Off-diagonal Bethe Ansatz on the $so(5)$ spin chain

TL;DR

This paper develops an off-diagonal Bethe Ansatz approach to solve the $so(5)$ quantum spin chain with various boundary conditions, deriving new spectral relations and extending the method to broader classes of models.

Contribution

It introduces a novel inhomogeneous $T-Q$ relation for the $so(5)$ chain with non-diagonal boundaries and generalizes the approach to $so(2n+1)$ chains.

Findings

Derived operator product identities for the $so(5)$ chain.

Constructed a new inhomogeneous $T-Q$ relation for non-diagonal boundaries.

Extended the method to $so(2n+1)$ quantum integrable spin chains.

Abstract

The $so(5)$ (i.e., $B_2$) quantum integrable spin chains with both periodic and non-diagonal boundaries are studied via the off-diagonal Bethe Ansatz method. By using the fusion technique, sufficient operator product identities (comparing to those in [1]) to determine the spectrum of the transfer matrices are derived. For the periodic case, we recover the results obtained in \cite{NYReshetikhin1}, while for the non-diagonal boundary case, a new inhomogeneous $T-Q$ relation is constructed. The present method can be directly generalized to deal with the $so(2n+1)$ (i.e., $B_n$) quantum integrable spin chains with general boundaries.

Equations

$$R_{12}^{vv}(u{)}_{kl}^{ij}=u(u+\frac{3}{2}){\delta }_{ik}{\delta }_{jl}+(u+\frac{3}{2}){\delta }_{il}{\delta }_{jk}-u{\delta }_{j\bar{i}}{\delta }_{k\bar{l}},$$

$$R_{12}^{vv}(u{)}_{ii}^{ii}=a_1(u)=(1+u)(u+\frac{3}{2}),i\ne 3,$$

$$R_{12}^{vv}(u{)}_{ij}^{ij}=b_1(u)=u(u+\frac{3}{2}),i\ne j,\bar{j},$$

$$R_{12}^{vv}(u{)}_{\bar{i}i}^{i\bar{i}}=c_1(u)=\frac{3}{2},i\ne \bar{i},$$

$$R_{12}^{vv}(u{)}_{j\bar{j}}^{i\bar{i}}=d_1(u)=-u,i\ne j,\bar{j},$$

$$R_{12}^{vv}(u{)}_{i\bar{i}}^{i\bar{i}}=e_1(u)=u(u+\frac{1}{2}),i\ne \bar{i},$$

$$R_{12}^{vv}(u{)}_{ii}^{ii}=f_1(u)=a_1(u)+d_1(u),i=3,$$

$$R_{12}^{vv}(u{)}_{ji}^{ij}=g_1(u)=u+\frac{3}{2},i\ne j,\bar{j}.$$

$$\mathrm{regularity}$$

$$\mathrm{unitarity}$$

$$\mathrm{crossing}-\mathrm{symmetry}$$

$$\displaystyle V=\left(\begin{array}[]{ccccc}&&&&1\\ &&&1&\\ &&1&&\\ &1&&&\\ 1&&&&\end{array}\right),\quad V^{2}={\rm id}.$$

$$\mathrm{crossing}-\mathrm{unitarity}$$

$$R_{12}^{vv}(u-v)R_{13}^{vv}(u)R_{23}^{vv}(v)=R_{23}^{vv}(v)R_{13}^{vv}(u)R_{12}^{vv}(u-v).$$

$$T_0^v(u)=R_{01}^{vv}(u-{\theta }_1)R_{02}^{vv}(u-{\theta }_2)\cdots R_{0N}^{vv}(u-{\theta }_N),$$

$$R_{12}^{vv}(u-v)T_1^v(u)T_2^v(v)=T_2^v(v)T_1^v(u)R_{12}^{vv}(u-v).$$

$$t^{(p)}(u)\stackrel{\mathrm{def}}{=}{t}_1^{(p)}(u)=t{r}_0{T}_0^v(u).$$

$$H_p=\frac{\partial \ln t^{(p)}(u)}{\partial u}{|}_{u=0,\{{\theta }_j\}=0}.$$

$$R_{12}^{ss}(u{)}_{ii}^{ii}=a_2(u)=(u+\frac{1}{2})(u+\frac{3}{2}),$$

$$R_{12}^{ss}(u{)}_{ij}^{ij}=b_2(u)=u(u+\frac{3}{2}),i\ne j,\bar{j},$$

$$R_{12}^{ss}(u{)}_{\bar{i}i}^{i\bar{i}}=c_2(u)=u+\frac{3}{4},$$

$${\xi }_i{\xi }_{\bar{j}}{R}_{12}^{ss}(u{)}_{j\bar{j}}^{i\bar{i}}=d_2(u)=-\frac{u}{2},i\ne j,\bar{j},$$

$$R_{12}^{ss}(u{)}_{i\bar{i}}^{i\bar{i}}=e_2(u)=u(u+1),$$

$$R_{12}^{ss}(u{)}_{ji}^{ij}=g_2(u)=\frac{u}{2}+\frac{3}{4},i\ne j,\bar{j},$$

$$\mathrm{regularity}$$

$$\mathrm{unitarity}$$

$$\mathrm{crossing}-\mathrm{unitarity}$$

$$R_{12}^{ss}(-\frac{1}{2})=P_{12}^{ss(5)}\times S,$$

$$P_{12}^{ss(5)}=\sum _{i=1}^5|{\tilde{\psi }}_i⟩⟨{\tilde{\psi }}_i|,$$

$$|{\tilde{\psi }}_1⟩=\frac{1}{\sqrt{2}}(|12{⟩}^{(s)}-|21{⟩}^{(s)}),|{\tilde{\psi }}_2⟩=\frac{1}{\sqrt{2}}(|31{⟩}^{(s)}-|13{⟩}^{(s)}),$$

$$|{\tilde{\psi }}_3⟩=\frac{1}{2}(|14{⟩}^{(s)}-|41{⟩}^{(s)}+|23{⟩}^{(s)}-|32{⟩}^{(s)}),|{\tilde{\psi }}_4⟩=\frac{1}{\sqrt{2}}(|24{⟩}^{(s)}-|42{⟩}^{(s)}),$$

$$|{\tilde{\psi }}_5⟩=\frac{1}{\sqrt{2}}(|34{⟩}^{(s)}-|43{⟩}^{(s)}).$$

$$R_{1⟨23⟩}^{sss}(u)=[(u-\frac{1}{4})(u+\frac{3}{4})(u+\frac{7}{4}){]}^{-1}{P}_{23}^{ss(5)}{R}_{12}^{ss}(u+\frac{1}{4})R_{13}^{ss}(u-\frac{1}{4})P_{23}^{ss(5)}.$$

$$|{\tilde{\psi }}_i⟩⟶|i⟩,i=1,\cdots ,5,$$

$$R^{sv}(u{)}_{11}^{11}=R^{sv}(u{)}_{12}^{12}=R^{sv}(u{)}_{21}^{21}=R^{sv}(u{)}_{24}^{24}=R^{sv}(u{)}_{32}^{32}=R^{sv}(u{)}_{35}^{35}$$

$$=R^{sv}(u{)}_{44}^{44}=R^{sv}(u{)}_{45}^{45}=a_3(u)=u+\frac{5}{4},$$

$$R^{sv}(u{)}_{14}^{14}=R^{sv}(u{)}_{15}^{15}=R^{sv}(u{)}_{22}^{22}=R^{sv}(u{)}_{25}^{25}=R^{sv}(u{)}_{31}^{31}=R^{sv}(u{)}_{34}^{34}$$

$$=R^{sv}(u{)}_{41}^{41}=R^{sv}(u{)}_{42}^{42}=b_3(u)=u+\frac{1}{4},$$

$$R^{sv}(u{)}_{41}^{14}=-R^{sv}(u{)}_{42}^{15}=R^{sv}(u{)}_{31}^{22}=-R^{sv}(u{)}_{34}^{25}=R^{sv}(u{)}_{22}^{31}=-R^{sv}(u{)}_{25}^{34}$$

$$=R^{sv}(u{)}_{14}^{41}=-R^{sv}(u{)}_{15}^{42}=-c_3(u)=-1,$$

$$R^{sv}(u{)}_{13}^{13}=R^{sv}(u{)}_{23}^{23}=R^{sv}(u{)}_{33}^{33}=R^{sv}(u{)}_{43}^{43}=e_3(u)=u+\frac{3}{4},$$

$$R^{sv}(u{)}_{22}^{13}=-R^{sv}(u{)}_{23}^{14}=R^{sv}(u{)}_{31}^{13}=-R^{sv}(u{)}_{33}^{15}=R^{sv}(u{)}_{13}^{22}=-R^{sv}(u{)}_{14}^{23}$$

Full text

PACS: 75.10.Pq, 02.30.Ik, 71.10.Pm

Keywords: Bethe Ansatz; Lattice Integrable Models; $T-Q$ Relation

1 Introduction

Both the algebraic and coordinate Bethe Ansatz are very powerful methods to construct exact solutions of quantum integrable models[2, 3, 4, 5, 6, 7]. Nevertheless, those methods still have their restrictions for depending on existence of an obvious reference state. An important issue is that when $U(1)$-symmetry is broken, the systems may not have obvious reference states. In such cases, the problem becomes more frustrated and many interesting efforts have been made in this direction [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] in the past several decades. A generic method named off-diagonal Bethe ansatz (ODBA) for solving quantum integrable models either with or without $U(1)$-symmetry was proposed in [26]. By constructing the inhomogeneous $T-Q$ relations based on some operator identities, several typical models are solved exactly [27]. With the resulting eigenvalues, the corresponding Bethe-type eigenstates can also be retrieved[28, 29]. The nested ODBA was initially proposed in studying the $su(n)$ (i.e., $A_{n}$) spin chain with generic boundaries [30, 31]. However, ODBA to approach high-rank quantum integrable models associated with $B_{n}$, $C_{n}$ and $D_{n}$ Lie algebras is still missing. We note that such kind of models with obvious $U(1)$-symmetry has been studied extensively. For example, with some functional relations and algebraic Bethe Ansatz analysis (the analytic Bethe Ansatz method), Reshetikhin derived the energy spectrum of the periodic quantum spin chains associated with $B_{n}$, $C_{n}$, $D_{n}$ and other Lie algebras [1, 32]. The algebraic Bethe Ansatz for those models with periodic boundary condition was constructed by Martins and Ramos [33], while the method for approaching such kind of models with diagonal open boundaries was developed by Li, Shi and Yue [34, 35].

In this paper, we develop a nested ODBA method to approach the quantum integrable $so(5)$ (i.e., $B_{2}$) spin chain with either periodic or non-diagonal open boundary condition. This method can be generalized to $so(2n+1)$ (i.e., $B_{n}$) case directly. The paper is organized as follows. In section 2, we study the $so(5)$ model with periodic boundary condition. Closed functional relations among the transfer matrices to determine the eigenvalues are constructed with fusion techniques. In section 3, we study the $so(5)$ model with an off-diagonal open boundary condition. By constructing some operator product identities, we derive the exact eigenvalues of the transfer matrix in terms of an inhomogeneous $T-Q$ relation. Section 4 is attributed to concluding remarks. Some detailed calculations are listed in Appendices A-C.

2 $so(5)$ spin chain with periodic boundary condition

2.1 The model

Let ${\rm V}$ denote a $5$-dimensional linear space with an orthonormal basis $\{|i\rangle|i=1,\cdots,5\}$ which endows the fundamental representation of the $so(5)$ (or $B_{2}$) algebra. The quantum spin chain associated with the $B_{2}$ algebra is described by a $25\times 25$ $R$-matrix $R^{vv}_{12}(u)$ defined in the $V\otimes V$ space with the matrix elements [33]

[TABLE]

where $\{i,j,k,l\}=\{1,2,3,4,5\}$, $i+\bar{i}=6$. We introduce the notation for simplicity

[TABLE]

The $R$-matrix satisfies the properties [36]

[TABLE]

where ${\cal P}_{12}$ is the permutation operator with the matrix elements $[{\cal P}_{12}]^{ij}_{kl}=\delta_{il}\delta_{jk}$, $t_{i}$ denotes the transposition in the $i$-th space, $R_{21}={\cal P}_{12}R_{12}{\cal P}_{12}$, and the crossing-matrix $V$ is

[TABLE]

Combining the crossing-symmetry and the unitarity of the $R$-matrix, one can derive the relation

[TABLE]

Here and below we adopt the standard notation: for any matrix $A\in{\rm End}({\rm V})$, $A_{j}$ is an embedding operator in the tensor space ${\rm V}\otimes{\rm V}\otimes\cdots$, which acts as $A$ on the $j$-th space and as an identity on the other factor spaces; $R_{ij}(u)$ is an embedding operator of $R$-matrix in the tensor space, which acts as an identity on the factor spaces except for the $i$-th and $j$-th ones. The $R$-matrix satisfies the Yang-Baxter equation

[TABLE]

For the periodic boundary condition, we introduce the monodromy matrix

[TABLE]

where the index [math] indicates the auxiliary space and the other tensor space $V^{\otimes N}$ is the physical or quantum space, $N$ is the number of sites and $\{\theta_{j}\}$ are the inhomogeneous parameters. The monodromy matrix satisfies the Yang-Baxter relation

[TABLE]

The transfer matrix is the trace of monodromy matrix in the auxiliary space

[TABLE]

From the Yang-Baxter relation, one can prove that the transfer matrices with different spectral parameters commute with each other, $[t^{(p)}(u),t^{(p)}(v)]=0$. Therefore, $t^{(p)}(u)$ serves as the generating function of all the conserved quantities of the system. The Hamiltonian is given by

[TABLE]

2.2 Spinorial $R$-matrix and the fused ones

In order to obtain closed operator product identities (see (2.50)-(2.56) below) which allow one to completely determine the eigenvalues of the transfer matrix $t^{(p)}(u)$, we need further an $R$-matrix associated with the spinorial representation of the $so(5)$ algebra. Let us denote the spinorial representation by ${\rm V^{(s)}}$ with an orthonormal basis $\{|i\rangle^{(s)}|i=1,\cdots,4\}$. The spinorial $16\times 16$ $R$-matrix has the following non-zero matrix elements [37]

[TABLE]

where $\{i,j\}=\{1,2,3,4\}$, $i+\bar{i}=5$, $\xi_{i}=1$ if $i\in\{1,2\}$ and $\xi_{i}=-1$ if $i\in\{3,4\}$. The spinorial $R$-matrix satisfies the properties

[TABLE]

where ${\cal P}^{(s)}_{12}$ is the permutation operator among the spinorial representation space (c.f., ${\cal P}_{12}$ in (2.10) ).

Following the fusion procedure [38, 39, 40, 41, 42, 43, 44, 45], we can construct another $R$-matrix $R^{sv}_{12}(u)$ defined in $V^{(s)}\otimes V$. It is easily to check that

[TABLE]

where $S$ is some non-degenerate constant matrix, and $P^{{ss}(5)}_{12}$ is a 5-dimensional projector operator with the form

[TABLE]

where the corresponding vectors are333We used a temporal notation $|ij\rangle^{(s)}=|i\rangle^{(s)}\otimes|j\rangle^{(s)}$.

[TABLE]

Let $V^{(\langle ss\rangle)}$ denote the projected subspace of $V^{(s)}\otimes V^{(s)}$ by the projector $P^{{ss}(5)}_{12}$. Namely, $V^{(\langle ss\rangle)}$ is a 5-dimensional subspace and spanned by $\{|\tilde{\psi}_{i}\rangle|i=1,\cdots,5\}$. Then we can construct a fused $R_{1\,\langle 23\rangle}^{sss}(u)$ matrix

[TABLE]

After taking the correspondence

[TABLE]

one has the identification: $V\equiv V^{(\langle ss\rangle)}$, which leads to an $R$-matrix $R^{sv}_{12}(u)$ defined in $V^{(s)}\otimes V$. The non-vanishing matrix elements of the resulting $R$-matrix are444Each matrix element of $R^{sv}_{12}(u)$, as a function of $u$, is a polynomial with degree up to one.

[TABLE]

It is easily to check that the fused $R_{12}^{sv}$ matrix has the properties

[TABLE]

Moreover, the fused $R_{1\,2}^{sv}$ also satisfy the Yang-Baxter equations

[TABLE]

and

[TABLE]

Similarly, one can reconstruct the $R$-matrix $R^{vv}_{12}(u)$ from the fused one $R^{sv}_{12}(u)$

[TABLE]

We have checked that the $R$-matrices $R^{sv}_{12}(u)$ and $R^{vv}_{12}(u)$ enjoy the properties:

[TABLE]

where $P^{{vv}(1)}_{12}$, $P_{12}$, $P_{123}$, $P_{1234}$ and $P^{{sv}(4)}_{12}$ are the projectors given by (A.14)-(A.19) in Appendix A and $S_{i}(i=1,2,\dots,5)$ are some irrelevant constant matrices. With the help of the above projectors and using the similar fusion procedure [38, 39, 40, 41, 42, 43, 44, 45], we can construct the fused $R$-matrices

[TABLE]

where

[TABLE]

Here we have used $\bar{V}$ (resp. $\tilde{V}$) to denote the projected subspace in $V\otimes V$ by $P_{21}$ (resp. the projected subspace in $V\otimes V\otimes V$ by $P_{321}$ ) and adopted the convention: $\bar{1}\equiv\langle 12\rangle$ and $\tilde{1}\equiv\langle 123\rangle$.

Some remarks are in order. It is shown that each matrix elements of the above fused $R$-matrices, as a function of $u$, is a polynomial with degree up to two. Due to the fact that the $16$-dimensional projected subspace in $V\otimes V\otimes V\otimes V$ by the projector $P^{{vv}}_{4321}$ is equivalent to the tensor space $V^{(s)}\otimes V^{(s)}$ according to the correspondence (A.9) below, we have the equivalence

[TABLE]

where the constant gauge transformation matrix $S_{12}$ is given by (A.11) below. Moreover, we have the identity

[TABLE]

2.3 Operator product identities

Besides the transfer matrix $t^{(p)}_{1}(u)$ given by (2.13), let us introduce $3$ fused transfer matrices:

[TABLE]

where ${\bar{T}}_{\langle 12\cdots m\rangle}^{v}(u)$ are the fused monodromy matrices

[TABLE]

and the projectors $\{P_{m\cdots 21}|m=2,3,4\}$ are given by (A.14)-(A.18) below. Moreover, in order to have closed ( or enough) operator product identities, we need to introduce an extra transfer matrix

[TABLE]

where the spinoral $R$-matrix $R^{sv}_{12}(u)$ is given by (2.21). It is easily to show that all the transfer matrices constitute a commutative family, namely,

[TABLE]

We have used the convention: ${\bar{t}}^{\,(p)}_{1}(u)=t^{(p)}_{1}(u)=t^{(p)}(u)$. Direct calculation shows that

[TABLE]

where we have introduced some normalized monodromy matrices:

[TABLE]

It is remarked that the quantum spaces of the above monodromy matrices are the same (i.e., $V^{\otimes N}$) and that the corresponding auxiliary spaces are $\bar{V}$ and $\tilde{V}$ with dimensions $11$ and $15$. Then the associated transfer matrices are given by

[TABLE]

The equivalence (2.36) and the relations (2.42)-(2.44) imply that

[TABLE]

Fowllowing the method developed in [30], we obtain the identities

[TABLE]

where the explicit expressions of the projectors $P^{{vv}(1)}_{21}$, $P_{21}$, $P_{321}$, $P_{4321}$, $P^{{\bar{v}v}(5)}_{\bar{1}2}$, $P^{{\tilde{v}v}(11)}_{\tilde{1}2}$ and $P^{{sv}(4)}_{{1}\,2}$ are given in Appendix A. With the help of the correspondences (2.36), (A.3), (A.8) and (A.13), we have that the transfer matrices satisfy the operator product identities:555Because for generic choices of inhomogeneous parameters $\{\theta_{j}\}$ these identities involve the different points values of the transfer matrices (for $t^{(p)}(u)$ at $\theta_{j},\theta_{j}-\frac{1}{2},\theta_{j}-1,\theta_{j}-\frac{3}{2};$ for $t_{2}^{(p)}(u)$ at $\theta_{j},\theta_{j}-\frac{1}{2},\theta_{j}-1,\theta_{j}-\frac{3}{2};$ for $t_{3}^{(p)}(u)$ at $\theta_{j}-\frac{1}{2},\theta_{j}-1,\theta_{j}-2;$ for $t_{s}^{(p)}(u)$ at $\theta_{j}-\frac{1}{4},\theta_{j}-\frac{5}{4},\theta_{j}-\frac{11}{4}$), the identities (2.50)-(2.56) are independent.

[TABLE]

Now, we consider the asymptotic behaviors of the fused transfer matrices. Direct calculation shows

[TABLE]

Let us denote the eigenvalues of the transfer matrices $t^{(p)}(u)$, $t^{(p)}_{2}(u)$ $t^{(p)}_{3}(u)$ and $t^{(p)}_{s}(u)$ as $\Lambda^{(p)}(u)$, $\Lambda^{(p)}_{2}(u)$, $\Lambda^{(p)}_{3}(u)$ and $\Lambda^{(p)}_{s}(u)$, respectively. From the operator product identities (2.50)-(2.56), we have the functional relations among the eigenvalues666It is remarked that only (2.58) and (2.64) were used to obtain $\Lambda^{(p)}(u)$ and $\Lambda^{(p)}_{s}(u)$ for the closed $B_{n}$ chain [1, 32]. Since that $\Lambda^{(p)}(u)$ (resp. $\Lambda^{(p)}_{s}(u)$) is a polynomial of $u$ with degree $2N$ (resp. degree $N$), one need $3N+2$ conditions to determine them completely while (2.58) and (2.64), together with the asymptotic behaviors, only give $2N+2$ conditions. Therefore, in order to close the functional relations, (2.58)-(2.64) are necessary.:

[TABLE]

The asymptotic behaviors (2.57) of the fused transfer matrices lead to the corresponding asymptotic behaviors of their eigenvalues:

[TABLE]

From the definitions (2.13), (2.40) and (2.47), we know that the eigenvalues $\Lambda^{(p)}(u)$, $\Lambda^{(p)}_{2}(u)$ and $\Lambda^{(p)}_{3}(u)$ are polynomials of $u$ with degree $2N$, while $\Lambda^{(p)}_{s}(u)$ is a polynomial of $u$ with degree $N$. Hence the functional relations (2.58)-(2.68) could completely determine the eigenvalues, which allows us to express them in terms of some homogeneous $T-Q$ relations in the next subsection.

2.4 $T-Q$ relations

Let us introduce some functions:

[TABLE]

The functional relations (2.58)-(2.68) enable us to parameterize the eigenvalues of the transfer matrices in terms of the $T-Q$ relations as follows:

[TABLE]

The regularity of the eigenvalue $\Lambda^{(p)}(u)$ given by (2.70) leads to that the parameters $\{\mu^{(m)}_{k}\}$ should satisfy the Bethe Ansatz equations (BAEs):

[TABLE]

We have verified that the above BAEs indeed guarantee all the $T-Q$ relations (2.70)-(2.4) are polynomials of $u$ with the required degrees. Moreover, these $T-Q$ relations also satisfy the function relations (2.58)-(2.64) and the asymptotic behaviors (2.65)-(2.68). Therefore, we conclude that $\Lambda^{(p)}(u)$, $\Lambda^{(p)}_{2}(u)$, $\Lambda^{(p)}_{3}(u)$ and $\Lambda^{(p)}_{s}(u)$ given by (2.70)-(2.4) are indeed the eigenvalues of the transfer matrices $t^{(p)}(u)$, $t^{(p)}_{2}(u)$, $t^{(p)}_{3}(u)$ and $t^{(p)}_{s}(u)$ provided that the $L_{1}+L_{2}$ parameters $\{\mu^{(m)}_{k}\}$ satisfy the associated BAEs (2.74)-(2.75). It is remarked that the $T-Q$ relations (2.70) and the associated BAEs (2.74)-(2.75) (after taking the homogeneous limit $\{\theta_{j}\to 0|j=1,2,\cdots,N\}$) coincide with those obtained previously via conventional Bethe Ansatz methods[1, 32, 33]. Comparing with the results of the structure for Bethe states [33], we conclude that the two no-negative integers $L_{1}$ and $L_{2}$ take values: $0\leq L_{1}\leq 2N,\quad 0\leq L_{2}\leq 2L_{1}$, which gives rise to the complete spectrum of the transfer matrix (2.13).

3 Off-diagonal open boundary case

3.1 Open chain

Integrable open chain can be constructed as follows [6, 7]. Let us introduce a pair of $K$-matrices $K^{v-}(u)$ and $K^{v+}(u)$. The former satisfies the reflection equation (RE)

[TABLE]

and the latter satisfies the dual RE

[TABLE]

For open spin chains, instead of the “row-to-row” monodromy matrix $T^{v}_{0}(u)$ (2.11), one needs to consider the “double-row” monodromy matrix as follows. Let us introduce another “row-to-row” monodromy matrix

[TABLE]

which satisfies the Yang-Baxter relation

[TABLE]

The transfer matrix $t(u)$ is defined as

[TABLE]

From the Yang-Baxter relation, reflection equation and its dual, one can prove that the transfer matrices with different spectral parameters commute with each other, $[t(u),t(v)]=0$. Therefore, $t(u)$ serves as the generating function of all the conserved quantities of the system. The Hamiltonian of the open chain can be obtained by taking the derivative of the logarithm of the transfer matrix

[TABLE]

with $K^{v-}(0)=\xi\times{\rm id}.$

In this paper, we consider an open chain associated with the off-diagonal $K$-matrix $K^{v-}(u)$ given by

[TABLE]

where the non-vanishing matrix elements are

[TABLE]

Here $c_{1}$ and $c_{2}$ are arbitrary boundary parameters. The dual reflection matrix $K^{v+}(u)$ is also an off-diagonal one and given by

[TABLE]

where $\tilde{c}_{1}$ and $\tilde{c}_{2}$ are the boundary parameters. For a generic choice of the four boundary parameters $\{c_{i},\,\tilde{c}_{i}|i=1,2\}$, it is easily to check that $[K^{v-}(u),\,K^{v+}(v)]\neq 0$. This implies that the $K^{\pm}(u)$ matrices cannot be diagonalized simultaneously. In this case, it is quite hard to derive solutions via conventional Bethe Ansatz methods due to the absence of a proper reference state. We will generalize the method developed in Section 2 to obtain eigenvalues of the transfer matrix $t(u)$ (3.5) specified by the $K$-matrices (3.12) and (3.14) in the following subsections.

Following the method developed in [46] and using the crossing-symmetry (2.3) of the $R$-matrix, the explicit expressions (3.12) and (3.14) of the K-matrices, we find that the transfer matrix (3.5) possesses the crossing symmetry

[TABLE]

3.2 Operator product relations

We define the dual fused monodromy matrices as

[TABLE]

where the $R$-matrices $R^{vs}_{21}(u)$ is defined by (2.22) and the others are defined by the relations

[TABLE]

We have checked that the $R$-matrices also enjoy the properties

[TABLE]

which are used to derive the functional relations (3.24) below.

Let us introduce the fused transfer matrices

[TABLE]

where the associated $K$-matrices $K^{s\pm}(u)$, $K^{\bar{v}\pm}(u)$ and $K^{\tilde{v}\pm}(u)$ are given by (B.7)-(B.12), (B.19)-(B.20) and (B.21)-(B.22) respectively. The Yang-Baxter relations, reflection equations and dual reflection equations allow one to show that all the transfer matrices constitute a commutative family [7], namely,

[TABLE]

where we have used the convention: $t_{1}(u)=t(u)$. Moreover, from the construction of the transfer matrices we know that $t_{s}(u)$, as a function of $u$, is a polynomial with degree $2N$, and that $t_{1}(u)$ is a polynomial of $u$ with degree $4N+2$, while $t_{2}(u)$ and $t_{3}(u)$ are polynomial of $u$ with degree $4N+4$. Thus one need to look for $14(N+1)$ independent conditions to completely determine them.

Using the method we have used in previous sections and with the help of the relations (C.2) and (C.6), we can obtain the operator product identities among the fused transfer matrices as777Due to the fact that for generic choices of inhomogeneous parameters $\{\theta_{j}\}$ the identities (3.23)-(3.24) involve the different points values of the transfer matrices, this implies that the identities should be independent.

[TABLE]

where

[TABLE]

By direct calculation, we have

[TABLE]

where we have used the identities:

[TABLE]

Moreover, we can derive the asymptotic behaviors of the transfer matrices

[TABLE]

Let us denote the eigenvalues of the fused transfer matrices $t(u)$, $t_{2}(u)$ $t_{3}(u)$ and $t_{s}(u)$ as $\Lambda(u)$, $\Lambda_{2}(u)$, $\Lambda_{3}(u)$ and $\Lambda_{s}(u)$, respectively. The above $14(N+1)$ operator production identities (3.23)-(3.26) directly imply that

[TABLE]

The above $14(N+1)$ conditions (3.27)-(3.45) could completely determine the eigenvalues $\Lambda(u)$, $\Lambda_{2}(u)$, $\Lambda_{3}(u)$ and $\Lambda_{s}(u)$ in terms of some inhomogeneous $T-Q$ relations.

3.3 Inhomogeneous $T-Q$ relations

Define

[TABLE]

and

[TABLE]

where $x$ is a parameter which will be determined later (see (3.53) below). The $14(N+1)$ functional relations (3.27)-(3.45) allow us to express the eigenvalues of the transfer matrices in terms of some inhomogeneous $T-Q$ relations as follows:

[TABLE]

All the eigenvalues are polynomials of $u$, the residues of right hand sides of Eq.(3.48) should be zero, which gives rise to the BAEs

[TABLE]

Considering the asymptotic behaviors of $\Lambda(u)$, $\Lambda_{2}(u)$, $\Lambda_{3}(u)$ and $\Lambda_{s}(u)$, we obtain a constraint between the two no-negative integers $L_{1}$ and $L_{2}$:

[TABLE]

which implies that only one of the two $U(1)$ conserved charges (related to the Cartan sub-algebra of $so(5)$) has been broken by the boundary terms in (3.6). Moreover, the value of parameter $x$ in the functions $\{f_{m}(u)\}$ given in (3.47) as

[TABLE]

Some remarks are in order. The expression of the inhomogeneous $T-Q$ relation (3.48) and the associated BAEs (3.50)- (3.51) imply that one of the two $U(1)$ conserved charges (corresponding to the non-negative integer $L_{1}$) is still survived, which means that $L_{1}$ can take some integers. Following the method developed in [47] and combining the results in [34], we know that $0\leq L_{1}\leq 2N$ and then $L_{2}$ is fixed by the constraint (3.52), which would give complete spectrum of the transfer matrix (3.5). However, for more generic $K$-matrices the two $U(1)$ charges might be all broken, then both non-negative integers $L_{1}$ and $L_{2}$ will be fixed888It has been confirmed for the $B_{1}$ model with generic boundary terms [31, 48]..

We have checked that the BAEs (3.50)-(3.51) indeed ensure that all the $T-Q$ relations (3.48)-(3.49) are polynomials of $u$ with the required degrees and also satisfy the required $14(N+1)$ function relations (3.27)-(3.45). Therefore, we conclude that $\Lambda(u)$, $\Lambda_{2}(u)$, $\Lambda_{3}(u)$ and $\Lambda_{s}(u)$ given by the inhomogeneous $T-Q$ relations (3.48)-(3.49) are indeed the eigenvalues of the transfer matrices $t(u)$, $t_{2}(u)$, $t_{3}(u)$ and $t_{s}(u)$ provided that the $L_{1}+L_{2}$ parameters $\{\lambda^{(m)}_{k}\}$ satisfy the associated BAEs (3.50)-(3.51). Moreover, when $c_{1}=c_{2}=\tilde{c}_{1}=\tilde{c}_{2}=0$, the boundary reflection matrices degenerate to diagonal ones and our results recover those obtained by the algebraic Bethe method [34].

4 Discussion

In this paper, we study the $so(5)$ quantum spin chains with integrable boundary conditions. By using the fusion technique, we obtain the closed operator product identities of the fused transfer matrices (2.50)-(2.56) for the periodic boundary condition and (3.23)-(3.24) for the off-diagonal boundary case. Based on them and the asymptotic behaviors as well as the special values at certain points, we obtain the exact solutions of the systems. For the periodic case, the eigenvalues of the transfer matrices are given in terms of the homogeneous $T-Q$ relations (2.70)-(2.73) and the associated BAEs are (2.74)-(2.75), which recover those obtained previously via conventional Bethe Ansatz methods[1, 32, 33]. For the open boundary case specified by the off-diagonal boundary $K$-matrices (3.12)-(3.14), (B.7)-(B.12), (B.19)-(B.20) and (B.21)-(B.22), the eigenvalues of the transfer matrices are given in terms of the inhomogeneous $T-Q$ relations (3.48)-(3.49) and the associated BAEs (3.50)-(3.51). The method and the results in this paper can be generalized to the $so(2n+1)$ (i.e., $B_{n}$) case directly.

Acknowledgments

The financial supports from the National Program for Basic Research of MOST (Grant Nos. 2016YFA0300600 and 2016YFA0302104), the National Natural Science Foundation of China (Grant Nos. 11434013, 11425522, 11547045, 11774397, 11775178, 11775177 and 91536115), the Major Basic Research Program of Natural Science of Shaanxi Province (Grant Nos. 2017KCT-12, 2017ZDJC-32), Australian Research Council (Grant No. DP 190101529) and the Strategic Priority Research Program of the Chinese Academy of Sciences, and the Double First-Class University Construction Project of Northwest University are gratefully acknowledged. GL Li acknowledges support from Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, Xian Jiaotong University. We would like to thank Prof. R. Nepomechie for drawing our attention on the crossing-symmetry (2.3) of the $R$-matrix which enables us to prove the crossing symmetry (3.15) of the fundamental transfer matrix $t(u)$.

Appendix A: Projectors and the fusion of the $R$-matrices

A.1: Fused $R$-matrices

From the definition (2.32) of the fused $R$-matrix $R_{\bar{1}2}^{\bar{v}v}(u)$, we have

[TABLE]

where $P^{{\bar{v}v}(5)}_{\bar{1}2}$ is a $5$-dimensional projector (see (A.23) below). The new projector allows us to construct a new fused $R$-matrix,

[TABLE]

Taking the correspondence

[TABLE]

where the basis $\{|\phi^{(5)}_{i}\rangle\}$ is given by (A.24) below, we have

[TABLE]

The definition (2.32) of the fused $R$-matrix $R_{\tilde{1}2}^{\tilde{v}v}(u)$ implies that

[TABLE]

where $P^{{\tilde{v}v}(11)}_{\tilde{1}2}$ is a $11$-dimensional projector (see below (A.26)) and $\tilde{S}$ is an irrelevant constant matrix. The projector allows us to construct the fused $R$-matrix

[TABLE]

where $S_{\bar{v}}$ is a $11\times 11$ diagonal constant matrix with the elements

[TABLE]

After taking the correspondence

[TABLE]

where the bases $\{|\phi^{(11)}_{i}\rangle\}$ and $\{|\varphi^{(11)}_{i}\rangle\}$ are given by (A.16) and (A.27) below, we have

[TABLE]

Taking the correspondence

[TABLE]

and after some calculations, we arrive at (2.36), namely,

[TABLE]

where $S_{12}$ is the $16\times 16$ gauge transformation on the tensor space $V^{(s)}\otimes V^{(s)}$ with the non-zero matrix elements

[TABLE]

With the help of the $4$-dimensional projector $P^{{sv}(4)}_{21}$ given by (A.19) and after taking the correspondence

[TABLE]

where the basis $\{|\psi^{\prime}_{i}\rangle\}$ is given by (A.22) below, we have

[TABLE]

A.2: Projectors

Here we list the projectors used in this paper. $P^{{vv}(1)}_{12}$ is the one-dimensional projector

[TABLE]

where

[TABLE]

$P_{12}$ is a 11-dimensional projector

[TABLE]

where

[TABLE]

$P_{123}$ is a 15-dimensional projector

[TABLE]

where

[TABLE]

$P_{1234}$ is a 16-dimensional projector

[TABLE]

where

[TABLE]

$P^{{sv}(4)}_{12}$ is a $4$-dimensional projector

[TABLE]

where

[TABLE]

Similarly, we can construct the projector $P^{{vs}(4)}_{21}$

[TABLE]

where

[TABLE]

$P^{{\bar{v}v}(5)}_{\bar{1}2}$ is the $5$-dimensional projector

[TABLE]

where

[TABLE]

Because the dimension of $V_{\bar{1}}$ and that of $V_{2}$ are not equal, we need introduce another $5$-dimensional projector

[TABLE]

where

[TABLE]

$P^{{\tilde{v}v}(11)}_{\tilde{1}2}$ is the 11-dimensional projector

[TABLE]

where

[TABLE]

Because the dimensions of spaces $V_{\tilde{1}}$ and $V_{2}$ in the operator $P^{{\tilde{v}v}(11)}_{\tilde{1}2}$ are not equal, we should introduce the related projector

[TABLE]

where

[TABLE]

Appendix B: Fusion of the $K$-matrices

B.1: Associated spinorial reflection matrices

We introduce the corresponding spinorial $K$-matrix $K^{s-}(u)$ and the dual reflecting matrix $K^{s+}(u)$, which satisfy the reflection equation

[TABLE]

and its dual

[TABLE]

respectively. The matrix forms of $K^{s\pm}(u)$ are

[TABLE]

With the help of the projector $P_{12}^{{ss}(5)}$ given by (2.18), one can construct the fused $K$-matrices [29] as follows:

[TABLE]

We remark that $P_{21}^{{ss}(5)}=P_{12}^{{ss}(5)}$. With the help of the equivalence (2.20), we have

[TABLE]

B.2: Other fused $K$-matrices

The one-dimensional projector $P_{21}^{vv(1)}$ given by (2.27) allows us to compute the quantum determinants ${\rm Det}_{q}(K^{v\pm}(u))$

[TABLE]

where

[TABLE]

Using the the $11$-dimensional projector $P_{21}^{vv}$ given by (A.15), we can construct the $11\times 11$ $K$-matrices $K^{\bar{v}\pm}(u)$

[TABLE]

The 15-dimensional projector $P_{321}$ given by (A.17) allows us to construct the $15\times 15$ $K$-matrices $K^{\tilde{v}\pm}(u)$

[TABLE]

All these $K$-matrices satisfy the associate reflection equations or the dual reflection equations.

Using the $5$-dimensional projectors $P_{\bar{1}2}^{\bar{v}v(5)}$ and $P_{2\bar{1}}^{\bar{v}v(5)}$ given by (A.25)-(A.26) and the correspondence (A.2), we have

[TABLE]

Similarly with the help of the $11$-dimensional projectors $P_{\tilde{1}2}^{\tilde{v}v(11)}$ and $P_{2\tilde{1}}^{\tilde{v}v(11)}$ given by (A.26)-(A.28) and the correspondence (A.7), we have

[TABLE]

where the $11\times 11$ diagonal matrix $S_{\bar{v}}$ is given by (A.6). Moreover the $16$-dimensional projector $P_{4321}$ and the correspondence (A.9) allow us to have the identifications:

[TABLE]

where the $16\times 16$ constant matrix is given by (A.11) and

[TABLE]

In order to complete the whole fusion of the $K$-matrices, we need the fusion between vectorial $K_{1}^{v-}(u)$ and spinorial $K_{2}^{s-}(u)$. The $4$-dimensional projectors $P_{12}^{sv(4)}$ and $P_{21}^{sv(4)}$ given by (A.19)-(A.21) and the correspondence (A.12) enable us to have the identifications:

[TABLE]

Appendix C: Operator identities for the open case

Similar to the periodic case (see (2.39)-(2.39)), let us introduce some fused transfer matrices as follows:

[TABLE]

where

[TABLE]

The fused reflecting monodromy matrices satisfy the fusion relations (c.f., (2.42)-(2.44) for the periodic case)

[TABLE]

The above relations imply that the fused transfer matrices are indeed proportional to those given by (3.21) by some polynomials (c.f., (2.48) for the periodic case)

[TABLE]

Similar to (2.49) for the periodic case, we can derive the relations among the reflecting monodromy matrices given by (3.16)

[TABLE]