An application of Pade approximants to Heisenberg ferromagnetism and antiferromagnetism By J. Gammel,* W. Marshall and L. Morgan Atomic Energy Research Establishment, Harwell, Berkshire, England {Communicated by R. E. Peierls, F.R.S.—Received 7 March 1963) The method of successive Pade approximants is applied to the high-temperature series for the susceptibility of Heisenberg systems with nearest-neighbour interactions only. It is concluded that the susceptibility of a ferromagnet diverges near the critical temperature with a law (T—Tc)-r, where r is either exactly f or indistinguishable from f by the method used with the power series at present available. The susceptibility of an antiferromagnet is also discussed. The specific heat near the critical point is considered but the results are inconclusive.
258 J. Gammel, W. Marshall and L. Morgan the (n,m) Pade approximant to f(x) is determined by expanding 2 j^l x+ b„ x +. x 2 71 as a new power series and equating coefficients term by term to a + a1x + a x2+. anxn 0 2 up to the term in xn+m. Terms involving higher powers of x are ignored; this gives n + m + 1 equations of which those coming from the coefficients of xn+x, ., xn+m determine bx, b ,. . . , bmand those from the coefficients of x°, xn determine 2 a0, an. Notice that if the power series (5) is given only up to a maximum term, say L t,hen the only Pade approximants which can be formed are those for which n + m < L. This procedure fits the function f(xot expressions of the form of (4) and by increasing n and m it seems likely that better and better approximations can be obtained. The power of the method is demonstrated most clearly when the problem can be formulated in such a way that the function has simple poles; because then the series of Pade approximants, which also have simple poles, con verges very rapidly.
Heisenberg ferromagnetism and antiferromagnetism 259 and that, in this case, the values of xc have an error of the order of \ % (the procedure described in § 3 gives a result which differs by this amount) but nevertheless the rate of convergence obtained is very satisfactory and gives us confidence that y0 does have a singularity of the form given by (6).
260 J. Gammel, W. Marshall and L. Morgan and the Kasteleijn-Van Kranendonk (1956) theories. The variation of r with S is slight and, apart from S = % which appears to be an exceptional case, varies smoothly from about T4 to 1-33 as S increases from unity to infinity. The r values are similar for the three lattices.
Heisenberg ferromagnetism and antiferromagnetism 261 a short power series, and we are therefore quite unable to give any strict judgement whether rsi a single constant applicable to all three-dimensional Heisenberg models or whether it varies slightly from one case to another. However, Baker has shown that it is plausible that r is exactly \ for the Ising two-dimensional lattices and £ for the Ising three-dimensional lattice and if we argue by analogy with his results that the Heisenberg result should also be some simple fraction then we would choose f because this is the value given by the most rapidly convergent cases. Therefore in later sections of this paper we shall assume that r is exactly § for all three-dimensional Heisenberg ferromagnets. We think this assumption is plausible but even if it is wrong then the deviations of r from an exact f value are so small that we cannot rely on our procedure to predict them.
262 J. Gammel, W. Marshall and L. Morgan Our remarks so far have referred to the f.c.c. and b.c.c. lattices; table 3 shows that for the s.c. lattice the results are harder to interpret. We first notice that the results of the Pf sequence for the s.c. lattice are quite unsatisfactory and no accurate estimate of the position or power of the singularity can be made from them. How ever, the P2 sequences appear satisfactory [except for where P\ locates Table 3. P oles in the Pade approximants to the logarithmic DERIVATIVE OP X FOR THE SIMPLE CUBIC LATTICE s =i S= 1 f 2 l A pole residue pole residue pole residue pole residue 0-5000 -0-7500 0-1711 -1-0143 0-08755 -1-0347 0-05372 -1-0391 0-4444 -0-5267 0-2064 -1-7819 0-1053 -1-8028 0-06420 -1-7731 0-5864 -1-4098 0-1723 -0-8655 0-08948 -0-9386 0-05518 -0-9676 1-5360 -203-04 0-2152 -2-6300 0-1062 -2-2109 0-06437 -2-0900 0-4384 -0-4982 0-1923 -1-3798 0-09723 -1-3557 0-05926 -1-3356 4-5616 -8-5018 -0-3102 0-2556 -0-1256 0-1327 -0-07191 0-1051 0-4808 -0-6846 0-1883 -1-2917 0-09662 -1-3296 0-05917 -1-3294 -0-2355 -0-002105 -0-1958 0-06820 -0-1095 0-08904 -0-06952 0-09526 r0-3206±l r—0-1346T1 0-1903 -1-3468 0-09728 -1-3664 0-05956 -1-3647 l0-09523i/ l 0-03290U -0-1549 0-02644 -0-09265 0-04582 -0-05855 0-04823 0-5214 -0-9315 0-1889 -1-3075 0-09664 -1-3308 0-05917 -1-3295 -0-6000±*> r0-4657 -0-1791 0-04716 -0-1088 0-08700 -0-06950 0-09515 . 0-9343i / \0-5017i J -1-5497 1-2603 -2-8908 1-2438 -7-6050 1-2344 1-9908 -5-2433 0-1897 -1-3286 0-09695 -1-3457 0-05924 -1-3348 r0-3230±^ r—0-005608 T -0-1409 0-01579 -0-06522 0-005134 -0-07243 0-1064 1 10-I734i J l 0-1315i J 0-6065 0-5362 -0-1494 0-1555 -0-01230 2-59 10“6 x 4 S=3 10 $ — oo A A < pole residue pole residue pole residue pole residue 0-03648 -1-0405 0-02645 -1-0410 0-002846 -1-0417 0-3125 -1-0417 0-04340 -1-7511 0-03137 -1-7364 0-003349 -1-6972 0-3673 -1-6920 0-03755 -0-9812 0-02725 -0-9886 0-002937 -1-0043 0-3226 -1-0060 0-04343 -2-0324 0-03137 -1-9994 0-003344 -1-9228 0-3668 -1-9135 0-04009 -1-3237 0-02900 -1-3164 0-003103 -1-2981 0-3405 -1-2958 -0-04737 0-09393 -0-03378 0-8811 -0-003511 0-07615 -0-3841 0-07486 0-04011 -1-3260 0-02905 -1-3231 0-003115 -1-3133 0-3419 -1-3119 -0-04800 0-09761 -0-3512 0-09866 -0-003857 0-1001 -0-4245 0-1002 0-04037 -1-3611 0-02924 -1-3581 0-003135 -1-3481 0-3441 -1-3467 -0-04015 0-04814 -0-02923 0-04774 -0-003166 0-04595 -0-3478 0-04567 0-04012 -1-3261 0-02905 -1-3232 0-003115 -1-3137 0-3419 -1-3123 -0-4799 0-09759 -0-3511 0-09851 -0-003849 0-09931 -0-4235 0-09928 13-6895 1-2285 3-4438 1-2246 0-1614 1-2144 16-7541 1-2130 0-4009 -1-3236 0-2899 -1-3152 0-003089 -1-2932 0-3387 -1-2911 -0-4741 0-09408 -0-03400 0-08981 -0-003617 0-08140 -0-3969 0-08062 0-003270 1-02 x 10”8 0-006884 6-56 xl0~* 0-001705 0-001890 0-1981 0-002977 complex poles] and locate a ferromagnetic pole with a residue approximately the same as found for the b.c.c. and f.c.c. lattices and a weak antiferromagnetic singu larity just as for the b.c.c. lattice. Notice even for this case, where the convergence is not thoroughly satisfactory, that nevertheless the P3 sequence gives the third pole with a very small residue indeed, indicating that only two poles are of physical significance. We do not understand why Pade approximants give poorer results for the simple cubic than for the face-centred and body-centred cubic lattices.
Heisenberg ferromagnetism and antiferromagnetism 263 examined the critical scattering of neutrons from iron and nickel. It is well known that at large distances the correlation function is of the form (S%S%> - (8) where V is the volume per unit cell, rx a microscopic length which is insensitive to 0 temperature and K goes to zero at the critical temperature. The rate at which K approaches zero is related to the divergence in y by the equation Xo ~ 1/r9() The neutron cross-section is sensitive to the parameter 2 and therefore from (9) can be used to measure the divergence in y. In this way Jacrot finds r = 1*30 for Fe, r = 1*32 for Ni.
264 J. Gammel, W. Marshall and L. Morgan distinguish two from three-dimensional lattices whereas the approach of Rushbrooke & Wood was not able to do so up to the approximation available.
Heisenberg ferromagnetism and antiferromagnetism 265 the pole was in good agreement with the other determinations and in table 5 we list the values of r given by this procedure.
266 J. Gammel, W. Marshall and L. Morgan singularity does not occur at |£| = 1. Formula (14) does not display any antiferro magnetic singularity and therefore gives good results only for temperatures some what larger than TN. The values of y determined in this way for ferromagnets and antiferromagnets are plotted graphically in figures 1 and 2, respectively, as a function of TJT. The antiferromagnetic results should be accurate only for the smaller values of TJT because the Neel singularity occurs at a value of TJT slightly larger than unity and this singularity is not properly treated by these Pade approxi- mants. Notice there is no clear evidence of a maximum in y at a temperature just above the Neel temperature: this maximum does occur for the Ising model.