other cases, the predictions are being checked by series- expansion methods [ 13].

In summary, many quasi-one-dimensional n-vector models have been studied by means of new crossover scaling hypothesis and new predictions made. Many scaling functions for the (1, 1) ~ (2, 1) case and for the n = ~ case have been calculated exactly. SmaU-x expansion o f × for the (1, 1) to (3, 1) case has been obtained by series methods. The remaining cases are being studied and the results will be reported else- where [13].

The author thanks Dr. B.K. Basu and Professor R. Vijayaraghavan for hospitality.

References

[1] M.E. Fisher, Rev. Mod. Phys. 46 (1974) 597; A. Aharony, in: Phase transitions and critical phenomena, Vol. 6, eds. C. Domb and M.S. Green (Academic Press, New York, 1976).

[ 2 ] H.E. Stanley, in: Phase transitions and critical phe- nomena, Vol. 3, eds. C. Domb and M.S. Green (Academic Press, New York, 1974).

[3] E.K. Riedel and F.J. Wegner, Z. Phys. 225 (1969) 195. [4] M.E. Fisher and D. Jasnow, in: Phase transitions and cri-

tical phenomena, Vol. 4, eds. C. Domb and M.S. Green, (Academic Press, New York, 1974); to be published.

[5] D.R. Nelson and E. Domany, Phys. Rev. BI3 (1976) 236; A.D. Bruce and D.J. Wallace, J. Phys. A9 (1976) 1117; C. Domb and M.S. Green, eds., Phase transitions and cri- tical phenomena, Vol. 6 (Academic Press, New York, 1976).

[6] F. Haxbus and H.E. Stanley, Phys. Rev. B8 (1973) 2268; P. Pfeuty, D. Jasnow and M.E. Fisher, Phys. Rev. B10 (1974) 2088; S. Singh and D. Jasnow, Phys. Rev. B12 (1975) 493; W.L. Basaiawmoit and S. Singh, Phys. Lett. 88A (1982) 251.

226

Volume 96A, number 5 PHYSICS LETTERS 4 July 1983

[7] L. Onsager~, Phys. Rev. 65 (1944) 117. [8] C.-Y. Weng, R.B. Griffiths and M.E. Fisher, Phys. Rev.

162 (1967) 475; M.E. Fisher, Phys. Rev. 162 (1967) 480.

[9] T. Graim and D.P. Landau, Phys. Rev. B24 (1981) 5156. [10] C.N. Yang, Phys. Rev. 85 (1952) 808. [ 11 ] B.M. McCoy and T.T. Wu, The two-dimensional Ising

model (Harvard Univ. Press, Cambridge, 1973). [12] J. Oitmaa and I.G. Enting, Phys. Lett. 36A (1971) 91;

F. Harbus and H.E. Stanley, Phys. Rev. B7 (1973) 365.

[ 13 ] S. Singh, to be published. [14] H.E. Stanley, Phys. Rev. 179 (1969) 567. [15] G.S. Joyce, in: Phase transitions and critical phenomena,

Vol. 2, eds. C. Domb and M.S. Green (Academic Press, New York, 1973).

[16] M.N. Barber and M.E. Fisher, Ann. Phys. N.Y. 77 (1973) 1.

227