Quantum Curie-Weiss Magnet Induced by Violation of Cluster Property

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 rela-tions 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.


Introduction
The concept of entanglement strongly contradicts with the classical one about H has the overall factor ( ) 2 u gN . Since Curie-Weiss model has to contain the factor 1/N in order that the thermodynamical properties are well-defined, the essential factor for the system is 2 u g . We conclude, therefore, that one would be able to observe the violation when the inverse temperature β is of order of ( ) Contents of this paper are as follows. In Section 2, we describe our model in some detail. The first subsection is devoted to a brief explanation of the spin 1/2 XXZ antiferromagnet on the square lattice. Also we collect the results related to the Hamiltonian , sq g H [26]. In the second subsection, we define ˆe x V which describes an extended part of the model. In Section 3, using the perturbation theory, we derive the effective Hamiltonian In Section 4, we calculate the energy and the specific heat of Curie-Weiss model. For this purpose, we use the mean field approximation, which is discussed in appendix C in detail. It should be noted that this method is absolutely reliable for the model when the system is infinitely large. In order to assure that our results are sufficiently accurate, we numerically calculate the specific heat on finite lattices. In Section 5, we investigate the thermodynamic properties of the effective Hamiltonian , eff ex H . The first subsection is to calculate the energy and the specific heat when the temperature is high. Here we employ the high temperature expansion described in appendix D. We find the effect by Nambu-Goldstone mode only in this region. In the second subsection, we calculate these thermodynamic properties at a low temperature. Here we employ the mean field approximation which is exact for the ferromagnet due to the degenerate states and reasonable for the one due to Nambu-Goldstone mode. The final section is devoted to summary and discussion.
Since many symbols are used in our paper, we list them in Table 1 for convenience.

Spin System on the Square Lattice
We will consider the quantum spin system on the square lattice. On each site i ( 1, , sq i N =  ) we have the spin operator ˆi S α ( , , x y z α = Here ( ) , i j denotes the nearest neighbor pair on the square lattice and λ is the parameter between 0 and 1.
Then we introduce the spin operator on each sub-lattice, In order to obtain the ground state, we introduce the symmetry breaking in- Then we have the Hamiltonian It is well known that in this system there exists Nambu-Goldstone mode, which can be described successfully by spin wave theory. On the other hand, adding the explicit symmetry breaking interaction into the Hamiltonian, we have obtained the lowest energy eigen state and the excited states which are linear combinations of the degenerate states [26] [34]. This leads us to consider two kinds of excited states, which are states due to degenerate states and those from Nambu-Goldstone mode. In order to describe these excited states we will employ two kinds of Hamiltonian  (9) Here α k is the annihilation operator of Nambu-Goldstone mode with the wave vector k , and NG E denotes the ground state energy. The effect due to the symmetry breaking interaction ˆg V is included in ω k , which is the energy of Nambu-Goldstone mode. Detailed expression of ω k is given in Appendix B.2.

Extended Spin System
Let us consider a new system which consists of the spin system on the square lattice and the one on ex N additional sites. The state for the additional sites is is spanned by these states. We will consider the spin system on the square lattice and the additional spin system. Whole vector space is sq Here ( ) a i is the additional site fixed by the site i as is shown in Figure 1. Note that the summation for i runs over A sub-lattice only.

Effective Hamiltonian of the Extended Spin System
In Appendix A we have derived the effective Hamiltonian using the perturbation theory. We apply it to our model, where 0 H is  We have two kinds of the excited states. One is the excited state l G ( 1 l ≥ ) that consists of the degenerate state and the other is the one-magnon state k with the wave vector k , which is Nambu-Goldstone mode. Following the discussions in the previous works [26] [34], we suppose that these excited states are independent.
We obtain the effective Hamiltonian From Appendix A we obtain We have calculated these coefficients in Appendix B. We obtain Here v denotes the expectation value of the spin operator in the ground state.
When we consider the terms of ( ) x y i j c y y i j = = (16) As for contributions by Nambu-Goldstone mode, we obtain  [26]. From (15), (16) and (17) the effective Hamiltonian for our model on the vector space ex V is given by The first term of , i j c has the factor ( ) and is independent of the site. We then come to an important conclusion that this effective Hamiltonian contains modified Curie-Weiss model induced by the degenerate state. In the next section, we will discuss this model in some detail.

Curie-Weiss Model
Curie-Weiss model [35] [36] [37] is defined by, with the site number CW N , In this model, we can exactly calculate the specific heat for the infinitely large lattice at any temperature by the mean field approximation. Since this fact is quite important we will make a numerical examination in this section. We compare the specific heat calculated by the eigen values on the large lattices with the result obtained from the mean field approximation.
In Curie-Weiss model, the partition function ( ) CW Z β with the inverse temperature β is given by Here CW N , which we suppose to be even, is the lattice size and Since no positive solution exists for β . We conclude, therefore, the mean field approximation for Curie-Weiss model on large lattices is satisfyingly reliable at any temperature. curve is calculated from the mean field approximation, which is given by (21).

High Temperature Region
Let us study the thermodynamical properties at very small β . It is known that the high temperature expansion described in Appendix D is a powerful tool in this region. We apply the results by this method to our effective Hamiltonian , eff ex H given in (18), for which 2 , Then we have . We obtain, with , x y Similarly, Therefore, we obtain ( ) Note that the first term of (26) In (27), we see the effect due to the first term of the effective Hamiltonian and that due to Nambu-Goldstone mode only.

Low Temperature Region
In order to calculate the thermodynamical properties at a low temperature, we employ the mean field approximation described in Appendix C to the effective 3 .
Let us study how the specific heat  g u ≤ , to observe this effect.
In Figure 3, we plot These results are also plotted in Figure 4 and Figure 5 for comparison. We see that the polynomial expansion of ex ex h ζ is reliable for . This suggests that the perturbation theory on ex h gives us the good approximation, which will be important in future study on the effective Hamiltonian.
Note that if the degenerate states are absent we should use 2 ex u v g ζ = instead of ( ) 2 3 2 u v g because the system is the ferromagnet induced by Nambu-Goldstone mode only. Measuring the specific heat, therefore, we would be able to judge if the degenerate states exist or not.
To summarize this section we present in Figure 6, a region formed by g and u, where one can observe the effect by Curie-Weiss model due to the degenerate states. The red curve in the figure gives the boundary for the validity of the perturbation theory. The black curve shows the boundary where we can observe the specific heat by this model. Therefore one can confirm the effect by the violation of the cluster property in the region between the red and the black curves.

Summary and Discussions
The cluster property is deeply connected with the classical concept about locality, but it is not trivial in quantum physics. 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. Our results indicate 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 20 10 N . The observation of VCP in experiments is not easy, therefore, because of its smallness.
In this paper, we proposed an extended spin system so that we find a better chance to observe the effect by VCP. We added a new spin system to the original spin system on the square lattice. The Hamiltonian is   contains Curie-Weiss model induced by the degenerate states. In order to calculate thermodynamic property of the effective Hamiltonian at a low temperature, we employed the mean field approximation, where the difference between the effect due to the degenerate states and that due to Nambu-Goldstone mode is found in the magnitude of the specific heat. Our conclusion is that it is possible to find the effect of the violation of cluster property in our extended model.
Our study in this paper is based on the effective Hamiltonian, which is derived by the perturbation theory. In order to examine the validity of the theory, we The agreement between both values is satisfactory to assure the validity of the perturbation theory. Several comments are in order for our calculations and results.  First let us discuss on effects of higher-order terms in the perturbation theory.
On large lattices the energy gap is of order of g and 2 excited state ground state ex V is of order of ( ) 1 g N . Then the next order term is of order of ( ) 2 u gN , but the factor N should be included into the Hamiltonian for the consistency. Therefore we conclude 2 u g should be small in order to neglect higher-order terms.
 Let us consider to estimate the parameters u and g in experiments. One can estimate u by measuring the specific heat at high temperature because, as we have seen in (27), the first term ( ) 2 uv dominates compared to the second T. Munehisa term with the factor 4 u v g . In order to estimate g, on the other hand, one should measure the correlation function of the spin operator which is given in (96) in Appendix D.  Next we discuss the qualitative difference between the effect due to the degenerate states and that due to Nambu-Goldstone mode. Since we do not see any effect due to the degenerate states at a high temperature, we need to examine the thermodynamic quantities at a low temperature. In this region, where the mean field approximation is valid, it is difficult to distinguish the effect due to the degenerate states from that due to Nambu-Goldstone mode.
Therefore we have to investigate the property connected with the excited states which cannot be calculated in the mean field approximation. This subject will be studied in a future work where we investigate the effective Hamiltonian in the extended system with SU(2) symmetry.  The last comment is about experimental realization of the proposed spin system. One idea to realize our model is following. In experiments for the spin system on the square lattice, the material contains multi layers. It will be possible to consider the material that has the sandwich structure where the magnetic layer and the quasi non-magnetic layer appear alternately. The magnetic layer realizes the spin system on the square lattice, while in the quasi non-magnetic layer we can partially add the magnetic elements such as Cu. In this additional system, the magnetic elements are sparse so that the coupling between spins on the additional system is weak. Therefore we can suppose that such material realizes ˆe Here 0 l ≥ , The eigen state of Ĥ is given by In the second order perturbation theory, we neglect the terms of 3 u in F.

Then the variation on
The variation on { } , a l s c is, on the other hand,

Appendix B
Here we calculate the inner product ) for the site i on the A sub-lattice.

Part 1
In this subsection we calculate the contributions due to the degenerate states.
In [26] we obtained the eigen state Then we obtain Then the inner product is given by Finally we obtain ( ) Next we discuss the matrix elements due to Nambu-Goldstone mode. We employ the results calculated in the previous work [26] based on spin wave theory.
Here the ground state NG G is 0 G and the excited state is the one magnon state with the wave vector k , which we denote by | 〉 k .
In spin wave theory, it is known that As for the operator ( )1 i P x i S − we obtain [26] ( ) ( ) Here we use the symbols defined by