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NEW SOLUTIONS FOR THE ANISOTROPIC
ANTIFERROMAGNETIC CHAIN
M. Lagos, G. Cabrera
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplkment au no 12, Tome 49, dkcembre 1988
NEW SOLUTIONS FOR THE ANISOTROPIC ANTIFERROMAGNETIC CHAIN
M. Lagos (I) and G . G. Cabrera (2)
(I) Facultad de Fisica, Universidad Cato'lica de Chile, Casilla 61 77, Santiago 22, Chile
( 2 ) Institute de Fisica "Gleb Wataghin"
,
Universidade Estadual de Campinas, CaCa Postal 6165, 13.081 Campinas, ,960 Paulo, BrazilAbstract. - Coherent states are proposed as solutions for the antiferromagnetic Heisenberg Hamiltonian in the quasi-Ising
asymptotic regime of high anisotropy. Since our solutions are given in closed analytical form, a deep insight concerning the structure of the ground state and its excitations is obtained. Our calculation is in excellent agreement with numerical simulations.
It has been suggested that the mechanisms in- volved in the recently discovered high-T, superconduc- tivity are related to the magnetic properties of these new compounds [I]. This fact has drifted a special attention to the study of the antiferromagnetic prop- erties of the insulating phase of these materials. As
a prototype copper oxyde, La2Cu04-s appears as a likely candidate for a two-dimensional antiferromag- netic system, displaying also superconductivity for a given range of doping [2]. At this stage, a reliable description of the highly correlated insulating phase in these low dimensional materials is desirable. The above regime can be modelled by an antiferromag- netic Heisenberg Hamiltonian, which in spite of its deceivingly simple appearance, displays highly non- trivial many-body effects. The exact solution for the ground state in one-dimension, including anistropy, is available in the literature [3], but no exact solution is known for higher dimensions. Even in one-dimension, the solution obtained through the Bethe ansatz [3] is cumbersome and difficult t o handle. Numerical results in higher dimensions are not conclusive and extrapo- late differently [4].
In the present contribution we put forward a general 1
solution of the anisotropic spin-- Heisenberg model
2
with antiferromagnetic coupling which is asymptoti- cally exact in the high correlation limit. We illustrate this solution with the linear chain, where large size (N
-
20) numerical simulations and exact results are available for comparison.The Heisenberg Hamiltonian for the linear chain reads
served in finite-size calculations for Heisenberg chains 151. As long as a remains smaller than 1, the ground state has a dominant component of the Ndel type. Next contributions come from configurations obtained from the NBel state by flipping one pair of spins, two pairs of them, and so on, in hierarchical order. Since the Ndel components dominate the overall quantum superposition, the ground state presents long-range order, but quantum fluctuations reduce considerably the staggered magnetization. This effect is similar to the one observed in LazCu04-6 by means of neutron diffraction experiments, where a reduction of the mag- netic moment ascribed t o C U + ~ ions is obtained [2].
The above considerations heuristically lead to the definition of the foilowing even and odd operators:
+ : = G
c
S+ (m+
1) S- (m)+
afi,
(2)m even
d.t=\/+
c
S+ (m) S- (m+
1)+
aT,
m odd
\IN
which have been constructed taking the Nee1 state S, (m) =
$
as a reference. It is then appar-ent that o t r treatment has a broken symmetry, since a similar construction can be done with the other Ndel
1
state S, (m) =
5
(-1)"". A a result we obtain that the ground state is a doublet (05
a<
1).In the quasi-Ising limit we replace the S, (m) opera- tor by -its eigenvalue for the dominant NBel component. This approximation yields the following algebra for the q5 's operators [6]
N
+
-
H =
J C
[sz ( m + 1)s. (m)+ [de 7 de]
-
[do, d?] = 1, [de, do] =[
Q
:
,
do] = 0, (3)m = ~ which are boson like commutation relations. Within
+:
{S+ (m+
1) S- (m)+
S+ (m) S- (m+
I)}],
(1) the same approximation, Hamiltonian (1) can be par- tially diagonalized in the subspace S, = 0, i.e. where a is the anisotropy parameter, being a = 0 the = J(4:
de+d$do) +Eg (a)+
H' ( a ),
(4) Ising limit and a = 1 the case of isotropic exchange. Ifwhere is the exchange of the original the departure point for our analysis is the Ising limit
( a = 0)
,
where we know that the ground state is of tonian,Eg
(a) is the energy for the ground state (ab- sence of quasiparticles), and H' (a) represents addi- the N&l type, the inclusion of the transverse part ofHamiltonian (1) can be visualized as a disordering pro- tional degrees of freedom which are not contained in cess where pairs of neighboring spins are flipped, the the manifold of total S, = 0. The ground state lg) is true ground state being contained in the manifold of the vacuum for the bbosons, i.e.
total S, = 0. This fact has been systematically ob- de 19) =
40
19) = 0,C8 - 1380 JOURNAL DE PHYSIQUE while the reference NBel state IN) satisfies
me
IN) =mo
IN) = % f i l N ) , 2showing that IN) is eigenvector of the annihilation op- erators
4,
and4,.
This implies that the ground state can be represented in the formwhich can be recognized as a quantum mechanical co- herent state [7]. The structure of the state (5) is ex-
tremely interesting. Fluctuations over the NBel state IN) are induced by the 4's operators, but the weight (and phase) of a given configuration is not arbitrary. The whole superposition defines a coherent state with a highly ordered structure and displaying long-range order. The ground state energy obtained in this way, up to second order in a, is given by
which compares with good accuracy to the exact solu- tion given by Orbach [3] in a wide range of variation of the anisotropy parameter a.
The result given by (6) is also more accurate than the zero-point energy obtained through the standard linearized spin-wave approximation in the whole range where our theory is valid (0
5
a5
0.5). Indeed, our calculation shows that the usual picture of sublat- tices in antiferromagnets can be saved, in spite of the complications introduced by the ground state struc- ture. As long as cu<
1, the ground state displays long-range order, but the zero-point sublattice mag- netization is smaller than the one estimated by spin- wave theory. In our approach we interpolate continu- ously from the Ising limit (which does not support spin waves) t o cases of lower anisotropies, thus presenting a novel view of this problem. Detailed comparison of our ground state, and the correlation functions obtained from it, with numerical simulations for very large sys- tems ( N 20),
shows a surprising agreement for cor- relation functions t o all orders [8]. It is concluded there that our asymptotic ground state is accurate to bet- ter than 1 % for 0 _< a<
0.5. Such a wide range of applicability is rather unexpected since our derivation always assumes the quasi-Ising limit. What this anal- ysis reveals is that the structure shown by the wave function (5) is essentially correct,The same type of solution can be given in higher di- mensions, assuming that the lattice has no geometric frustration to the antiferromagnetic ordering, hence it can be separated into two crystallographic equivalent sublattices. One of them, associated to up spins, is de- scribed by a set of vectors R. The other one, associated
to down spins, is characterized by vectors R+S, where the 6-vectors connect each lattice ;site with its nearest neighbors. We now introduce the set of operators
in a similar way to expression (2). In formula (7) z
is the lattice coordination number, and hence we have as many operators as nearest neighbors in the lattice. A parallel derivation as the one outlined above for the linear chain can be done. In the q~uasi-Ising limit the set {q56) of operators is bosonlike, and the ground state is a generalized coherent state for a many modes system [7]. The ground state energy to second order in a is given by
E. (a) =
EN
(1)
:
+
:
,
where EN is the corresponding energy for the NBel state. Equation (8) shows that th.e correction to the NBel state diminishes with the coordination number z,
and hence with the dimensionality. The lower the di- mensionality the more important the role of quantum fluctuations. Thus, the linear chaih is the worst case and a better agreement with our theory is expected for higher dimensional systems [9].
[I] Anderson, P. W., Science 235 (1987) 1196; Baskaran, G., Zou, Z. and Anderson, P. W., Solid
State Commun. 63 (1987) 973;
Emery, V. J., Phys. Rev. Lett. 58 (1987) 2794. [2] Vaknin, D., Sinha, S. K., Moricton, D. E., John-
ston, D. C., Newsam, J. M., Sizfinya, C. R., King, Jr., H. E., Phys. Rev. Lett. 513 (1987) 2802;
Mitsuda, S., Shirane, G., Sinha, S. K., Johnston, D. C., Alvarez, M. S., Vaknin, D. and Moncton, D. E., Phys. Rev. B 36 (1987) 822.
[3] Bethe, H., 2. Phys. 71 (1931) 205;
HulthBn, L., Ark. Mat. Astmn.. Fys. A 26 (1938) 1;
Orbach, R., Phys. Rev. 112 (1958) 309;
des Cloizeaux, J. and Gaudin, M., J. Math. Phys.
7 (1966) 1384.
[4] Oitmaa, J. and Betts, D. D., Can. J. Phys. 56
(1978) 897;
Reger, J. D. and Young, A. P., Phys. Rev. B 37. (1988) 5978.
[5] Medeiros, D., Cabrera, G. G. and Lagos, M., un- published.
[6] Lagos, M. and Cabrera, G. G., Solid State Com- mun., in press; Phys. Rev. B, to be published. [7] Klauder, J. R. and Shagerstam, B., Coherent
States: Applications in Physics and Mathemat- ical Physics (World-Scientific, Singapore) 1985. [8] Lagos, M., Kiwi, M., Gagliano, E. R. and Cabr-
