ZERO TEMPERATURE CRITICAL BEHAVIOUR OF THE ONE DIMENSIONAL X-Y MODEL WITH RANDOM COUPLING CONSTANTS

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ZERO TEMPERATURE CRITICAL BEHAVIOUR OF THE ONE DIMENSIONAL X-Y MODEL WITH

RANDOM COUPLING CONSTANTS

E. Smith

To cite this version:

E. Smith. ZERO TEMPERATURE CRITICAL BEHAVIOUR OF THE ONE DIMENSIONAL X- Y MODEL WITH RANDOM COUPLING CONSTANTS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-1010-C1-1011. �10.1051/jphyscol:19711360�. �jpa-00214397�

JOURNAL DE PHYSIQUE Colbque C 1, supplkment au ^no 2-3, Tome 32, Fkurier-Mars 1971, page C 1 - 1010

ZERO TEMPERATURE CRITICAL BEHAVIOUR OF THE ONE DIMENSIONAL X-Y MODEL

WITH RANDOM COUPLING CONSTANTS

E. R. SMITH

Mathematics Department, Imperial College, London, S. W. 7

R6sum6. - On a analyse le modkle X-Y B une dimension avec les constantes d'6change aleatoires dans un champ magnBtique suivant l'axe Oz. Pour le systkme non aleatoire B la temperature zero il y a une singularit6 thermodynamique en champ fini, qui correspond au changement d'un Btat partiellement ordonnB en un Btat ordonne complktement. Quelques quantit6s thermodynamiques divergent et la longueur de correlation devient infinie. Pour le systkme alBatoire on peut montrer que toutes les quantites thermodynamiques varient analytiquement et la longueur de corrBlation est k i e au point critique.

Abstract. - We analyse the one dimensional X-Y model with random coupling constants in a z-axis magnetic field.

For the non-random system at zero temperature, there is a thermodynamic singularity at finite field, corresponding to a change in the type of spin ordering. Some thermodynamic quantities diverge and the correlation length becomes infinite.

For the random system however, we can show that all the thermodynamic quantities are smoothly varying and the corre- lation length is bounded above at the critical point.

Recent experiments on critical phenomena have indicated that some quantities which had hitherto been thought to diverge at the critical point are in fact finite at the critical point [I]. One explanation of this behaviour is that it is caused by randomly distributed impurities in the sample. The impurities are thought to change the local properties of the lattice only slightly.

To examine this idea we examine the one dimensional X-Y model of Lieb, Schultz and Mattis [2]. We have the Hamiltonian

in which the J,,, are independent variables of mean 2 and variance 4/n, with n a large integer.

This Hamiltonian may be reduced to a quadratic form in fermion operators [2]. The diagonalization of the quadratic form is the same as that performed by Dyson [3] for the problem of the disordered harmonic chain and we are able to get exact results for the den- sity of states of the Hamiltonian (1) by using his analysis. If all the Jm are the same and equal to 2 (the non-random case) then the density of states is given by (112 n) (1 - y2/4)-% for - 2 < y < 2 , elsewhere . ⁽²⁾

For the random case we consider, the density of states is only very slightly different from that in (2) except in the region n2I3 1 ^{(1 - y2/4)} 1 ^< 1, for which we have D2Q)

-

At the critical point, we can define critical exponents for the non-random system :

For susceptibility,

x ( T = 0 ) - - A I ( 2 - h ) - " for h 5 2 , (4)

For specific heat,

lim _T-+O - Ch(T) _T

-

For magnetization,

M ( T = O ) - + - F ( 2 - h ) % for h 5 2 . (8) All of this critical behaviour derives directly from the divergence in the density of states at y = 2 as shown in (2). For the system with random coupling constants, exactly the same sort of behaviour is seen a t a small but finite distance from the critical point.

However if ( (2 - h) 1 ^{n2/3 < 1} and kTn2I3 < 1 (very close to the critical point) the thermodynamic beha- viour is quite different from that of the non-random system because of the change in the density of states given in (3). The diverging quantities are bounded and infinitely differentiable in the vicinity of the critical point. Thus all the exponents are zero (except + ^{1 for}

the magnetization') and this tells us very little about the critical behaviour of the random system.

If we look at the z-component correlation functions for the non-random system we find

that is, the density of states has a parabolic peak of

width 0 ( n e 2 I 3 ) and height 0(n1I3) instead of the < ^Sg %+* > ⁼

divergence shown in (2) for the case with all the coupl- 4 ing constants the same. For this case, there is a phase

transition at T = 0, h = J. For fields h > J, at zero - (nr)-' sin2 (r oos-I

:)I.

temperature, all the spins align along the z-axis. For

fields h < J, this complete ordering at zero tempera- In the random system, the correlation functions are not ture is disrupted. The z-axis magnetization decreases translationally invariant, but have a probability distri- to zero as h decreases to zero. bution which is. We must calculate the mean of the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711360

ZERO TEMPERATURE CRITICAL BEHAVIOUR OF THE ONE DIMENSIONAL X-Y MODEL C 1 - ^{101 1} distribution and its standard deviation. We expect to

find a correlation length L, for the decay of the mean, and a length L, for the increase of the standard devia- tion with respect to the mean [4]. At the critical point we have, for r < n1I3, mean

and

S. d. (< S i S;,, >)

-

At the critical point the random lattice is, on average, not as strongly correlated as the non-random one, and some parts of the random lattice are more sfrongly correlated than others. The mean correlation length L1 is, from (lo), L, -- O(n1I3). The length L, for the increase of the standard deviation of the correlation function is, from (1 I), L,

-

spin separations r > L,, then we cannot say how the two spins are correlated. At the critical point, the system tries to make its correlation length as large as possible, but it cannot become any larger than L,.

The result is that L,

-

-

This suggests that the lattice is split up into ((islands >> of length O(L,) so that the length L, is an upper bound on the mean correlation length L,.

Suzuki [5] has shown that the eigenvectors of the diagonalization of the Hamiltonian (1) are localized in the random case. The above discussion suggests that the eigenvectors are localized over a length O(n113) of the lattice, which would account for the behaviour of the correlation functions.

If the effects described here generalize to random systems of higher dimensionality, then randomness in the coupling constants may be considered a partial explanation of the recent experiments showing round- ing of critical behaviour in some systems.

References

[I] VAN DER HOEVEN (B.), TEANEY (D.) and MORUZZI (V.), _{[3] DYSON} (I?.), Phys. Rev., 1953, 92, 1331.

Phys. Rev. Letts., 1968, 20, 719. [4] SMITH (E.) and GUTTMANN (A.), J. Phys. C. (Sol.

[2] LIEB (E.), SCHULTZ (T.) and MATTIS (D.), Ann. Phys. St. Phys.), 1970, 3, L 109.

(N. Y.) 1961, 16, 407. [S] SUZUKI (M.), Preprint, 1970.