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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.

Entanglement [

In previous researches [

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 [

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 this purpose in mind we now study a quantum nonlinear sigma model, which has been derived as the effective model for two-dimensional Heisenberg antiferromagnets [

In our study of the cluster property with spontaneous symmetry breaking in the system on a lattice with N s sites, the key observation is the quasi-degenerated states | n Q 〉 whose energy E n Q is the lowest one for a quantum number n Q related to the symmetry. In the spin systems, it has been well known that the energy gap E n Q − E 0 is proportional to n Q 2 / N s [

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 action [

In the second subsection, we define our model on the square lattice where the unitary operators U ^ q i and U ^ p i are defined at each site i. The Hamiltonian of the model has two terms named A-term and B-term, which correspond with the kinetic term and the potential term in the ordinary nonlinear sigma model respectively. Here we introduce q-representation where the B-term of the Hamiltonian is diagonalized, and p-representation where the A-term is diagonalized. The Hamiltonian introduced in [

When L d is quite large, we suppose the variable of our model becomes the angle variable. Discussing the A-term of our Hamiltonian in the large L d case, we show that this term becomes the kinetic term which is the differential operators in q-representation. It means that A-term ∑ i ( U ^ p i + U ^ p i † ) becomes ∑ i ∂ 2 / ∂ θ i 2 in the large L d limit, where 2 π n / L d becomes θ .

Sections 3 is devoted to the theoretical discussion for the lowest energy E n Q ( L d , N s ) with a fixed value of n Q , while Sections 4 is to the numerical investigation for this energy. Our purpose of our work is to show that the energy gap E n Q ( L d , N s ) − E n Q = 0 ( L d , N s ) is proportional to n Q 2 / N s . In section 3, we give the theoretical arguments for this form, neglecting the effect due to the B-term. Through this discussion, we find correction terms for the energy gap which would be observed on small lattices. Section 4 is to show our numerical results on lattices with N s = 5 , 9, 16, 36 and 64 sites. In the first subsection, the results of N s = 5 and 9 are presented. For the N s = 5 lattice we employ the diagonalization so that we are not bothered by the numerical error. The results on the N s = 9 lattice are obtained by stochastic state selection method [

In the last section, we conclude that the quasi-degenerate states exist in the quantum nonlinear sigma model by summarizing the theoretical studies and the extensive examinations by numerical approaches. Also we make comments on the dependence of the interaction strength on our conclusion, as well as those on future studies on the violation of the cluster property and the extension of our work to the model with SU(2) symmetry.

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 [

First we introduce a variable ω ^ for which the eigenvalue is ω and the eigenstate is | ω 〉 .

ω ^ | ω 〉 = | ω 〉 ω , 〈 ω | ω ′ 〉 = δ ( ω − ω ′ ) . (1)

The value of ω is continuous and is limited to the range [ 0,2 π ] , since ω ^ expresses U(1) symmetry.

We also introduce a conjugate operator of ω ^ , which we denote p ^ ω .

[ ω ^ , p ^ ω ] = i . (2)

This commutation relation implies that

〈 ω | ψ 〉 = ψ ( ω ) , 〈 ω | p ^ ω | ψ 〉 = − i d d ω ψ ( ω ) . (3)

The eigenvalue of p ^ ω should be discrete because, for the eigenstate | p ω 〉 of p ^ ω , the inner product 〈 ω | p ω 〉 = exp ( i ω p ω ) has the same value at ω = 0 and ω = 2 π . Therefore we denote | p ω 〉 = | n 〉 hereafter,

p ^ ω | n 〉 = | n 〉 n , n = 0 , ± 1 , ± 2 , ⋯ . (4)

For models on the lattice, we introduce operators ω ^ j and p ^ ω j at each site j, where j = 0 , 1 , ⋯ , N s − 1 for the lattice size N s . They satisfy the following commutation relations.

[ ω ^ j , p ^ ω l ] = i δ j l , [ ω ^ j , ω ^ l ] = 0, [ p ^ ω j , p ^ ω l ] = 0. (5)

Using these operators given at every site, we define the Hamiltonian for a nonlinear sigma model on the square lattice by

H ^ L a t t i c e ≡ f ∑ j = 0 N s − 1 ( p ^ ω j ) 2 − g ∑ ( l , n ) cos ( ω ^ l − ω ^ n ) . (6)

Here j denotes the site number, while ( l , n ) denotes the nearest neighbor on the square lattice. For this Hamiltonian, we can introduce the generator Q ^ l a t t i c e defined by

Q ^ L a t t i c e ≡ ∑ j = 0 N s − 1 p ^ ω j . (7)

Using Equations (5) and (6) it is easy to see that

[ H ^ L a t t i c e , Q ^ L a t t i c e ] = − g ∑ ( l , n ) [ cos ( ω ^ l − ω ^ n ) , Q ^ L a t t i c e ] = − g ∑ ( l , n ) [ cos ( ω ^ l − ω ^ n ) , p ^ ω l + p ^ ω n ] = − i g ∑ ( l , n ) { sin ( ω ^ l − ω ^ n ) − sin ( ω ^ l − ω ^ n ) } = 0. (8)

We would like to obtain the energy for the quantum number of the generator Q ^ L a t t i c e 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 H ^ L a t t i c e 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 U ^ p j and U ^ q j at each lattice site j, and impose the following Weyl relation to them.

U ^ p j U ^ q j = U ^ q j U ^ p j exp ( i δ d ) , δ d ≡ 2 π / L d ,

[ U ^ q j , U ^ q l ] = 0 , [ U ^ p j , U ^ p l ] = 0 , [ U ^ p j , U ^ q l ] = 0 , for j ≠ l . (9)

Assuming the existence of an eigenstate | 0 〉 q j of U ^ q j and the relation (9) we obtain, for n = 0 , 1 , ⋯ , L d − 1 , the eigenstates of U ^ q j or U ^ p j .

| n 〉 q j ≡ ( U ^ p j ) n | 0 〉 q j , U ^ q j | n 〉 q j = | n 〉 q j exp ( − i n δ d ) ,

| n 〉 p j ≡ ( U ^ q j ) n | 0 〉 p j , U ^ p j | n 〉 p j = | n 〉 p j exp ( i n δ d ) . (10)

Here | 0 〉 p j is defined by | 0 〉 q j following Equation (63) in Appendix. We also obtain

U ^ p j | n 〉 q j = ( U ^ p j ) n + 1 | 0 〉 q j = | n + 1 〉 q j , U ^ q j | n 〉 p j = ( U ^ q j ) n + 1 | 0 〉 p j = | n + 1 〉 p j . (11)

Here | L d 〉 q j = | 0 〉 q j and | L d 〉 p j = | 0 〉 p j , as is shown in Appendix. Also the inner product 〈 n | m 〉 q j p j is given by

〈 n | m 〉 q j p j = ( 〈 m | n 〉 p j q j ) * = 1 L d exp ( − i n m δ d ) . (12)

The state in q-representation on the lattice is defined by

| n 0 , n 1 , ⋯ , n N s − 1 〉 q ≡ | n 0 〉 q 0 ⊗ | n 1 〉 q 1 ⊗ ⋯ ⊗ | n N s − 1 〉 q N s − 1 . (13)

Similarly the state in p-representation on the lattice is defined by

| n 0 , n 1 , ⋯ , n N s − 1 〉 p ≡ | n 0 〉 p 0 ⊗ | n 1 〉 p 1 ⊗ ⋯ ⊗ | n N s − 1 〉 p N s − 1 . (14)

We then define a Hamiltonian H ^ D for the discrete variables on the lattice by

H ^ D ≡ H ^ A + H ^ B ,

H ^ A ≡ − A ∑ l ( U ^ p l + U ^ p l † − 2 ) , H ^ B ≡ − B ∑ ( l , n ) ( U ^ q l † U ^ q n + U ^ q n † U ^ q l ) . (15)

For H ^ D we can introduce an increment operator Q ^ D defined by

Q ^ D ≡ ∏ l U ^ p l , Q ^ D U ^ q j Q ^ D † = U ^ q j exp ( i δ d ) . (16)

We can obtain the eigenstate which is common to H ^ D and Q ^ D , because

[ H ^ D , Q ^ D ] = − B ∑ ( l , n ) [ ( U ^ q l † U ^ q n + U ^ q n † U ^ q l ) , Q ^ D ] = − B ∑ ( l , n ) [ ( U ^ q l † U ^ q n + U ^ q n † U ^ q l ) , U ^ p l U ^ p n ] ∏ j ≠ l , n U ^ p j = 0. (17)

In the last equation of Equation (17), note that, for l ≠ n , we have

U ^ q l † U ^ q n U ^ p l U ^ p n = U ^ q l † U ^ p l U ^ q n U ^ p n = { U ^ p l U ^ q l † exp ( i δ d ) } { U ^ p n U ^ q n exp ( − i δ d ) } = U ^ p l U ^ q l † U ^ p n U ^ q n = U ^ p l U ^ p n U ^ q l † U ^ q n ,

U ^ q n † U ^ q l U ^ p l U ^ p n = U ^ q n † U ^ q l U ^ p n U ^ p l = U ^ p n U ^ p l U ^ q n † U ^ q l = U ^ p l U ^ p n U ^ q n † U ^ q l . (18)

Operating Q ^ D to the state | n 0 , n 1 , ⋯ , n N s − 1 〉 p given by the definition (14) we obtain

Q ^ D | n 0 , n 1 , ⋯ , n N s − 1 〉 p = ∏ j = 0 N s − 1 U ^ p j | n 0 , n 1 , ⋯ , n N s − 1 〉 p = | n 0 , n 1 , ⋯ , n N s − 1 〉 p ∏ j = 0 N s − 1 exp ( i δ d n j ) = | n 0 , n 1 , ⋯ , n N s − 1 〉 p exp ( i δ d ∑ j = 0 N s − 1 n j ) . (19)

As described in the previous section, the eigenvalue of U ^ q j is discrete. When L d is quite large, however, the continuous eigenvalue of U ^ q j would be realized so that the physical quantities of our model become good approximations to those of H ^ L a t t i c e . Consider the case δ d = 2 π / L d ≪ 1 . If we introduce a notation θ = δ d n = 2 π n / L d , we can replace | n 〉 q j by | θ 〉 q j . Then we have

U ^ q j | θ 〉 q j = | θ 〉 q j exp ( − i θ ) . (20)

Let us operate ( U ^ p j + U ^ p j † − 2 ) to a state of | Ψ 〉 q j ≡ ∑ θ | θ 〉 q j ψ ( θ ) at a site j of the lattice. Then

( U ^ p j + U ^ p j † − 2 ) | Ψ 〉 q j = ∑ θ ( U ^ p j + U ^ p j † − 2 ) | θ 〉 q j ψ ( θ ) = ∑ θ ( | θ + δ d 〉 q j + | θ − δ d 〉 q j − 2 | θ 〉 q j ) ψ ( θ ) = ∑ θ | θ 〉 q j { ψ ( θ − δ d ) + ψ ( θ + δ d ) − 2 ψ ( θ ) } ∼ ∑ θ | θ 〉 q j δ d 2 { d 2 ψ ( θ ) d θ 2 } . (21)

We apply this discussion to the whole state | θ 0 , θ 1 , ⋯ , θ N s − 1 〉 q (13). The state | Φ 〉 q is given by

| Φ 〉 q ≡ ∑ { θ j } | θ 0 , θ 1 , ⋯ , θ N s − 1 〉 q ϕ q ( θ 0 , θ 1 , ⋯ , θ N s − 1 ) . (22)

Hereafter we abbreviate ϕ q ( θ 0 , θ 1 , ⋯ , θ N s − 1 ) as ϕ q ( { θ j } ) . Operating H ^ A in Equation (15) to this state we obtain

H ^ A | Φ 〉 q ∼ − A ∑ { θ j } | θ 0 , θ 2 , ⋯ , θ N s − 1 〉 q ∑ l = 0 N s − 1 δ d 2 ∂ 2 ϕ q ( { θ j } ) ∂ θ l 2 . (23)

As for H ^ B note that, for one nearest neighbor pair ( l , n ) ,

( U ^ q l † U ^ q n + U ^ q n † U ^ q l ) | θ 0 , θ 2 , ⋯ , θ N s − 1 〉 q = { exp ( i θ l ) exp ( − i θ n ) + exp ( i θ n ) exp ( − i θ l ) } | θ 0 , θ 2 , ⋯ , θ N s − 1 〉 q = 2 cos ( θ l − θ n ) | θ 0 , θ 2 , ⋯ , θ N s − 1 〉 q . (24)

Therefore

H ^ B | Φ 〉 q = ∑ { θ j } | θ 0 , θ 2 , ⋯ , θ N s − 1 〉 q ϕ q ( { θ j } ) { − 2 B ∑ ( l , n ) cos ( θ l − θ n ) } . (25)

With Equations (23) and (25) we obtain

H ^ D | Φ 〉 q ∼ ∑ { θ j } | θ 0 , θ 2 , ⋯ , θ N s − 1 〉 q × { − A δ d 2 ∑ l = 0 N s − 1 ∂ 2 ϕ q ( { θ j } ) ∂ θ l 2 − 2 B ϕ q ( { θ j } ) ∑ ( l , n ) cos ( θ l − θ n ) } . (26)

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

Here we use new operators V ^ q l and V ^ p l instead of U ^ q j and U ^ p j . We will show that we can express the increment operator Q ^ D (16) by the single operator V ^ p 0 . In addition, we express H ^ B in Equation (15) by V ^ q l ( l ≥ 1 ). As for H ^ A in Equation (15) we present an expression where Q ^ D is included in an explicit form.

First we consider an operator m ^ j which is defined by U ^ p j as

exp ( i δ d m ^ j ) ≡ U ^ p j . (27)

Note that m ^ j is hermitian. For a set of operators { m ^ j } ( j = 0,1, ⋯ , N s − 1 ) we can introduce a set of new operators { ζ ^ l } .

ζ ^ l ≡ ∑ j = 0 N s − 1 a l j m ^ j , m ^ j = ∑ l = 0 N s − 1 b j l ζ ^ l , ∑ l = 0 N s − 1 b j l a l k = δ j k , ∑ j = 0 N s − 1 a l j b j n = δ l n . (28)

Let us assume that the matrix [ a l j ] is orthonormal, i.e. b j l = a l j and they are real. Therefore ∑ j a l j a n j = δ l n and ∑ l a l j a l k = δ j k . Also we assume a 0 j = 1 / N s for any j, which means that b j 0 = a 0 j = 1 / N s and ∑ j a 0 j b j 0 = ∑ j ( 1 / N s ) = 1 . Using a l j and b j l we define unitary operators V ^ p l and V ^ q l ,

V ^ q l ≡ ∏ j = 0 N s − 1 ( U ^ q j ) b j l , V ^ p l ≡ ∏ j = 0 N s − 1 ( U ^ p j ) a l j . (29)

Then we express U ^ q j and U ^ p j by V ^ q l and V ^ p l .

∏ l = 0 N s − 1 ( V ^ q l ) a l j = ∏ l = 0 N s − 1 { ∏ k = 0 N s − 1 ( U ^ q k ) b k l } a l j = ∏ k = 0 N s − 1 ( U ^ q k ) ∑ l = 0 N s − 1 b k l a l j = ∏ k = 0 N s − 1 ( U ^ q k ) δ k j = U ^ q j ,

∏ l = 0 N s − 1 ( V ^ p l ) b j l = ∏ l = 0 N s − 1 { ∏ k = 0 N s − 1 ( U ^ p k ) a l k } b j l = ∏ k = 0 N s − 1 ( U ^ p k ) ∑ l = 0 N s − 1 a l k b j l = ∏ k = 0 N s − 1 ( U ^ p k ) δ k j = U ^ p j . (30)

Next we show that a set of operators { V ^ p l , V ^ q l } has the same Weyl relations as those of { U ^ p j , U ^ q j } given by Equation (9). Namely,

V ^ p l V ^ q l = V ^ q l V ^ p l exp ( i δ d ) ,

[ V ^ q l , V ^ q n ] = 0 , [ V ^ p l , V ^ p n ] = 0 , [ V ^ p l , V ^ q n ] = 0 , for l ≠ n . (31)

The first Weyl relation is verified by notifying

V ^ p l V ^ q n = ∏ j = 0 N s − 1 ( U ^ p j ) a l j ∏ k = 0 N s − 1 ( U ^ q k ) b k n = ∏ j = 0 N s − 1 ( U ^ p j ) a l j ( U ^ q j ) b j n = ∏ j = 0 N s − 1 ( U ^ q j ) b j n ( U ^ p j ) a l j exp ( i δ d a l j b j n ) = V ^ q n V ^ p l exp ( i δ d ∑ j a l j b j n ) = V ^ q n V ^ p l exp ( i δ d δ l n ) . (32)

The rest of relations are trivial from Equations (9) and (29). By this proof we confirm that the set of { V ^ p l , V ^ q l } is independent and complete.

Then we will express H ^ B in Equation (15) by V ^ q l . For this purpose, we calculate U ^ q l U ^ q n † , noting U ^ q n † = U ^ q n − 1 and a 0 l = a 0 n = 1 / N s ,

U ^ q l U ^ q n † = ∏ j = 0 N s − 1 ( V ^ q j ) a j l ∏ k = 0 N s − 1 ( V ^ q k ) − a k n = ∏ j = 0 N s − 1 ( V ^ q j ) a j l − a j n = ∏ j = 1 N s − 1 ( V ^ q j ) a j l − a j n . (33)

It should be noted that U ^ q l U ^ q n † contains no V ^ q 0 . Therefore we can express H ^ B by ( N s − 1 ) operators V ^ q j ( j ≥ 1 ).

H ^ B = − B ∑ ( l , n ) { ∏ j = 1 N s − 1 ( V ^ q j ) a j l − a j n + ∏ j = 1 N s − 1 ( V ^ q j ) a j n − a j l } . (34)

Next we express H ^ A in Equation (15) by means of new operators ζ ^ l defined in Equation (28). It should be noted that, from Equations (27) and (28), we obtain

ζ ^ 0 = n ^ Q N s , Q ^ D = ∏ j = 0 N s − 1 U ^ p j = exp ( i δ d n ^ Q ) , n ^ Q ≡ ∑ j = 0 N s − 1 m ^ j . (35)

Then we can express H ^ A in Equation (15) using n ^ Q and ζ ^ l ( l ≥ 1 ) because

U ^ p l = exp ( i δ d m ^ l ) = exp { i δ d ( ∑ n = 1 N s − 1 b l n ζ ^ n + ζ ^ 0 N s ) } = exp { i δ d ( ∑ n = 1 N s − 1 b l n ζ ^ n + n ^ Q N s ) }

U ^ p l † = exp { − i δ d ( ∑ n = 1 N s − 1 b l n ζ ^ n + n ^ Q N s ) } . (36)

The result is, with η ^ l ≡ ∑ n = 1 N s − 1 b l n ζ ^ n ,

H ^ A = − A ∑ l = 0 N s − 1 { exp [ i δ d ( η ^ l + n ^ Q N s ) ] + exp [ − i δ d ( η ^ l + n ^ Q N s ) ] − 2 } = − 2 A ∑ l = 0 N s − 1 { cos [ δ d ( η ^ l + n ^ Q N s ) ] − 1 } . (37)

Now consider the case δ d ≪ 1 , where we expand H ^ A by δ d .

H ^ A = ∑ k = 0 ∞ h ^ A ( k ) δ d k = − 2 A ∑ l = 0 N s − 1 { ∑ k = 0 ∞ [ δ d ( η ^ l + n ^ Q N s ) ] 2 k ( − 1 ) k ( 2 k ) ! − 1 } = − 2 A { ∑ k = 1 ∞ δ d 2 k ( − 1 ) k ( 2 k ) ! ∑ l = 0 N s − 1 ( η ^ l + n ^ Q N s ) 2 k } . (38)

It should be noted that h ^ A ( 0 ) = 0 due to the definition of H ^ A and h ^ A ( 2 k + 1 ) = 0 because H ^ A is hermitian. In the second order of δ d we obtain H ^ A ( 2 ) ,

H ^ A ( 2 ) ≡ h ^ A ( 2 ) δ d 2 = − 2 A δ d 2 ( − 1 ) 2 ! ∑ l = 0 N s − 1 ( η ^ l + n ^ Q N s ) 2 = A δ d 2 ∑ l = 0 N s − 1 { η ^ l 2 + 2 η ^ l n ^ Q N s + ( n ^ Q N s ) 2 } = A δ d 2 ( ∑ l = 0 N s − 1 η ^ l 2 + n ^ Q 2 N s ) . (39)

Here note that

∑ l = 0 N s − 1 η ^ l = ∑ l = 0 N s − 1 ∑ n = 1 N s − 1 b l n ζ ^ n = ∑ n = 1 N s − 1 ( ∑ l = 0 N s − 1 b l n ) ζ ^ n = N s ∑ n = 1 N s − 1 ( ∑ l = 0 N s − 1 b l n 1 N s ) ζ ^ n = N s ∑ n = 1 N s − 1 ( ∑ l = 0 N s − 1 b l n b l 0 ) ζ ^ n = N s ∑ n = 1 N s − 1 δ 0 n ζ ^ n = 0. (40)

For later use we also calculate H ^ A ( 4 ) ≡ h ^ A ( 2 ) δ d 2 + h ^ A ( 4 ) δ d 4 . We obtain

H ^ A ( 4 ) = H ^ A ( 2 ) − 2 A δ d 4 1 4 ! ∑ l = 0 N s − 1 ( η ^ l + n ^ Q N s ) 4 = H ^ A ( 2 ) − A 12 δ d 4 ∑ l = 0 N s − 1 { η ^ l 4 + 4 η ^ l 3 n ^ Q N s + 6 η ^ l 2 ( n ^ Q N s ) 2 + 4 η ^ l ( n ^ Q N s ) 3 + ( n ^ Q N s ) 4 } = H ^ A ( 2 ) − A δ d 4 ∑ l = 0 N s − 1 { 1 12 ∑ l = 0 N s − 1 η ^ l 4 + n ^ Q 3 N s ∑ l = 0 N s − 1 η ^ l 3 + n ^ Q 2 2 N s 2 ∑ l = 0 N s − 1 η ^ l 2 + n ^ Q 4 12 N s 3 } . (41)

In Equation (41) we dropped the term of ∑ l η ^ l because of Equation (40).

In order of δ d 2 the Hamiltonian H ^ D ( 2 ) is given by

H ^ D ( 2 ) ≡ H ^ A ( 2 ) + H ^ B = H ^ D 0 ( 2 ) + A δ d 2 n ^ Q 2 N s ,

H ^ D 0 ( 2 ) ≡ A δ d 2 ∑ l = 0 N s − 1 η ^ l 2 − B ∑ ( l , n ) { ∏ j = 1 N s − 1 ( V ^ q j ) a j l − a j n + ∏ j = 1 N s − 1 ( V ^ q j ) a j n − a j l } . (42)

When we calculate the energy in p-representation, where the basic state is | m 0 , m 1 , m 2 , ⋯ , m N s − 1 〉 p (14), the eigenvalue of the operator n ^ Q is n Q = ∑ j m j . Therefore the lowest energy with a fixed value of n Q is given by

E n Q ( 2 ) = E 0 ( 2 ) + A δ d 2 n Q 2 N s . (43)

Here E 0 ( 2 ) is the lowest energy from H ^ D 0 ( 2 ) . We conclude that the energy gap Δ n Q t h ,2 with a fixed value of n Q is given by

Δ n Q t h ,2 ≡ E n Q ( 2 ) − E 0 ( 2 ) = A δ d 2 n Q 2 N s . (44)

Then let us discuss energy up to δ d 4 . The effective Hamiltonian H ^ D ( 4 ) is given by

H ^ D ( 4 ) ≡ H ^ A ( 4 ) + H ^ B = H ^ D 0 ( 2 ) + A δ d 2 n ^ Q 2 N s − A δ d 4 { e ^ 0 + e ^ 1 n ^ Q + e ^ 2 n ^ Q 2 + e ^ 4 n ^ Q 4 } ,

e ^ 0 ≡ 1 12 ∑ l = 0 N s − 1 η ^ l 4 , e ^ 1 ≡ 1 3 N s ∑ l = 0 N s − 1 η ^ l 3 , e ^ 2 ≡ 1 2 N s 2 ∑ l = 0 N s − 1 η ^ l 2 , e ^ 4 ≡ 1 12 N s 3 . (45)

As for terms with e ^ k we estimate their contributions by the first-order perturbation theory. When the eigenstate of H ^ D 0 ( 2 ) with the eigen energy E 0 ( 2 ) is | E 0 ( 2 ) 〉 , the energy is given by

E n Q ( 4 ) ≡ E n Q ( 2 ) + 〈 E 0 ( 2 ) | { H ^ D ( 1 ) − H ^ D ( 2 ) } | E 0 ( 2 ) 〉 = ( E 0 ( 2 ) + A δ d 2 n Q 2 N s ) − A δ d 4 ( N s 12 η ¯ 4 + n Q 3 η ¯ 3 + n Q 2 2 N s η ¯ 2 + n Q 4 12 N s 3 ) ,

η ¯ k ≡ 1 N s 〈 E 0 ( 2 ) | ∑ l = 0 N s − 1 η ^ l k | E 0 ( 2 ) 〉 . (46)

Here we see that η ¯ 3 vanishes. The reason is that H ^ D is invariant under the exchange of U ^ q l and U ^ q l † as well as the exchange of U ^ p l and U ^ p l † , which means to replace η ^ l by − η ^ l . In conclusion, the energy gap up to δ d 4 is given by

Δ n Q t h , 4 ≡ E n Q ( 4 ) − E 0 ( 4 ) = A δ d 2 a s q t h , 2 n Q 2 N s − A δ d 4 n Q 4 12 N s 3 ,

E 0 ( 4 ) = E 0 ( 2 ) − A δ d 4 N s 12 η ¯ 4 , a s q t h , 2 ≡ 1 − 1 2 δ d 2 η ¯ 2 . (47)

Note that the value of a s q t h ,2 depends on not only H ^ A but also H ^ B .

Now we present our numerical results of the lowest energy from the Hamiltonian H ^ D (15) with a fixed value of n Q , which we denote E n Q ( L d , N s ) , for several values of L d on the square lattice whose size is N s . In order to obtain the energy for each value of n Q we employ the basis states in p-representation, which consist of | m 0 , m 1 , ⋯ , m N s − 1 〉 p (14) with ∑ l m l = n Q . We examine the numerically obtained energy gap defined by

Δ n Q ( L d , N s ) ≡ E n Q ( L d , N s ) − E 0 ( L d , N s ) . (48)

Throughout this section we fix A in H ^ D to be δ d − 2 = ( 2 π / L d ) − 2 , so that A δ d 2 = 1 , and B = 1 . Note that the energy gap scarcely depends on B. Our study is carried out on N s = 5 , 9, 16, 36 and 64 lattices with L d = 32 , 36 or 64. The fixed value ranges from n Q = 0 to n Q = L d / 2 . Note that the result for L d − n Q , which means the result for − n Q , is the same as the result for n Q because of the periodicity of the operators.

We will show that Δ n Q ( L d , N s ) is well described by Δ n Q t h ,2 (44) or Δ n Q t h ,4 (47) which we discussed in the previous section. For this purpose, we introduce three ratios,

D 1 n Q ( L d , N s ) ≡ { Δ n Q ( L d , N s ) − n Q 2 N s } / Δ n Q ( L d , N s ) ,

D 2 n Q ( L d , N s ) ≡ { Δ n Q ( L d , N s ) − a s q t h , 2 n Q 2 N s } / Δ n Q ( L d , N s ) ,

D 3 n Q ( L d , N s ) ≡ { Δ n Q ( L d , N s ) − Δ n Q t h , 4 } / Δ n Q ( L d , N s ) . (49)

Here D 1 n Q ( L d , N s ) is useful to compare numerical results with Δ n Q t h ,2 , while D 2 n Q ( L d , N s ) is for the comparison with the first term of Δ n Q t h ,4 . Since η ¯ 2 = 〈 E 0 ( 2 ) | ∑ l η ^ l 2 | E 0 ( 2 ) 〉 / N s in a s q t h ,2 of the expression (47) is beyond analytical arguments, we numerically estimate this expectation value by 〈 E 0 | ∑ l η ^ l 2 | E 0 〉 / N s , where | E 0 〉 is the state of the lowest energy with n Q = 0 .

In this subsection, we present the numerical results of Δ n Q ( L d = 32 , N s = 5 ) and Δ n Q ( L d = 32 , N s = 9 ) .

The edge vectors for the N s = 5 lattice are ( 2,1 ) and ( − 1,2 ) . On this lattice we calculate the lowest energy with n Q = 0 to 16 by means of the diagonalization so that we can obtain precise results to start with. The number of the states we should consider amounts to 32 5 = 2 25 ∼ 3.4 × 10 7 . For the N s = 9 lattice, which is already too large to apply the diagonalization, we employ stochastic state selection method [

For larger lattices with N s = 16 , 36 , 64 we estimate the energy gaps by means of quantum Monte Carlo methods [

What we want to examine is whether our results agree with Δ n Q t h ,4 within the statistical error, which we will denote ε N s hereafter. It should be noted that ε N s is scarcely dependent on values of n Q . Then we need to estimate values of n Q for which we can see the effect of the correction terms in Δ n Q t h ,4 . From discussions in the previous section we see that, with A δ d 2 = 1 ,

Δ n Q t h , 4 − Δ n Q t h , 2 = − δ d 2 ( c 1 η ¯ 2 + c 2 ) , c 1 ≡ n Q 2 2 N s , c 2 ≡ n Q 4 12 N s 3 . (50)

Note that the first correction term in Equation (50), which enables us to distinguish D 2 n Q ( L d , N s ) from D 1 n Q ( L d , N s ) , is observable in the range of n Q to satisfy δ d 2 c 1 η ¯ 2 ≥ ε N s . Similarly we should search for values of n Q which satisfy the condition δ d 2 c 2 ≥ ε N s in order to find difference between D 3 n Q ( L d , N s ) and D 2 n Q ( L d , N s ) .

Now let us first present our results on the N s = 16 lattice.

a s q M C , 2 ≡ 1 − δ d 2 2 N s 〈 E 0 | ∑ l η ^ l 2 | E 0 〉 . (51)

β | l t | Δ n Q = 16 |
---|---|---|

3.8 | 164 | 15.70 ± 0.02 |

3.8 | 140 | 15.70 ± 0.02 |

3.8 | 120 | 15.70 ± 0.02 |

3.5 | 152 | 15.69 ± 0.02 |

3.5 | 132 | 15.70 ± 0.02 |

3.5 | 112 | 15.70 ± 0.02 |

Next we present results of D k n Q ( L d = 64 , N s = 36 ) ( k = 1 , 2 ) in

Finally

Conclusively speaking, therefore, numerical results presented in this section on N s = 5 , 9 , 16 , 32 and 64 lattices strongly support our discussions on the energy gap in section 3.

L d | N s | max ( δ d 2 c 1 ) | max ( δ d 2 c 2 ) | ε N s | 〈 E 0 | ∑ l η ^ l 2 | E 0 〉 / N s | a s q M C ,2 |
---|---|---|---|---|---|---|

32 | 16 | 0.31 | 0.05 | 0.02 | 0.818 | 0.98423 |

64 | 16 | 0.31 | 0.21 | 0.02 | 0.814 | 0.99606 |

64 | 36 | 0.14 | 0.03 | 0.03 | 0.820 | 0.99605 |

64 | 64 | 0.08 | 0.003 | 0.03 | 0.820 | 0.99605 |

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 [

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 relation [

A few comments are in order now.

First let us comment on the parameter B in our Hamiltonian H ^ D (15), which we chose to be 1 in section 4. Although the estimated value of η ¯ 2 increases as B becomes large, the second term δ d 2 η ¯ 2 / 2 in a s q t h ,2 in the expression (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 additional calculations on the N s = 5 lattice with L d = 32 , increasing the value of B up to 50. The result is 〈 E 0 | ∑ l η ^ l 2 | E 0 〉 / N s = 0.79 (6.2) when B = 1 (50). Then the difference D 3 n Q = 1 ( L d , N s ) in Equation (49) becomes ~3 × 10^{−3} for B = 50 , which should be compared with the value ~4 × 10^{−4} for B = 1 . Summarizing the results for B = 50 , we confirmed that D 3 n Q ( L d , N s ) ≤ 0.015 for all values of N Q .

The next comment is on the violation of the cluster property in the nonlinear sigma model with U(1) symmetry. Based on discussions in the previous work [

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 n Q . 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.

I wish to thank Dr. Yasuko Munehisa who made the stimulating suggestion on the operator rearrangement in Section 3, and did valuable comments on the manuscript through her critical review.

The author declares no conflicts of interest regarding the publication of this paper.

Munehisa, T. (2021) Degenerate States in Nonlinear Sigma Model with U(1) Symmetry. World Journal of Condensed Matter Physics, 11, 29-52. https://doi.org/10.4236/wjcmp.2021.113003

Here we present a brief description for the Weyl relation. In quantum mechanics for one particle the commutation relation of the hermite operator of the position q ^ and the operator of the momentum p ^ is a starting point.

[ q ^ , p ^ ] = i , q ^ | q 〉 = | q 〉 q , p ^ | p 〉 = | p 〉 p , 〈 p | q 〉 = exp ( − i p q ) . (52)

In the Weyl relation, we introduce unitary operators defined by

U ^ q ( t ) ≡ exp ( i q ^ t ) , U ^ p ( s ) ≡ exp ( i p ^ s ) . (53)

Here s and t are real numbers. In this representation, we define the Weyl relation by

U ^ q ( t ) U ^ p ( s ) = U ^ p ( s ) U ^ q ( t ) exp ( − i s t ) , U ^ p ( s ) U ^ q ( t ) = U ^ q ( t ) U ^ p ( s ) exp ( i s t ) . (54)

Using these operators we have

U ^ q ( t ) | q 〉 = | q 〉 exp ( i t q ) , U ^ p ( s ) | p 〉 = | p 〉 exp ( i s p ) ,

U ^ p ( s ) | q 〉 = | q − s 〉 , U ^ q ( t ) | p 〉 = | p + t 〉 . (55)

The third equation is led by

U ^ p ( s ) | q 〉 = ∫ d p 1 2 π U ^ p ( s ) | p 〉 〈 p | q 〉 = ∫ d p 1 2 π | p 〉 exp ( i s p ) exp ( − i p q ) = | q − s 〉 . (56)

A2. Wely Relation for Discrete VariableIn our work, we introduce an unitary operator U ^ q with a discrete value n where n = 0,1, ⋯ , L d − 1 for a finite integer L d . We then introduce another unitary operator U ^ p which satisfies the following Weyl relation.

U ^ p U ^ q = U ^ q U ^ p exp ( i δ d ) , δ d ≡ 2 π / L d . (57)

Note that

U ^ q ( U ^ p ) n = U ^ q U ^ p ( U ^ p ) n − 1 = U ^ p U ^ q ( U ^ p ) n − 1 exp ( − i δ d ) = ⋯ = ( U ^ p ) m U ^ q ( U ^ p ) n − m exp ( − i m δ d ) = ⋯ = ( U ^ p ) n U ^ q exp ( − i n δ d ) . (58)

We suppose that U ^ q has an eigenstate | n 0 〉 q whose eigenvalue is λ 0 = exp ( i γ ) with a real number γ .

U ^ q | n 0 〉 q = | n 0 〉 q λ 0 . (59)

Here we can make λ 0 = 1 by using U ^ q exp ( − i γ ) instead of U ^ q . Then, with this re-defined U ^ q , we obtain

U ^ q | 0 〉 q = | 0 〉 q ,

U ^ q ( U ^ p ) n | 0 〉 q = ( U ^ p ) n U ^ q | 0 〉 q exp ( − i n δ d ) = ( U ^ p ) n | 0 〉 q exp ( − i n δ d ) . (60)

The state | n 〉 q ≡ ( U ^ p ) n | 0 〉 q is therefore the eigenstate of the unitary operator U ^ q with the eigenvalue λ n ≡ exp ( − i n δ d ) .

Let us consider the state | L d 〉 q ≡ U ^ p | L d − 1 〉 q . Then

U ^ q | L d 〉 q = U ^ q U ^ p | L d − 1 〉 q = U ^ p U ^ q exp ( − i δ d ) | L d − 1 〉 q = U ^ p | L d − 1 〉 q exp ( − i δ d ) exp { − i ( L d − 1 ) δ d } = | L d 〉 q ⋅ 1. (61)

Therefore | L d 〉 q is the eigenstate of U ^ q with the eigenvalue 1 so that | L d 〉 q = | 0 〉 q e i β holds for a real number β . Re-defining U ^ p e − i β / L d as U ^ p we obtain

| L d 〉 q = | 0 〉 q . (62)

Let us next make a new state | 0 〉 p defined by

| 0 〉 p ≡ 1 L d ∑ k = 0 L d − 1 ( U ^ p ) k | 0 〉 q = 1 L d ∑ k = 0 L d − 1 | k 〉 q . (63)

This state is the eigenstate of U ^ p with the eigenvalue 1, because

U ^ p | 0 〉 p = 1 L d ∑ k = 0 L d − 1 ( U ^ p ) k + 1 | 0 〉 q = 1 L d ∑ k = 1 L d ( U ^ p ) k | 0 〉 q = 1 L d { ∑ k = 1 L d − 1 ( U ^ p ) k + ( U ^ p ) L d } | 0 〉 q = 1 L d { ∑ k = 1 L d − 1 ( U ^ p ) k + I ^ } | 0 〉 q = | 0 〉 p . (64)

We then see that | m 〉 p ≡ ( U ^ q ) m | 0 〉 p is the eigenstate of U ^ p with the eigenvalue exp ( i m δ d ) , since

U ^ p | m 〉 p = U ^ p ( U ^ q ) m | 0 〉 p = ( U ^ q ) m U ^ p | 0 〉 p exp ( i m δ d ) = ( U ^ q ) m | 0 〉 p exp ( i m δ d ) = | m 〉 p exp ( i m δ d ) . (65)

Finally we calculate the inner product 〈 n | m 〉 q p . Note that

U ^ p † U ^ q = { U ^ p † U ^ q } { U ^ p U ^ p † } = U ^ p † { U ^ q U ^ p } U ^ p † = U ^ p † { U ^ p U ^ q exp ( − i δ d ) } U ^ p † = { U ^ p † U ^ p } { U ^ q U ^ p † } exp ( − i δ d ) = U ^ q U ^ p † exp ( − i δ d ) . (66)

Using (64) and (66) we obtain

〈 n | m 〉 q p = 〈 0 | q ( U ^ p † ) n ( U ^ q ) m | 0 〉 p = 〈 0 | q ( U ^ q ) m ( U ^ p † ) n | 0 〉 p exp ( − i n m δ d ) = 〈 0 | 0 〉 q p exp ( i n m δ d ) = 〈 0 | q 1 L d ( ∑ k = 0 L d − 1 | k 〉 q ) exp ( − i n m δ d ) = 〈 0 | q 1 L d | 0 〉 q exp ( − i n m δ d ) = 1 L d exp ( − i n m δ d ) . (67)