Honeycomb lattice Kitaev model with Wen-Toric-code interactions, and anyon excitations

The honeycomb lattice Kitaev model H_{K} with two kinds of Wen-Toric-code four-body interactions H_{WT} is investigated exactly using a new fermionization method, and the ground state phase diagram is obtained. Six kinds of three-body interactions are also considered. A Hamiltonian equivalent to the honeycomb lattice Kitaev model is also introduced. The fermionization method is generalized to two-dimensional systems, and the two-dimensional Jordan-Wigner transformation is obtained as a special case of this formula. The model H_{K}+H_{WT} is symmetric in four-dimensional space of coupling constants, and the anyon type excitations appear in each phase.


Introduction
Recently, a new fermionization formula was introduced [1], in which solvable Hamiltonians and the transformations to diagonalize them can be obtained simultaneously. The one-dimensional transverse Ising model, XY model, cluster model, the two-dimensional square lattice Ising model, and an infinite number of unsolved models were diagonalized by this formula. The Jordan-Wigner transformation is obtained as a special case of this tratment. [1] [2] The formula is summarized as follows: Let us consider a series of operators {η j } (j = 1, 2, . . . , M ). The operators η j and η k are called 'adjacent' when (j, k) = (j, j + 1) (1 ≤ j ≤ M − 1), or (j, k) = (M, 1). If the operators η j satisfy the relations then we can introduce a solvable Hamiltonian which can be mapped to the free fermion system by the transformation where η 0 is an initial operator satisfying η 2 0 = −1, η 0 η 1 = −η 1 η 0 , and η 0 η k = η k η 0 (2 ≤ k ≤ M ). The operators ϕ j satisfy (−2i)ϕ j ϕ j+1 = η j+1 , and Hence the Hamiltonian (2) is expressed as a sum of two-body products of the fermion operators ϕ j , and can be diagonalized. The transformation (3) is automatically generated from the series of operators {η j }, and only the algebraic relations (1), together with the translational invariance, are needed to obtain the free energy. This procedure can be applied to any systems written by the operators that satisfy (1).
The one-dimensional XY model, and its generalizations [4] can be solved by this formula. In these cases, the transformation (3) results in the Jordan-Wigner transformation.
The one-dimensional cluster model with the next-nearest-neighbor interaction cannot be diagonalized by the Jordan-Wigner transformation. This model, however, can be decoupled into H = H even + H odd , where j =even in H even , and j =odd in H odd , respectively. They satisfy [H even , H odd ] = 0, and H even , for example, is obtained from a series of operators which satisfy (1). In this case, the transformation (3) becomes which is apparently different from the Jordan-Wigner transformation, and the Hamiltonian (5) is diagonalized through this formula. [2] [3] In this paper, this formula is applied to two-dimensional systems. The transformation (3) is generally formulated for the square lattice. The Hamiltonian is transformed to the fermion system, and the ground state of H K + H W T is exactly specified; here H K denotes the Hamiltonian of the honeycomb lattice Kitaev model, H 3 consists of the six kinds of three-body interactions, H W T denotes the Hamiltonian of the Wen model which is equivalent to the Kitaev toric-code model. A Hamiltonian, which consists of the cluster-type chains coupled by the Ising interactions, is obtained as an system equivalent to H K . The ground-state phase diagram of H K + H W T is obtained exactly, and it is depicted that the phase structure of gapped phases and gapless phases change with the rates of the interactions. The symmetry of the system is investigated, and it is derived that the system is symmetric in four-dimensional space of coupling constants. The anyon excitations exist in each phase. In section 2, the honeycomb lattice Kitaev model and the three-body interactions are introduced. The Wen model is also introduced and the relation with the Kitaev toric-code model is considered. In section 3, the transformation (3) is generally formulated for the two-dimensional square lattice. A specific series of operators is then introduced to obtain and diagonalize the Hamiltonian (8). The transformation (3) in this case is found to be the two-dimensional Jordan-Wigner transformation. In section 4, the interactions are expressed by Majorana fermion operators. Operators that commute with the Hamiltonian are also introduced. In section 5, the series of operators is rearranged, and in section 6, a Hamiltonian equivalent to the honeycomb lattice Kitaev model is introduced. In section 7, H K + H W T is diagonalized in a subspace containing one of the ground states. In section 8, the gapless condition is derived, and in section 9, the ground state phase diagram is obtained. Symmetries of the model is investigated and it is pointed out that the anyon excitations appear in each phase.

Hamiltonian
Let us consider the brick-wall lattice, shown in Fig.1 (see also Fig.5), with the interactions where M 1 is even and the summation is taken over all odd j. Hamiltonian (9) is the Kitaev model on the honeycomb lattice shown in Fig.2. The Kitaev model is introduced in [5], in which the ground state is specified, the phase diagram is obtained, and abelian anyon excitations in gapped phases, and non-abelian anyon excitations in gapless phases are found.
We will also introduce six kinds of three-body interactions shown in Fig.3 as These three-body interactions are already investigated by several authors. Lee et al. [6] and Shi et al. [7] introduced the interactions K 1 and K 2 , and Yu and Wang [8] and Yu [9] introduced from K 3 to K 6 . Yu [9] also introduced various kinds of four-body and six-body interactions.
Let us here consider the Wen model [10]. The Hamiltonian is given by and the interactions are shown in Fig.3. Wen originally introduced the case L 1 = L 2 , and investigated the ground state quantum orders. We will also consider the Kitaev toric-code model [11], which consists of two types of interactions, as shown in Fig.4. The spin variables are located on each edge. Let us consider the spins on the vertical edges and consider a canonical transformation σ x jl → σ z jl , σ z jl → σ x jl , σ y jl → −σ y jl , and next another transformation of all spins σ x jl → σ y jl , σ y jl → σ z jl , σ z jl → σ x jl , then we find the Wen model on the square lattice rotated by π/4 from the original square lattice. Thus these two models are in this sense equivalent (see also sec.7.2 of [5]). In the case of the honeycomb lattice Kitaev model having only two-body interactions, the Hamiltonian commutes with the following operators associated to each hexagon Each W jl has the eigenvalues w jl = ±1. It is easy to demonstrate that the Hamiltonian with the three-body interactions (10) and with the Wen-Toriccode four-body interactions (11) also commute with all W jl . The eigenstates of the Hamiltonian may thus be labelled by the set of eigenvalues of W jl , and the total Hilbert space is divided into subspaces labelled by {w jl }.
It should be noted that the interactions L 1 and L 2 are not independent. When we consider the product of the four-body terms, we find the following relation

Transformation
We will generalize (3), and formulate the fermionization transformation for the two-dimensional lattice. Let us introduce operators η kl on each row l. The operators η kl with fixed l satisfy the condition (1), and η kl on different row l commute with each other. The series of operators on the first row is with an initial operator η 01 . The transformation is introduced as where j = 1, 2, . . . , M . From (13) we obtain At the end of the first row, we find The operator (−1) i M η 11 · · · η M1 commute with the Hamiltonian (2), is hermitian and has the eigenvalues ±1. The Hilbert space is divided into two subspaces corresponding to the eigenvalues +1 and −1. We assume the periodic boundary condition for η j1 , and thus introduce the boundary condition for ϕ j1 as in each eigenspace.
Next, let us consider the transformation for the second row. We will introduce the following factor that comes from the first row as Then the transformation for the second row is defined as The transformation (15) is schematically written as Note that η 02 is introduced in (16), though η 01 is not introduced in H(1). The boundary condition for the second row is obtained from (14) replacing ϕ j1 by ϕ j2 , and η j1 by η j2 . There is no boundary in the first row. Generally for the l-th row, the transformation is defined as where From (17) we obtain The boundary condition for the l-th row is obtained from (14) replacing ϕ j1 by ϕ jl , and η j1 by η jl . It is easy to convince from (17) that Let us here consider a specific series of operators together with the initial operators η 0l = iσ x 1l . The index j runs 1 ≤ j ≤ N , where N is the number of sites in a row, and in this case we have M = 2N . Then the transformations, for example with l = 2 and 1, are schematically written as One may readily verify that the anti-commutation relation {ϕ jl , ϕ km } = δ jk δ lm comes from the anti-commutation relations {σ y j2 , σ x j2 } = 0 and {σ z k1 , σ x k1 } = 0. This is the Jordan-Wigner transformation in two-dimension [12]- [14], i.e. the two-dimensional Jordan-Wigner transformation is obtained as a special case of (17).

Equivalent Hamiltonian
As an example of (26), let us consider (24), which is a special case of (26) with ρ = 1, l = 2, and k = 0. Let us consider the series of operators (18) in Table 1, and in this case we havē and generallyη The initial operators can be chosen asη 0l = iσ x 1l . Following (9) and (21), we will introduce the Hamiltonians whereφ α (j, l) are obtained from (3) and (20) replacing η jl byη jl . The interactions are shown in Fig.8. The sum H x + H y is the Hamiltonian of parallel spin chains with the cluster-type interactions, and H z is the Hamiltonian of the Ising interactions between these chains. The total Hamiltonian H K2 = H x + H y + H z is equivalent to the honeycomb lattice Kitaev model (9) (there is no difference when two Hamiltonians are written in terms of ϕ α (j, l) and ϕ α (j, l)). We can also find other equivalent Hamiltonians from the series of operators (6) in Table 1.

Diagonalization
We will derive the phase structure of the case with the interactions K x , K y , K z , and L 1 and L 2 . In this case, Lieb's theorem [15] applies and it is proved that one of the ground states is found in the subspace where Ψ jl = −i/2 for all j and l. In this subspace, the translational invariance of Ψ jl enables us to derive the ground state energy explicitly.

Gapless condition
We will consider the gapless condition that √ 4kk * in (34) becomes zero with some q 1 and q 2 . Let K x = βJ x , K y = βJ y , K z = βJ z , and L = βJ 4 . Four terms from (35) , J x e iq1/2 + J y e −iq1/2 and J z e −i(q1/2+q2) + J 4 e i(q1/2+q2) , form two ellipses on the complex plane of the variable z = x + iy. The first two terms including J x and J y form where 0 ≤ q 1 /2 < π/2 corresponds to a part of the ellipse. The latter two terms including J z and J 4 form where 0 ≤ q 1 /2 < π/2 and −π ≤ q 2 < π corresponds to the full ellipse. The condition is satisfied if (36) and (37) are simultaneously satisfied with some real (x, y). From (36) and (37) The condition (41) determines the gapless region. From (36) and (37), we find that all the boundaries of the gapless region determined by (41) are gapless. We will here consider the following two cases: Case I. J x ≥ 0, J y ≥ 0, J z ≥ 0, J x + J y + J z = 1, and J 4 ≥ 0. In this case (41) is written as and When J 4 = 0, we find the phase diagram obtained by Kitaev [5].
((1 − J 4 )/2, (1 − J 4 )/2, J 4 ), both (36) and (37) become finite intervals on the real axis, and in these two cases, the ground state is also highly degenerate. When J 4 = 1/3, we have a symmetric point J x = J y = J z = J 4 = 1/3, where above three points become identical, as shown in the third diagram in Fig10. In this case, (36) and (37) become finite intervals on the real axis.
When we consider the case with uniform Ψ ij , we can find the ground state in this subspace, and because of the translational invariance, the Hamiltonian can be diagonalized in the momentum representation. In this subspace, the Hamiltonian is expressed symmetrically as the sum in (30) and sums coming from other interactions. Let us consider the replacement of the variables and accordingly Then the range of the summation −π ≤ q 1 < π and −π ≤ q 2 < π in (30) is invariant, andc 1 andc 2 satisfy the fermion anticommutation relations. We find from h 21 and h 12 in (32) that the Hamiltonian H(J x , J y , J z , J 4 ) and H(J y , J x , J 4 , J z ) are equivalent. This is the particle-hole transformation.
In this sense, the model with the interactions J x = J y , J z = 0, J 4 = 0, and the model with J x = J y , J z = 0, J 4 = 0 are equivalent.
Kitaev [5] considered the large J z limit of the honeycomb lattice Kitaev model, and derived an effective Hamiltonian that consists of the Wen-type fourbody interaction J 4 (see (37) in [5]). In this effective Hamiltonian, vortices are generated by two kinds of string operators, and an additional sign appears from each cross point of the strings when one interchange the positions of two excitations. In this sense the excitations are regarded as anyons.
This fact is consistent with our argument that the large J z region is equivalent to the large J 4 region, and we thus also find that the abelian anyons appear in the large J 4 region as well as in the large J z region.
Let us again consider the replacement of the variables that and accordingly It is easy to check that the summation over the region −π ≤ q 1 < π and −π ≤ q 2 < π is equivalent to the summation over −π ≤ q 1 < π and −π ≤ q < π, because of the periodic structure of the system with period 2π. The operators c 1 andc 2 satisfy the fermion anticommutation relations, and we find from (32) that the Hamiltonian H(J x , J y , J z , J 4 ) and H(J z , J 4 , J x , J y ) are equivalent.
In this sense, from (43) and (44), the model with the interactions J y = J z = J 4 , J x = 0, and the model with J x = J z = J 4 , J y = 0, are equivalent to the case J x = J y = J z , J 4 = 0.
In the original Kitaev model, in the gapless phases, an external field opens an energy gap, and the string operators generate vortices that behave as anyons. For the purpose to investigate this phenomena, let us consider the Fourier transformation in whole the Hilbert space. (Note that the Fourier transformation itself is always possible even if Ψ jl are not uniform, though the Hamiltonian H cannot be simplified in the subspace where H does not have translational invariance.) It can be verified that the operators ϕ α (j, l) are transformed as ϕ α (j, l) → ϕ α (−j, −l) and ϕ α (j, l) → ϕ α (j − l/2, l) with the transformations (43) and (44), respectively. So (43) and (44) correspond to change of locations in real space. The spin operators can be expressed by ϕ α (j, l) as The Zeeman term and the string operators are, therefore, transformed together with ϕ α (j, l). In the subspace where Ψ jl are uniform, the Hamiltonian H is decomposed as (33) according to the wave numbers. The Zeeman term and the string operators do not commute with Ψ jl , thus they are not simple in this momentum bases, they change their locations, and generate anyons.

Conclusion
At last we would like to note an interesting methodology presented in [16] and [17], in which isomorphisms of algebras that are generated from interactions are considered, and equivalences and mappings are investigated. In [16], results of [5] and [14] on the honeycomb lattice Kitaev model was rederived by the algebraic isomorphism, and in [17], the Jordan-Wigner transformation is generated in an iterative way in the case of the XY chain. The basic idea in these papers that the algebraic structure of interactions determine the spectrum of the model is common to our formula. In the present paper, however, the transformation (3) is explicitly given for the series of operators that satisfy (1), and the two-dimensional systems are investigated.
In summary, we obtain the exact ground state phase diagram of the honeycomb lattice Kitaev model with the Wen-Toric-code four-body interactions, and find that the structure of the system is symmetric in four-dimensional space (J x , J y , J z , J 4 ). The fermionization transformation (3) is generally formulated for two-dimensional systems. The construction of the series of operators that satisfy (1) is also generalized, and a model equivalent to the Kitaev model is introduced. We also find that the anyon excitations appear in all of the regions shown in the phase diagram, they can be transformed each other.