Abstract
There are some concepts that are accepted in our daily life but are not trivial in physics. One of them is the cluster property that means there exist no relations between two events which are sufficiently separated. In the works recently published by the author, the extensive and quantitative examination has been made about the violation of cluster property in the correlation function of the spin operator for the quantum spin system. These works have shown that, when we include the symmetry breaking interaction, the effect by the violation is proportional to the inverse of the system size. Therefore this effect is tinny since the system size is quite large. In order to find the effect due to the violation even when the size is large, we propose a new system where additional spins couple with the spin system on the square lattice, where the coupling constant between these systems being assumed to be small. Applying the perturbation theory, we obtain the effective Hamiltonian for the additional system. This Hamiltonian includes Curie-Weiss model that is induced by the violation of the cluster property. Then we find that this effective Hamiltonian has the factor which is the inverse of the system size. Since Curie-Weiss model, which is known to be exactly soluble, has to contain this factor so that the thermodynamical properties are well-defined, the essential factor for the Hamiltonian is determined by the coupling and the strength of the symmetry breaking interaction. Our conclusion is, therefore, that it is possible to observe the effect by the violation of the cluster property at the inverse temperature whose order is given by these parameters.
Highlights
( ) magnitude of the violation is order of 1 g N, where g is the strength of the explicit symmetry breaking interaction and N is the size of the system
In the previous papers [26] [34], we showed the violation of the cluster property (VCP) in spin 1/2 XXZ antiferromagnet and Heisenberg antiferromagnet on the square lattice
( ) cate that the magnitude of VCP is order of 1 g N, where g is the strength of the explicit symmetry breaking interaction and N is the size of the system, which we suppose N ~ 1020
Summary
In the previous paper [26] we have investigated the cluster property of spin 1/2 XXZ antiferromagnet on the square lattice For this antiferromagnet, the ground state realizes semi-classical Neel order [27], in other words, spontaneous symmetry breaking (SSB) [17] [28] of U(1) symmetry. Applying the perturbation theory with small u, we obtain the effective Hamiltonian Heff ,ex for the spins in the additional system We see that it includes Curie-Weiss model. We show that the effective Hamiltonian contains Curie-Weiss model, whose Hamiltonian is the square of the sum of all spin operators on the extended sites. For this purpose, we use the mean field approximation, which is discussed in appendix C in detail. Since many symbols are used in our paper, we list them in Table 1 for convenience
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