Abstract
The cluster property is one of fundamental properties in physics. This property means that there are no relations between two events that are sufficiently separated. Because the cluster property is directly connected with entanglement in quantum field theory and in many-body systems, theoretical and experimental progress on entanglement stimulates us to study this property deeply. In this paper we investigate the cluster property in the spin 1/2 XXZ antiferromagnet on the square lattice with an explicitly symmetry breaking interaction of strength g. In this model spontaneous symmetry breaking occurs when the lattice size N is infinitely large. On the other hand, we have to make g zero in order to obtain quantities in the XXZ model with no symmetry breaking interaction. Since some results depend on the sequence of limit operations — and , it is difficult to draw a clear conclusion in these limits. Therefore we study the model with finite g on the finite lattice, whose size N is supposed to be 1020, for our quantitative calculations. Then we can obtain the concrete ground state. In order to study the cluster property we calculate the spin correlation function. It is known that the function due to Nambu-Goldstone mode (gapless mode), which is calculated using linear spin wave theory, satisfies this property. In this paper we show that almost degenerate states also induce the spin correlation. We assume that the spin correlation function in measurements is a sum of the function due to Nambu-Goldstone mode and one due to these degenerate states. Then we examine whether the additional correlation function violates the cluster property or not. Our conclusion is that this function is finite at any distance, which means the violation of the cluster property, and it is of order of . Except for the case of extremely small g, this violation is the fine effect. Therefore the correlation function due to the degenerate states can be observed only when it is larger than the spin correlation function due to Nambu-Goldstone mode. We show that g required for this condition depends on the distance between positions of two spin operators.
Highlights
It has been widely understood that entanglement is one of the fundamental concepts in quantum physics
The energy En ( N ) in (15) is given by the expression (6) [24] [28]. Since this expression is quite important for our study on the cluster property, we present numerical results on N = 36 lattice in Figure 1, which are calculated by the exact diagonalization
In conclusion we find that the spin correlation function due to the quasi-degenerate states violates the cluster property
Summary
It has been widely understood that entanglement is one of the fundamental concepts in quantum physics. It is well known that the ground states of models on this system realize semi-classical Neel order [24], in other words, SSB of SU(2) or U(1) symmetry This realization has been strongly supported by spin wave theory (SWT) [25] as well as numerical studies [26] [27]. One important fact clarified by the previous numerical study [24] is that the eigen energy characterized by the quantum number of U(1) symmetry is almost degenerate This implies that we have strictly degenerate states when N becomes infinitely large, while we do not have any degenerate states on the finite lattice. In Appendix B we will show the basis of our assumption that the spin correlation function is the sum of the function due to Nambu-Goldstone mode and the one due to the quasi-degenerate states
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