Exact surface energy of the $D^{(1)}_2$ spin chain with generic non-diagonal boundary reflections

The exact solution of the $D^{(1)}_2$ quantum spin chain with generic non-diagonal boundary reflections is obtained. It is found that the generating functional of conserved quantities of the system can be factorized as the product of transfer matrices of two anisotropic $XXZ$ spin chains with open boundary conditions. By using the factorization identities and the fusion technique, the eigenvalues and the Bethe ansatz equations of the model are obtained. The eigenvalues are also parameterized by the zero roots of the transfer matrix, and the patterns of root distributions are obtained. Based on them, ground states energy and the surface energies induced by the twisted boundary magnetic fields in the thermodynamic limit are obtained. These results are checked by the numerical calculations. The corresponding isotropic limit is also discussed. The results given in this paper are the foundation to study the exact physical properties of high rank $D^{(1)}_{n}$ model by using the nested processes.


Introduction
The D (1) 2 spin chain is a typical one-dimensional quantum integrable system and has many applications in the high energy, topological and mathematical physics. The corresponding exact solution is the foundation to exactly solve the high rank D (1) n spin chain by the nested methods. Each spin of the D (1) 2 chain has four components thus the integrable D (1) 2 model is characterized by the 16 × 16 R-matrix [1][2][3], which is the solution of Yang-Baxter equation. Staring from the R-matrix, the transfer matrix and conserved quantities including the Hamiltonian of the system with periodic boundary condition can be constructed by using the quantum inverse scattering method. Eigenvalues of the transfer matrix of the periodic D (1) n model are obtained by using the analytical Bethe ansatz [4,5] and then by the algebraic Bethe ansatz [6].
Later, it was found that the D  [7][8][9]. For the open boundary case, the conserved quantities are generated by the transfer matrix consisted of the R-matrices and reflection matrices [10][11][12]. If the boundary reflection matrices only have the diagonal elements, the particle numbers of each spin-component are conserved. In this case, the quasivacuum (or the reference) state is easy to be constructed, and the eigenvalues and Bethe-type eigenstates of the transfer matrix and Bethe ansatz equations can be obtained by the nested algebraic Bethe ansatz [13][14][15][16][17]. Then based on the Bethe states, the correlation function, norm, form factor and other scalar products can be calculated. The interesting thing is that how to diagonalize the transfer matrix if the reflection matrices are non-diagonal. Due to the fact that the reflection matrices at the two ends can not be diagonalized simultaneously, the U(1) symmetry of the system is broken. Then the traditional nested algebraic Bethe ansatz does not work.
Recently, in an interesting work [18], Robertson, Pawelkiewicz, Jacobsen and Saleur showed that the R-matrix of the D 2 spin chains [26].
In this paper, we study the exact solution of the D (1) 2 spin chain with non-diagonal boundary reflections. We find that the transfer matrix of the model can be factorized as the product of transfer matrices of two six-vertex models with generic integrable open boundary condition. With the help of this factorization identity and using the fusion, we obtain the eigenvalues expressed by the inhomogeneous T − Q relation and the Bethe ansatz equations of the model. In order to study the physical properties of the system, we also use the t − W scheme and obtain the patterns of zero roots distributions. Based on them, we obtain the ground state energy and the surface energy induced by the non-diagonal boundary magnetic fields in the thermodynamic limit and check these results numerically. The results of both the anisotropic and isotropic D (1) 2 spin chains are given. This paper is organized as follows. In section 2, we give a brief description of the anisotropic D where u is the spectral parameter, the non-zero matrix elements are and η is the crossing or anisotropic parameter. The R-matrix (2.1) is defined in the tensor space V 1 ⊗V 2 , where V 1 and V 2 are two four-dimensional linear spaces, and has the properties Crossing unitarity : where P 12 is the permutation operator with the matrix elements [P 12 ] αγ βδ = δ αδ δ βγ , R 21 (u) = P 12 R 12 (u)P 12 , t 1 (or t 2 ) denotes the transposition in the subspace V 1 (or V 2 ) and M 1 is the Besides, the R-matrix (2.1) satisfies the Yang-Baxter equation which means that the scattering processes among the quasi-particle do not depend on the paths.
The boundary reflection at one end is characterized by the reflection matrix where matrix elements are and {t, t 1 , t 2 , s, s 1 , s 2 } are six free boundary parameters 4 . The reflection matrix (2.6) satisfies the reflection equation [10,11] where K 1 (u) = K(u) ⊗ I, K 2 (u) = I ⊗ K(u) and I is the 4 × 4 unit matrix. The boundary reflection at the other end is quantified by the dual reflection matrix The K-matrix (2.7) is a new K-matrix in the sense which has more non-vanishing matric elements and more boundary parameters than those of the K-matrix obtained by A. Lima-Santos [7,8].
where {t ′ , t ′ 1 , t ′ 2 , s ′ , s ′ 1 , s ′ 2 } are the free boundary parameters. The dual reflection matrix (2.9) satisfies the dual reflection equation [10][11][12] Now, we are ready to construct the quantum many-body system with interactions. The conserved quantities including the model Hamiltonian of D 2 spin chain is generated by the transfer matrix t(u) (2.11) Here the subscript 0 means the four-dimensional auxiliary space V 0 and tr 0 means taking trace only in the auxiliary space. T 0 (u) andT 0 (u) are the monodromy matrix and the reflecting one, respectively, T 0 (u) = R N 0 (u + θ N ) · · · R 20 (u + θ 2 )R 10 (u + θ 1 ), (2.12) where {θ j |j = 1, · · · , N} are the inhomogeneity parameters and N is the number of sites.
T 0 (u) andT 0 (u) are defined in the tensor space V 0 ⊗V q and V q = ⊗ N j=1 V j is the physical space. From the Yang-Baxter relation (2.5), reflection equation (2.8) and the dual one (2.10), we can prove that the transfer matrices with different spectral parameters commutate with each (2.13) Thus the system is integrable. Expanding t(u) with respect to u, all the coefficients and their combinations are the conserved quantities. The model Hamiltonian of the integrable quantum spin chain is obtained by taking the derivative of logarithm of the transfer matrix t(u) with the homogeneous limit {θ j } = 0 where The next task is to diagonalize the transfer matrix t(u). However, the reflection matrices K(u) andK(u) have the non-diagonal elements and can not be diagonalized simultaneously.
It is very hard to construct the reference state and to solve the eigen-equation of t(u) [27].

Factorizations
In order to obtain the eigenvalues of the transfer matrix (2.11), here we adopt the factorization method [18,26]. The four-dimensional space can be regarded as the tensor of two equivalent two-dimensional subspaces. For example, the structures of V 1 ′ and V 1 ′′ are the same. Then the R-matrix (2.1) can be decomposed into We shall note that the spaces of two R-matrices in the factorization (3.1) are different. The
By using the factorizations (3.1) and (3.7), we find that the transfer matrix of D (1) 2 spin chain can be factorized as the product of transfer matrices of two independent XXZ spin chains where S = ⊗ N S and t s± (u) are defined as We should note that the physical spaces of t s+ (u) and t s− (u) in Eq.(3.11) are different, which means that they are two independent operators, and their tensor consists the physical space 2 spin chain. The monodromy matrix T s 0 ′ and reflecting oneT s 0 ′ (u) (3.12) satisfy the Yang-Baxter relations By using the Yang-Baxter relation (3.13) and reflection equations (3.9), we have

Exact solution
The transfer matrices t s+ (u) and t s− (u) can be diagonalized separately. We first consider t s+ (u). Following the method developed in [27], we adopt the fusion technique [28][29][30][31][32][33] to diagonalize the transfer matrix t(u) given by (2.11). The R-matrix (3.3) at the point of 1 ′ 2 ′ is the one-dimensional projector operator and {|1 ′ , |2 ′ } are the orthogonal bases of 2-dimensional linear space V 1 ′ (and V 2 ′ ). Taking the fusion among the R-matrices, we obtain The fusion of reflection matrices gives where the related constants are defined as Yang-Baxter relations (3.13) at certain points gives By using the fusion relations (4.3)-(4.6), we obtain Besides, the values of t s+ (u) at the points of u = 0, 4η, iπ can be calculated directly The asymptotic behavior of t s+ (u) is From the definition, we know that the transfer matrix t s+ (u) is an operator polynomial of e u 2 with the degree 2N + 4, which can be completely determined by 2N + 5 constraints. Thus above 2N fusion identities (4.7) and 6 additional conditions (4.9)-(4.10) give us sufficient information to determine the eigenvalue Λ s+ (u) of t s+ (u). After some algebra, we express the eigenvalue Λ s+ (u) as the inhomogeneous T − Q relation Because Λ s+ (u) is a polynomial of e u 2 , the singularities of right hand side of Eq.(4.11) should be cancelled with each other, which give that the Bethe roots {µ l } should satisfy the Bethe ansatz equations The eigenvalue Λ s− (u) of t s− (u) can be obtained by the mapping 2 spin chain as Some remarks are in order. The solutions of algebraic equations (4.12) gives the values of Bethe roots {µ l }. Substituting these values into the inhomogeneous T − Q relation (4.11), we obtain the eigenvalue Λ s+ (u). As proven in [34,35], the T −Q relation (4.11) can generate all the values of Λ s+ (u). These results are also valid for Λ s− (u). Then we conclude that the expression (4.14) can give the complete spectrum of the transfer matrix t(u) of the D 2 spin chain. All the above results have the well-defined homogeneous limit {θ j } = 0.
With the help of Eq.(2.14), we obtain the Hamiltonian of D where σ ± j = (σ x j ± iσ y j )/2, τ ± j = (τ x j ± iτ y j )/2, σ α j and τ α j are the Pauli matrices at j-th site and α = x, y, z. We should note that the Hamiltonian (4.15) is the direct summation of two anisotropic XXZ spin-1/2 chains with non-diagonal boundary magnetic fields up to the where with S = ⊗ N S and S = 1 2 (1 − σ z + τ z + σ z τ z ). The conclusion (4.16) is consistent with the fact that the corresponding R-matrix and reflection matrices in the transfer matrix t(u) of the D where c + = s ′ and c − = t ′ .
The eigenvalue of the Hamiltonian of D (1) In the derivation, we have used the useful relations Substituting the eigenvalue (4.14) into (4.20), we obtain the eigen-energy of the D to overcome this difficulty, we use the t − W scheme [42,43].
Due to the fact that the D 2 model can be factorized as two XXZ spin-1/2 chains, we consider the spin-1/2 chain first. Rewrite the Hamiltonian (4.17) as where the strengthes of boundary magnetic fields are quantified as In this paper, we consider the case that the anisotropic parameter η > 0. The hermitian of the Hamiltonian (5.1) requires that h z 1 and h z N are real, h + 1 = h − * 1 and h + N = h − * N , where the superscript * means the complex conjugate. According to the parametrization (4.5), the hermitian requires that the boundary parameters α 1 and α ′ 1 are real, α 2 =ᾱ 2 + πi and α ′ 2 =ᾱ ′ 2 + πi, whereᾱ 2 andᾱ ′ 2 are real. These conclusions are obtained as follows. The real h z N gives that s is real or s =s + iπ 2 wheres is real. For the case of real s, from Eq.(4.5) we have β > 1 + α 2 2 and cosh α 2 < −1. Thus we denote α 2 =ᾱ 2 + πi, whereᾱ 2 is real. For the case of s =s + iπ 2 , we obtain the same results. The constraints of boundary parameters α 1 , α ′ 1 and α ′ 2 can be obtained similarly. With the help of Eq.(4.19), the energy of the Hamiltonian (5.1) is Because the eigenvalue Λ s+ (u) is a polynomial of e u 2 with the degree 2N +4, we parameterize it by the 2N + 4 zero points {z k } instead of the Bethe roots as where the coefficient Λ 0 = −4(e −2η s 1 s ′ 2 + e 2η s 2 s ′ 1 ), which can be calculated directly from the asymptotic behavior of transfer matrix t s+ (u) with u → ∞. We note that the contributions and

Patterns of zero points
Now, we should determine the solutions of Bethe-ansatz-like equations (5.6)-(5.8). From the definition of transfer matrix t s+ (u) and using the crossing unitarity (3.4), we find that the t s+ (u) and its eigenvalue Λ s+ (u) satisfy the crossing symmetry Thus we conclude if z k is a zero root of Λ s+ (u), then −z k must be the root.
If the inhomogeneity parameters {θ k } are pure imaginary or zero, by using the intrinsic properties of R-matrix (3.3), one can easily prove that which means that if z k is one zero root of Λ s+ (u), then −z k , z * k and −z * k must be the roots. Due to the periodicity of Λ s+ (u), we fix the real parts of {z j } in the interval (−π, π]. We focus that {θ k } are pure imaginary. The inhomogeneity parameters are imaginary to keep the transfer matrix t s+ (u) hermitian for real η case. From the numerical solutions of The above four kinds of zero roots are valid for the whole energy spectrum. From now, we focus on the ground state. Only the configurations (I), (III) and (IV) of zero roots appear at the ground state. The further analysis gives that the boundary strings take the form of where parameters α 1 , α 2 , α ′ 1 , α ′ 2 are defined in Eq.(4.5). We should note that the fixed boundary magnetic fields can give the positive or negative values of Re(α y x ) due to the property of hyperbolic cosine function. Please see the parametrization (4.5). However, from Eq.(5. 13) we know that if the signs of Re(α y x ) are different, the lengthes of corresponding boundary strings are different. Thus we need some selection rules to give the correct boundary strings.
If h z 1 h z N > 0, the selection rule is that all the Re(α y x ) take the positive value. While if h z 1 h z N < 0, the selection rule is that the smaller Re(α y x ) in the set α y x ∈ {α 1 , α ′ 1 } takes the negative value and the others remain positive.
Another thing we should remark is that if Re(α y x ) < 2η, there exist the boundary strings determined by the boundary parameter α y x . If Re(α y x ) > 2η, the corresponding boundary string vanishes. Because α y x ∈ {α 1 , α 2 , α ′ 1 , α ′ 2 }, the number of boundary strings varies from zero to four.   Fig.1. Fig.1(a) shows the case that there is no the boundary string. The pattern of zero roots includes the real roots and two additional roots at the boundary. The pattern of Fig.1(b) includes the real roots, four boundary strings (two of them at the origin and two at the boundary) and two additional roots at the boundary. We note that the imaginary inhomogeneity parameters almost do not affect the imaginary parts of zero roots but the positions in the real axis.

Surface energy of XXZ spin-1/2 chain
We first consider the case that there is no the boundary string, where all Re(α y x ) > 2η. The zero roots at the ground state include the real roots {z k } and the additional roots {π ± z a i}, where z a is real. In the thermodynamic limit, where the system size N tends to infinity, the distribution of zero roots can be characterized by a density per site ρ(z).
Furthermore, we assume that the inhomogeneity parameters also has a continuum density per site σ(θ j ) ∼ 1/N(θ j −θ j−1 ), whereθ j = −iθ j . Thus the density of zero roots ρ(z) can be derived with the help of an auxiliary function σ(θ) which is a given density of the inhomogeneity.
Substituting the pattern of zero roots at the ground state into the Bethe-ansatz-like equations (5.6), taking the logarithm, making the difference of the resulted equations for θ j and θ j−1 , in the thermodynamic limit, we obtain where the functions a n (u) and b n (u) are defined as a n (u) = (ln sin Eq.(5.14) is a standard convolution integral equation and can be solved by the Fourier transformation. The Fourier transformations of Eq.(5.14) is whereã n (k) andb n (k) are calculated as a n (k) = π −π a n (u)e −iku du = −2πi sign(k)e −|nηk| , (5.17) In the homogeneous limit, we take σ(θ) = δ(θ). Thusσ(k) = 1. From Eq.(5.16), we obtain the density of zero roots as The inverse Fourier transformation gives Thus the ground state energy of the Hamiltonian (5.1) is Now, we prove that the z a tends to infinity in the thermodynamic limit. Taking the logarithm of Eq.(5.7) and considering the homogeneous limit {θ j = 0}, we have log sin z k − 2ηi 2 = log 2 + log cosh(2η) + log sinh s + log sinh s ′ + 2N log sinh(2η). (5.21) In the thermodynamic limit, Eq.(5.21) reads where C ′ is a constant irrelevant with N. Thus z a tends to infinity if N → ∞. The same result can also be obtained by using the Bethe-ansatz-like equations (5.8) with the similar method.
Next, we check the conclusion (5.23) numerically. By using the exact numerical diagonalization method, we obtain the values of z a with certain model parameters. In the homogeneous limit, the values of z a for different system sizes are shown in Fig.2. The data can be fitted as aN +b. From the fitting, we see that a = 2η and b = 1.758 which is irrelevant with N.
where E s+ g is the ground state energy given by (5.20) and is the ground state energy of the system with periodic boundary condition Considering z a → ∞ in the thermodynamic limit, we obtain the surface energy as Next, we consider the case that there exists one boundary string. For example, we choose the boundary parameters in the regime of α 1 < 2η, α ′ 1 > 2η, Re(α 2 ) > 2η and Re(α ′ 2 ) > 2η. In this case, only the boundary string π ± (2η − α 1 )i exists. This boundary string would affect the distribution of zero roots and make the density ρ(z) has a deviation. After some calculations, we obtain the deviation δρ(k) as The contribution of boundary string to the ground state energy is The ground state energy of the Hamiltonian H s− can be obtained similarly where the boundary parameters are defined as The selection rules of γ y x ∈ {γ 1 , γ ′ 1 , γ 2 , γ ′ 2 } are that all the Re(γ y x ) take positive values if coth s coth s ′ < 0; or the smaller Re(γ y x ) in the set γ y x ∈ {γ 1 , γ ′ 1 } takes the negative value and the others remain positive if coth s coth s ′ > 0. Similar to α 2 and α ′ 2 , we denote γ 2 =γ 2 + πi and γ ′ 2 =γ ′ 2 + πi, whereγ 2 andγ ′ 2 are real.

Surface energy of the anisotropic D
The Hamiltonian of anisotropic D 2 spin chain with periodic boundary condition is The ground state energy of the Hamiltonian (5.33) is Thus the surface energy of anisotropic D 2 spin chain reads where s r , s i , t r , t i are real, and the superscript * means the complex conjugate).
We check the conclusion (5.35) numerically. The surface energy can also be calculated by using the exact diagonalization method with the help of finite size scaling analysis. Here, the exact diagonalization is performed with the anisotropic η = 1.2 and the system size N is set from 4 to 9. The results are shown in Fig.3. The exact diagonalization results with the fixed boundary parameters are shown in Fig.3(a). The date can be fitted as E s = −0.778485e −2.44378N − 0.566456. When N tends to infinity, the finite size scaling analysis gives the value −0.566456, which is exactly the same as the surface energy calculated from the analytic expression (5.35). In Fig.3(b), we show the surface energy versus one free boundary parameter. Again, the analytic results obtained from (5.35) and the numerical ones calculated from exact diagonalization are consistent with each other very well.
The reflection matrices (6.2)-(6.3) can be factorized as where the matrix S is given by Eq.(3.2) and The R-matrix (6.4) can be decomposed intõ (6.9) Based on the above decompositions, we obtain the eigenvalue of transfer matrix (6.1) as Λ(u) =Λ s+ (u)Λ s− (u), (6.10) whereΛ (u − θ j + 1)(u + θ j + 1) =d(u + 1), The Bethe roots {u l } in Eq.(6.11) should satisfy the Bethe ansatz equations The Hamiltonian of isotropic D 2 spin chain generated by the transfer matrix (6.1) is We find that the Hamiltonian (6.14) is the direct summation of two XXX spin-1/2 chains with non-diagonal boundary magnetic fields up to the similar transformation S with S = ⊗ N S and S = 1 2 (1 − σ z + τ z + σ z τ z ). The conclusion (6.15) is consistent with the fact that the corresponding R-matrix and reflection matrices in the transfer matrixt(u) of the system can be factorized. The eigenvalue of the Hamiltonian (6.14) is whereΛ(u) is given by Eq.(6.10). In the derivation, the identity tr 0K0 (0) ′ = 0 is used.

Symmetry
In the Hamiltonian (6.16), there are six free boundary parameters. However, due to the fact that the interactions in the bulk are isotropic and have the su(2) symmetry, the free boundary parameters can be reduced. The method is as follows. First, we choose the direction of magnetic field at the left side as the z-direction, which can be achieved by acting the transformation matrix ⊗ N j=1C 1 j on the Hamiltonian (6.16). Then, we take a rotation of the resulted Hamiltonian in the xy plane to make the direction of magnetic field at the right side lying in the xz plane. This can be achieved by using the transition matrix ⊗ N j=1C 2 j . We note that the interactions in the bulk do not change after the transformations.

The final result is
(σ x j σ x j+1 + σ y j σ y j+1 + σ z j σ z j+1 ) + Here the transformation matricesC 1 j andC 2 j arẽ , (6.20) and the matrix elements are , The boundary parameters p, q and ξ in the Hamiltonian (6.19) reads We see that only three free boundary parameters are left. The hermitian of Hamiltonian (6.19) requires that all the p, q and ξ are real. The HamiltoniansH s+ andH s+ have the same eigen-energies. Thus in the following, we derive the eigenvalues of the Hamiltonians H s+ instead ofH s+ .
Similar with the anisotropic case, we consider that the inhomogeneity parameter are pure imaginary or zero. We denote θ k = iθ k , whereθ k is real. From the properties of R-matrix (6.9), it is easy to prove thatΛ(u) has the crossing symmetrȳ Λ(u) =Λ(−u − 1). ( which means that if z k is one zero root ofΛ(u), then −z k , z * k and −z * k must be the roots. The distribution of zero roots with odd N is slightly different from that with even N at the ground state. Because the odd and even N give the same physical properties in the thermodynamic limit, we focus on even N. We consider the case that p > 1/2 and q/ 1 + ξ 2 < −1/2. From the numerical solutions of Bethe-ansatz-like equations with finite system sizes and the singularity analysis in the thermodynamic limit, we obtain that the roots include the conjugate pairs {z j + i,z j − i} and the additional roots ±z a i, wherez j and z a are real. Because the spin exchanging interactions in HamiltonianH s+ (6.19) are antiferromagnetic, most zero roots form the bulk 2-strings rather than the real roots at the ground state.
Substituting the patterns of zero roots into the Bethe-ansatz-like equations (6.32), taking the logarithm, and making the difference of the resulted equations for θ j and θ j−1 , in the thermodynamic limit, we have (6.39) In the homogeneous limit, we set the densityσ(θ) of inhomogeneity parameters as the delta function,σ(θ) = δ(θ). Then, by using the Fourier transformation, we obtain the density of zero roots asρ We see that the additional roots do not have the contribution to the ground state energy.
The ground state energy of the XXX spin chain with periodic boundary condition is N(1 − 2 ln 2) [27]. Thus the surface energy of the system (6. 16)

Discussion
We have studied the exact solution of anisotropic D 2 quantum spin chain with generic nondiagonal boundary magnetic fields. We find that the transfer matrix of the system can be constructed by two six-vertex models with boundary reflections. Based on the factorization identities, we obtain the eigenvalues and the Bethe-ansatz-like equations of the model. By using the t − W scheme, we obtain the patterns of zero roots of the transfer matrix. Based on them, the thermodynamic limit, ground states energy and surface energy are obtained.
We also give the results for the isotropic couplings case.
Starting from the obtained eigenvalues (4.14) and (6.10), the corresponding eigenstates can be retrieved by using the separation of variables for the integrable systems [36][37][38][39] or the off-diagonal Bethe ansatz [40,41]. Then the correlation functions, norm, form factors and other interesting scalar products can be calculated. Based on the patterns of zero roots, other physical quantities such as the elementary excitations and the free energy at finite temperature can be studied. We also note that the results given in this paper are the foundation to exactly solve the high rank D (1) n model with the nested method.