Exact solution of cluster model with next-nearest-neighbor interaction

A one-dimensional cluster model with next-nearest-neighbor interactions and two additional composite interactions is solved; the free energy is obtained and a correlation function is derived exactly. The model is diagonalized by a transformation obtained automatically from its interactions, which is an algebraic generalization of the Jordan-Wigner transformation. The gapless condition is expressed as a condition on the roots of a cubic equation, and the phase diagram is obtained exactly. We also find that the distribution of roots for this algebraic equation determines the existence of long-range order, and we again obtain the ground-state phase diagram. Finally, we note that our results are universally valid for an infinite number of solvable spin chains whose interactions obey the same algebraic relations.


Introduction
Quantum spin models have been widely investigated as a basic theme in statistical physics, mathematical physics, and condensed matter physics. In onedimension, there exist many solvable systems, and there also exsit general relationships with two-dimensional classical systems. Specifically, a one-dimensional XY-model was introduced in [1]- [4] and was later found to be equivalent to the two-dimensional rectangular Ising model [5].
In the 2000s, the cluster models attracted wide attention. The ground state of this model is called the cluster state, which is a candidate of a resource for measurement-based quantum computation(MBQC) [6][7] [8]. The onedimensional cluster model was first introduced and solved by Suzuki [5] and has been investigated by numerous researchers [9]- [22].
A new method of fermionization has recently been introduced [23], and an infinite number of new solvable models have been reported in [23]- [25]. In this method, the transformation that diagonalizes the system is obtained automatically from the interactions of the model. The Hamiltonian is diagonalized using only algebraic relations of the interactions, and therefore, the equivalences of the models are immediately understood. In the case of the XY model, the transformation results in the Jordan-Wigner transformation [26] [2]; thus, this new transformation can be regarded as the algebraic generalization of the Jordan-Wigner transformation. 1 In this paper, we investigate the cluster model with next-nearest-neighbor interactions and two additional composite interactions. The Hamiltonian is where 1 j is the identity operator at site j. When K 2 = K −1 = 0, this model becomes the cluster model with next-nearest-neighbor interactions and found to be equivalent [24] to the XY chain, which can be diagonalized by the Jordan-Wigner transformation. However, the model (1) cannot be diagonalized by the Jordan-Wigner transformation when K 2 = 0 or K −1 = 0. Our formula can diagonalize this Hamiltonian. The formula in this paper is summarized as follows. Let the number of sites be N = 2M , where M is even, and let us assume a cyclic boundary condition σ k N +i = σ k i ( k = x, y, z ). Let us consider two series of operators {η j } and {ζ j }, which are defined as and Note that the operators η j and ζ k are the interactions found in the Hamiltonian (1), and the operators η j and ζ k commute with each other for all j and k. Next, following equation (2.9) in [23], we introduce transformations ϕ 1 (j), ϕ 2 (j), and ϕ 3 (j), ϕ 4 (j) ( 1 j M ) as where are the initial operators. Then, the operators ϕ l (j) satisfy {ϕ 1 (j), ϕ 2 (k)} = δ jk , {ϕ 3 (j), ϕ 4 (k)} = δ jk , and [ϕ l (j), ϕ m (k)] = 0 ( l = 1, 2 , m = 3, 4) for all j and k. Therefore, the operators ϕ l (j) form two series of Majorana fermions. The Hamiltonian (1) is expressed as the sum of two-body products of ϕ l (j). Hence, the Hamiltonian (1) can be diagonalized. The transformations from η j to ϕ 1 and ϕ 2 ( ζ j to ϕ 3 and ϕ 4 ) are clearly different from the Jordan-Wigner transformation as shown in (12)- (15). From the transformations (4), we can derive the free energy and a correlation function exactly. Next, let us consider the algebraic equation where α −1 = K −1 /K 0 , α 1 = K 1 /K 0 , α 2 = K 2 /K 0 , and also consider three subsets of the complex plane We find that there is no energy gap above the ground state if and only if at least one root of equation (6) belongs to the unit circle C u . In another way, we find that long-range order exists if and only if one root of the cubic equation (6) belongs to D O , with the other two roots belonging to D I . Clearly, the boundary of the phase in terms of the long-range order satisfies the gapless condition; therefore, we obtain a consistent phase diagram from two different procedures. We also find that the phase diagram has a symmetry under a shift of indices from j to j + 1 for K j . These results are obtained using only the algebraic relations of the interactions. Hence, the results we obtained are universally valid for models consisting of the interactions that obey the same algebraic relations.
In section 2, we diagonalize the Hamiltonian (1) exactly by applying the transformation (4) and obtain the free energy. In section 3, we consider the gapless condition and obtain a corresponding phase diagram. In section 4, a correlation function is introduced, and the asymptotic limit is obtained exactly. In section 5, we demonstrate that long-range order exists if and only if the roots of equation (6) satisfy the condition described above, and again, we obtain the same phase diagram. In section 6, we consider the symmetry of the phase diagram, and illustrate that the results obtained in this paper are valid for an infinite number of Hamiltonians that satisfy our condition.

Diagonalization and the free energy
In this section, we diagonalize the Hamiltonian (1) and derive the free enegy. The operators η j and η k ( ζ j and ζ k ) are called adjacent if (j, k) = (j, j + 1) (1 ≤ j ≤ N − 1) or (j, k) = (N, 1) . Then, the operators {η j } in (2) satisfy the relations and {ζ j } satisfy the same relations replacing η j by ζ j in (8). Note that Other interactions in (1) are obtained from η j and ζ k as Then the Hamiltonian (1) is written in terms of η j and ζ k as Next, let us consider the transformations ϕ k (j) ( k = 1, 2, 3, 4 , 1 ≤ j ≤ M ) introduced in (4). They are written in terms of the Pauli operators as They are clearly different from the Jordan-Wigner transformation. The initial are not necessary to diagonalize the Hamiltonian, but introduced to avoid boundary terms in (12) - (15). The initial operators satisfy the relations From (9) and (16), we find that and ϕ l (j) (l = 1, 2) and ϕ m (k) (m = 3, 4) commute with each other. Moreover, we find Then the Hamiltonian is written as which is the sum of two-body products of the Majorana fermion operators ϕ l (j).
For the purpose to check the boundary condition, let us consider the boundary terms Then the cyclic boundary condition σ k N +i = σ k i yields, for l = 1 and 2, that Next let us introduce the Fourier transformation where and C † l (p), C l (p) (l = 1, 2) and C † m (q), C m (q) (m = 3, 4) commute with each other. From (23) and (24), we find Without loss of generality, we can assume (28). The Hamiltonian is then expressed as where and Here L(q) * is the complex conjugate of L(q) . For each q > 0, the Hamiltonian is therfore the sum of two commutative operators. We obtain which is real and non-negative. With respect to the basis |0 , and its eigenvalues are found to be 0, 0, ±2 A(q). In the same way, the eigenvalues of W 34 (q) are found to be 0, 0, ±2 A(q).
The partition function is obtained as where Λ(q) = 2 A(q) . The free energy is where we have used that ∆q = 2 N/2 π.
When K 2 = K −1 = 0, the free energy (35) becomes identical to that of the XY chain. In fact, if we introduce

Gapless condtion and the phase diagram
Let us consider the gapless condition that A(q) = |L(q)| 2 = 0 with some q. The condition L(q) = 0 is equivalent to Thus the model is gapless if and only if the algebraic equation (37) has a root that belongs to C u = {z ∈ C | |z| = 1 }. We find that the condition is equivalent to (X1) or (X2) or (X3), where A derivation is given in Appndix A. These (X1) -(X3) form the boundary of each phase in the diagram shown in Figure 1 and 2. Figure 1: The phase diagram on α −1 -α 1 plane for each α 2 ≥ 0. Two thin lines denote α 1 = −α −1 ± (α 2 − 1), and a thick curve denotes Regions characterized by I(∞) > 0 are colored by red. Figure 2: The phase diagram for each α 2 < 0. The blue segment corresponds to (X3).

Correlation function and the ground state phase transition
Next let us investigate a ground-state correlation function and its asymptotic behavior. The correlation function we consider is where 0 is the ground-state expectation. In the thermodynamic limit, we obtain the exact expression where A derivation of (40) and (41) is given in Appendix B. The determinant (40) is a Toeplitz determinant, i.e. (j, k) elements depend only on k − j = r . We can therefore apply the Szegö's theorem to obtain the asymptotic limit n → ∞ of the determinant. Let C(r) = 2 i D(r + 1) , then (40) is expressed as C(0) C(1) · · · C(n − 1) C(−1) C(0) · · · C(n − 2) . . . . . . . . . . . .
Let us introduce f (p) by the relation e −ipr f (p)dp.
With the use of the equality where t is a real solution of the equation α −1 − t + α 1 t 2 + α 2 t 3 = 0, and the factor A is independent of p, we obtain Then we consider the following four cases: (i) |a 1 | < 1 and |a 2 | < 1 and |a 3 | < 1, (−1) n−1 n (a n 1 − a n 2 − a n 3 ) (n > 0).
Then from (44) we obtain (53) case(ii) Let |b| > 1, then 1 2π π −π e −inp log(1 + be −ip )dp = (−1) n n + (56) The limit lim n→∞ I(2n) in the case (iii) is obtained from (56) with a 2 and a 3 replaced by a 3 and a 2 , respectively. In the case (iv), it is easy to derive that Next let us consider the ground state phase diagram. From (47), we obtain Therefore, −1/a 1 , −a 2 and −a 3 are the roots of the cubic equation Note that the ground state phase diagram is already obtained in [24] for K 2 = 0. It is easy to convince that the phase diagram in Figure 1 with α 2 = 0 is identical to the diagram in [24].

Symmetry and Generalizations
Let us first consider the symmetry of the phase diagram. When K 0 = 0 and α 2 = 0, the gapless condition is simplified and expressed as When K 1 = 0 and α −1 = 0, let us consider the variables β −1 = K −1 /K 1 , β 0 = K 0 /K 1 , and β 2 = K 2 /K 1 , i.e. the normalization by K 1 instead of K 0 . In this case the condition is expressed as From (20), we find the natural coupling constants of this model are K −1 , −K 0 , K 1 , K 2 . Thus when we consider the shift of indices of K j from j to j + 1, the correspondences of α j and β j are When we replace α j with corresponding β j in (60), we find that the condition (60) becomes (61). In both cases, the phase diagrams are also identical to that of the XY chain with an external field. Next we will consider a generalization of the results obtained in Section 2-5. We solve the model (1) using only the algebraic relations of the operators (8). Hence the model (11) generated from the operators which satisfy (8) can be simultaneously diagonalized, and its string order parameter (39) yields the same phase diagram as shown in Figure 1 and 2. For example, from the operators (36), the model (63) is generated from (11), its string order parameter (39) in this case is and this model results in the free energy (35) and the phase diagram shown in Figure 1 and 2. We introduce, in this paper, two series of operators {η j } and {ζ j }, which commute with each other. As a result, our model factorizes into two commutative spin chains as discussed in [24] and [27]. One series of operators, however, can yield the free energy (35) and the phase diagram shown in Figure 1 and 2. For example, the series of operators, yields the Hamiltonian This Hamiltonian is composed of the XY interactions, an external field and the cluster interactions. The string order parameter (39) in this case is and this model also results in (35) and the diagram in Figure 1 and 2. In case of both (36) and (65), the transformation (4) results in the Jordan-Wigner transformation, and in case of (2) and (3), the transformation becomes (12) - (15). Generally, an infinite number of operators that satisfy the condition (8), including (36) and (65), are given in Table 1 and 2 in [24]. Our present results are universally valid for these infinite number of solvable spin chains.

Conclusion
In this paper, we obtain the exact solution of the model (1), which is a cluster model that considers next-nearest-neighbor interactions and two additional compsite interactions. We introduce the series of operators (2) and (3), which satisfy the algebraic relation (8). Then, we introduce the transformations (4), and the model (1) is diagonalized. We derive the free energy (35), consider a correlation function (39), and derive its exact expressions (40) and (41). We obtained the ground-state phase diagram, as shown in Figure 1 and 2, from the gapless condition, and from the asymptotic behavior of the correlation function, which is classified by the location of the roots of an algebraic equation (58). The exact solution is obtained using only the algebraic relation (8) of the interactions. Finally, we note that an infinite number of solvable models, generated from operators that satisfy (8), yield exactly the same results obtained in this paper. Our transformation can be regarded as an algebraic generalization of the Jordan-Wigner transformation. This paper provides a nontrivial example that cannot be solved by the Jordan-Wigner transformation but can be solved by our method.