Degenerate States in Nonlinear Sigma Model with U(1) Symmetry —For Study on Violation of Cluster Property

Entanglement in quantum theory is a concept that has confused many scientists. This concept implies that the cluster property, which means no relations between sufficiently separated two events, is non-trivial. In the works for some quantum spin systems, which have been recently published by the author, extensive and quantitative examinations were made about the violation of cluster property in the correlation function of the spin operator. The previous study of these quantum antiferromagnets showed that this violation is induced by the degenerate states in the systems where the continuous symmetry spontaneously breaks. Since this breaking is found in many materials such as the high temperature superconductors and the superfluidity, it is an important question whether we can observe the violation of the cluster property in them. As a step to answer this question we study a quantum nonlinear sigma model with U(1) symmetry in this paper. It is well known that this model, which has been derived as an effective model of the quantum spin systems, can also be applied to investigations of many materials. Notifying that the existence of the degenerate states is essential for the violation, we made numerical calculations in addition to theoretical arguments to find these states in the nonlinear sigma model. Then, successfully finding the degenerate states in the model, we came to a conclusion that there is a chance to observe the violation of cluster property in many materials to which the nonlinear sigma model applies.


Introduction
Entanglement [1] [2] [3] is quite difficult concept even for researchers [4], because it contradicts the classical concept on the locality [5]. This difficulty has not been reduced when a huge number of researchers apply entanglement to quantum information [6] [7] [8] after the birth of the concept of quantum computer [9] that originates from Deutsch [10]. Entanglement is the correlation of quantum objects, which can not be explained by the classical statistics. When the entangled correlation is found at the infinitely large distance, it leads to violation of the cluster decomposition [11], or the cluster property [12], which is a fundamental concept of physics that there is no relation between two events occurring infinitely apart from each other. Since this violation was found in toy models or in the academic models [12], there are active studies on the cluster property in the many-body systems [13] [14] [15] and in quantum field theory [16] [17] including QCD [18] [19].
In previous researches [20] [21] [22] we studied two-dimensional antiferromagnetic quantum spin systems on the square lattice with U(1) or SU(2) symmetry. We presented the methods to observe the violation of the cluster property on the system where the continuous symmetry breaks spontaneously [11] [23].
In the case of U(1) symmetry [20], we introduced, in order to make the ground state unique, an explicitly symmetry breaking interaction whose strength is g.
Then we pointed out that the violation is the order of ( ) 1 s g N on the finite lattice with s N sites. For the quantum spin system with SU(2) symmetry [21], we discussed the observation of the violation introducing two kinds of explicitly symmetry breaking interactions. Our results there also indicated the possibility to observe the violation, although the effect of the violation contains the factor 1 s N so that fine experiments would be necessary. In the following work [22], however, we found this situation changes when the spin system couples with another spin system. We showed that the Hamiltonian in this work includes Curie-Weiss model [24] [25] [26] induced by the violation of the cluster property.
Then we found that the effective Hamiltonian has the factor 1 s N , which is needed for the thermodynamical properties to be well-defined. We concluded that it is possible to find the effect from the violation of the cluster property through observing the thermodynamical properties given by Curie-Weiss model.
In these studies where our models are quantum antiferromagnets, we recognized that the degenerate states due to the spontaneous symmetry breaking induce this violation. As is well known this breaking is found in many materials [27] [28] [29], including the high temperature superconductor. It is quite important to examine whether we can observe the violation of the cluster property in such materials. Therefore we are sure that to observe the violation of the cluster property is not only a theoretical concern but also a subject to be experimentally investigated in many systems.
In order to make quantitative discussions, we need to find a model which effectively helps us to study the low energy behaviors in many systems. Keeping  [32] and has been applied to various spin systems [33] [34] [35] [36]. We also find many applications of this model in particle physics [37] [38] [39], since the chiral Lagrangian that contains it has given fruitful results on hadron physics [40]. The reason for such wide applications is that this model realizes the symmetry by the minimum degrees of freedom when the continuous symmetry is spontaneously broken.
In our study of the cluster property with spontaneous symmetry breaking in the system on a lattice with s N sites, the key observation is the quasi-degenerated states Q n whose energy Q n E is the lowest one for a quantum number Q n related to the symmetry. In the spin systems, it has been well known that the energy gap [27]. In our work on the nonlinear sigma model, we will reveal the existence of the quasi-degenerated states. We then show that the energy gap in this model is also proportional to Let us describe the plan of this paper.
In the next section, we describe our quantum nonlinear sigma model on the square lattice with U(1) symmetry. The first subsection is devoted to comments on the model which has the continuous symmetry. Then we clarify the quantum property of our model with the discrete symmetry in the second subsection. The third subsection discusses relations between these two models.
In many researches, the nonlinear sigma model is defined by the effective ac-

Continuous Model
In many literatures, the quantum nonlinear sigma model has been defined in the form of the effective action. In this work, however, we define it in the form of Hamiltonian following to [31]. Instead of SU(2) symmetry which is supposed in [31] we employ U(1) symmetry for simplicity.
First we introduce a variable ω for which the eigenvalue is ω and the eigenstate is ω .
We also introduce a conjugate operator of ω , which we denote p ω .
This commutation relation implies that The eigenvalue of p ω should be discrete because, for the eigenstate p ω of p ω , the inner product For models on the lattice, we introduce operators ˆj ω and ˆj p ω at each site j, where 0,1, , 1 for the lattice size s N . They satisfy the following commutation relations.
Using these operators given at every site, we define the Hamiltonian for a nonlinear sigma model on the square lattice by Using Equations (5) and (6) it is easy to see that World Journal of Condensed Matter Physics

Discrete Model
We would like to obtain the energy for the quantum number of the generator ˆL attice Q in numerical calculations by the diagonalization or quantum Monte Carlo methods. Since these methods are formulated through a finitely dimensional linear algebra, we employ the discrete variable instead of the continuous one. For this purpose, the commutation relation (5) is not suitable, because it can not apply to the quantum theory of the discrete variable. In order to make a model that has the discrete variable and that is a good approximation to the model with ˆL attice H of the angle variable ω , we would like to make our model to satisfy the Weyl relation.
Based on the discussion in Appendix A2, we introduce two kinds of unitary operators ˆp j U and ˆq j U at each lattice site j, and impose the following Weyl relation to them.
Assuming the existence of an eigenstate 0 qj of ˆq j U and the relation (9) we obtain, for 0,1, , 1 Here 0 pj is defined by 0 qj following Equation (63) The state in q-representation on the lattice is defined by Similarly the state in p-representation on the lattice is defined by We then define a Hamiltonian ˆD H for the discrete variables on the lattice by For ˆD H we can introduce an increment operator ˆD Q defined by We can obtain the eigenstate which is common to ˆD H and ˆD Q , because In the last equation of Equation (17), note that, for l n ≠ , we have

Hamiltonian for Large Ld
We apply this discussion to the whole state 0 1 1 , , , Hereafter we abbreviate As for ˆB H note that, for one nearest neighbor pair ( ) With Equations (23) and (25) we obtain

Energy with a Fixed Number nQ
In this section, we present a theoretical argument about the lowest energy with a fixed number Q n . In the first subsection, we discuss the effective Hamiltonian where the operator ˆQ n is clearly separated. The second subsection is to estimate the energy gap using this effective Hamiltonian.

Effective Hamiltonian
Here we can introduce a set of new operators { }l ζ .
The first Weyl relation is verified by notifying The rest of relations are trivial from Equations (9) and (29). By this proof we confirm that the set of { }, pl ql V V is independent and complete.
Then we will express ˆB H in Equation (15) It should be noted that †q l qn U U contains no 0q V . Therefore we can express The result is, with It should be noted that Here note that In Equation (41)

Energy with a Fixed Value of Q n
When we calculate the energy in p-representation, where the basic state is Therefore the lowest energy with a fixed value of Q n is given by As for terms with ˆk e we estimate their contributions by the first-order perturbation theory. When the eigenstate of ( )

Numerical Results
Now we present our numerical results of the lowest energy from the Hamilto-

Results on Ns = 5, 9 Lattices
In this subsection, we present the numerical results of ( )  These results tell us that the second term of

Results on Ns = 16, 36, 64 Lattices
For larger lattices with 16,36, 64 s N = we estimate the energy gaps by means of quantum Monte Carlo methods [52] [53] [54]. The reasons why we employ these methods are that we can easily apply them to the study on these lattices and that we can obtain reliable results on the energy. In quantum Monte Carlo methods we have two technical parameters, which are inverse temperature β and Trotter number t l . For the lowest energy, we need large β as well as large t l . Since our concern is the energy gap, we judge that β and t l are large enough if the gap calculated with some values of β and t l does not change for slightly smaller or larger values of β and t l . Table 1 shows the results for ( ) 16

32, 16
Note that the first correction term in Equation (50) Table 2, the correction from the 2 c term on this lattice is too small to distinguish As is shown in Figure 5

Conclusions and Comment to Future Study
In this paper, we studied the quasi-degenerate states, which is essential on the violation of the cluster property, in the quantum nonlinear sigma model with U(1) symmetry. Here we present our conclusion on the quasi-degenerate states by summarizing previous sections. Also in addition to the influence of the interaction strength on these states, we comment on the observation of the violation and the extension to the model with SU(2) symmetry.
In previous researches [20] [21] [22] we have shown that it is possible to observe the violation of the cluster property in spin systems when the continuous symmetry breaks spontaneously. The quite important question is whether we can observe the violation in other systems. It is specially interesting to examine whether the nonlinear sigma model shows the violation or not, because this model can be used as the effective model in the low energy region for the system with the spontaneous symmetry breaking. The study on the spin system showed that the existence of the quasi-degenerate states is the key for the violation. If there exist the quasi-degenerate states whose energies are proportional to the squared value of the quantum number, we can apply the same discussion as that in the spin system to the nonlinear sigma model. Therefore in this paper, we have presented the extensive study on the energy of this model.
In this work we have considered a quantum model defined on a lattice, introducing discrete and finite variables instead of the continuous angle variables. In order to justify these discrete variables, our discussion has started from the Weyl  (47) will still stay small compared to the first term 1. We therefore expect that our numerical results in this paper will not be largely changed even if we use larger values of B. In order to confirm this expectation, we carried out several addi- The final comment is on an extension of our work to the nonlinear sigma model with SU(2) symmetry. The essential element of our present work is founded on the formulation of the model in p-representation, where we can fix the quantum number Q n . In addition, we introduced discrete variables so that we can calculate the energy using the finite dimensional matrices for the Hamiltonian. Can we apply our ideas to the study of the model with SU (2) symmetry?
The answer is perhaps yes, but more technical improvement would be required.
The reason is the following. The nonlinear sigma model has been defined by fixing the magnitude of the scalar field whose Hamiltonian is the same as that of the free field. Then we have the variables with SU(2) symmetry only, which are the angles in the polar coordinate. It is difficult, however, to define the conjugate operators corresponding to these angle variables. Therefore we have no naive method to construct the nonlinear sigma model in p-representation. The technical improvement to solve this problem is under study now. World Journal of Condensed Matter Physics manuscript through her critical review.