Novel integrability in string theory from automorphic symmetries

Abstract

We develop a technique based on the boost automorphism for finding new lattice integrable models with various dimensions of local Hilbert spaces. We initiate the method by implementing it in two-dimensional models and resolve a classification problem, which not only confirms the known vertex model solution space but also extends to the new $\mathfrak{sl}_2$ deformed sector. A generalization of the approach to integrable string backgrounds is provided and allows finding new integrable deformations and associated $R$-matrices. The new integrable solutions appear to be of a nondifference or pseudo-difference form admitting $AdS_2$ and $AdS_3$ $S$-matrices as special cases (embeddings), which also includes a map of the double-deformed sigma model $R$-matrix. The corresponding braiding and conjugation operators of the novel models are derived. We also demonstrate implications of the obtained free-fermion analogue for $AdS$ deformations.

Appendix A: Difference and nondifference boost automorphic symmetry

In this Appendix, we elaborate on the algebraic structure and analytic properties of the boost automorphism on the lattice and its implementation for integrable models, whose scattering operators exhibit an arbitrary spectral dependence (including a pseudodifference one). Certain relevant details of the elliptic apparatus are also provided.

Boost automorphism on a lattice

We analyze the structure of the boost operator on a lattice systems and see how the automorphic condition emerges [13], [15]. It is known that the generating property of the boost automorphism is based on integrable master symmetries (initially noted for constant integrable models). We note that it is useful to consider the $R\mathcal L\mathcal L$ relation along the lines of QISM [4], [6], which can be written in the form

$$ R_{12}(v)\mathcal L_{10}(u+v)\mathcal L_{20}(u)=\mathcal L_{20}(u)\mathcal L_{10}(u+v)R_{12}(v).$$

(A.1)

It is instructive to study the regular points of the $R$-matrix. For this, we can take derivative with respect to one of the spectral parameters, e.g., $v$, and subsequently set it $v=0$, which after some transformations yields

$$ i[\mathcal L_{10}\mathcal L_{20},\mathcal H_{12}]=\mathcal L_{10}\mathcal L'_{20} -\mathcal L_{10}'\mathcal L_{20},$$

(A.2)

where $\mathcal L_{i0}\equiv\mathcal L_{i0}(u)$ and the conditions

$$ R_{12}^{}(0)=P_{12},\qquad R'_{j,j+1}(0)=d_uR_{j,j+1}^{}=(u)\big|_{u=0}$$

(A.3)

and

$$ \mathcal H_{j,j+1}^{}=iP_{j,j+1}^{}=R'_{j,j+1}(0)$$

(A.4)

are satisfied. In (A.2), we now change from the 1st and 2nd spaces some $k$th and $(k + 1)$th,

$$ i[\mathcal L_{k,0}^{}\mathcal L_{k+1,0}^{},\mathcal H_{k,k+1}^{}]=\mathcal L_{k,0}^{}\mathcal L'_{k+1,0}-\mathcal L_{k,0}'\mathcal L_{k+1,0}^{},$$

(A.5)

and multiply by the monodromic complements from left and right, whence

$$\begin{aligned} \, &i\prod_{j=1}^{k-1}\mathcal L_{0,j}[\mathcal L_{0,k}\mathcal L_{0,k+1},\mathcal H_{k,k+1}]\prod_{n=k+2}^{L}\mathcal L_{0,n}= \nonumber\\ &= \prod_{j=1}^{k-1}\mathcal L_{0,j}^{}\mathcal L_{0,k}^{}\mathcal L'_{0,k+1}\prod_{n=k+2}^{L}\mathcal L_{0,n}^{}- \prod_{j=1}^{k-1}\mathcal L_{0,j}^{}\mathcal L_{0,k}'\mathcal L_{0,k+1}^{}\prod_{n=k+2}^{L}\mathcal L_{0,n}^{}. \end{aligned}$$

(A.6)

which after rearranging the monodromies gives

$$ i\biggl[\,\prod_{j=1}^{L}\mathcal L_{0,j}^{},\mathcal H_{k,k+1}^{}\biggr]= \prod_{j=1}^{k}\mathcal L_{0,j}^{}\mathcal L'_{0,k+1}\prod_{n=k+2}^{L}\mathcal L_{0,n}^{}- \prod_{j=1}^{k-1}\mathcal L_{0,j}^{}\mathcal L_{0,k}'\prod_{n=k+1}^{L}\mathcal L_{0,n}^{}.$$

(A.7)

We multiply this equation by $k$ and sum over $k$ from $1$ to $L$:

$$\begin{aligned} \, i\biggl[\,\prod_{j=1}^{L}\mathcal L_{0,j},&\sum_{k=1}^{L}k\mathcal H_{k,k+1}\biggr]= \sum_{k=1}^{L}k\prod_{j=1}^{k}\mathcal L_{0,j}^{}\mathcal L'_{0,k+1}\prod_{n=k+2}^{L}\mathcal L_{0,n}^{}- \sum_{k=1}^{L}k\prod_{j=1}^{k-1}\mathcal L_{0,j}^{}\mathcal L_{0,k}'\prod_{n=k+1}^{L}\mathcal L_{0,n}^{}. \end{aligned}$$

(A.8)

Recalling the $\mathcal L$-monodromy definition and analyzing the intertwined structure in the right-hand side of (A.8)

$$\begin{aligned} \, i\biggl[\,\sum_{k=1}^{L}k\mathcal H_{k,k+1}^{},T(u)\biggr]&= \sum_{k=1}^{L}k\prod_{j=1}^{k-1}\mathcal L_{0,j}^{}\mathcal L'_{0,k}\!\!\prod_{n=k+1}^{L}\!\!\mathcal L_{0,n}^{}- \sum_{k=1}^{L}k\prod_{j=1}^{k}\mathcal L_{0,j}^{}\mathcal L'_{0,k+1}\!\!\prod_{n=k+2}^{L}\!\!\mathcal L_{0,n}^{}= \nonumber\\ &=d_uT(u)+0\cdot\mathcal L_{0,1}'\prod_{j=2}^{L}\mathcal L_{0,j}^{}-L\cdot\prod_{j=1}^{L}\mathcal L_{0,j}^{}\mathcal L'_{0,L+1} \end{aligned}$$

(A.9)

with the monodromy $T(u)$ and the boundary terms represented by the last line in (A.9). This expression depends on our choice of the initial (1) and final ($L$) site of the spin chain. By considering a consistent infinite limit, i.e., shifting the boundaries to $-\infty\leftarrow 1$, $L\rightarrow+\infty$, we obtain

(A.10)

As expected, the boundary terms vanish and we are left with the boost recursion formula

$$ i[\mathcal B,T]=\dot T,\qquad\mathcal B\equiv\sum_{k=-\infty}^{+\infty}k\mathcal H_{k,k+1}$$

(A.11)

or

$$ i[\mathcal B,t]=\dot t,$$

(A.12)

where $\dot T\equiv d_uT(u)$ and (A.12) corresponds to an analogue of the transfer matrix $t(u)$. We note that (A.11) is nothing but a discrete form (see [15]) of the field theory boost symmetry [13] with a standard discretization scheme $\int x\,dx\mapsto\sum_k k$.Footnote

Importantly, the boost for systems on a lattice is associated to the Drinfeld boost automorphism of $\mathcal Y[\mathfrak{g}]$ [42], [17].

Importantly, the boost for systems on a lattice is associated to the Drinfeld boost automorphism of $\mathcal Y[\mathfrak{g}]$ [42], [17].

A nondifference generalization

We can consider a generalization of the boost symmetry to the case where scattering describing the operator $S$- and $R$-matrix has an arbitrary spectral dependence. To achieve that, we can perform analysis following the steps close to those taken above and derive the corresponding boost expression. More specifically, we recall the coupled differential structure (2.5), which in turn was derived by differentiating the $RTT$ and qYBE relations,

$$ [R_{13} R_{12},\mathcal H_{23}(v)]=R_{13}R_{12,v}-R_{13,v}R_{12},\qquad R_{ij}\equiv R_{ij}(u,v).$$

(A.13)

By the shifts $1\to a$ (auxiliary), $2\to k$, and $3\to k+1$, we generalize this formula to

$$ [R_{a,k+1}R_{a,k},\mathcal H_{k,k+1}]=R_{a,k+1}R'_{a,k}-R'_{a,k+1}R_{a,k}.$$

(A.14)

We can then analyze $R$-monodromies with the appropriate product expansions

$$\begin{aligned} \, &\prod_{j=L}^{k+2}R_{a,j}[R_{a,k+1}R_{a,k},\mathcal H_{k,k+1}]\prod_{n=k-1}^1R_{a,n}= \nonumber\\ &=\prod_{j=L}^{k+2}R_{a,j} R_{a,k+1}R'_{a,k}\prod_{n=k-1}^1R_{a,n}-\prod_{j=L}^{k+2}R_{a,j}R'_{a,k+1}R_{a,k}\prod_{n=k-1}^1R_{a,n}. \end{aligned}$$

(A.15)

After a rearrangement, we have

$$ \biggl[\,\prod_{j=L}^1 R_{a,j},\mathcal H_{k,k+1}\biggr]= \prod_{j=L}^{k+1}R_{a,j} R'_{a,k}\prod_{n=k-1}^1R_{a,n}-\prod_{j=L}^{k+2}R_{a,j} R'_{a,k+1}\prod_{n=k}^1R_{a,n}.$$

(A.16)

We immediately note that there are internal mutual cancellations analogous to those in (A.9). We multiply by $k$ and sum so as to obtain the boost:

$$ \biggl[\,\prod_{j=L}^1 R_{a,j},\sum_k k\mathcal H_{k,k+1}\biggr]= \sum_k k\prod_{j=L}^{k+1}R_{a,j} R'_{a,k}\prod_{n=k-1}^1R_{a,n}-\sum_k k\prod_{j=L}^{k+2}R_{a,j}R'_{a,k+1}\prod_{n=k}^1R_{a,n}.$$

(A.17)

We next take the infinite limit $-\infty\to 1,L\to\infty$ and specifying the sums accordingly, we obtain

$$ \biggl[T_a,\sum_{k=-\infty}^{+\infty}k\mathcal H_{k,k+1}\biggr]=d_v T_a+ \underbrace{1\cdot\prod_{j=L}^2 R_{a,j} R'_{a,1}-L\cdot R'_{a,L+1}\prod_{n=L}^1 R_{a,n}}_{\lim_{\{1,L\}\to\mp\infty} R'_{a,L+1}\prod_{n=L}^1R_{a,n}\to L\cdot\prod_{j=L}^2R_{a,j}R'_{a,1}\,\Rightarrow 0}\kern-21pt.$$

(A.18)

We observe that due to the closed spin-chain periodicity, there are boundary terms that become commutative (boundary factors can swap, i.e., $L+1\mapsto 1$); however, strictly speaking, their absence is also guaranteed at infinity. For finite spin chains and local short-range charges, a consistent unified boost automorphism can be written as

$$ [T_a,\mathcal B]=\dot T_a,\qquad T_a\equiv T_a(u,v),\quad\mathcal H_{k,k+1}\equiv\mathcal H_{k,k+1}(v),$$

(A.19)

where for transfer matrix $t$, we obtain

$$ \biggl[t,\sum_{k=-\infty}^{+\infty}k\mathcal H_{k,k+1}\biggr]=\dot t,$$

(A.20)

with $\dot t\equiv d_vt(u,v)$. The perturbation of $t$ gives rise to

$$ \ln t(u,v)=\mathbb{Q}_1+\mathbb{Q}_2(u-v)+\frac{\mathbb{Q}_3}{2}(u-v)^2+\mathcal O[(u-v)^3],$$

(A.21)

with $\mathbb{Q}_i\equiv\mathbb{Q}_i(v)$. We must take the existence of special spectral points and expansion of the underlying transfer matrix into account. From the automorphic structure of the boost, we can derive, with (A.20) and (A.21), that at each level of the recursion the following contributions arise:

$$ \mathbb{Q}_{r+1}=[\mathcal B,\mathbb{Q}_r(v)]+\alpha_1\,\partial_v\mathbb{Q}(v),\qquad r\ge 2,$$

(A.22)

where $r=1$ is fixed, but (A.22) is applicable in that case as well. Importantly, it is possible to construct a long-range boost operator using the notion of bilocal charges [15], which allow generating dressing contributions to operators (e.g., dilatation of $\mathcal N=4$ SYM [14], [19]) and deriving a long-range deformation of the charges of the associated spin chains, including the Hamiltonians $\mathcal Q_2=\mathcal H$.

Appendix B: $$AdS$$ integrable structures and relations

B.1. $AdS$-$R$-matrix

To compare the $AdS_n$ structures [9], [27], [44] and our models, we need to make manifest the decomposition into chiral blocks of our classes, leading to a consistent framework. We start with a regular $R$-matrix given by a building block for the $16\times 16$ embedding

$$ R^{\mathcal X}(u,v)=\sigma \begin{pmatrix} r_1 & 0 & 0 & r_8 \\ 0 & r_2 & r_6 & 0 \\ 0 & r_5 & r_3 & 0 \\ r_7 & 0 & 0 & r_4 \\ \end{pmatrix},$$

(B.1)

where in comparison with (3.1), we allow an extra free scalar factor $\sigma$ (dressing),Footnote

The block dressing factor is in general identified such that the (corresponding) blocks and the full $R$-matrix obey the braiding unitarity, crossing symmetric relations, and possibly specific auxiliary constraints.

and $\mathcal X$ indicates the corresponding chirality of the block. We have four blocks: two in the pure sector, (LL, RR) and two in the mixed one, (LR, RL). The regularity of the pure blocks also must hold, i.e.,

$$ R^{\text{LL}}(u,u)\sim P,\qquad R^{\text{RR}}(u,u)\sim P,$$

(B.2)

where $P$ is a suitable dimension permutation operator. Shown for the full $16\times 16$ embedding [44], it can be shown that

The block dressing factor is in general identified such that the (corresponding) blocks and the full $R$-matrix obey the braiding unitarity, crossing symmetric relations, and possibly specific auxiliary constraints.

where all entries have acquired a chiral sector label stemming from the chirality of the associated block (B.1). The content of the full $16\times 16$ $R$ [29] will contain $r_k^{\mathcal X}\equiv r_k^{\mathcal X}(u,v)$ entries with chiral sectors denoted by $\mathcal X\in\{\text{LL},\text{RR},\text{LR},\text{RL}\}$. These elements must satisfy the qYBE, which implies that the underlying combinations of chiral blocks satisfy the equations

$$ R_{12}^{\mathrm x_1\mathrm x_2}(u,v)R_{13}^{\mathrm x_1\mathrm x_3}(u,w)R_{23}^{\mathrm x_2\mathrm x_3}(v,w)= R_{23}^{\mathrm x_2\mathrm x_3}(v,w)R_{13}^{\mathrm x_1\mathrm x_3}(z_1,z_3)R_{12}^{\mathrm x_1\mathrm x_2}(z_1,z_2),$$

(B.3)

where each space $\mathbb{V}$ is spanned by the chiralities $\mathrm x_i\in\{\text{L},\text{R}\}$, which hence leads to eight independent qYBEs for $R_{ij}^{\mathrm x_i\mathrm x_j}$.

Based on the above, it is possible to obtain the full $R$ (or the $S$-matrix). For that, we need to solve for all $r_k^{\mathcal X}$, or more appropriately, for the $r_k^{\mathcal X}$ in the mixed sectors [39]. The pure $R$-blocks then serves as an input for a complete solution of (B.3). In this regard, we are left with six independent qYBEs, because in the case where all chiralities are the same (all equal to L or R), the original YBEs are satisfied automatically. It is then natural to consider the chiral combinations $\mathrm x_1=\mathrm x_2=\text{L}$, $\mathrm x_3=\text{R}$, where we solve the associated YBEs with $\text{L}\leftrightarrow\text{R}$ for $r_k^{\text{LR}}$ and $r_k^{\text{RL}}$ accordingly.

The remaining YBEs are used to identify some functions of one parameter. After these are solved for, the corresponding $r_k^{\mathcal X}$ allow assembling the full $R$-matrix and performing the qYBE check.

B.2. Elliptic functions and properties

We describe the apparatus of Jacobi elliptic functions (JEFs), their structure and properties, that are used in the derivation of the 6vB/8vB $AdS_2$- and $AdS_3$ deformations. These functions are also needed in the new deformations obtained by solving differential systems, for the proof of free-fermion property, and for other string constraints.

A meromorphic function that is double-periodic is called an elliptic function. An elliptic integral is any integral of the class

$$ \int\frac{A+B}{C+D\sqrt{S}}\,dx,$$

(B.4)

where $A$, $B$, $C$, and $D$ are polynomials in $x$, but $S$ is a polynomial in $x$ of degree three or four. The class of elliptic integrals can be viewed as a generalization of the inverse trigonometric functions.

The first-kind elliptic integral can be formulated as the incomplete Legendre elliptic integral

$$ F(\phi,k)=\int_{0}^{\phi}\frac{1}{\sqrt{1-k^2\sin^2\omega}}\,d\omega,\qquad 0\le k^2\le 1,\quad 0\le\phi\le\frac{\pi}{2},$$

(B.5)

where $k$ is elliptic modulus, and at $\phi=\pi/2$ it is called complete. Similarly, the second-kind elliptic integral is

$$ E(\phi, k)=\int_{0}^{\phi}\sqrt{1-k^2\sin^2\omega}\,d\omega.$$

(B.6)

It useful to define the quarter periods [43] through the 1st of elliptic integral

$$ \begin{aligned} \, &K(k)=\int_{0}^{\pi/2}\frac{1}{\sqrt{1-k^2\sin^2\omega}}\,d\omega, \\ &iK'(k)=i\int_{0}^{\pi/2}\frac{1}{\sqrt{1-k^{\prime\,2}\sin^2\omega}}\,d\omega,\qquad k+k'=1, \end{aligned}$$

(B.7)

where $k+k'=1$, $K(k)=K'(k')=K'(1-k)$ and $K,K'\in\mathbb{R}$ are called real and imaginary periods.

We can identify the vertices of the rectangle as $0$, $K$, $K+iK'$, and $iK'$ by s, c, d, n (Argand diagram). The corresponding translations within the rectangle by $\alpha K$ and $i\beta K'$ (with $\alpha,\beta\lessgtr 0$) induce

$$ \begin{array}{c|c|c|c} \mathrm s & \mathrm c & \mathrm s & \mathrm c\\ \hline \mathrm n & \mathrm d & \mathrm n & \mathrm d\\ \hline \mathrm s & \mathrm c & \mathrm s & \mathrm c\\ \hline \mathrm n & \mathrm d & \mathrm n & \mathrm d \end{array}\;\;,$$

(B.8)

where any $p,q\in\{\mathrm s,\mathrm c,\mathrm d,\mathrm n\}$ define a Jacobi elliptic function $pq(u)$ with the following properties:

• • $pq(u)$ has a simple zero $p$ and a simple pole at $q$;

• • the step $p\to q$ is a half period of $pq(u)$;

• • the leading coefficient of the $pq(u)$ expansion in $u$ in the vicinity of zero is unity (the leading term is $u$, $u^{-1}$, or $1$ if and only if $u=0$ corresponds to a zero, a pole, or a regular point).

In terms of the first-kind integral,. the JEFs are

$$ u=\int_{0}^{\phi}\frac{1}{\sqrt{1-k^2\sin^2\omega}}\,d\omega,$$

(B.9)

where the $\phi=\operatorname{am}(u)$ is an amplitude. The standard definition is

$$ \operatorname{sn} {u}=\sin\phi,\qquad \operatorname{cn} {u}=\cos\phi,\qquad \operatorname{dn} {u}=\sqrt{1-k^2\sin^2\phi}.$$

(B.10)

Similarly, all JEFs can expressed through $\phi$. Moreover, the rest of the $pq(u)$ functions can be defined as

$$ pq(u)=\dfrac{pr(u)}{qr(u)}$$

(B.11)

with $p,q,r\in\{\mathrm s,\mathrm c,\mathrm d,\mathrm n\}$ and for the coinciding symbols, the function is unity. The properties of the squares can be given as

$$ \begin{alignedat}{3} & \operatorname{sn} ^2+ \operatorname{cn} ^2=1,&\qquad &k^2 \operatorname{cn} ^2+k^{\prime\,2}= \operatorname{dn} ^2, \\[2mm] &k^2 \operatorname{sn} ^2+ \operatorname{dn} ^2=1,&\qquad & \operatorname{cn} ^2+k^{\prime\,2} \operatorname{sn} ^2= \operatorname{dn} ^2, \end{alignedat}$$

(B.12)

where $\mathrm{xn}\equiv\mathrm{xn}(u)$. The period shifts are

$$ \begin{alignedat}{3} & \operatorname{sn} (u+K)=\frac{ \operatorname{cn} (u)}{ \operatorname{dn} (u)},&\qquad & \operatorname{sn} (u+2K)=- \operatorname{sn} (u), \\[2mm] & \operatorname{cn} (u+K)=-\frac{k' \operatorname{sn} (u)}{ \operatorname{dn} (u)},&\qquad & \operatorname{cn} (u+2K)=- \operatorname{cn} (u), \\[2mm] & \operatorname{dn} (u+K)=\frac{k'}{ \operatorname{dn} (u)},&\qquad & \operatorname{dn} (u+2K)= \operatorname{dn} (u), \end{alignedat}$$

(B.13)

and the the formulas for a half period and a double period are

$$ \begin{alignedat}{3} & \operatorname{sn} \biggl(\frac{K}{2}\biggr)=(1+k')^{-1/2},&\qquad & \operatorname{sn} (2u)=\frac{2 \operatorname{sn} (u) \operatorname{cn} (u) \operatorname{dn} (u)}{1-k^2 \operatorname{sn} ^4(u)}, \\[2mm] & \operatorname{cn} \biggl(\frac{K}{2}\biggr)=\biggl(\frac{k'}{1+k'}\biggr)^{\!1/2},&\qquad & \operatorname{cn} (2u)=\frac{1-2 \operatorname{sn} ^2(u)+k^2 \operatorname{sn} ^4(u)}{1-k^2 \operatorname{sn} ^4(u)}, \\[2mm] & \operatorname{dn} \biggl(\frac{K}{2}\biggr)=k^{\prime\,1/2},&\qquad & \operatorname{dn} (2u)=\frac{1-2k^2 \operatorname{sn} ^2(u)+k^2 \operatorname{sn} ^4(u)}{1-k^2 \operatorname{sn} ^4(u)}. \end{alignedat}$$

(B.14)

Appendix C: The $$AdS_3\times S^3\times\mathcal M^4$$ string sector

Here, we explicitly give $AdS_3\times S^3\times T^4$ [44], [32], [34] and $AdS_3\times S^3\times S^3\times S^1$ [35] in the form that allows a comparative analysis compatible with the 6vB/8vB model deformations. In the first case, we give the $S$- to $R$-matrices where all quantum constraints are explicitly satisfied ($\gamma(p)$ corresponds to $\eta(p)$ in [44], [32]–[35]).

C.1. The case $AdS_3\times S^3\times T^4$

From the $S$-/$R$-matrix in [34], [44], we can work out a block of the form

$$ R^\chi=\zeta^\chi \begin{pmatrix} r_1^\chi & 0 & 0 & r_8^\chi \\ 0 & r_2^\chi & r_6^\chi & 0 \\ 0 & r_5^\chi & r_3^\chi & 0 \\ r_7^\chi & 0 & 0 & r_4^\chi \\ \end{pmatrix}, \qquad r_i^\chi\equiv r_i^\chi(p,q),\quad \zeta^\chi\equiv\zeta^\chi(p,q),$$

(C.1)

where $\chi$ labels chiral sectors $\chi\in\{\text{LL},\text{RR},\text{LR},\text{RL}\}$. We present all the four blocks from which the full $R$-matrix is assembled. The pure blocks (LL, RR) are

$$\begin{aligned} \, &r_1^{\text{LL}}=-\sqrt{\frac{x^{-}(q)}{x^{+}(q)}\frac{x^{+}(p)}{x^{-}(p)}}\frac{x^{-}(p)-x^{+}(q)}{x^{-}(q)-x^{+}(p)},\qquad r_2^{\text{LL}}=\sqrt{\frac{x^{-}(q)}{x^{+}(q)}}\frac{x^{+}(p)-x^{+}(q)}{x^{+}(p)-x^{-}(q)}, \\ &r_3^{\text{LL}}=\sqrt{\frac{x^{+}(p)}{x^{-}(p)}}\frac{x^{-}(p)-x^{-}(q)}{x^{+}(p)-x^{-}(q)},\qquad r_4^{\text{LL}}=-1,\qquad \\ &r_5^{\text{LL}}=-\biggl(\frac{x^{-}(q)}{x^{+}(q)}\frac{x^{+}(p)}{x^{-}(p)}\biggr)^{\!1/4} \frac{x^{+}(q)-x^{-}(q)}{x^{+}(p)-x^{-}(q)}\frac{\gamma(p)}{\gamma(q)}, \\ &r_6^{\text{LL}}=-\biggl(\frac{x^{-}(q)}{x^{+}(q)}\frac{x^{+}(p)}{x^{-}(p)}\biggr)^{\!1/4} \frac{x^{+}(p)-x^{-}(p)}{x^{+}(p)-x^{-}(q)}\frac{\gamma(q)}{\gamma(p)}; \\ &r_1^{\text{RR}}=-\sqrt{\frac{x^{-}(q)}{x^{+}(q)}\frac{x^{+}(p)}{x^{-}(p)}}\frac{x^{-}(p)-x^{+}(q)}{x^{-}(q)-x^{+}(p)},\vphantom{r_1^{\bigg|}} \\ &r_2^{\text{RR}}=\sqrt{\frac{x^{-}(q)}{x^{+}(q)}}\frac{x^{+}(p)-x^{+}(q)}{x^{+}(p)-x^{-}(q)},\qquad r_3^{\text{RR}}=\sqrt{\frac{x^{+}(p)}{x^{-}(p)}}\frac{x^{-}(p)-x^{-}(q)}{x^{+}(p)-x^{-}(q)}, \\ &r_4^{\text{RR}}=-1,\qquad r_5^{\text{RR}}=-\biggl(\frac{x^{-}(q)}{x^{+}(q)}\frac{x^{+}(p)}{x^{-}(p)}\biggr)^{\!3/4}\frac{x^{+}(p)-x^{-}(p)}{x^{+}(p)-x^{-}(q)}\frac{\gamma(q)}{\gamma(p)}, \\ &r_6^{\text{RR}}=-\biggl(\frac{x^{-}(p)}{x^{+}(p)}\frac{x^{+}(q)}{x^{-}(q)}\biggr)^{\!1/4}\frac{x^{+}(q)-x^{-}(q)}{x^{+}(p)-x^{-}(q)}\frac{\gamma(p)}{\gamma(q)}, \\ &r_7^{\text{RR}}=0,\qquad r_8^{\text{RR}}=0.\vphantom{|^{\big|^*}} \end{aligned}$$

For the mixed blocks (LR, RL), we have

$$\begin{aligned} \, &r_1^{\text{LR}}=\sqrt{\frac{x^{+}(p)}{x^{-}(p)}\frac{x^{+}(q)}{x^{-}(q)}}\frac{1-x^{+}(p)x^{-}(q)}{1-x^{+}(p)x^{+}(q)},\qquad r_2^{\text{LR}}=\frac{x^{+}(p)}{x^{-}(p)}\sqrt{\frac{x^{+}(q)}{x^{-}(q)}}\frac{1-x^{-}(p)x^{-}(q)}{1-x^{+}(p)x^{+}(q)}, \\ &r_3^{\text{LR}}=\sqrt{\frac{x^{+}(p)}{x^{-}(p)}},\qquad r_4^{\text{LR}}=-\frac{x^{+}(p)}{x^{-}(p)}\frac{1-x^{+}(q)x^{-}(p)}{1-x^{+}(p)x^{+}(q)},\qquad r_5^{\text{LR}}=0,\qquad r_6^{\text{LR}}=0, \\ &r_7^{\text{LR}}=\biggl(\frac{x^{+}(p)}{x^{-}(p)}\frac{x^{+}(q)}{x^{-}(q)}\biggr)^{\!3/4} \frac{x^{+}(q)-x^{-}(q)}{1-x^{+}(p)x^{+}(q)}\frac{\gamma(p)}{\gamma(q)}, \\ &r_8^{\text{LR}}=-\biggl(\frac{x^{-}(q)}{x^{+}(q)}\biggr)^{\!1/4} \biggl(\frac{x^{+}(p)}{x^{-}(p)}\biggr)^{\!3/4}\frac{x^{+}(p)-x^{-}(p)}{1-x^{+}(p)x^{+}(q)}\frac{\gamma(q)}{\gamma(p)}; \\ &r_1^{\text{RL}}=\sqrt{\frac{x^{-}(p)}{x^{+}(p)}\frac{x^{-}(q)}{x^{+}(q)}}\frac{1-x^{+}(p)x^{-}(q)}{1-x^{-}(p)x^{-}(q)},\vphantom{r_1^{\bigg|}} \qquad r_2^{\text{RL}}=\frac{x^{-}(q)}{x^{+}(q)}\sqrt{\frac{x^{-}(p)}{x^{+}(p)}}\frac{1-x^{+}(p)x^{+}(q)}{1-x^{-}(p)x^{-}(q)}, \\ &r_3^{\text{RL}}=\sqrt{\frac{x^{-}(q)}{x^{+}(q)}},\qquad r_4^{\text{RL}}=-\frac{x^{-}(q)}{x^{+}(q)}\frac{1-x^{+}(q)x^{-}(p)}{1-x^{-}(p)x^{-}(q)},\qquad r_5^{\text{RL}}=0,\qquad r_6^{\text{RL}}=0, \\ &r_7^{\text{RL}}=\biggl(\frac{x^{-}(q)}{x^{+}(q)}\biggr)^{\!3/4} \biggl(\frac{x^{+}(p)}{x^{-}(p)}\biggr)^{\!1/4}\frac{x^{+}(p)-x^{-}(p)}{1-x^{-}(p)x^{-}(q)}\frac{\gamma(q)}{\gamma(p)}, \\ &r_8^{\text{RL}}=-\biggl(\frac{x^{-}(p)}{x^{+}(p)}\frac{x^{-}(q)}{x^{+}(q)}\biggr)^{\!3/4} \frac{x^{+}(q)-x^{-}(q)}{1-x^{-}(p)x^{-}(q)}\frac{\gamma(p)}{\gamma(q)}. \end{aligned}$$

It is important to note that the formulas are different from those in [34], [44]. Specifically, for the above blocks to be completely in the Zhukovski space, the identification

$$ e^{ip}=\frac{x^{+}(p)}{x^{-}(p)}$$

(C.2)

was implemented. Blocks (C.1) satisfy the qYBE for any $\gamma(p)$, and therefore

$$ \gamma(p)=e^{\frac{ip}{4}}\sqrt{\frac{ih}{2}(x^{-}(p)-x^{+}(p))}.$$

(C.3)

This restores the $R$-matrix in [44].

C.2. The case $AdS_3\times S^3\times S^3\times S^1$

For $\mathcal M^4=S^3\times S^1$, along the lines of [35], we also pass to the proceed blocks of the form

$$ R^\chi=\rho^\chi\upsilon^\chi \begin{pmatrix} r_1^\chi & 0 & 0 & r_8^\chi \\ 0 & r_2^\chi & r_6^\chi & 0 \\ 0 & r_5^\chi & r_3^\chi & 0 \\ r_7^\chi & 0 & 0 & r_4^\chi \end{pmatrix}, \qquad \begin{aligned} \, r_i^\chi&\equiv r_i^\chi(p,q),\\ \rho^\chi&\equiv\rho^\chi(p,q),\\ \upsilon^\chi&\equiv\upsilon^\chi(p,q), \end{aligned}$$

(C.4)

where again $\chi\in\{\text{LL},\text{RR},\text{LR},\text{RL}\}$. Hence, in the LL and RR sectors, we have

$$\begin{aligned} \, &r_1^{\text{LL}}=1,\qquad r_2^{\text{LL}}=\sqrt{\frac{x^{-}_{{\text L}}(p)}{x^{+}_{{\text L}}(p)}} \frac{x^{+}_{{\text L}}(p)-x^{+}_{{\text L}}(q)}{x^{-}_{{\text L}}(p)-x^{-}_{{\text L}}(q)},\qquad r_3^{\text{LL}}=\sqrt{\frac{x^{+}_{{\text L}}(q)}{x^{-}_{{\text L}}(q)}} \frac{x^{-}_{{\text L}}(p)-x^{-}_{{\text L}}(q)}{x^{-}_{{\text L}}(p)-x^{+}_{{\text L}}(q)}, \\ &r_4^{\text{LL}}= \sqrt{\frac{x^{-}_{{\text L}}(p)}{x^{+}_{{\text L}}(p)} \frac{x^{+}_{{\text L}}(q)}{x^{-}_{{\text L}}(q)}} \frac{x^{-}_{{\text L}}(q)-x^{+}_{{\text L}}(p)}{x^{-}_{{\text L}}(p)-x^{+}_{{\text L}}(q)},\qquad r_5^{\text{LL}}=\sqrt{\frac{x^{-}_{{\text L}}(p)}{x^{+}_{{\text L}}(p)} \frac{x^{+}_{{\text L}}(q)}{x^{-}_{{\text L}}(q)}} \frac{x^{-}_{{\text L}}(q)-x^{+}_{{\text L}}(q)}{x^{-}_{{\text L}}(p)-x^{+}_{{\text L}}(q)} \frac{\gamma^{\text{L}}(p)}{\gamma^{\text{L}}(q)}, \\ &r_6^{\text{LL}}= \frac{x^{-}_{{\text L}}(p)-x^{+}_{{\text L}}(p)}{x^{-}_{{\text L}}(p)-x^{+}_{{\text L}}(q)} \frac{\gamma^{\text{L}}(q)}{\gamma^{\text{L}}(p)},\qquad r_7^{\text{LL}}=0,\qquad r_8^{\text{LL}}=0; \\ &r_1^{\text{RR}}=1,\qquad\vphantom{r_1^{\bigg|}} r_2^{\text{RR}}= \sqrt{\frac{x^{-}_{{\text R}}(p)}{x^{+}_{{\text R}}(p)}} \frac{x^{+}_{{\text R}}(p)-x^{+}_{{\text R}}(q)}{x^{-}_{{\text R}}(p)-x^{-}_{{\text R}}(q)},\qquad r_3^{\text{RR}}= \sqrt{\frac{x^{+}_{{\text R}}(q)}{x^{-}_{{\text R}}(q)}}\frac{x^{-}_{{\text R}}(p)-x^{-}_{{\text R}}(q)}{x^{-}_{{\text R}}(p)-x^{+}_{{\text R}}(q)}, \\ &r_4^{\text{RR}}= \sqrt{\frac{x^{-}_{{\text R}}(p)}{x^{+}_{{\text R}}(p)}\frac{x^{+}_{{\text R}}(q)}{x^{-}_{{\text R}}(q)}} \frac{x^{-}_{{\text R}}(q)-x^{+}_{{\text R}}(p)}{x^{-}_{{\text R}}(p)-x^{+}_{{\text R}}(q)},\qquad r_5^{\text{RR}}=\frac{x^{-}_{{\text R}}(p)-x^{+}_{{\text R}}(p)}{x^{-}_{{\text R}}(p)-x^{+}_{{\text R}}(q)} \frac{\gamma^{\text{R}}(q)}{\gamma^{\text{R}}(p)}, \\ &r_6^{\text{RR}}= \sqrt{\frac{x^{-}_{{\text R}}(p)}{x^{+}_{{\text R}}(p)}\frac{x^{+}_{{\text R}}(q)}{x^{-}_{{\text R}}(q)}} \frac{x^{-}_{{\text R}}(q)-x^{+}_{{\text R}}(q)}{x^{-}_{{\text R}}(p)-x^{+}_{{\text R}}(q)} \frac{\gamma^{\text{R}}(p)}{\gamma^{\text{R}}(q)},\qquad r_7^{\text{RR}}=0,\qquad r_8^{\text{RR}}=0. \end{aligned}$$

In the LR and RL sectors, accordingly,

$$\begin{aligned} \, &r_1^{\text{LR}}=\sqrt{\frac{x^{-}_{{\text L}}(p)}{x^{+}_{{\text L}}(p)}} \frac{1-x^{+}_{{\text L}}(p)x^{-}_{{\text R}}(q)}{1-x^{-}_{{\text L}}(p)x^{-}_{{\text R}}(q)},\qquad r_2^{\text{LR}}=1, \\ &r_3^{\text{LR}}=\sqrt{\frac{x^{-}_{{\text L}}(p)}{x^{+}_{{\text L}}(p)}\frac{x^{-}_{{\text R}}(q)}{x^{+}_{{\text R}}(q)}} \frac{1-x^{+}_{{\text L}}(p)x^{+}_{{\text R}}(q)}{1-x^{-}_{{\text L}}(p)x^{-}_{{\text R}}(q)},\qquad r_4^{\text{LR}}=-\sqrt{\frac{x^{-}_{{\text R}}(q)}{x^{+}_{{\text R}}(q)}} \frac{1-x^{-}_{{\text L}}(p)x^{+}_{{\text R}}(q)}{1-x^{-}_{{\text L}}(p)x^{-}_{{\text R}}(q)}, \\ &r_5^{\text{LR}}=0,\qquad r_6^{\text{LR}}=0, \\ &r_7^{\text{LR}}=\sqrt{\frac{x^{-}_{{\text L}}(p)}{x^{+}_{{\text L}}(p)}} \frac{x_{{\text R}}^{+}(q)-x_{{\text R}}^{-}(q)}{1-x_{{\text L}}^{-}(p)x_{{\text R}}^{-}(q)} \frac{\gamma^{\text{ L}}(p)}{\gamma^{\text{R}}(q)},\qquad r_8^{\text{LR}}=-\sqrt{\frac{x^{-}_{{\text R}}(q)}{x^{+}_{{\text R}}(q)}} \frac{x_{{\text L}}^{+}(p)-x_{{\text L}}^{-}(p)}{1-x_{{\text L}}^{-}(p)x_{{\text R}}^{-}(q)} \frac{\gamma^{\text{ R}}(q)}{\gamma^{\text{L}}(p)};\\ \\ &r_1^{\text{RL}}=\sqrt{\frac{x^{-}_{{\text R}}(p)}{x^{+}_{{\text R}}(p)}} \frac{1-x^{+}_{{\text R}}(p)x^{-}_{{\text L}}(q)}{1-x^{-}_{{\text R}}(p)x^{-}_{{\text L}}(q)},\qquad r_2^{\text{RL}}=1, \\ &r_3^{\text{RL}}=\sqrt{\frac{x^{-}_{{\text R}}(p)}{x^{+}_{{\text R}}(p)}\frac{x^{-}_{{\text L}}(q)}{x^{+}_{{\text L}}(q)}} \frac{1-x^{+}_{{\text R}}(p)x^{+}_{{\text L}}(q)}{1-x^{-}_{{\text R}}(p)x^{-}_{{\text L}}(q)},\qquad r_4^{\text{RL}}=-\sqrt{\frac{x^{-}_{{\text L}}(q)}{x^{+}_{{\text L}}(q)}} \frac{1-x^{-}_{{\text R}}(p)x^{+}_{{\text L}}(q)}{1-x^{-}_{{\text R}}(p)x^{-}_{{\text L}}(q)}, \\ &r_5^{\text{RL}}=0,\qquad r_6^{\text{RL}}=0, \\ &r_7^{\text{RL}}=\sqrt{\frac{x^{-}_{{\text L}}(q)}{x^{+}_{{\text L}}(q)}} \frac{x_{{\text R}}^{+}(p)-x_{{\text R}}^{-}(p)}{1-x_{{\text R}}^{-}(p)x_{{\text L}}^{-}(q)} \frac{\gamma^{\text{ L}}(q)}{\gamma^{\text{R}}(p)},\qquad r_8^{\text{RL}}=-\sqrt{\frac{x^{-}_{{\text R}}(p)}{x^{+}_{{\text R}}(p)}} \frac{x_{{\text L}}^{+}(q)-x_{{\text L}}^{-}(q)}{1-x_{{\text R}}^{-}(p)x_{{\text L}}^{-}(q)} \frac{\gamma^{\text{ R}}(p)}{\gamma^{\text{L}}(q)}. \end{aligned}$$

Let

$$ \upsilon(p,q)^{\text{LR}}= \biggl(\frac{x_{\text{L}}^{+}(p)}{x_{\text{L}}^{-}(p)}\biggr)^{-1/4} \biggl(\frac{x_{\text{R}}^{+}(q)}{x_{\text{R}}^{-}(q)}\biggr)^{-1/4} \biggl(\frac{1-\frac{1}{x_{\text{L}}^{-}(p)x_{\text{R}}^{-}(q)}}{1-\frac{1}{x_{\text{L}}^{+}(p)x_{\text{R}}^{+}(q)}}\biggr)$$

(C.5)

and let a similar function labeled RL be obtained by $\text{L}\leftrightarrow\text{R}$, and in the pure sectors,

$$ \upsilon^{\text{LL}}=\upsilon^{\text{RR}}=1.$$

(C.6)

It can be shown that the corresponding sectors for $\gamma$ and the Zhukovski variables are related by

$$ \gamma^{\text{L}}(p+\omega)=i\frac{\gamma^{\text{R}}(p)}{x_{\text{R}}^{+}(p)},\qquad x_{\text{L}}^{\pm}(p+\omega)=\frac{1}{x_{\text{R}}^{\pm}(p)},$$

(C.7)

where $p+\omega\to\bar{p}$ in [35]. In the R sector, formulas follow by $L\leftrightarrow R$.

C.3. The case $AdS_3\times S^3\times S^3\times S^1\mapsto AdS_3\times S^3\times T^4$

It can be shown that by the set of restrictions and maps, we can obtain the $AdS_3\times S^3\times T^4$ $R$-matrix from the one for $AdS_3\times S^3\times S^3\times S^1$. We first redefine variables in the Zhukovski space as

$$ x_{\text{L}}^{-}\mapsto x^{-},\qquad x_{\text{L}}^{+}\mapsto x^{+},\qquad\quad x_{\text{R}}^{-}\mapsto x^{-},\qquad x_{\text{R}}^{+}\mapsto x^{+}$$

(C.8)

and define $\gamma^{\text{L}}$ and $\gamma^{\text{R}}$ as

$$ \gamma^{\text{L}}(p)=a\biggl(\frac{x^{+}(p)}{x^{-}(p)}\biggr)^{\!1/4}\gamma(p),\qquad \gamma^{\text{R}}(p)=a\biggl(\frac{x^{-}(p)}{x^{+}(p)}\biggr)^{\!1/4}\gamma(p),$$

(C.9)

where $a=\text{const}$ and $\rho^\chi$ is related to $\zeta^\chi$ by

$$ \begin{aligned} \, &\rho^{\text{LL}}(p,q)=-\sqrt{\frac{x^{+}(p)}{x^{-}(p)}\frac{x^{-}(q)}{x^{+}(q)}}\frac{x^{-}(p)-x^{+}(q)}{x^{-}(q)-x^{+}(p)}\zeta^{\text{LL}}(p,q), \\ &\rho^{\text{RR}}(p,q)=-\sqrt{\frac{x^{+}(p)}{x^{-}(p)}\frac{x^{-}(q)}{x^{+}(q)}}\frac{x^{-}(p)-x^{+}(q)}{x^{-}(q)-x^{+}(p)}\zeta^{\text{RR}}(p,q), \\ &\rho^{\text{LR}}(p,q)=\frac{x^{+}(p)}{x^{-}(p)}\sqrt{\frac{x^{+}(q)}{x^{-}(q)}} \frac{1-x^{-}(p)x^{-}(q)}{1-x^{+}(p)x^{+}(q)}\frac{\zeta^{\text{LR}}(p,q)}{\upsilon^{\text{ LR}}(p,q)}, \\ &\rho^{\text{RL}}(p,q)=\sqrt{\frac{x^{-}(q)}{x^{+}(q)}}\frac{\zeta^{\text{RL}}(p,q)}{\upsilon^{\text{RL}}(p,q)}. \end{aligned}$$

(C.10)

Appendix D: Properties of free fermions: Recursion for the $$AdS_3$$ - $$T$$ -matrix

We give recursive formulas for the $N$-site transfer matrix $t_{N}$ that do not exploit an auxiliary Bethe system. We note a special form of $R_{AdS_3}$ with a Ramond–Ramond flux,

$$ R_{0i}(\theta_0-\theta_i)=\cosh\frac{\theta_{0i}}{2}\cdot(1-2\eta_{0i}^\dagger\eta_{0i})\equiv\cosh\frac{\theta_{0i}}{2}\cdot(1-2 N_{0i});$$

(D.1)

where the notation $\eta_{0,i}$ and $\eta_{0,i}^{\dagger}$ emphasizes that inhomogeneities are present in sites $0$ and $i$. We note that this gives rise to a nontrivial structure of the $R$-matrix. For the $t_N$ matrix, we have

$$\begin{aligned} \, t_N&=\prod_{i=1}^N\cosh\frac{\theta_{0i}}{2}\cdot \operatorname{str}_0[1-2N_{01}]\ldots[1-2N_{0N}]= \nonumber\\ &=\prod_{i=1}^N\cosh\frac{\theta_{0i}}{2}\bigl(\langle 0_0|[1-2N_{01}]\ldots[1-2N_{0N}]|0_0\rangle-\langle 0_0|c_0[1-2N_{01}]\ldots[1-2N_{0N}]c^\dagger_0|0_0\rangle\bigl), \end{aligned}$$

(D.2)

where $|0_{0}\rangle$ is a state $|\phi\rangle$ in the $0$th auxiliary space. Clearly, in order to compute (D.2), we need to find

$$ \langle 0_0|N_{01}\ldots N_{0m}|0_0\rangle,\qquad \langle 0_0|c_0 N_{01}\ldots N_{0m}c^\dagger_0|0_0\rangle,\qquad m=\overline{1,N}.$$

(D.3)

We first show that recursions for the elements above can be derived. At the first, we use (D.3) to find the elements

$$ X_m\equiv\langle 0_0|N_{01}\ldots N_{0m}|0_0\rangle,\qquad Y_m\equiv\langle 0_0|N_{01}\ldots N_{0m}c_0^\dagger|0_0\rangle.$$

(D.4)

It follows that

$$\begin{aligned} \, \langle 0_0|N_{01}\ldots N_{0m}|0_0\rangle&=\langle 0_0|N_{01}\ldots N_{0,m-1}\eta^\dagger_m\eta_m |0_0\rangle= \\ &=\langle 0_0|N_{01}\ldots N_{0,m-1}\eta^\dagger_m (\alpha_m c_0+\beta_m c_m)|0_0\rangle= \\ &=\beta_m\langle 0_0|N_{01}\ldots N_{0,m-1}(\alpha_m c_0^\dagger+\beta_m c_m^\dagger)|0_0\rangle c_m= \\ &=\alpha_m\beta_m\langle 0_0|N_{01}\ldots N_{0,m-1}c_0^\dagger|0_0\rangle c_m+\beta_m^2\langle 0_0|N_{01}\ldots N_{0,m-1}|0_0\rangle n_m. \end{aligned}$$

whence we have

$$ \alpha_i\equiv\cos\alpha_{0i},\qquad\beta_i\equiv\sin\alpha_{0i},\qquad\cot 2\alpha_{0i}=\sinh\frac{\theta_{0i}}{2}.$$

(D.5)

As a result, a recursion for $X_{m}$ follows,

$$ X_m^{}=\alpha_m^{}\beta_m^{}Y_{m-1}^{}c_m^{}\beta_m^2X_{m-1}^{}n_m^{}.$$

(D.6)

Similarly by the observations above, but now for $\langle 0_0|c_0 N_{01}\ldots N_{0m}c^\dagger_0|0_0\rangle$, we can derive a recursive relation for $Y_m$,

$$ Y_m^{}=\alpha_m^2Y_{m-1}^{}+\alpha_m^{}\beta_m^{}X_{m-1}^{}c^\dagger_m+\beta_m^2Y_{m-1}^{}n_m.^{}$$

(D.7)

The recursive relations can be solved by

$$ X_m^{}=x_{m-1}^{}\nu_m^{}+\sum_{i=1}^{m-1} X_{i-1}^{}\psi^\dagger_i\xi_{im}^{},\qquad m\ge 1,\qquad X_0=1,$$

(D.8)

and can be expressed through the block molecules $\Gamma_{ij}=\psi^\dagger_i\xi_{ij}$ for a given $m$. However, a universal expression can also be written. After redefining operators and the index algebra, system of equations (D.6) can be solved: we have

$$\begin{aligned} \, &X_m=\prod_{i=0}^{m-1}\Omega_{2,i+1}+ \sum_{i=0}^{m-1}\Omega_{1,i+1}\biggl(\,\prod_{j=i}^{m-2}\Omega_{2,j+2}\biggr) \sum_{\mu=0}^{i-1}\Omega_{3,\mu+1}\biggl(\,\prod_{\nu=\mu}^{i-2}(\Omega_{4,\nu+2}+\Omega_{2,\nu+2})\biggr)X_{\mu}, \\ &Y_m=\sum_{i=0}^{m-1}\Omega_{3,i+1}\biggl(\,\prod_{j=i}^{m-2}(\Omega_{4,j+2}+\Omega_{2,j+2})\biggr) \biggl(\,\prod_{\alpha=0}^{i-1}\Omega_{2,\alpha+1}+ \sum_{\alpha=0}^{i-1}\Omega_{1,\alpha+1}\biggl(\,\prod_{\beta=\alpha}^{i-2}\Omega_{2,\beta+2}\biggr)Y_\alpha\biggr) \end{aligned}$$

for $m\ge 0$, where

$$\Omega_{1,q}^{}=\alpha_q^{}\beta_q^{}c_q^{},\qquad \Omega_{2,q}^{}=\beta_q^2n_q^{},\qquad \Omega_{3,q}^{}=\alpha_q^{}\beta_q^{}c_q^{\dagger},\qquad \Omega_{4,q}^{}=\alpha_q^2,$$

For $m=0$, these quantities satisfy the same initial conditions.

We now recall that in calculating (D.2), we are left with elements of the form

$$ Z_m^{}\equiv\langle 0_0^{}|c_0^{}N_{01}^{}\ldots N_{0m}^{}c_0^\dagger|0_0^{}\rangle,\qquad L_m^{}\equiv\langle 0_0^{}|c_0^{}N_{01}^{}\ldots N_{0m}^{}c_0^{}c_0^\dagger|0_0^{}\rangle,$$

(D.9)

which have the same structure as above with extra operators acting on both sides. By a similar prescription, we can define a closed recursion system for the operators $L_m$ and $Z_m$:

$$ \begin{cases} L_m^{}=\alpha_m^{}\beta_m^{}Z_{m-1}^{}c_m^{}+\beta_m^2L_{m-1}^{}n_m^{}, \\ Z_m^{}=\alpha_m^2Z_{m-1}^{}+\alpha_m^{}\beta_m^{}L_{m-1}^{}c_m^\dagger+\beta_m^2Z_{m-1}^{}n_m^{}. \end{cases}$$

(D.10)

Now we can solve this system by using the set of parameters $\Xi_{i,s}$, $i=\overline{1,4}$:

$$ \begin{aligned} \, L_m&=\sum_{i=0}^{m-1}\Xi_{4,i+1}\prod_{j=i}^{m-2}\Xi_{3,j+2}\biggl(\,\prod_{\alpha=0}^{i-1}(\Xi_{1,\alpha+1}+\Xi_{3,\alpha+1})+ \sum_{\beta=1}^{i-1}\Xi_{2,\beta+1}\prod_{\alpha=\beta}^{i-2}(\Xi_{1,\alpha+2}+\Xi_{3,\alpha+2})U_\beta\biggr), \\ Z_m&=\prod_{i=0}^{m-1}(\Xi_{1,i+1}+\Xi_{3,i+1})+\sum_{j=1}^{m-1}\Xi_{2,j+1}\biggl(\,\prod_{i=j}^{m-2}(\Xi_{1,i+2}+\Xi_{3,i+2})\biggr) \sum_{\mu=0}^{j-1}\Xi_{4,\mu+1}\biggl(\,\prod_{\nu=\mu}^{j-2}\Xi_{3,\nu+2}\biggr)Z_{\mu} \end{aligned}$$

(D.11)

for $m\ge 0$, where

$$\Xi_{1,s}^{}=\alpha_s^2,\qquad\Xi_{2,s}^{}=\alpha_s^{}\beta_s^{}c_s^{\dagger},\qquad \Xi_{3,s}^{}=\beta_s^2n_s^{},\qquad\Xi_{4,s}^{}=\alpha_s^{}\beta_s^{}c_s^{}.$$

The block-molecule form the recursion is also applicable for general $m$.

In fact, we have obtained a closed efficient way to generate $t_{N}$. Clearly, it is still necessary to arrange the operator combinations into a certain compact expression of the type of (D.2), which apriori is not obvious. Already at small values of $N$, the operator structure develops nontrivially. For instance, for $N = 2$ we have

$$ t_2=\cosh\frac{\theta_{01}}{2}\cosh\frac{\theta_{02}}{2}\bigl[1-2X_1-2\widetilde X_1+4X_2-(1-2Z_1-2\widetilde Z_1+4Z_2)\bigr],$$

(D.12)

where $\widetilde X_i=\langle 0_{0}|N_{0i}|0_{0}\rangle$, which follows from the initial conditions as Eqs. (D.6) and (D.7). For $t_3$, we then have

$$\begin{aligned} \, &t_3\propto 2\sum_{i=1}^3\alpha_i^2+ 4\bigl[\alpha_1^{}\beta_1^{}\alpha_2^{}\beta_2^{} \bigl(c_1^\dagger c_2^{}(1-2\beta_3^2n_3^{})-c_1^{}c_2^\dagger(1-2\alpha_3^2-2\beta_3^2n_3^{})\bigr)+{} \\ &\kern76pt +\alpha_1^{}\beta_1^{}\alpha_3^{}\beta_3^{} \bigl(c_1^\dagger c_3^{}(1-2\alpha_2^2-2\beta_2^2n_2^{})-c_1^{}c_3^\dagger(1-2\beta_2^2n_2^{})\bigr)+{} \\ &\kern76pt +\alpha_2^{}\beta_2^{}\alpha_3^{}\beta_3^{} \bigl(c_2^\dagger c_3^{}(1-2\beta_1^2n_1^{})-c_2^{}c_3^\dagger(1-2\alpha_1^2-2\beta_1^2n_1^{})\bigr)\bigr]-{} \\ &-8\bigl[\beta_1^2\beta_2^2\beta_3^2n_1^{}n_2^{}n_3^{}- (\alpha_1^2+\beta_1^2n_1^{})(\alpha_2^2+\beta_2^2n_2^{})(\alpha_3^2+\beta_3^2n_3^{})\bigr]- \\ &-4\bigl[(\alpha_1^2+\beta_1^2n_1^{})(\alpha_2^2+\beta_2^2n_2^{})+ (\alpha_1^2+\beta_1^2n_1^{})(\alpha_3^2+\beta_3^2n_3^{})+ (\alpha_2^2+\beta_2^2n_2^{})(\alpha_3^2+\beta_3^2n_3^{})\bigr], \end{aligned}$$

and there is no universal identification in terms of developing $X$-type combinations starting with (D.12). Such a structure requires either a grouping scheme for the expansion or a more efficient recursion analytic solution for $t_N$.