Towards the solution of an integrable $D_2^{(2)}$ spin chain

TL;DR

This paper proposes a Bethe ansatz solution for a specific branch of the $D_2^{(2)}$ spin chain, addressing a longstanding gap in understanding its eigenvalues, especially for the branch that resisted previous solutions.

Contribution

It introduces a Bethe ansatz approach for the epsilon=1 branch of the $D_2^{(2)}$ spin chain, including a proposal for the missing eigenvalues, advancing the understanding of this integrable model.

Findings

Proposed a Bethe ansatz solution for the epsilon=1 branch.

Identified limitations of the solution, as it only covers eigenvalues with odd degeneracy.

Suggested a method to find the missing eigenvalues.

Abstract

Two branches of integrable open quantum-group invariant $D_{n+1}^{(2)}$ quantum spin chains are known. For one branch (epsilon=0), a complete Bethe ansatz solution has been proposed. However, the other branch (epsilon=1) has so far resisted solution. In an effort to address this problem, we consider here the simplest case n=1. We propose a Bethe ansatz solution, which however is not complete, as it describes only the transfer-matrix eigenvalues with odd degeneracy. We also consider a proposal for the missing eigenvalues.

Tables

Table 1. Table 1: Fraction of eigenvalues with odd degeneracy

$N$	1	2	3	4	5
${𝒩}_{odd}/4^N$	1	0.625	0.375	0.210938	0.117188

Table 2. Table 2: N = 1 𝑁 1 N=1

degeneracy	multiplicity	$m$
3	1	0
1	1	2

Table 3. Table 3: N = 2 𝑁 2 N=2

degeneracy	multiplicity	$m$
5	1	0
1	2	2
3	1	4
6	1	?

Table 4. Table 4: N = 3 𝑁 3 N=3

degeneracy	multiplicity	$m$
7	1	0
3	3	2
1	3	4
5	1	6
10	2	?
6	3	?
2	1	?

Table 5. Table 5: N = 4 𝑁 4 N=4

degeneracy	multiplicity	$m$
9	1	0
5	4	2
1	6	4
3	4	6
7	1	8
14	3	?
10	8	?
6	12	?
2	4	?

Table 6. Table 6: N = 5 𝑁 5 N=5

degeneracy	multiplicity	$m$
11	1	0
7	5	2
3	10	4
1	10	6
5	5	8
9	1	10
18	4	?
14	15	?
10	35	?
6	40	?
2	16	?

Equations

$$R_{12}(u)R_{21}(-u)=\zeta (u),\zeta (u)=16{\sinh }^2(u+2\eta ){\sinh }^2(u-2\eta ),$$

$$M_1^{-1}{R}_{12}(-u-2\rho {)}^{t_1}{M}_1{R}_{21}(u{)}^{t_1}=\zeta (u+\rho ),\rho =-2\eta ,$$

$$M=\mathrm{diag}\left(e^{2\eta },1,1,e^{-2\eta }\right).$$

$$\displaystyle K^{R}(u)=\left(\begin{array}[]{c|c|cc|c|c}k_{-}(u)\,\mathbb{I}_{p\times p}&&&&&\\ \hline\cr&g(u)\,\mathbb{I}_{(n-p)\times(n-p)}&&&&\\ \hline\cr&&k_{1}(u)&k_{2}(u)&&\\ &&k_{2}(u)&k_{1}(u)&&\\ \hline\cr&&&&g(u)\,\mathbb{I}_{(n-p)\times(n-p)}&\\ \hline\cr&&&&&k_{+}(u)\,\mathbb{I}_{p\times p}\end{array}\right)\,,$$

$$k_{\mp }(u)$$

$$g(u)$$

$$k_1(u)$$

$$k_2(u)$$

$$K^L(u)=K^R(-u-\rho {)}^{t}M,$$

$$t(u)={\mathrm{tr}}_a{K}_a^L(u)T_a(u;\{{\theta }_j\})K_a^R(u){\hat{T}}_a(u;\{{\theta }_j\}),$$

$$T_a(u;\{{\theta }_j\})$$

$${\hat{T}}_a(u;\{{\theta }_j\})$$

$$\left[t(u),t(v)\right]=0.$$

$$t({\theta }_j)t({\theta }_j+2\eta )=\Delta ({\theta }_j),j=1,\dots ,N,$$

$$h^{(R)}(u)$$

$$\times {\sinh }^2(2u-6\eta )\sinh (2u-4\eta )\sinh (u-\eta ),$$

$$h^{(L)}(u)$$

$$\times {\sinh }^2(2u-8\eta ){\sinh }^2(2u-4\eta )\sinh (2u-12\eta )\sinh (u-3\eta ),$$

$$h(u)$$

$$\Delta (u)$$

$$t(u)=t(u+i\pi ),$$

$$t(u)=t(-u+2\eta ).$$

$$\displaystyle\lim_{u\rightarrow 0}\frac{t(u)}{\sinh u}\Big{|}_{\{\theta_{j}\}=0}=2^{4N}\sinh^{4N-3}(2\eta)\sinh^{2}(4\eta)\sinh(\eta)\mathop{\rm csch}\nolimits(3\eta)\,.$$

$$\Lambda ({\theta }_j)\Lambda ({\theta }_j+2\eta )=\Delta ({\theta }_j),j=1,\dots ,N,$$

$$\Lambda (u)=\Lambda (u+i\pi ),$$

$$\Lambda (u)=\Lambda (-u+2\eta ),$$

$$\displaystyle\lim_{u\rightarrow 0}\frac{\Lambda(u)}{\sinh u}\Big{|}_{\{\theta_{j}\}=0}=2^{4N}\sinh^{4N-3}(2\eta)\sinh^{2}(4\eta)\sinh(\eta)\mathop{\rm csch}\nolimits(3\eta)\,.$$

$$\Lambda (u)\sim 2e^{\pm 4N(u-\eta )}\left[\cosh (4\eta (N-m+\frac{1}{2}))+1\right]\text{mbox}foru\to \pm \infty ,$$

$$\Lambda (u)=\varphi (u)\lambda (u),\varphi (u)=\frac{\sinh (u)\sinh (u-2\eta )}{\sinh (u+\eta )\sinh (u-3\eta )}.$$

$$\lambda ({\theta }_j)\lambda ({\theta }_j+2\eta )=\frac{\Delta ({\theta }_j)}{\varphi ({\theta }_j)\varphi ({\theta }_j+2\eta )},j=1,\dots ,N,$$

$$\lambda (u)$$

$$\displaystyle\lambda(0)\Big{|}_{\{\theta_{j}\}=0}$$

$$\lambda (u)$$

$$\lambda (u)=\sum _{k=-2N}^{2N}{\lambda }_k{e}^{2ku},$$

$$\lambda (u)=Z_1(u)+Z_2(u)+Z_3(u)+Z_4(u),$$

Full text

UMTG–299

Towards the solution

of an integrable $D_{2}^{(2)}$ spin chain

Rafael I. Nepomechie 111Physics Department, P.O. Box 248046, University of Miami, Coral Gables, FL 33124 USA, nepomechie@miami.edu, Rodrigo A. Pimenta 222Institut Denis-Poisson CNRS/UMR 7013 - Université de Tours - Université d’Orléans, Parc de Grammont, 37200 Tours, France*,333CAPES Foundation, Ministry of Education of Brazil, Brasilia - DF, Zip code 70.040-020, Brazil,*444Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13566-590, São Carlos, SP, Brazil, pimenta@ifsc.usp.br and Ana L. Retore 555Instituto de Física Teórica-UNESP, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II 01140-070, São Paulo, Brazil, ana.retore@unesp.br

Two branches of integrable open quantum-group invariant $D_{n+1}^{(2)}$ quantum spin chains are known. For one branch ($\varepsilon=0$), a complete Bethe ansatz solution has been proposed. However, the other branch ($\varepsilon=1$) has so far resisted solution. In an effort to address this problem, we consider here the simplest case $n=1$. We propose a Bethe ansatz solution, which however is not complete, as it describes only the transfer-matrix eigenvalues with odd degeneracy. We also consider a proposal for the missing eigenvalues.

1 Introduction

Among the non-exceptional trigonometric solutions of the Yang-Baxter equation (R-matrices) [1, 2, 3, 4], the R-matrices associated with $D_{n+1}^{(2)}$ are – by far – the most complicated. It is therefore not surprising that relatively few results are known about the corresponding integrable quantum spin chains. Bethe ansatz solutions for the closed $D_{n+1}^{(2)}$ chains with periodic boundary conditions were proposed by Reshetikhin [5]. Following the pioneering work of Sklyanin [6], the study of open integrable $D_{n+1}^{(2)}$ chains was initiated in [7], and was pursued further in [8, 9].

New families of solutions of the $D_{n+1}^{(2)}$ boundary Yang-Baxter equation (K-matrices) were recently proposed in [10]. These K-matrices depend on the discrete parameters $p$ (which can take $n+1$ possible values $p=0,\ldots,n$) and $\varepsilon$ (which can take two possible values $\varepsilon=0,1$), see (2.11). The open spin chains constructed with these K-matrices have quantum group symmetry corresponding to removing the $p^{th}$ node from the $D_{n+1}^{(2)}$ Dynkin diagram, namely, $U_{q}(B_{n-p})\otimes U_{q}(B_{p})$ (for both $\varepsilon=0,1$). These spin chains also have a $p\leftrightarrow n-p$ duality symmetry. Bethe ansatz solutions for the open $D_{n+1}^{(2)}$ spin chains with $\varepsilon=0$ (and all the possible values of $p$) were proposed in [9, 11]. However, the open $D_{n+1}^{(2)}$ spin chains with $\varepsilon=1$ have so far resisted solution.

In an effort to address this problem, we consider here the simplest case $n=1$; that is, we consider the open $U_{q}(B_{1})$-invariant $D_{2}^{(2)}$ spin chain with $\varepsilon=1$ and the two possible values of $p$ (namely, 0 and 1). This model has potential applications to black hole physics [12]. We propose a Bethe ansatz solution that is similar to the one for $\varepsilon=0$; however, unlike the latter solution, it is not complete: this solution describes only the transfer-matrix eigenvalues with odd degeneracy. It remains a challenge to account for the eigenvalues with even degeneracy, which may be related to a higher symmetry of the transfer matrix.

The outline of this paper is as follows. In Sec. 2, we briefly review the construction of the transfer matrix and list some of its useful properties. In Sec. 3, we try to determine the eigenvalues of the transfer matrix with $p=0$. We arrive at a compact expression for the eigenvalues (3.26), (3.27) and corresponding Bethe equations (3.28), which unfortunately do not give all the eigenvalues. However, for the eigenvalues which can be described in this way, we find even simpler expressions for the eigenvalues (3.32) and Bethe equations (3.33), which closely resemble those of the XXZ chain. We then argue that this Bethe ansatz describes the eigenvalues with odd degeneracy. In Sec. 4, we consider the case $p=1$. We consider a proposal for the missing eigenvalues in Sec. 5, which is motivated by a preliminary algebraic Bethe analysis presented in an appendix. Our brief conclusions are in Sec. 6.

2 Basics

In this section, we briefly review the construction of the transfer matrix for the integrable open $U_{q}(B_{1})$-invariant $D_{2}^{(2)}$ spin chain, with a 4-dimensional vector space at each site. We also list some useful properties of this transfer matrix. We begin by recalling its two basic building blocks: an R-matrix and a K-matrix.

2.1 R-matrix

For the $16\times 16$ R-matrix $R(u)$, we use the expression for the $D_{n+1}^{(2)}$ R-matrix in the fundamental (vector) representation given in Appendix A of [9] with $n=1$. This R-matrix depends on the anisotropy parameter $\eta$. In addition to the Yang-Baxter equation, it satisfies the unitarity relation

[TABLE]

and the crossing-unitarity relation

[TABLE]

where $M$ is the diagonal $4\times 4$ matrix

[TABLE]

2.2 K-matrix

For the right K-matrix $K^{R}(u)$, we take [10]

[TABLE]

with

[TABLE]

see also [7]. Since we restrict our attention here to $D_{2}^{(2)}$, which corresponds to $n=1$, the matrix $K^{R}(u)$ is $4\times 4$. There are two possible values of $p$ (namely, 0 and 1), and we now set $p=0$. (We consider the $p=1$ case in Sec. 4.) As emphasized in the Introduction, we focus in this paper on the case $\varepsilon=1$.

For the left K-matrix $K^{L}(u)$, we take [10]

[TABLE]

so that the transfer matrix has quantum-group symmetry.

2.3 Transfer matrix

The transfer matrix $t(u)$ for an open integrable quantum spin chain of length $N$ is given by [6, 10]

[TABLE]

where the monodromy matrices with inhomogeneities $\{\theta_{1}\,,\ldots\,,\theta_{N}\}$ are given by

[TABLE]

and the trace in (2.13) is over the (4-dimensional) auxiliary space.

2.4 Properties of the transfer

matrix

By construction, the transfer matrix satisfies the commutativity property

[TABLE]

The transfer matrix also obeys the functional relations (see [9] and references therein)

[TABLE]

where

[TABLE]

and $\zeta(u)$ is defined in (2.1). The transfer matrix also has $i\pi$ periodicity

[TABLE]

as well as crossing symmetry

[TABLE]

Finally, the transfer matrix has the particular value

[TABLE]

3 Eigenvalues of the transfer matrix

This section, which is devoted to determining the transfer-matrix eigenvalues, contains most of our new results. We first show in Sec. 3.1 that the transfer-matrix properties listed in Sec. 2.4 do not suffice to determine the eigenvalues. We then formulate in Sec. 3.2 a conjecture for the eigenvalues, which is developed further in Sec. 3.3.

3.1 Properties of the eigenvalues

Let $\Lambda(u)$ denote the eigenvalues of the transfer matrix $t(u)$. It follows from the transfer-matrix properties (2.16) - (2.23) that the eigenvalues satisfy similar properties:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Moreover, the eigenvalues have the asymptotic behavior

[TABLE]

where $m$ is a non-negative integer.

In order to proceed further, it is convenient to consider rescaled eigenvalues $\lambda(u)$, defined such that

[TABLE]

The rescaled eigenvalues $\lambda(u)$ do not have any poles for finite $u$, and do not have zeros at $u=0,2\eta$. The rescaled eigenvalues have the properties

[TABLE]

and

[TABLE]

The periodicity (3.8) and asymptotic behavior (3.11) imply that the eigenvalues have the form

[TABLE]

where $\lambda_{k}$ are $u$-independent coefficients, of which there are $4N+1$. However, the crossing symmetry (3.9) relates the coefficients $\lambda_{k>0}$ to $\lambda_{k<0}$. Hence, there are $2N+1$ independent coefficients for $\lambda(u)$.

The functional relations (3.7) provide $N$ constraints. The asymptotic behavior (3.11) provides one constraint (the behavior at $-\infty$ follows from the behavior at $+\infty$ together with crossing symmetry), and (3.10) provides one more constraint, for a total of only $N+2$ constraints. Therefore, for $N>1$, these constraints do not suffice to determine $\lambda(u)$.

We have tried to obtain additional constraints by formulating functional relations involving fused transfer matrices, as in e.g. [13, 14]. However, this introduces even more unknown coefficients (to describe the eigenvalues of the fused transfer matrices, similarly to (3.12)), and does not seem to help.

In the next subsection, we conjecture an expression for $\lambda(u)$ that is compatible with the above constraints.

3.2 Formulating a conjecture for

$\lambda(u)$

In view of the result for $\varepsilon=0$ [9, 11], let us assume that the rescaled eigenvalues $\lambda(u)$ have the form

[TABLE]

where

[TABLE]

and

[TABLE]

where the functions $a(u)$ and $b(u)$ are still to be determined.

The function $a(u)$ can readily be seen to be given by

[TABLE]

either from the functional relation (3.7), or by explicitly computing the reference-state eigenvalue for small values of $N$ and with values of the inhomogeneities chosen such that only $Z_{1}(u)$ is nonzero, as explained in detail in [11]. Note that $a(u)$ (3.16) has a double-pole at $u=\eta+\frac{i\pi}{2}$.

The function $b(u)$ must have the same double-pole as $a(u)$ in order for $\lambda(u)$ (3.13) to be analytic. We therefore set

[TABLE]

where $c(u)$ is finite at $u=\eta+\frac{i\pi}{2}$. The function $b(u)$ must also satisfy

[TABLE]

in order to ensure that $\lambda(u)$ is correct for the reference state, for which $Q(u)=1$. Therefore, $c(u)$ satisfies

[TABLE]

The condition that the residue of $\lambda(u)$ (3.13) at the double-pole vanishes implies

[TABLE]

where prime denotes differentiation. Finally, let us assume that $b(u)$ (and therefore also $c(u)$) is $i\pi$ periodic 111The weaker assumption

$b(-u+2\eta)=b(u+i\pi)\,,\qquad b(u)=b(u+2i\pi)\,,$

is also compatible with the $i\pi$ periodicity of $\lambda(u)$ (3.8), and leads to

$b(u)=\frac{\cosh(u)\cosh(u-2\eta)}{\cosh^{2}(u-\eta)}+\beta\frac{\sinh(u-\eta)}{\cosh^{2}(u-\eta)}\,,$

where $\beta$ is a free parameter, cf. (3.25). However, even for $N=2$, we cannot find any value of $\beta$ for which (3.13) gives all the eigenvalues.

[TABLE]

and has the asymptotic behavior

[TABLE]

(which is compatible with (3.11)), which imply that $c(u)$ has the form

[TABLE]

where $c_{k}$ are independent of $u$. The constraints (3.19) and (3.20) then uniquely determine $c(u)$ to be given by

[TABLE]

It follows from (3.17) that $b(u)$ is given by

[TABLE]

In summary, we conjecture that the rescaled eigenvalues $\lambda(u)$ are given by (3.13) and (3.14), where $Q(u)$, $a(u)$ and $b(u)$ given by (3.15), (3.16) and (3.25), respectively. This ansatz satisfies all the constraints (3.7) - (3.10).

3.3 Bethe ansatz

We observe that this expression for $\lambda(u)$ can be factored as follows 222For the case $\varepsilon=0$ [9, 11], and presumably also for the periodic chain [5], such a factorization is possible for all the eigenvalues, hence it may hold at the level of the transfer matrix.

[TABLE]

where $\chi(u)$ is defined by

[TABLE]

which satisfies $\chi(-u+2\eta)=\chi(u)$. The requirement that the residues of $\chi(u)$ vanish at $u=u_{j}+\eta$ leads to the Bethe equations for $\{u_{1},\ldots,u_{m}\}$

[TABLE]

Unfortunately, this Bethe ansatz solution is not complete: we have checked numerically for small values of $N$ that this solution gives some, but not all, of the transfer-matrix eigenvalues. However, for every eigenvalue that we do find, the number of Bethe roots ($m$) is even, and all the Bethe roots come in pairs separated by exactly $i\pi$

[TABLE]

Assuming that the Bethe roots always form pairs of the form (3.29), then the Q-function (3.15) becomes (up to an irrelevant overall factor)

[TABLE]

and therefore $Q(u)$ becomes $i\pi$ periodic

[TABLE]

It follows that $\chi(u)$ (3.27) also becomes $i\pi$ periodic, and therefore $\lambda(u)$ (3.26) becomes a perfect square

[TABLE]

The requirement that the residues of $\chi(u)$ vanish at $u=u_{j}+\eta$ now leads to the simplified Bethe equations

[TABLE]

Interestingly, these Bethe equations are similar to those for the spin-1/2 XXZ chain.

We have solved the simplified Bethe equations (3.33) with all $\theta_{l}=0$ numerically (for some generic value of anisotropy parameter $\eta$) for $N=1,2,\ldots,5$; we have then computed the corresponding eigenvalues (for some generic value of spectral parameter $u$) using (3.6), (3.27) and (3.32); and we have compared with the eigenvalues obtained by direct diagonalization of the transfer matrix (2.13). The results are summarized in Tables 2 - 6. For a given value of $N$, each table reports the degeneracy (the number of times a given eigenvalue appears), the multiplicity (the number of times a given degeneracy appears), and $m$ (twice the number of Bethe roots of the simplified Bethe equations (3.33) that are needed to describe an eigenvalue with the given degeneracy). A question mark (?) means that an eigenvalue with the given degeneracy cannot be described by this Bethe ansatz. For example, from Table 3, we can see that for $N=2$, there is one eigenvalue with degeneracy 5 which corresponds to the reference state ($m=0$); there are two eigenvalues with degeneracy 1 which are each described by 1 Bethe root ($m=2$); there is one eigenvalue with degeneracy 3 which is described by 2 Bethe roots ($m=4$); and there is one eigenvalue with degeneracy 6 which cannot be described by this Bethe ansatz. Note that $m$ takes even values from 0 to $2N$. (We do not report the actual Bethe roots and eigenvalues in order to avoid having prohibitively large tables.)

An inspection of these tables shows that our Bethe ansatz describes all the eigenvalues with odd degeneracy, but does not describe any of the eigenvalues with even degeneracy. We conjecture that this is true for generic values of $\eta$ and for all values of $N$.

For a given value of $N$, let ${\cal N}_{odd}$ and ${\cal N}_{even}$ denote the total number of eigenvalues (given by the product degeneracy $\times$ multiplicity) with odd and even degeneracy, respectively. Clearly,

[TABLE]

From Tables 2-6, we can see that the fraction of eigenvalues with odd degeneracy rapidly decreases as $N$ increases, as summarized in Table 1.

We expect that the “missing” eigenvalues (i.e., the eigenvalues with even degeneracy, which cannot be described by this Bethe ansatz) cannot be expressed as perfect squares, as in (3.32). We have verified this for $N=2$, in which case all the eigenvalues can be explicitly computed as functions of $u$ and $\eta$.

3.3.1 Degeneracies and symmetries

On the basis of $U_{q}(B_{1})$ symmetry alone, one would expect that every eigenvalue of the transfer matrix has odd degeneracy [9, 10]. For example, for $N=2$:

[TABLE]

and for $N=3$:

[TABLE]

However, we can easily see from Tables 3 and 4 that the actual degeneracies are higher: for $N=2$, one pair of ${\bf 3}$’s becomes degenerate (giving a 6-fold degenerate eigenvalue); and for $N=3$, two pairs of ${\bf 5}$’s become degenerate (giving two 10-fold degenerate eigenvalues), three pairs of ${\bf 3}$’s become degenerate (giving three 6-fold degenerate eigenvalues), and one pair of ${\bf 1}$’s becomes degenerate (giving a 2-fold degenerate eigenvalue).

We have conjectured in [9, 10] that these higher (even) degeneracies are due to an additional symmetry of the transfer matrix that doubles the degeneracy of certain eigenvalues, for both $\varepsilon=0$ and $\varepsilon=1$. Indeed, for $N=2$, we have explicitly constructed an involutory matrix that maps one ${\bf 3}$ to another ${\bf 3}$, commutes with all the $U_{q}(B_{1})$ generators, and commutes with $t(u)$. However, an extension of this construction to general values of $N$ is still not known.

Our new observation here is that the “missing” eigenvalues are precisely those that would become degenerate as the result of this additional symmetry.

4 The case $p=1$

The results discussed so far in Secs. 2.4 and 3 are for $p=0$. We now consider the case $p=1$. To this end, it is convenient to now change notation so that the dependence on $p$ becomes manifest, e.g. $K^{R,L}(u)\mapsto K^{R,L}(u,p)$, $t(u)\mapsto t(u,p)$, $\Lambda(u)\mapsto\Lambda(u,p)$, etc. In particular, the result (3.6) becomes

[TABLE]

For $p=1$, we obtain in a similar way

[TABLE]

where $\lambda(u)$ is again given by (3.13), (3.26), etc. The Bethe equations are therefore also the same as before. In other words, only the overall factor changes.

This result is consistent with the $p\leftrightarrow n-p$ duality symmetry that was mentioned in the Introduction. Indeed, the transfer matrix has the symmetry [10]

[TABLE]

where ${\cal U}$ is a certain operator acting in the quantum space, and $f(u,p)$ is a scalar function given by

[TABLE]

with

[TABLE]

It follows that the corresponding eigenvalues are related by

[TABLE]

Substituting (4.1) and (4.2) into (4.6) with $n=p=1$ leads to the constraint

[TABLE]

which is indeed consistent with (4.4) for $\varepsilon=1$.

5 An ansatz for the missing eigenvalues?

Let us now consider the following ansatz for the “missing” eigenvalues with $p=0$

[TABLE]

where the functions $Z_{1}(u),\ldots,Z_{4}(u)$ are given as before by (3.14), and

[TABLE]

This ansatz is very similar to the previous one (3.13), except for some signs and the shift of all the eigenvalues by $Z_{5}(u)$, which does not depend on the Q-function. This ansatz also satisfies the constraints (3.7)-(3.10). It is motivated by a preliminary algebraic Bethe ansatz analysis, which is presented in Appendix A.

The expression (5.1)-(5.2) for the eigenvalues, up to the shift, can be factored as follows

[TABLE]

where $\tilde{\chi}(u)$ is defined by

[TABLE]

which satisfies $\tilde{\chi}(-u+2\eta)=-\tilde{\chi}(u)$. Requiring that the residues of $\tilde{\chi}(u)$ vanish at $u=u_{j}+\eta$ leads to the following Bethe equations

[TABLE]

These Bethe equations are unusual, as they involve $\cosh$ instead of $\sinh$ on the RHS. (For the $\varepsilon=0$ case [7, 9, 11], the Bethe equations are the same as (5.5) except with $\sinh$ on the RHS.)

The ansatz (5.3)-(5.5) is correct for $m=1$. Indeed, we prove it in Appendix A, and we have confirmed numerically that this ansatz with $m=1$ correctly describes eigenvalues with even degeneracy for $N=2$ (degeneracy 6), $N=3$ (degeneracy 10) and $N=4$ (degeneracy 14). We expect that, for general $N$, this ansatz with $m=1$ describes eigenvalues with degeneracy $4N-2$.

We also find numerically that this ansatz with $m=3$ describes the eigenvalue for $N=3$ with degeneracy 2. Unfortunately, we have not succeeded to find more examples of even-degeneracy eigenvalues with $m>1$. Hence, it appears that this ansatz cannot account for all the missing eigenvalues.

6 Conclusions

One of the aims of this paper is to draw attention to the unexpected difficulty in solving the integrable quantum-group invariant $D_{n+1}^{(2)}$ spin chains with $\varepsilon=1$. We focused for simplicity on the case $n=1$. Using standard assumptions, we arrived at a Bethe ansatz solution (3.26)-(3.28) that is not complete. Indeed, the argument in Sec. 3.2 could be regarded as a “no-go theorem”, which we hope will motivate others to find a better approach.

We believe that we did succeed to describe a subset of the transfer-matrix eigenvalues, namely, those with odd degeneracy. The remarkably simple solution (3.32)-(3.33) suggests that there may be a connection to the XXZ model. Unfortunately, as $N$ increases, the fraction of eigenvalues with odd degeneracy rapidly decreases.

The “missing” eigenvalues (namely, those with even degeneracy) may be related to a higher symmetry of the transfer matrix, as discussed in Sec. 3.3.1. It would be interesting to also understand this symmetry better. The ansatz (5.3)-(5.5) may also provide a hint about the eventual complete solution.

Acknowledgments

We thank Niall Robertson and Hubert Saleur for encouragement, and Nicolas Crampé for discussions. RN also thanks Wen-Li Yang for valuable discussions, and for his warm hospitality at the Institute of Modern Physics at Northwest University in Xian. AR also thanks Marius de Leeuw, Anton Pribytok and Paul Ryan for helpful discussions. RN was supported in part by the Chinese Academy of Sciences President’s International Fellowship Initiative Grant No. 2018VMA0017, and by a Cooper fellowship. AR was supported by the São Paulo Research Foundation FAPESP under the process # 2017/03072-3 and # 2015/00025-9. RP thanks the Institut Denis Poisson for hospitality and the support from FAPESP and the Coordination for the Improvement of Higher Education Personnel (CAPES), process # 2017/02987-8 and #88881.171877/2018-01

Appendix A First steps of the algebraic Bethe ansatz

In this appendix we construct 1-particle states of the transfer matrix (2.13) by means of the algebraic Bethe ansatz. We restrict our attention to the case $\varepsilon=1$ with $p=0$; and, for simplicity, we set the inhomogeneities to zero, $\theta_{j}=0$. The results are consistent with the $m=1$ case of the ansatz (5.1).

We start by setting the following representation for the double-row monodromy matrix in the auxiliary space

[TABLE]

whose elements satisfy the reflection algebra

[TABLE]

It is convenient to also define the following double-row operators

[TABLE]

From the form of the R and K-matrices, it follows that the action of the double-row operators on the reference state |0\rangle=\left(\begin{array}[]{c}1\\ 0\\ 0\\ 0\end{array}\right)^{\otimes N} is given by

[TABLE]

where

[TABLE]

It follows from (A.5) that the transfer matrix $t(u)=\mathop{\rm tr}\nolimits_{a}K_{a}^{L}(u)U_{a}(u)$ can be written as

[TABLE]

from which it follows, using (A.8), that $|0\rangle$ is an eigenstate of the transfer matrix with eigenvalue given by $m=0$ in (3.13) or (5.1).

Having settled the reference state, the next step is to identify a 1-particle creation operator. We find that $\mathcal{B}_{1}(v)$ is suitable for that purpose. Indeed, we can use the reflection algebra (A.6) to obtain the following relations333Specifically, we use the following matrix elements of the reflection algebra (A.6): from the entry $(1,5)$, we obtain $\mathcal{D}_{1}(u)\mathcal{B}_{1}(v)|0\rangle$. Next, we use entries $(1,9)$ and $(5,13)$ to extract $\mathcal{D}_{1}(u)\mathcal{B}_{2}(v)|0\rangle$ and $\mathcal{D}_{1}(u)\mathcal{B}_{5}(v)|0\rangle$, respectively. This last step allows us to obtain $\mathcal{D}_{2}(u)\mathcal{B}_{1}(v)|0\rangle$ from $(5,6)$. From the entry $(9,13)$, we obtain $\mathcal{D}_{1}(u)\mathcal{B}_{6}(v)|0\rangle$, and then $\mathcal{D}_{3}(u)\mathcal{B}_{1}(v)|0\rangle$ from $(9,10)$. Finally, we obtain $\mathcal{B}(u)\mathcal{B}_{1}(v)|0\rangle$, $\mathcal{C}(u)\mathcal{B}_{1}(v)|0\rangle$ and $\mathcal{D}_{4}(u)\mathcal{B}_{1}(v)|0\rangle$ from the entries $(5,10)$, $(9,6)$ and $(13,14)$, respectively.

[TABLE]

where

[TABLE]

Acting with the transfer matrix (A) on $\mathcal{B}_{1}(v)|0\rangle$ and using the relations (A.13)-(A.15), we obtain the $m=1$ off-shell equation

[TABLE]

where

[TABLE]

and

[TABLE]

Setting $v=u_{1}+\eta$, we identify the first and second terms in (A.18) (involving $\Lambda_{1}(u)$ and $\Lambda_{4}(u)$) as $\phi(u)Z_{1}(u)$ and $\phi(u)Z_{4}(u)$ in (5.1), respectively. Moreover, by noticing that for $m=1$ the function $f_{2}(u,v)$ can be written as

[TABLE]

as well as the identity (valid for $m=1$)

[TABLE]

we conclude that the T-Q equation (A.18) is consistent with the ansatz (5.1) for $m=1$. In addition, we see that the Bethe equations (5.5) for $m=1$ correspond to one of the possible branches of $E(u_{1}+\eta)=0$. We also remark that (A.18) has the form of an inhomogeneous T-Q equation.