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Submitted on 1 Jan 1978
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ZERO TEMPERATURE PROPERTIES OF THE S =
1/2 HEISENBERG ANTIFERROMAGNET AND XY
FERROMAGNET IN THREE DIMENSIONS
J. Oitmaa, D. Betts
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-817
ZERO TEMPERATURE PROPERTIES OF THE S = 1/2 HEISENBERG ANTIFERROMAGNET AND XY FERROMAGNET IN THREE DIMENSIONS
J . Oitmaa and D.D. Betts
School of Physios, University of New South Wales, Kensington, Australia Department of Physios, University of Alberta, Edmonton, Canada
Résumé.- L'énergie de l'état fondamental et le paramètre d'ordre de 1'antiferromagnétique de Heisen-berg et du ferromagnétique XY ont été calculés exactement dans des cellules symétriques à 8 et 16
spins sur les réseaux s.c. et b.c.c. avec les conditions des limites périodiques. L'énergie obtenue par extrapolation est en bon accord avec les autres estimations mais la déviation du spin est très grande•
Abstract.- The ground state energy and order parameter for the Heisenberg antiferromagnet and XY fer-romagnet are calculated exactly for symmetric 8 and 16 spin cells with periodic boundary conditions on s.c. and b.c.c. lattices. Extrapolation yields estimates of the energy in good agreement with other estimates but the spin deviation is very large.
In this paper we discuss the zero temperature (ground state) properties of two basic spin models
on the simple cubic and body centred cubic lattices. The isotropic spin one half Heisenberg antiferroma-gnet (HA) is defined by the interaction Hamiltonian
1 = | ^ < o * o *+o W + o?a?> (1)
™ <ij> J J J
with J>0. The a's are the usual Pauli matrices and the sum is over nearest neighbour pairs of lattice
sites. The S = 1/2 H magnet is defined by
i-}jl
<oW
+0**)
(2)
<ij> J J
The first Hamiltonian provides a good descrip-tion of antif erromagnetic insulators such as CuCS,-. 2H.0 while the second Hamiltonian serves to descri-be antiferromagnets such as CoCCJUNO) (CSLO ) . The XY model also serves as a model for liquid ''He near the lamda transition.
For the XY ferromagnet (J>0) the order para-meter is the transverse magnetization
N
M = E 0Z. (3)
i=l L
For the Heisenberg antlferromagnet the order para-meter is the staggered magnetization (a=x, y or z, e.=+l or -1 for A or B sublattice)
N
N = I cr.e. (4)
a i-i x x
In neither model does the order parameter commute with Hamiltonian so in neither case is the ground
state completly ordered. The ground state proper-ties of (1) on two and three dimensional lattices
have received considerable attention since 1952 /l/, but similar interest in (2) is very recent /2,4/.
We have previously studied both models in two dimensions by computing essentially exactly the ground state wave function and properties derived therefrom for a sequence of symmetric cells of N<18 sites and extrapolating linearly in 1/N to es-timate the infinite lattice properties 111. We have found that E /NJ, <M2>/N2 and <N2>/N2 quickly settle down to very linear behaviour as a function of 1/N. Our estimates for the ground state energy of the Heinsenberg antiferromagnet were in very good agree-ment with previous estimates but we concluded that the previous, rather scattered, estimates for the spin deviation were in general too small by a fac-tor of 2 to 3. Concurrent calculations for the XY model /3,4/ gave very good agreement with our
fini-te lattice method for both energy and magnetization (or spin deviation).
Whereas the square lattice can be tiled by square cells of N = 2, 4, 8, 10, 16, 18.'.. sites the simple cubic and b.c.c. lattices can be filled by symmetric cells containing only N = 8, 16, 32... sites. Thus in three dimensions it is feasible to compute the ground state properties of only the first two cells and extrapolation is uncertain. For N = 8 and 16 on both lattices using periodic boun-dary conditions we have computed for both models all inequivalent two spin correlations and the in-teresting properties arising thereform.
Table I gives all the inequivalent
tions for N = 16 on the s.c. lattice. Results for the other three cells show a similar pattern.
Table I
Spin-spin correlations in the sixteen spin cell of the simple cubic lattice
r/6 magnet antiferromagnet Heisenberg <uXaX> <azaz> <$aa>
O E O Z o g
The values of the energy and square of the order parameter computed from the correlations are listed in Table 11.
Table I1
Values for the ground energy and square of the or- der parameter on the simple cubic and body centred
cubic lattices
Lattice XY magnet antiferromagnet Heisenberg
s.c.,N=8 s.c. ,N=16 s.c.,N- (l /N extrapolant) (bounds) b.c.c.,N=8 b.c.c.,N=16 b.c.c.,N- (I/N extrapolant) (bounds)
Linear extrapolation in 1/N yields energy es- timates which are barely outside established bounds /5b. We therefore take as our best estimate the a- verage of the extrapolant and the sixteen cell re- sult in each case. Thus for the energy per site of the XY model we get 1.55 and 2.06 for,the s.c. and b.c.c. lattices while Suzuki and Miyashita / 4 / ob- tain 1.57 and 2.07.
For the Heisenberg antiferromagnet our results are..compared with some previous estimates in table
111. The spin deviation is defined by
ASZ = (1
-
*/~)/2 ( 5 )Within a 10 X uncertainty our estimates for the energy agree with previous estimates.
Our most striking conclusion is that the spin deviation is a factor of 3 or 4 greater than estima- ted by previous authors ! Earlier estimates of ASZ have been much smaller because the NBel state (with AS =0) has been taken as the first approximation to the true ground state.
Table I11
Comparison of present finite lattice estimates of ground state energy and spin deviation with some previous estimates by other methods.
Author Simple cubic' lattice (date) -Eo/NJ Anderson ASZ (1952).
...
1.79 0.078 Marshall (1955)...
1.81-
Davis (1960)...
1.80 0.064 Oguchi (1963)....
1.77 0.045 Bartkowski (1972)...
1.80 0.059 Present tJork...l.7+0.2 0.2040.02 lattice ASz 0.059-
0.047 0.033 0.051 0.19*0.02 References/I/ Bartkowski,R.R.. Phys. Rev. (1972) 4536 and references therein
/2/ Betts,D.D. and Oitmaa,J., Phys. Lett. (1977) 277. Oitmaa,
J.
and Betts, D.D., Can. J. Phys. (in press)/3/ Pearson,R.B., Phys. Rev. (1977) 1109 /4/ Suzuki,M. and Miyashita,S., Can. J. Phys. (in
press)
/5/ Anderson,P.W., Phys. Rev.
