Q-operators for higher spin eight vertex models with a rational anisotropy parameter

Abstract

Q-operators for generalised eight vertex models associated with higher spin representations of the Sklyanin algebra are constructed by Baxter’s first method and Fabricius’s method, when the anisotropy parameter is rational.

Appendices

Sklyanin algebra

In this Appendix, we recall several facts on the Sklyanin algebra and its representations from [26, 27].

The Sklyanin algebra is an associative algebra generated by four generators $S^a$ ($a=0,\ldots ,3$) subject to the following relations:

$$\begin{aligned} L_{12}(v) L_{13}(u) R_{23}(u-v) = R_{23}(u-v) L_{13}(u) L_{12}(v). \end{aligned}$$

(A.1)

Here, the symbols are defined as follows:

• the L-operatorL(u) with a complex parameter u is defined by

  $$\begin{aligned} L(u) = \sum _{a=0}^3 W_a^L(u) S^a \otimes \sigma ^a, \end{aligned}$$

  (A.2)

  where

  $$\begin{aligned} \begin{aligned} W_0^L(u)&= \frac{\theta _{11}(u,\tau )}{\theta _{11}(\eta ,\tau )},&W_1^L(u)&= \frac{\theta _{10}(u,\tau )}{\theta _{10}(\eta ,\tau )},\\ W_2^L(u)&= \frac{\theta _{00}(u,\tau )}{\theta _{00}(\eta ,\tau )},&W_3^L(u)&= \frac{\theta _{01}(u,\tau )}{\theta _{01}(\eta ,\tau )}. \end{aligned} \end{aligned}$$

  (A.3)

• The matrix R(u) is Baxter’s R-matrix defined by

  $$\begin{aligned} R(u) = \sum _{a=0}^3 W_a^R(u) \sigma ^a \otimes \sigma ^a,\qquad W_a^R(u) := W_a^L(u + \eta ). \end{aligned}$$

  (A.4)

  Explicitly, it has the form (cf. [29] “Appendix A”)

  $$\begin{aligned} R(u) = \begin{pmatrix} a(u) &{}\quad 0 &{}\quad 0 &{}\quad d(u)\\ 0 &{}\quad b(u) &{}\quad c(u) &{}\quad 0 \\ 0 &{}\quad c(u) &{}\quad b(u) &{}\quad 0 \\ d(u) &{}\quad 0 &{}\quad 0 &{}\quad a(u) \end{pmatrix}, \end{aligned}$$

  (A.5)

  where

  $$\begin{aligned} a(u)&= C \theta _{01}(2it\eta ,2it)\,\theta _{01}(itu,2it) \,\theta _{11}(it(u+2\eta ),2it) \\ b(u)&= C \theta _{01}(2it\eta ,2it)\,\theta _{11}(itu,2it) \,\theta _{01}(it(u+2\eta ),2it) \\ c(u)&= C \theta _{11}(2it\eta ,2it)\,\theta _{01}(itu,2it) \,\theta _{01}(it(u+2\eta ),2it) \\ d(u)&= C \theta _{11}(2it\eta ,2it)\,\theta _{11}(itu,2it) \,\theta _{11}(it(u+2\eta ),2it) \\ C&= \frac{-2 \text {e}^{-\pi t u(u+2\eta )}}{\theta _{01}(0,2it)\, \theta _{01}(2it\eta ,2it)\,\theta _{11}(2it\eta ,2it)},\quad t=\frac{i}{\tau }. \end{aligned}$$

• The indices designate the spaces on which operators act non-trivially. For example,

  $$\begin{aligned} L_{12}(u) = \sum _{a=0}^3 W_a^L(u) S^a \otimes \sigma ^a \otimes 1,\qquad R_{23}(u) = \sum _{a=0}^3 W_a^R(u) 1 \otimes \sigma ^a \otimes \sigma ^a. \end{aligned}$$

Although the relation (A.1) contains parameters u and v, the commutation relations among $S^a$ ($a=0, \dots , 3$) do not depend on them:

$$\begin{aligned}{}[S^\alpha , S^0 ]_- = -i J_{\alpha ,\beta } [S^\beta ,S^\gamma ]_+, \qquad [S^\alpha , S^\beta ]_- = i [S^0, S^\gamma ]_+, \end{aligned}$$

(A.6)

where $(\alpha , \beta , \gamma )$ stands for an arbitrary cyclic permutation of (1, 2, 3) and $[A,B]_\pm$ are the (anti-)commutator $AB\pm BA$. The structure constants $J_{\alpha ,\beta } = ((W^L_\alpha )^2-(W^L_\beta )^2)/((W^L_\gamma )^2-(W^L_0)^2)$ depend on $\tau$ and $\eta$ but not on u.

Let l be a positive half integer. The spin l representation$\rho ^{l}$ of the Sklyanin algebra is defined as follows: The representation space is a space of entire functions,

$$\begin{aligned} \varTheta ^{4l+}_{00}:= & {} \{f(z) \, |\, \nonumber \\ f(z+1)= & {} f(-z) = f(z), f(z+\tau )=\exp ^{-4l\pi i(2z+\tau )}f(z) \}, \end{aligned}$$

(A.7)

which is of dimension $2l+1$. The generator $S^a$ of the Sklyanin algebra acts as a difference operator on this space:

$$\begin{aligned} (\rho ^l(S^a) f)(z) = \frac{s_a(z-l\eta )f(z+\eta )-s_a(-z-l\eta )f(z-\eta )}{\theta _{11}(2z,\tau )}, \end{aligned}$$

(A.8)

where

$$\begin{aligned} s_0(z)&= \theta _{11}(\eta ,\tau ) \theta _{11}(2z,\tau ),\quad&s_1(z)&= \theta _{10}(\eta ,\tau ) \theta _{10}(2z,\tau ),\\ s_2(z)&= i\theta _{00}(\eta ,\tau ) \theta _{00}(2z,\tau ),\quad&s_3(z)&= \theta _{01}(\eta ,\tau ) \theta _{01}(2z,\tau ). \end{aligned}$$

In the simplest case $l= 1/2$, $\rho ^{1/2}(S^a)$ are expressed by the Pauli matrices $\sigma ^a$. We can identify $\varTheta ^{2+}_{00}$ and $\mathbb {C}^2$ by

$$\begin{aligned} \begin{aligned} \theta _{00}(2z,2\tau )-\theta _{10}(2z,2\tau )&\longleftrightarrow \begin{pmatrix} 1 \\ 0 \end{pmatrix},\\ \theta _{00}(2z,2\tau )+\theta _{10}(2z,2\tau )&\longleftrightarrow \begin{pmatrix} 0 \\ 1 \end{pmatrix}. \end{aligned} \end{aligned}$$

(A.9)

Under this identification $S^a$ have matrix forms

$$\begin{aligned} \rho ^{1/2}(S^a) = \theta _{11}(2\eta ,\tau ) \sigma ^a. \end{aligned}$$

(A.10)

The representation space $\varTheta ^{4l+}_{00}$ has a natural Hermitian structure defined by the following Sklyanin form:

$$\begin{aligned} \left\langle f(z), g(z)\right\rangle := \int _0^1 \text {d}x \int _0^{\tau /i} \text {d}y\, \overline{f(z)} g(z) \mu (z,\bar{z}), \end{aligned}$$

(A.11)

where $z=x+iy$ and the kernel function $\mu (z,w)$ is defined by

$$\begin{aligned} \mu (z,w) := \frac{\theta _{11}(2z,\tau ) \theta _{11}(2w,\tau )}{ \prod _{j=0}^{2l+1} \theta _{00}(z+w+(2j-2l-1)\eta ,\tau )\theta _{00}(z-w+(2j-2l-1)\eta ,\tau ) }. \end{aligned}$$

(A.12)

The most important property of this sesquilinear positive definite scalar product is that the generators $S^a$ of the Sklyanin algebra become self-adjoint:

$$\begin{aligned} (S^a)^* = S^a,\text { namely, } \langle f(z), S^a g(z) \rangle = \langle S^a f(z), g(z) \rangle . \end{aligned}$$

(A.13)

In [27] Sklyanin also defined involutive automorphisms:

$$\begin{aligned} X_a: (S^0, S^a, S^b, S^c) \mapsto (S^0, S^a, -S^b, -S^c), \end{aligned}$$

(A.14)

for $a=1,2,3$, where (a, b, c) is a cyclic permutation of (1, 2, 3). The unitary operators $U_a$ defined by

$$\begin{aligned} \begin{aligned} U_1:&\varTheta ^{4\ell +}_{00} \ni f(z) \mapsto&(U_1 f)(z) = \text {e}^{\pi i \ell } f\left( z + \frac{1}{2} \right) ,\\ U_3:&\varTheta ^{4\ell +}_{00} \ni f(z) \mapsto&(U_3 f)(z) = \text {e}^{\pi i \ell } \text {e}^{\pi i \ell (4z+\tau )} f\left( z + \frac{\tau }{2} \right) , \end{aligned} \end{aligned}$$

(A.15)

and $U_2= U_3 U_1$, intertwine representations $\rho ^\ell \circ X_a$ and $\rho ^{\ell }$: $\rho ^\ell (X_a(S^b)) = U_a^{-1} \rho ^\ell (S^b) U_a$. Operators $U_a$ satisfy the relations: $U_a^2 = (-1)^{2\ell }$, $U_a U_b = (-1)^{2\ell } U_b U_a = U_c$.

Values of Sklyanin forms of elements of $\varTheta ^{4l++}_{00}$

In Appendix B of [31] [equation (B.14)], we have computed the Sklyanin form of two shifted products of theta functions, using the results by Rosengren, [24, 25] (see also Konno’s work [17]):

$$\begin{aligned} \begin{aligned}&\langle [z;\alpha ]_N, [z;\gamma ]_N \rangle \\&\quad ={} C_{N} \text {e}^{\pi i N \tau /2} \prod _{j=0}^{N-1} \theta _{00}(\gamma -\bar{\alpha }+ (2j-N+1)\eta ,\tau ) \theta _{00}(\gamma +\bar{\alpha }{+} (2j{+}N{-}1)\eta ,\tau ), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} C_N = \frac{-2\eta \text {e}^{3\pi i \tau /4}}{[2(N+1)\eta ] \prod _{j=1}^\infty (1-\text {e}^{2j \pi i \tau })^3}. \end{aligned}$$

Hence, the Sklyanin form of $f_\varepsilon (\lambda ,-\bar{u},z)$ and $f_{\varepsilon '}(\lambda ',v,z)$ ($\varepsilon ,\varepsilon '=\pm$, $\lambda ,\lambda '\in \mathbb {R}$, cf. (3.13)) has the following form:

$$\begin{aligned} \begin{aligned}&\langle f_\varepsilon (\lambda ,-\bar{u},z), f_{\varepsilon '}(\lambda ',v,z) \rangle \\&\quad ={} F\left( \frac{\varepsilon '\lambda -\varepsilon \lambda }{2}+\frac{v+u}{2} \right) G\left( \frac{\varepsilon '\lambda +\varepsilon \lambda }{2}+\frac{v-u}{2} + 2(2l-1)\eta \right) , \end{aligned} \end{aligned}$$

(B.1)

where F and G are defined by

$$\begin{aligned} F(z)&:= C_{2l} \text {e}^{\pi i l \tau } \prod _{j=0}^{2l-1} \theta _{00}(z + (2j-2l+1)\eta ,\tau ), \end{aligned}$$

(B.2)

$$\begin{aligned} G(z)&:= \prod _{j=0}^{2l-1} \theta _{00}(z + (2j+2l-1)\eta - 2(2l-1)\eta ,\tau ). \end{aligned}$$

(B.3)

We shifted the argument in G(z) so that G(z) becomes an even function:

$$\begin{aligned} G(-z) = G(z). \end{aligned}$$

(B.4)

It has also the periodicity:

$$\begin{aligned} G(z+1)=G(z), \end{aligned}$$

(B.5)

because of the periodicity of $\theta _{00}$.

In Sect. 4.2, we need the Sklyanin form among $\omega _\lambda (u,v)$’s defined by (4.4). The following formula is useful.

$$\begin{aligned} \begin{aligned}&\langle \omega _{\sigma \lambda }(-\bar{u},\sigma v ), \omega _{\sigma '\lambda '}( u',\sigma ' v') \rangle \\&\quad = C'_{2l}\, \theta ^{(2l)}_{00} \left( \frac{(\lambda '-v')-\overline{(\lambda -v)}}{2}+ \frac{\sigma 'u'+\sigma u}{2}+(\sigma '-\sigma )l\eta \right) \\&\qquad \times \theta ^{(2l)}_{00} \left( \frac{(\lambda '-v')+\overline{(\lambda -v)}}{2}+ \frac{\sigma 'u'-\sigma u}{2}+(\sigma '+\sigma )l\eta \right) , \end{aligned} \end{aligned}$$

(B.6)

where

$$\begin{aligned} \begin{aligned} C'_{2l}&:=C_{2l} \text {e}^{\pi i l \tau },\\ \theta ^{(2l)}_{00}(u)&:= \prod _{j=0}^{2l-1} \theta _{00}( u + (2j-2l+1)\eta , \tau ). \end{aligned} \end{aligned}$$

(B.7)