functionally independent RG invariants. Thus one arrives at the hypothesis
that for sufficiently large $n$ and given $c_{\mu,j}$, it is always possible
to organize the scaling limit such that the scaled Bethe roots are described
by the ODE (6.1).

The specification of the RG invariants $\mathfrak{a}_{s}$ and their relation
to the coefficients $c_{\mu,j}$ can be deduced, in principle, for any given
value of $r$ and $A=1,2,\ldots,r-2$ similar to how it was done previously.
Unfortunately, we did not find a convenient form for these relations that
works in general. For odd $r$, $A=\frac{r-1}{2}$ and $n>0$ they are given by
eqs. (2.83),(2.73) and (2.74). Below we present the explicit formulae for even
$r$ and $A=\frac{r}{2}-1,\,\frac{r}{2}$ as well as arbitrary $r$ with
$A=1,r-2$.

###  6.1 The case $A=1$

The definition (6.2) for the set of admissible values of $(\mu,j)$,
specialized to $A=1$, yields

$\bm{\Xi}_{r,1}=\big{\\{}(\mu,j)\,:\ \
\mu=1,\ldots\,,\big{[}\tfrac{r-1}{2}\big{]}\ \ \&\ \ j=0\,\big{\\}}\ ,$ (6.5)

so that the ODE (6.1) takes the form

$\displaystyle\Bigg{[}-\partial_{y}^{2}+p^{2}+\mbox{e}^{(n+r)y}+E^{r}\
\mbox{e}^{ry}-\sum_{\mu=1}^{[\frac{r-1}{2}]}c_{\mu}\,E^{\mu}\
\mbox{e}^{\frac{n+2r}{r}\,\mu y}\,\Bigg{]}\,\psi=0\ .$ (6.6)

One should distinguish between the case when $r$ is an odd or even integer.
For the former, a computation shows that the exponents for the RG invariants
$\mathfrak{a}_{s}$ are given by

$\displaystyle d_{s}=\frac{2}{r}\,o_{s}\ \ \ \ \ \ \ \ \ \ \ (r-{\rm odd})\ ,$
(6.7)

where

$\displaystyle o_{s}\equiv\begin{cases}r-2s\ \ \ \ \ &{\rm for}\ \ \
s=1,2,\ldots,\frac{r-1}{2}\\\\[2.84544pt] 2r-2s\ \ \ \ \ &{\rm for}\ \ \
s=\frac{r+1}{2},\,\frac{r+3}{2},\,\ldots,r-1\end{cases}\ .$ (6.8)

The values of ${\mathfrak{a}}_{s}$ are related to the coefficients of the ODE
(6.6) as

$\displaystyle{\mathfrak{a}}_{s}=(-1)^{\frac{(r+1)}{2}o_{s}}\ \ \frac{1}{s}\
\sum_{k=1}^{o_{s}}(-1)^{k}\ \frac{\Gamma(s\nu+k)}{k!\Gamma(s\nu)}\
\sum_{j_{1}+j_{2}\ldots+j_{k}=o_{s}-k\atop j_{1},\ldots,j_{k}\geq
0}R_{2j_{1}+1}\ldots R_{2j_{k}+1}$ (6.9)

with $\nu=\frac{2n}{n+r}$ and we use the same notation as in (2.73). Namely

$R_{2j+1}=\frac{1}{r}\sum_{\alpha_{1}+2\alpha_{2}+\ldots+(j+1)\alpha_{j+1}=j+1\atop\alpha_{1},\ldots,\,\alpha_{j+1}\geq
0}\frac{(-1)^{\alpha_{1}+\ldots+\alpha_{j+1}}}{\alpha_{1}!\alpha_{2}!\ldots\alpha_{j+1}!}\
\frac{\Gamma(\alpha_{1}+\ldots+\alpha_{j+1}-\frac{2j+1}{r})}{\Gamma(1-\frac{2j+1}{r})}\
G_{2}^{\alpha_{1}}G_{4}^{\alpha_{2}}\dots G_{2j+2}^{\alpha_{j+1}}$ (6.10)

where now

$\displaystyle
G_{2m}=\frac{\Gamma(\frac{3}{2}+\frac{r}{2n})}{\Gamma(\frac{r}{2n})}\
\sum_{k=1}^{m}\mbox{e}^{\frac{{\rm i}\pi}{2}(m-k)}\
\frac{\Gamma\big{(}\frac{k}{2}-(\frac{1}{2r}+\frac{1}{n})\,m+\frac{r}{2n}\big{)}}{k!\Gamma\big{(}\frac{3}{2}-\frac{k}{2}-(\frac{1}{2r}+\frac{1}{n})\,m+\frac{r}{2n}\big{)}}\sum_{\mu_{1}+\ldots+\mu_{k}=\frac{1}{2}(kr-m)\atop
1\leq\mu_{1},\ldots,\mu_{k}\leq\frac{r-1}{2}}c_{\mu_{1}}\cdots c_{\mu_{k}}\,.$
(6.11)

Our numerical work shows that the value of the RG invariant $\mathfrak{a}_{s}$
with $s=\frac{r+1}{2}$, whose exponent is $d_{s}=2-\frac{2}{r}$, has no effect
on the scaling limit. Thus it can be set to zero and the number of non-trivial
RG invariants is equal to $r-2$.

In the case of even $r$ the analogous relations read as follows. The exponents
corresponding to ${\mathfrak{a}}_{s}$ are given by

$\displaystyle d_{s}=\frac{4}{r}\,e_{s}\ \ \ \ \ \ \ \ \ \ \ (r-{\rm even})\
,$ (6.12)

where we use

$\displaystyle e_{s}\equiv\begin{cases}\frac{r}{2}-s\ \ \ \ \ &{\rm for}\ \ \
s=1,\ldots\frac{r}{2}-1\\\\[2.84544pt] r-s\ \ \ \ \ &{\rm for}\ \ \
s=\frac{r}{2},\ldots,r-1\end{cases}\ .$ (6.13)

As for the $\mathfrak{a}_{s}$ — $c_{\mu}$ relation, one has

$\displaystyle{\mathfrak{a}}_{s}=(-1)^{e_{s}-1}\ \
\frac{1}{s}\times\begin{cases}\frac{1}{2{\rm i}}\,\
\big{(}F^{(+)}_{s}-F^{(-)}_{s}\big{)}\ \ \ \ &{\rm for}\ \ \ \ \ \ \
s=1,\ldots,\tfrac{r}{2}-1\\\\[2.84544pt]
\frac{1}{2}\,\big{(}F^{(+)}_{s}+F^{(-)}_{s}\big{)}\ \ \ \ \ &{\rm for}\ \ \ \
\ \ \ s=\frac{r}{2},\ldots,r-1\end{cases}\ .$ (6.14)

Here

$\displaystyle F^{(\pm)}_{s}=\sum_{k=1}^{e_{s}}(-1)^{k}\
\frac{\Gamma(\frac{s\nu}{2}+k)}{k!\Gamma(\frac{s\nu}{2})}\
\sum_{j_{1}+j_{2}\ldots+j_{k}=e_{s}-k\atop j_{1},\ldots,j_{k}\geq
0}R^{(\pm)}_{2j_{1}+1}\ldots R^{(\pm)}_{2j_{k}+1}$ (6.15)

with

$\displaystyle{R}^{(\pm)}_{2j+1}$ $\displaystyle=$
$\displaystyle\frac{2}{r}\sum_{\alpha_{1}+2\alpha_{2}+\ldots+(j+1)\alpha_{j+1}=j+1\atop\alpha_{1},\ldots,\,\alpha_{j+1}\geq
0}\frac{(-1)^{\alpha_{1}+\ldots+\alpha_{j+1}}}{\alpha_{1}!\alpha_{2}!\ldots\alpha_{j+1}!}\
\frac{\Gamma(\alpha_{1}+\ldots+\alpha_{j+1}-\frac{2}{r}\,(2j+1))}{\Gamma(1-\frac{2}{r}\,(2j+1))}$
(6.16) $\displaystyle\times$
$\displaystyle\big{(}G^{(\pm)}_{2}\big{)}^{\alpha_{1}}\big{(}G^{(\pm)}_{4}\big{)}^{\alpha_{2}}\dots\big{(}G^{(\pm)}_{2j+2}\big{)}^{\alpha_{j+1}}$
(6.17)

and

$\displaystyle
G^{(\pm)}_{2m}=\frac{\Gamma(\frac{3}{2}+\frac{r}{2n})}{\Gamma(\frac{r}{2n})}\
\sum_{k=1}^{m}\mbox{e}^{\frac{{\rm i}\pi}{2}(r\pm 1)k}\
\frac{\Gamma\big{(}\frac{k}{2}-(\frac{1}{r}+\frac{2}{n})\,m+\frac{r}{2n}\big{)}}{k!\Gamma\big{(}\frac{3}{2}-\frac{k}{2}-(\frac{1}{r}+\frac{2}{n})\,m+\frac{r}{2n}\big{)}}\sum_{\mu_{1}+\ldots+\mu_{k}=\frac{r}{2}k-m\atop
1\leq\mu_{1},\ldots,\mu_{k}\leq\frac{r}{2}-1}c_{\mu_{1}}\cdots c_{\mu_{k}}\ .$
(6.18)

The following comment is in order. Formula (6.14) yields that
$\mathfrak{a}_{r-1}\equiv 0$. Moreover, the RG invariant
$\mathfrak{a}_{\frac{r}{2}}$ has exponent $d_{\frac{r}{2}}=2$ and its value
does not affect the scaling limit. In fact, for even $r$ the set of $r-3$ RG
invariants turns out to not be minimal in general. For example, we found that
in the cases with $r\leq 10$ the scaled Bethe roots are also insensitive to
the value of $\mathfrak{a}_{\frac{r}{2}+1}$, in spite that the corresponding
exponent $d_{\frac{r}{2}+1}<2$. This is similar to what happens for the RG
invariant $\mathfrak{a}_{\frac{r-1}{2}}$ in the case of odd $r$.

Finally, the above formulae for both odd and even $r$ are literally applicable
provided

$n>\frac{1}{2}\,r\,(r-4)\ .$ (6.19)

###  6.2 The case even $r$ and $A=\frac{r}{2}-1$

Since

$\bm{\Xi}_{r,\frac{r}{2}-1}=\big{\\{}(\mu,j)\,:\ \ \mu=2j+1\ \ \&\ \
j=0,1,\ldots,\big{[}\tfrac{r}{4}\big{]}-1\,\big{\\}}\ ,$ (6.20)

the corresponding ODE reads explicitly as

$\displaystyle\Bigg{[}-\partial_{y}^{2}+p^{2}+\mbox{e}^{(n+r)y}+(-1)^{\frac{r}{2}}\,E^{r}\
\mbox{e}^{ry}-\sum_{j=0}^{[\frac{r}{4}]-1}c_{2j+1}\,E^{2j+1}\
\mbox{e}^{\big{(}\frac{n+r}{2}-(2j+1)\,\frac{n}{r}\big{)}y}\,\Bigg{]}\,\psi=0\
.$ (6.21)

Based on an analysis of the differential equation and a numerical study of the
scaling limit, we expect that there are only the $[\frac{r}{4}]$ non-trivial
RG invariants:

$\displaystyle{\mathfrak{a}}_{2j+1}=\frac{1}{2j+1}\
\bigg{(}\frac{N}{rN_{0}}\bigg{)}\ \frac{1}{r}\
\sum_{\ell=1}^{r}(\eta_{\ell})^{-2j-1}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}0\leq j\leq\big{[}\tfrac{r}{4}\big{]}-1\big{)}\ .$ (6.22)

The relation between their values and the coefficients $c_{2j+1}$ is
especially simple

$\displaystyle{\mathfrak{a}}_{2j+1}=(-1)^{j}\
\frac{\Gamma(\frac{1}{2}+\frac{r}{2n})}{r\Gamma(\frac{r}{2n})}\
\frac{\Gamma\big{(}\frac{1}{2}-\frac{2j+1}{r}\big{)}}{\Gamma\big{(}1-\frac{2j+1}{r}\big{)}}\
c_{2j+1}\ .$ (6.23)

The differential equation (6.21) describes the scaling limit of the ground
state Bethe roots for any $n>0$.

###  6.3 The case even $r$ and $A=\frac{r}{2}$

The set $\bm{\Xi}_{r,\frac{r}{2}}$ coincides with $\bm{\Xi}_{r,\frac{r}{2}-1}$
(6.20):

$\bm{\Xi}_{r,\frac{r}{2}}=\big{\\{}(\mu,j)\,:\ \ \mu=2j+1\ \ \&\ \
j=0,1,\ldots,\big{[}\tfrac{r}{4}\big{]}-1\,\big{\\}}$ (6.24)

and the corresponding ODE takes the form

$\displaystyle\Bigg{[}-\partial_{y}^{2}+p^{2}+\mbox{e}^{(n+r)y}-(-1)^{\frac{r}{2}}\,E^{r}\
\mbox{e}^{ry}-\sum_{j=0}^{[\frac{r}{4}]-1}c_{2j+1}\,E^{2j+1}\
\mbox{e}^{\big{(}\frac{n+r}{2}+2j+1\big{)}y}\,\Bigg{]}\,\psi=0\ .$ (6.25)

The specification of the RG invariants and the $\mathfrak{a}_{s}$ — $c_{2j+1}$
relation looks similar to those for the case $A=1$ and we’ll use the analogous
notations. However, now one should distinguish between even and odd
$\frac{r}{2}$. For even $\frac{r}{2}$ the exponents for the RG invariants
$\mathfrak{a}_{s}$ are given by

$\displaystyle d_{s}=\frac{2}{r}\,o_{s}\ \ \ \ \ \ \ \ \ \ \ \big{(}r/2-{\rm
even}\big{)}\ ,$ (6.26)

where

$\displaystyle o_{s}\equiv\begin{cases}\frac{r}{2}-s\ \ \ \ \ &{\rm for}\ \ \
s=1,3,\ldots,\frac{r}{2}-1\\\\[2.84544pt] r-s\ \ \ \ \ &{\rm for}\ \ \
s=\frac{r}{2},\,\frac{r}{2}+2,\,\ldots,r-2\end{cases}\ .$ (6.27)

The values of ${\mathfrak{a}}_{s}$ are related to the coefficients of the ODE
(6.25) as

$\displaystyle{\mathfrak{a}}_{s}=(-1)^{\scriptscriptstyle{[\frac{o_{s}}{2}]}-1}\
\ \frac{1}{s}\,\sum_{k=1}^{o_{s}}(-1)^{k}\
\frac{\Gamma(s\nu+k)}{k!\Gamma(s\nu)}\
\sum_{j_{1}+j_{2}\ldots+j_{k}=o_{s}-k\atop j_{1},\ldots,j_{k}\geq
0}R_{2j_{1}+1}\ldots R_{2j_{k}+1}$ (6.28)

with $\nu=\frac{2n}{n+r}$ and we use the same notation (6.10) where now

$\displaystyle
G_{2m}=\frac{\Gamma(\frac{3}{2}+\frac{r}{2n})}{\Gamma(\frac{r}{2n})}\
\sum_{k=1}^{m}\mbox{e}^{-\frac{{\rm
i}\pi}{4}rk}\frac{\Gamma\big{(}\frac{k}{2}-\frac{m}{n}+\frac{r}{2n}\big{)}}{k!\Gamma\big{(}\frac{3}{2}-\frac{k}{2}-\frac{m}{n}+\frac{r}{2n}\big{)}}\
\sum_{{j_{1}+\ldots+j_{k}=\frac{r-2}{4}\,k-\frac{m}{2}}\atop 0\leq
j_{1},\ldots,j_{k}\leq\frac{r}{4}-1}c_{2j_{1}+1}\cdots c_{2j_{k}+1}\ .$ (6.29)

In the case of odd $\frac{r}{2}$ the exponents are given by

$\displaystyle d_{s}=\frac{4}{r}\,e_{s}\ \ \ \ \ \ \ \ \ \ \ \big{(}r/2-{\rm
odd}\big{)}\ ,$ (6.30)

where

$\displaystyle e_{s}\equiv\begin{cases}\frac{1}{4}\,(r-2s)\ \ \ \ \ &{\rm
for}\ \ \ s=1,\,3,\,\ldots,\,\frac{r}{2}-2\\\\[2.84544pt] \frac{1}{2}\,(r-s)\
\ \ \ \ &{\rm for}\ \ \
s=\frac{r}{2}+1,\,\frac{r}{2}+3,\,\ldots,\,r-4\end{cases}\ .$ (6.31)

The $\mathfrak{a}_{s}$ — $c_{2j+1}$ relation reads as

$\displaystyle{\mathfrak{a}}_{s}=(-1)^{e_{s}-1}\ \
\frac{1}{s}\times\begin{cases}\frac{1}{2{\rm
i}}\,\big{(}F^{(-)}_{s}-F^{(+)}_{s}\big{)}\ \ \ \ &{\rm for}\ \ \ \ \ \ \
s=1,\,3,\,\ldots,\,\frac{r}{2}-2\\\\[2.84544pt]
\frac{1}{2}\,\big{(}F^{(+)}_{s}+F^{(-)}_{s}\big{)}\ \ \ \ \ &{\rm for}\ \ \ \
\ \ \ s=\frac{r}{2}+1,\,\frac{r}{2}+3,\,\ldots,\,r-4\end{cases}\ .$ (6.32)

Here the functions $F^{(\pm)}_{s}$ are defined by eqs.(6.15) and (6.16) but
now

$\displaystyle
G^{(\pm)}_{2m}=\frac{\Gamma(\frac{3}{2}+\frac{r}{2n})}{\Gamma(\frac{r}{2n})}\
\sum_{k=1}^{m}\mbox{e}^{\frac{{\rm i}\pi}{2}(\frac{r-2}{2}\pm 1)k}\
\frac{\Gamma\big{(}\frac{k}{2}-\frac{2m}{n}+\frac{r}{2n}\big{)}}{k!\Gamma\big{(}\frac{3}{2}-\frac{k}{2}-\frac{2m}{n}+\frac{r}{2n}\big{)}}\
\sum_{{j_{1}+\ldots+j_{k}=\frac{r-2}{4}\,k-m}\atop 0\leq j_{1},\ldots
j_{k}\leq\frac{r-2}{4}-1}c_{2j_{1}+1}\cdots c_{2j_{k}+1}\ .$ (6.33)

The differential equation (6.25) describes the scaling limit of the ground
state Bethe roots for any $n>0$.

###  6.4 The case $A=r-2$

The set of admissible values of $(\mu,j)$ (6.2) are given by

$\bm{\Xi}_{r,r-2}=\big{\\{}(\mu,j)\,:\ \ \mu=j+1\ \ \&\ \
j=0,1,\ldots,\big{[}\tfrac{r-1}{2}\big{]}-1\big{\\}}$ (6.34)

and the ODE becomes

$\displaystyle\Bigg{[}-\partial_{y}^{2}+p^{2}+\mbox{e}^{(n+r)y}-(-E)^{r}\
\mbox{e}^{ry}-\sum_{\mu=1}^{[\frac{r-1}{2}]}c_{\mu}\,E^{\mu}\
\mbox{e}^{\big{(}n+r-\frac{2n+r}{r}\,\mu\big{)}y}\,\Bigg{]}\,\psi=0\ .$ (6.35)

We have found that for any $n>0$ the differential equation describes the
scaling limit of the Bethe roots for the ground state if the minimal set of
non-trivial RG invariants consists of the $[\frac{r}{2}]$ members

$\displaystyle{\mathfrak{a}}_{s}=\frac{1}{s}\
\bigg{(}\frac{N}{rN_{0}}\bigg{)}^{\frac{2s}{r}}\ \frac{1}{r}\
\sum_{\ell=1}^{r}(\eta_{\ell})^{-s}\ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}s=1,\ldots,\big{[}\tfrac{r}{2}\big{]}\big{)}\ .$ (6.36)

Their values are related to the coefficients of eq.(6.35) as

$\displaystyle{\mathfrak{a}}_{s}=\frac{1}{s}\ \sum_{k=1}^{s}(-1)^{k}\
\frac{\Gamma(s\nu+k)}{k!\Gamma(s\nu)}\ \sum_{j_{1}+j_{2}\ldots+j_{k}=s-k\atop
j_{1},\ldots,j_{k}\geq 0}R_{2j_{1}+1}\ldots R_{2j_{k}+1}\ .$ (6.37)

Here $\nu=\frac{2n}{n+r}$ and again we use the notation $R_{2j+1}$ (6.10) with

$\displaystyle
G_{2m}=\frac{\Gamma(\frac{3}{2}+\frac{r}{2n})}{\Gamma(\frac{r}{2n})}\
\sum_{k=1}^{m}\frac{\Gamma\big{(}k-(\frac{2}{r}+\frac{1}{n})\,m+\frac{r}{2n}\big{)}}{k!\,\Gamma\big{(}\frac{3}{2}-(\frac{2}{r}+\frac{1}{n})\,m+\frac{r}{2n}\big{)}}\
\sum_{\mu_{1}+\ldots+\mu_{k}=m\atop
1\leq\mu_{1},\ldots,\mu_{k}\leq[\frac{r-1}{2}]}c_{\mu_{1}}\cdots c_{\mu_{k}}\
.$ (6.38)

##  7  ODEs for $A=0$

As $A=0$ the integer $\mu$ may take any value from $1$ to $r-1$, while $j=0$
(see (3.23)). Then the ODE (3.3), (3.4) becomes

$\displaystyle\bigg{[}-\partial_{y}^{2}+p^{2}+\mbox{e}^{(n+r)y}-E^{r}\,\mbox{e}^{ry}-\sum_{\mu=1}^{r-1}c_{\mu}\,E^{\mu}\,\mbox{e}^{\mu
y}\,\bigg{]}\,\psi=0\ .$ (7.1)

This is a more general version of equation (5.10), which describes the scaling
limit of the Bethe roots for the ground state when $A\mu=rj>0$. The latter is
connected to the case $A=0$ since, as was explained in sec. 5.2, the ground
state Bethe roots for $A\mu=rj$ and $A=0$ are simply related. In view of this,
one may expect that in performing the scaling limit in the regime
$\gamma\in(0,\frac{\pi}{r})$ one should take the RG invariants
$\mathfrak{a}_{s}$ with $\frac{r}{2}\leq s\leq r-1$ to be non-vanishing with
the corresponding exponents given by

$d_{s}=\tfrac{2}{r}\,(r-s)\qquad\qquad\qquad\big{(}s=\big{[}\tfrac{r+1}{2}\big{]},\ldots,r-1\big{)}\,.$
(7.2)

The $\mathfrak{a}_{s}$ — $c_{\mu}$ relation is obtained along the similar
lines as before yielding that

$\displaystyle{\mathfrak{a}}_{s}=(-1)^{s-1}\ \ \frac{1}{s}\
\sum_{k=1}^{r-s}(-1)^{k}\ \frac{\Gamma(s\nu+k)}{k!\Gamma(s\nu)}\
\sum_{j_{1}+j_{2}\ldots+j_{k}=r-s-k\atop j_{1},\ldots,j_{k}\geq
0}R_{2j_{1}+1}\ldots R_{2j_{k}+1}\ \ \ \ \ \big{(}\tfrac{r}{2}<s\leq
r-1\big{)}\,,$ (7.3)

where $\nu=\frac{2n}{n+r}$ and we use the notation $R_{2j+1}$ (6.10) with

$\displaystyle
G_{2m}=\frac{\Gamma(\frac{3}{2}+\frac{r}{2n})}{\Gamma(\frac{r}{2n})}\sum_{k=1}^{m}\frac{\Gamma\big{(}\frac{r-2m}{2n}\big{)}}{k!\Gamma\big{(}\frac{3}{2}-k+\frac{r-2m}{2n}\big{)}}\
\sum_{\mu_{1}+\ldots+\mu_{k}=rk-m}c_{\mu_{1}}\cdots c_{\mu_{k}}\ .$ (7.4)

These formulae are literally applicable for $\frac{r}{2}<s\leq r-1$. If $r$ is
even and $s=\frac{r}{2}$ the summand in the r.h.s. of (7.3) at $k=1$ coincides
with $\nu R_{r-1}$, which contains a divergent term $G_{r}$. As was discussed,
this signals that the definition of the RG invariant
$\mathfrak{a}_{\frac{r}{2}}$ needs to be modified:

${\mathfrak{a}}_{\frac{r}{2}}\equiv\frac{2}{r^{2}}\
\frac{1}{\log\big{(}\frac{N}{rN_{0}}\big{)}}\
\bigg{(}\frac{N}{rN_{0}}\bigg{)}\
\sum_{\ell=1}^{r}(\eta_{\ell})^{-\frac{r}{2}}$ (7.5)

and, similar to (5.5), we expect that its value is expressed through the
coefficients of the ODE as

${\mathfrak{a}}_{\frac{r}{2}}=\frac{2\Gamma(\frac{1}{2}+\frac{r}{2n})}{\sqrt{\pi}r\Gamma(1+\frac{r}{2n})}\
\sum_{k=1}^{\frac{r}{2}}(-1)^{\frac{r}{2}-k}\
\frac{\Gamma\big{(}k-\frac{1}{2}\big{)}}{\sqrt{\pi}k!}\
\sum_{\mu_{1}+\ldots+\mu_{k}=r(k-\frac{1}{2})}c_{\mu_{1}}\cdots c_{\mu_{k}}\
.$ (7.6)

In the case under consideration, in taking the scaling limit, the
$[\frac{r}{2}]$ non-trivial RG invariants can be treated as the independent
parameters. A quick inspection shows that formulae (7.3) - (7.6) involve the
$[\frac{r}{2}]$ coefficients $c_{\mu}$, where
$\mu=[\frac{r+1}{2}],[\frac{r+1}{2}]+1,\ldots,r-1$ only. These relations can
be inverted to express $c_{[\frac{r+1}{2}]},\ldots,c_{r-1}$ in terms of
$\mathfrak{a}_{[\frac{r+1}{2}]},\ldots,\mathfrak{a}_{r-1}$. The remaining
coefficients of the ODE $c_{\mu}$ with $\mu=1,\ldots,[\frac{r-1}{2}]$ depend
on the RG invariants and the anisotropy parameter $\gamma=\frac{\pi}{n+r}$.
Contrary to $c_{\mu}$ with $\mu\geq\frac{r}{2}$, they also turn out to be
complicated functions of $p=\frac{1}{2}\,(n+r)\,{\tt k}$, where ${\tt k}$ is
the twist parameter. While there are no simple analytical expressions for
these $c_{\mu}$, in principle, they can be determined numerically. One way of
doing so is by studying the scaling behaviour of the sums of inverse powers of
the Bethe roots, i.e., the sum rules. In practice, however, obtaining
$c_{\mu}$ becomes a difficult computational task for $r\geq 6$. In this
regard, we found it also useful to consider the products

${\cal K}^{(\ell)}=\mbox{e}^{{\rm i}\pi{\tt
k}}\,q^{-\frac{N}{2}}\,\prod_{m=1}^{M}\frac{\zeta_{m}+\eta_{\ell}\,q^{+1}}{\zeta_{m}+\eta_{\ell}\,q^{-1}}\qquad\qquad(\ell=1,2,\ldots,r)\,,$
(7.7)

where $M=N/2-S^{z}$ so that in the case of the ground state $M=N/2$. Such
products are the eigenvalues of the so-called quasi-shift operators, which are
members of the commuting family that includes the spin chain Hamiltonian (for
details see, e.g., sec. 6.2 of ref.[25]). Note that $\prod_{\ell=1}^{r}{\cal
K}^{(\ell)}$ coincides with the eigenvalue of the lattice translation operator
(2.102). Having at hand the numerical values of the Bethe roots
$\\{\zeta_{m}\\}_{m=1}^{N/2}$ corresponding to the ground state and computing
(7.7) for increasing numbers of lattice sites, we observed that the following
limits exist

$\displaystyle{b}_{\mu}=\lim_{N\to\infty}\Bigg{[}\bigg{(}\frac{N}{rN_{0}}\bigg{)}^{\frac{r-2\mu}{r}}\frac{1}{2\pi{\rm
i}r}\ \sum_{\ell=1}^{r}\mbox{e}^{\frac{{\rm i}\pi}{r}\mu(r+1-2\ell)}\
\log\big{(}{\cal K}^{(\ell)}\big{)}\Bigg{]}$ (7.8)

(here the branches of the logarithms are chosen in such a way that $\log({\cal
K}^{(\ell)})$ vary continuously with the values of the RG invariants and
vanish when all $\mathfrak{a}_{s}=0$). Moreover, it was checked for $r=3,4,5$
that a relation of the form

$\displaystyle
c_{\mu}=\sum_{k=1}^{r-\mu}\sum_{\mu_{1}+\ldots+\mu_{k}=(k-1)r+\mu\atop
1\leq\mu_{1},\ldots,\mu_{k}\leq
r-1}B^{(\mu)}_{\mu_{1}\ldots\mu_{k}}\,{b}_{\mu_{1}}\cdots{b}_{\mu_{k}}\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ (\mu=1,\ldots,r-1)$ (7.9)

is satisfied by the $r-1$ numbers ${b}_{\mu}$ and the coefficients $c_{\mu}$.
Here $B_{\mu_{1}\ldots\mu_{k}}^{(\mu)}$ depends on $n$ and $r$, but not on the
twist parameter ${\tt k}$.

Let’s illustrate the above on the simplest example with $r=3$. In this case,
the value of the RG invariant $\mathfrak{a}_{1}$ may be set to zero. The
coefficient $c_{2}$ in the differential equation

$\displaystyle\bigg{[}-\partial_{y}^{2}+p^{2}+\mbox{e}^{(n+3)y}-E^{3}\,\mbox{e}^{3y}-c_{1}\,E\,\mbox{e}^{y}-c_{2}\,E^{2}\,\mbox{e}^{2y}\,\bigg{]}\,\psi=0$
(7.10)

is related to $\mathfrak{a}_{2}$ as

${\mathfrak{a}}_{2}=-\frac{\Gamma(\frac{1}{2}+\frac{3}{2n})}{3\Gamma(\frac{3}{2n})}\
\frac{\Gamma(\frac{1}{2n})}{\Gamma(\frac{1}{2}+\frac{1}{2n})}\ c_{2}\ .$
(7.11)

Formula (7.9), specialized to the case at hand, becomes

$\displaystyle c_{1}=B_{1}^{(1)}\,b_{1}+B_{22}^{(1)}\,(b_{2})^{2}\ ,\ \ \ \ \
\ \ \ \qquad\qquad c_{2}=B_{2}^{(2)}\,b_{2}\ .$ (7.12)

Numerical values of $B_{22}^{(1)}$ and $B_{2}^{(2)}$ for different $n$ are
presented in tab. 3. As for $B_{1}^{(1)}$, we found that the analytical
expression

$\displaystyle
B^{(1)}_{1}=\frac{\sqrt{\pi}\,\Gamma(\frac{1}{2n})}{\Gamma(\frac{1}{2}+\frac{1}{2n})}$
(7.13)

is in agreement with the data within the accuracy of our computations (at
least $\sim 10^{-6}$ relative error).

$n$ | $B_{22}^{(1)}$ | $B_{2}^{(2)}$ | $n$ | $B_{22}^{(1)}$ | $B_{2}^{(2)}$
---|---|---|---|---|---
$0.5$ | $-24.31181$ | $6.335556$ | $5.5$ | $-214.2092$ | $17.56490$
$1.0$ | $-33.51001$ | $8.000000$ | $6.0$ | $-244.4040$ | $18.57161$
$1.5$ | $-45.26519$ | $9.248695$ | $6.5$ | $-276.6034$ | $19.57701$
$2.0$ | $-59.22168$ | $10.37020$ | $7.0$ | $-310.8065$ | $20.58141$
$2.5$ | $-75.27314$ | $11.43904$ | $7.5$ | $-347.0127$ | $21.58504$
$3.0$ | $-93.37681$ | $12.48197$ | $8.0$ | $-385.2213$ | $22.58808$
$3.5$ | $-113.5123$ | $13.51060$ | $8.5$ | $-425.4320$ | $23.59065$
$4.0$ | $-135.6686$ | $14.53070$ | $9.0$ | $-467.6444$ | $24.59284$
$4.5$ | $-159.8393$ | $15.54535$ | $9.5$ | $-511.8583$ | $25.59473$
$5.0$ | $-186.0204$ | $16.55638$ | $10.0$ | $-558.0735$ | $26.59636$
Table 3: Listed are the numerical values of $B_{22}^{(1)}$ and $B_{2}^{(2)}$
from the relation (7.12), which expresses the coefficients $c_{1}$, $c_{2}$ in
the differential equation (7.10) in terms of $b_{1}$, $b_{2}$ defined by
(7.7), (7.8) with $r=3$.

The numerical computation of the limits $b_{\mu}$ (7.8) is straightforward for
any $r$. Hence, knowledge of $B^{(\mu)}_{\mu_{1}\ldots\mu_{k}}$ in the
relation (7.9) would provide an effective way of determining all $r-1$
coefficients $c_{\mu}$ in the differential equation (7.1). Unfortunately, at
the current moment the analytical formula for
$B^{(\mu)}_{\mu_{1}\ldots\mu_{k}}$ is not known.

The following comment is in order here. In sec. 2.4, devoted to the case
$A=\frac{r-1}{2}$ with odd $r$, it was mentioned that the Bethe roots for a
low energy state develop a scaling behaviour not only in the vicinity of
$\zeta=0$, but also $\zeta=\infty$. This is manifest in that the limits

$\bar{E}_{m}^{(a)}=\lim_{N\to\infty\atop m-{\rm
fixed}}\,\bigg{(}\frac{N}{rN_{0}}\bigg{)}^{\frac{2n}{r(n+r)}}\Big{(}\zeta_{M_{a}-m}^{(a)}\Big{)}^{-1}$
(7.14)

exist and are non-vanishing (for the ground state $S^{z}=0$ and
$M_{a}=N/(2r)$). In the case $A=0$, we found that if the scaling limit is
performed, where the set of non-trivial RG invariants is taken to be
$\\{\mathfrak{a}_{s}\\}_{s=[\frac{r+1}{2}]}^{r-1}$ with corresponding
exponents $d_{s}=\tfrac{2}{r}\,(r-s)$, then the scaled Bethe roots
$\bar{E}_{m}^{(a)}$ are described in terms of the ODE of the form

$\displaystyle\Bigg{[}-\partial_{\bar{y}}^{2}+{\bar{p}}^{2}+\mbox{e}^{(n+r){\bar{y}}}-{\bar{E}}^{r}\,\mbox{e}^{r{\bar{y}}}-\sum_{\mu=1}^{[\frac{r}{2}]}{\bar{c}}_{\mu}\,{\bar{E}}^{\mu}\,\mbox{e}^{\mu{\bar{y}}}\,\Bigg{]}\,{\bar{\psi}}=0\
.$ (7.15)

The coefficients $\bar{c}_{\mu}$, entering therein, depend on the values of
the RG invariants, the anisotropy $n$ and $\bar{p}=-\frac{1}{2}\,(n+r)\,{\tt
k}$.

For given values of $\mathfrak{a}_{s}$ and $N$ the relations (4.17), along
with the normalization condition (4.21), may be treated as a system of
algebraic equations, which determines the set of inhomogeneities
$\\{\eta_{\ell}\\}_{\ell=1}^{r}$. For large $N$ it is not difficult to show
that

$\displaystyle\frac{1}{r}\,\sum_{\ell=1}^{r}(\eta_{\ell})^{r}=(-1)^{r-1}+O(N^{-2})\
,$ (7.16)

while

$\displaystyle\frac{1}{s}\ \bigg{(}\frac{N}{rN_{0}}\bigg{)}^{{\bar{d}}_{s}}\
\frac{1}{r}\,\sum_{\ell=1}^{r}(\eta_{\ell})^{s}={\bar{\mathfrak{a}}}_{s}+O(N^{-2})\
\ \ \ \ \ \ \ \ \ \ (s=1,\ldots,r-1)\ .$ (7.17)

Here

$\displaystyle{\bar{d}}_{s}=\frac{2s}{r}$ (7.18)

and the limiting values ${\bar{\mathfrak{a}}}_{s}$ are certain polynomials of
the RG invariants ${\mathfrak{a}}_{1},\ldots,{\mathfrak{a}}_{r-1}$. In
particular,

$\displaystyle{\bar{\mathfrak{a}}}_{1}$ $\displaystyle=$
$\displaystyle(-1)^{r}\,{\mathfrak{a}}_{r-1}$
$\displaystyle{\bar{\mathfrak{a}}}_{2}$ $\displaystyle=$
$\displaystyle(-1)^{r}\,{\mathfrak{a}}_{r-2}+\frac{r}{2}\
({\mathfrak{a}}_{r-1})^{2}$ (7.19) $\displaystyle{\bar{\mathfrak{a}}}_{3}$
$\displaystyle=$
$\displaystyle(-1)^{r}\,{\mathfrak{a}}_{r-3}+r\,{\mathfrak{a}}_{r-2}\,{\mathfrak{a}}_{r-1}+(-1)^{r}\
\frac{r^{2}}{3}\ ({\mathfrak{a}}_{r-1})^{3}\ \ \ \ \ \ \ \ (r>3)\ .$

The scaling limit can be performed with the set of RG invariants chosen to be
$\\{\bar{\mathfrak{a}}_{s}\\}$. The latter would be defined by formula (7.17)
with the term $O(N^{-2})$ ignored, and are kept independent of $N$. Also, one
may impose the normalization condition (7.16), again, without the remainder
term. The two schemes based on the different sets of RG invariants lead to the
same limiting values $E_{m}^{(a)}$ and $\bar{E}_{m}^{(a)}$, provided that
${\mathfrak{a}}_{s}$ and $\bar{\mathfrak{a}}_{s}$ are related as in (7). The
scaled Bethe roots would be described by the ODEs (7.1) and (7.15). Since the
Bethe Ansatz equations (1.11) are invariant w.r.t. the simultaneous inversion
$\zeta_{j}\mapsto\zeta_{j}^{-1}$, $\eta_{\ell}\mapsto(\eta_{\ell})^{-1}$ and
$\omega\mapsto\omega^{-1}$ (${\tt k}\mapsto-{\tt k}$), the results for
$E_{m}^{(a)}$ carry over to $\bar{E}_{m}^{(a)}$ with a change in nomenclature
only.

### Comment on the ${\cal Z}_{r}$ invariant case with
$\eta_{\ell}=(-1)^{r}\,\mbox{e}^{\frac{{\rm i}\pi}{r}\,(2\ell-1)}$

In the work [31], the scaling limit for the low energy states of the ${\cal
Z}_{r}$ invariant spin chain in the regime $\gamma\in(0,\frac{\pi}{r})$ was
considered. It was found that such states can be labelled by $S^{z}$, the
winding number ${\tt w}\in\mathbb{Z}$, the pair of non-negative integers
$({\tt L},\bar{\tt L})$; and also, for $r$ even, $s$ which may take any real
value. The corresponding eigenvalues of the CFT Hamiltonian, defined according
to eqs. (2.105), (2.106), are given by

${\cal E}_{\rm CFT}=\frac{p^{2}+\bar{p}^{2}}{n+r}-\frac{r}{12}\,+{\tt
L}+\bar{{\tt L}}+\begin{cases}\dfrac{s^{2}}{2n}&\ \ \ {\rm for}\ \ \ r\ {\rm
even}\\\\[5.69046pt] 0&\ \ \ {\rm for}\ \ \ r\ {\rm odd}\end{cases}\,,$ (7.20)

where

$\displaystyle p=\frac{S^{z}}{2}+\frac{n+r}{2}\ ({\tt k}+{\tt w})\ ,\ \ \ \ \
\ \ {\bar{p}}=\frac{S^{z}}{2}-\frac{n+r}{2}\ ({\tt k}+{\tt w})\ .$ (7.21)

For any low energy Bethe state, the differential equations were proposed,
which describe the scaling limit of the Bethe roots. In particular, the ODEs
for the so-called “primary” Bethe states, for which the integers ${\tt
L}=\bar{\tt L}=0$, take the form

$\displaystyle\Bigg{[}-\partial^{2}_{y}+p^{2}+\mbox{e}^{(n+r)y}-E^{r}\,\mbox{e}^{ry}+\sum_{\mu=1}^{[\frac{r}{2}]}(-1)^{\mu}\,s_{\mu}\,E^{\mu}\,\mbox{e}^{\mu
y}\,\Bigg{]}\,\psi=0$ (7.22a)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad.$
$\displaystyle\Bigg{[}-\partial_{\bar{y}}^{2}+{\bar{p}}^{2}+\mbox{e}^{(n+r){\bar{y}}}-{\bar{E}}^{r}\,\mbox{e}^{r{\bar{y}}}-\sum_{\mu=1}^{[\frac{r}{2}]}(-1)^{\mu}\,{\bar{s}}_{\mu}\,{\bar{E}}^{\mu}\,\mbox{e}^{\mu{\bar{y}}}\,\Bigg{]}\,{\bar{\psi}}=0$
(7.22b)

The coefficients $s_{\mu}$, $\bar{s}_{\mu}$ are expressed in terms of the
eigenvalues of the quasi-shift operators (7.7):

$\displaystyle s_{\mu}$ $\displaystyle=$ $\displaystyle+{B_{\mu}}\
\lim_{N\to\infty}\Bigg{[}\bigg{(}\frac{N}{rN_{0}}\bigg{)}^{\frac{r-2\mu}{r}}\frac{1}{2\pi{\rm
i}r}\ \sum_{\ell=1}^{r}\mbox{e}^{+\frac{{\rm i}\pi}{r}\mu(r+1-2\ell)}\
\log\big{(}{\cal K}^{(\ell)}\big{)}\Bigg{]}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\
\ ,$ $\displaystyle{\bar{s}}_{\mu}$ $\displaystyle=$ $\displaystyle-{B_{\mu}}\
\lim_{N\to\infty}\Bigg{[}\bigg{(}\frac{N}{rN_{0}}\bigg{)}^{\frac{r-2\mu}{r}}\frac{1}{2\pi{\rm
i}r}\ \sum_{\ell=1}^{r}\mbox{e}^{-\frac{{\rm i}\pi}{r}\mu(r+1-2\ell)}\
\log\big{(}{\cal K}^{(\ell)}\big{)}\Bigg{]}$ (7.23b)

where

$\displaystyle B_{\mu}=(-1)^{(r-1)\mu}\ 2n\
\frac{\sqrt{\pi}\,\Gamma(1+\frac{r-2\mu}{2n})}{\Gamma(\frac{1}{2}+\frac{r-2\mu}{2n})}\qquad\qquad\qquad\qquad\big{(}\,\mu=1,\ldots,[\tfrac{r}{2}]\,\big{)}\
.$ (7.24)

Note that when $r$ is even, $s_{\frac{r}{2}}=\bar{s}_{\frac{r}{2}}$ and
coincides with $s$, which appears in formula (7.20).

The pair (7.22) resembles the differential equations (7.1) and (7.15).
However, it describes the scaling limit of a class of low energy excited
states, not just the ground state. The values of $s_{\mu}$ and $\bar{s}_{\mu}$
specify the primary Bethe state and are not arbitrary. For odd $r$ they belong
to a certain discrete set, which is determined through a so-called
“quantization condition”. In the case of even $r$, the parameter $s\equiv
s_{\frac{r}{2}}=\bar{s}_{\frac{r}{2}}$ may be any real number, but once it is
fixed, the remaining $s_{\mu}$, $\bar{s}_{\mu}$ with $\mu<\frac{r}{2}$ take a
discrete set of admissible values, which is also defined by the quantization
condition. In contrast, the pair of ODEs (7.1) and (7.15) correspond to the
ground state. The coefficients $c_{\mu}$ with $\mu\geq\frac{r}{2}$ are related
to the values of the RG invariants through eqs. (7.3)-(7.6). We expect that
the values of the remaining $c_{\mu}$ with $\mu<\frac{r}{2}$ as well as
$\bar{c}_{\mu}$ in (7.15) are a particular solution of a certain quantization
condition. Determining the latter would be crucial for describing the scaling
limit of the low energy spectrum for the spin chain with softly broken ${\cal
Z}_{r}$ symmetry in the regime $\gamma\in(0,\frac{\pi}{r})$. However, this
problem lies outside the scope of the paper.

##  8 Concluding remarks

In this work, we described within the ODE/IQFT approach the scaling behaviour
of the Bethe roots for just the ground state of the spin - $\frac{1}{2}$ chain
associated with the inhomogeneous six-vertex model. Even so, the results
indicate the presence of a remarkable variety of multiparametric integrable
structures in the underlying field theories. Further development of the
ODE/IQFT correspondence would require obtaining the class of differential
equations that describe the scaling limit of all the low energy Bethe states.
In the ${\cal Z}_{r}$ invariant case with anisotropy parameter
$\gamma\in(0,\frac{\pi}{r})$ this was done in ref.[31]. For the more general
setup discussed in the present work, a preliminary analysis shows that a
creative application of the original ideas from ref.[10] is required. An
important physical question is uncovering the field theories governing the
critical behaviours for $0<\gamma<\pi$. It is relatively well understood now
in the case $r=2$ and for the domain $\pi(1-\frac{1}{r})<\gamma<\pi$ with
arbitrary $r$ due to the works [35, 23, 36, 37, 38, 24, 39] and [21],
respectively.

Among the most interesting directions of research in the area of the ODE/IQFT
correspondence is its extension to the spin chain associated with the higher
spin - $j$ generalization of the inhomogeneous six-vertex model and especially
the $j\to\infty$ limit for the isotropic (XXX) spin chain, i.e., at the
boundary of the domain of criticality, $\gamma=0$ and $\gamma=\pi$. This may
be useful for exploring the isotropic spin chain built from (infinite
dimensional) unitary representations of $\mathfrak{sl}(2,\mathbb{C})$. The
latter is relevant to high energy physics since, as was discovered in the
pioneering paper of Lipatov [40], this spin chain (in a certain setup)
describes the wave functions of compound states of reggeized gluons in
multicolour QCD ($N_{c}\to\infty$) in the generalized leading logarithmic
approximation. The idea was further developed by Faddeev and Korchemsky in
[41]. In the works [42, 43] the spectrum of the spin chain was studied for
small lattice sizes $N$, which is interpreted as the number of gluons. The
analysis becomes complicated for increasing $N$. This is where, one may hope,
that the ODE/IQFT approach would be useful.

## Acknowledgments

The authors acknowledge discussions with Andreas Klümper and thank him for
providing us with a copy of the paper [44] before it was published. SG and GK
are grateful to Holger Frahm and Márcio Martins for stimulating conversations.

The research of SG is funded by the Deutsche Forschungsgemeinschaft under
grant No. Fr 737/9-2. Part of this work was carried out during GK’s visits to
the NHETC at Rutgers University. He is grateful for the support and
hospitality he received during the stays.

The research of SL is supported by the NSF under grant number NSF-PHY-2210187.

##  Appendix A Auxiliary functions appearing in the sum rules

Here we present the explicit formulae for $f_{1}$, $f_{2}$ and $f_{3}$
entering into eq.(2.2). We use the same notation as in ref.[39], where these
functions have previously appeared.

The function $f_{1}$ is defined as

$\displaystyle f_{1}(h,g)$ $\displaystyle=$
$\displaystyle\frac{\pi\Gamma(1-2g)}{\sin(\pi g)}\
\frac{\Gamma(g+2h)}{\Gamma(1-g+2h)}\ .$ (A.1)

As for $f_{2}$ and $f_{3}$, the expression for these is more complicated. In
particular, $f_{2}$ is given by the integral

$f_{2}(h,g)=2^{1-4g}\,\frac{\Gamma^{2}(1-g)}{\Gamma^{2}(\frac{1}{2}+g)}\
\frac{\Gamma(2g+2h)}{\Gamma(1-2g+2h)}\,\int_{-\infty}^{\infty}\frac{\mbox{d}x}{2\pi}\,\frac{S_{1}(x)}{x+{\rm
i}h}\ \ \ \ \ \ \big{(}0<g<\tfrac{1}{2},\ \Re e(h)>0\big{)}$ (A.2)

with

$S_{1}(x)=\sinh(2\pi x)\,\Gamma(1-2g+2{\rm i}x)\,\Gamma(1-2g-2{\rm
i}x)\,\big{(}\Gamma(g+2{\rm i}x)\Gamma(g-2{\rm i}x)\big{)}^{2}\ .$ (A.3)

The above is applicable for $0<g<\frac{1}{2}$. Its analytic continuation to
the interval $\tfrac{1}{2}<g<1$, yields

$\displaystyle f_{2}(h,g)$ $\displaystyle=$ $\displaystyle
2^{1-4g}\,\frac{\Gamma^{2}(1-g)}{\Gamma^{2}(\frac{1}{2}+g)}\
\frac{\Gamma(2g+2h)}{\Gamma(1-2g+2h)}\,\Bigg{(}\int_{-\infty}^{\infty}\frac{\mbox{d}x}{2\pi}\,\frac{S_{1}(x)}{x+{\rm
i}h}$ $\displaystyle-$ $\displaystyle\frac{\sin(2\pi
g)\Gamma(3-4g)\Gamma^{2}(1-g)\Gamma^{2}(3g-1)}{(2h+1-2g)(2h-1+2g)}\Bigg{)}\qquad\qquad\big{(}\,\tfrac{1}{2}<g<1,\
\Re e(h)>0\,\big{)}\ .$

In the case of $f_{3}$, the following expression is valid for
$0<g<\tfrac{1}{2}$ and $\Re e(h)>0$:

$\displaystyle f_{3}(h,g)$ $\displaystyle=$ $\displaystyle
2^{2-6g}\,\sqrt{\pi}\ \frac{\Gamma^{3}(1-g)}{\Gamma^{3}(\frac{1}{2}+g)}\
\frac{\Gamma(3g-1+2h)}{\Gamma(2-3g+2h)}$ $\displaystyle\times$
$\displaystyle\Bigg{(}\,-{\sin(4\pi
g)\over\pi^{2}}\,\int_{-\infty}^{+\infty}\frac{\mbox{d}y}{2\pi}\int_{-\infty}^{+\infty}\frac{\mbox{d}x}{2\pi}\,\frac{S_{2}(x,y)}{(y+{\rm
i}h)(x-y-{\rm
i}0)}\,+\frac{1}{3}\,\int_{-\infty}^{+\infty}\frac{\mbox{d}x}{2\pi}\,\frac{S_{3}(x)}{x+{\rm
i}h}\,\Bigg{)},$

where the functions $S_{2}$ and $S_{3}$ read as

$\displaystyle S_{2}(x,y)$ $\displaystyle=$ $\displaystyle\sinh(2\pi
y)\,\sinh(2\pi x)\,\Gamma(g+2{\rm i}y)\,\Gamma(g-2{\rm i}y)\,\Gamma(2g+2{\rm
i}y)\,\Gamma(2g-2{\rm i}y)$ (A.6) $\displaystyle\times$
$\displaystyle\Gamma(2-3g+2{\rm i}y)\,\Gamma(2-3g-2{\rm
i}y)\,\Gamma(1-2g+2{\rm i}x)\,\Gamma(1-2g-2{\rm i}x)$ $\displaystyle\times$
$\displaystyle\big{(}\Gamma(g+2{\rm i}x)\,\Gamma(g-2{\rm i}x)\big{)}^{2}$
$\displaystyle S_{3}(x)$ $\displaystyle=$ $\displaystyle\sinh(2\pi
x)\,\Gamma(2-3g+2{\rm i}x)\,\Gamma(2-3g-2{\rm i}x)\,\big{(}\Gamma(g+2{\rm
i}x)\,\Gamma(g-2{\rm i}x)\big{)}^{3}$ $\displaystyle\times$
$\displaystyle\frac{\sin(4{\rm i}\pi x+2\pi g)-2\sin(2\pi g)}{\sin(2{\rm i}\pi
x+2\pi g)}\ .$

For $\tfrac{1}{2}<g<\tfrac{2}{3}$ and $\Re e(h)>0$ one has

$\displaystyle f_{3}(h,g)$ $\displaystyle=$ $\displaystyle
2^{2-6g}\,\sqrt{\pi}\ \frac{\Gamma^{3}(1-g)}{\Gamma^{3}(\frac{1}{2}+g)}\
\frac{\Gamma(3g-1+2h)}{\Gamma(2-3g+2h)}$ $\displaystyle\times$
$\displaystyle\Bigg{[}\,-{\sin(4\pi
g)\over\pi^{2}}\,\int_{-\infty}^{+\infty}\frac{\mbox{d}y}{2\pi}\int_{-\infty}^{+\infty}\frac{\mbox{d}x}{2\pi}\,\frac{S_{2}(x,y)}{(y+{\rm
i}h)(x-y-{\rm
i}0)}\,+\frac{1}{3}\,\int_{-\infty}^{+\infty}\frac{\mbox{d}x}{2\pi}\,\frac{\tilde{S}_{3}(x)}{x+{\rm
i}h}\,\Bigg{]},$

where

$\displaystyle\tilde{S}_{3}(x)$ $\displaystyle=$ $\displaystyle
S_{3}(x)-3\pi^{-2}\sin(4\pi g)\sin(2\pi g)\sinh(2\pi
x)\Gamma(3-4g)\Gamma^{2}(1-g)\Gamma^{2}(3g-1)$ $\displaystyle\times$
$\displaystyle\Gamma(2-3g-2{\rm i}x)\Gamma(2-3g+2{\rm i}x)$
$\displaystyle\times$ $\displaystyle\Gamma(g+2{\rm i}x)\Gamma(g-2{\rm
i}x)\Gamma(2g+2{\rm i}x-1)\Gamma(2g-2{\rm i}x-1)\,.$

Finally if $\tfrac{2}{3}<g<1$ and $\Re e(h)>0$, the analytic continuation of
(A) gives

$\displaystyle f_{3}(h,g)$ $\displaystyle=$ $\displaystyle
2^{2-6g}\,\sqrt{\pi}\ \frac{\Gamma^{3}(1-g)}{\Gamma^{3}(\frac{1}{2}+g)}\
\frac{\Gamma(3g-1+2h)}{\Gamma(2-3g+2h)}$ $\displaystyle\times$
$\displaystyle\Bigg{[}\,-{\sin(4\pi
g)\over\pi^{2}}\,\int_{-\infty}^{+\infty}\frac{\mbox{d}y}{2\pi}\int_{-\infty}^{+\infty}\frac{\mbox{d}x}{2\pi}\,\frac{S_{2}(x,y)}{(y+{\rm
i}h)(x-y-{\rm
i}0)}\,+\frac{1}{3}\,\int_{-\infty}^{+\infty}\frac{\mbox{d}x}{2\pi}\,\frac{\tilde{S}_{3}(x)}{x+{\rm
i}h}\,+S_{4}\Bigg{]},$

where $S_{4}$ is defined as

$\displaystyle S_{4}$ $\displaystyle=$ $\displaystyle\frac{\sin(2\pi
g)\sin(3\pi
g)\Gamma(5-6g)\Gamma(2-2g)\Gamma^{3}(1-g)\Gamma^{2}(3g-1)\Gamma(5g-3)}{\pi\big{(}(2-3g)^{2}-4h^{2}\big{)}}$
$\displaystyle+$ $\displaystyle\frac{4\sin(3\pi g)\cos(\pi
g)\Gamma(4-6g)\Gamma^{3}(2-2g)\Gamma^{3}(4g-2)}{3\big{(}(2-3g)^{2}-4h^{2}\big{)}\,\big{(}2\cos(2\pi
g)+2\cos(4\pi g)+1\big{)}}$ $\displaystyle\times$
$\displaystyle\big{(}3g-6h-2+(15g-6h-10)\cos(2\pi g)+(9g-6h-6)\cos(4\pi
g)+2(3g-2)\cos(6\pi g)\big{)}\ .$

##  Appendix B Lagrange formula

Consider the algebraic equation

$\displaystyle X^{r}+\sum_{m=1}^{r-1}G_{m}X^{r-m}={Y}^{r}\ .$ (B.1)

Let $X(Y)$ be the solution such that

$X(Y)\to Y\qquad\qquad{\rm as}\qquad\qquad Y\to\infty\ .$ (B.2)

In fact, $X(Y)$ is a multivalued function of $Y$ and this condition specifies
its branch for sufficiently large $Y$. It was found by Lagrange [32], see also
[33], that $X(Y)$ admits a convergent power series expansion of the form

$\displaystyle X(Y)=Y+\sum_{k=1\atop k\not=0\,{\rm mod}\,r}^{\infty}R_{k}\
Y^{-k}\ ,$ (B.3)

where

$\displaystyle R_{k}=\frac{1}{r}\sum_{\alpha_{1},\ldots\alpha_{n-1}\geq
0\atop\alpha_{1}+2\alpha_{2}+\ldots+(r-1)\alpha_{r-1}=k+1}\frac{(-1)^{\alpha_{1}+\ldots+\alpha_{r-1}}}{\alpha_{1}!\alpha_{2}!\ldots\alpha_{r-1}!}\
\frac{\Gamma(\alpha_{1}+\ldots+\alpha_{r-1}-\frac{k}{r})}{\Gamma(1-\frac{k}{r})}\
G_{1}^{\alpha_{1}}G_{2}^{\alpha_{2}}\dots G_{r-1}^{\alpha_{r-1}}\ .$ (B.4)

##  Appendix C Relations between $\mathfrak{a}_{2j+1}$ and $c_{2j+1}$ for
$r=3,5,7$ and $A=\frac{r-1}{2}$

For $r$ odd and $A=\frac{r-1}{2}$ the formula (2.83) was obtained in the main
body of the text that expresses the RG invariants $\mathfrak{a}_{2j+1}$ (2.33)
with $j=1,2,\ldots,\frac{r-1}{2}$ in terms of the coefficients $c_{2j+1}$ of
the ODE (2.18). The inversion of this relation leads to eq. (2.44). Here we
write it out explicitly for $r=3,5,7$.

In the case $r=3$ there is only a single non-trivial RG invariant
$\mathfrak{a}_{1}$. The coefficient $c_{1}$ entering into the differential
equation is simply proportional to it:

$\displaystyle c_{1}=C_{0}^{(0)}\,{\mathfrak{a}}_{1}\ ,$ (C.1)

where we use the notation $C_{j}^{(j)}$ from eq.(2.84), namely,

$C_{j}^{(j)}=(-1)^{j}\,\frac{r\Gamma\big{(}\frac{r}{2n}\big{)}\Gamma\big{(}1-\frac{(2j+1)(n-r)}{2rn}\big{)}}{\Gamma\big{(}\frac{1}{2}+\frac{r}{2n}\big{)}\Gamma\big{(}\frac{1}{2}-\frac{(2j+1)(n-r)}{2rn}\big{)}}\
.$ (C.2)

For $r=5$ the two coefficients $c_{1}$ and $c_{3}$ are expressed in terms of
the two RG invariants $\mathfrak{a}_{1}$ and $\mathfrak{a}_{3}$ as:

$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle
C_{0}^{(0)}\,{\mathfrak{a}}_{1}+\frac{n-5}{20n}\
\Big{(}\big{(}C_{1}^{(1)}\big{)}^{2}-5C_{0}^{(0)}\Big{)}\,{\mathfrak{a}}^{2}_{3}$
$\displaystyle c_{3}$ $\displaystyle=$ $\displaystyle
C_{1}^{(1)}\,{\mathfrak{a}}_{3}\ .$ (C.3)

For $r=7$, one has

$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle
C_{0}^{(0)}\,{\mathfrak{a}}_{1}-\frac{n-7}{14n}\
\Big{(}7C_{0}^{(0)}+C_{1}^{(1)}C_{2}^{(2)}\Big{)}\
{\mathfrak{a}}_{3}{\mathfrak{a}}_{5}$ $\displaystyle+$
$\displaystyle\frac{n-7}{n^{2}}\ \bigg{(}\,\frac{15n-7}{24}\
C_{0}^{(0)}+\frac{3(n-7)}{56}\ C_{1}^{(1)}\,C_{2}^{(2)}-\frac{3n+28}{588}\
\big{(}C_{2}^{(2)}\big{)}^{3}\bigg{)}\ {\mathfrak{a}}_{5}^{3}$ $\displaystyle
c_{3}$ $\displaystyle=$ $\displaystyle
C_{1}^{(1)}\,{\mathfrak{a}}_{3}-\frac{3(n-7)}{28n}\
\Big{(}7C_{1}^{(1)}+\big{(}C_{2}^{(2)}\big{)}^{2}\Big{)}\
{\mathfrak{a}}_{5}^{2}$ (C.4) $\displaystyle c_{5}$ $\displaystyle=$
$\displaystyle C_{2}^{(2)}\,{\mathfrak{a}}_{5}\ .$

##  Appendix D Scaling limit at the free fermion point with
$\eta_{\ell}=(-1)^{r}\,\mbox{e}^{\frac{{\rm i}\pi}{r}\,(2\ell-1)}$

Here we focus on the case with $r$ odd and $q={\rm i}$. In addition, the
inhomogeneities are set to be $\eta_{\ell}=(-1)^{r}\,\mbox{e}^{\frac{{\rm
i}\pi}{r}\,(2\ell-1)}$ for which the spin chain possesses ${\cal Z}_{r}$
invariance. In this case the Bethe Ansatz equations take the especially simple
form

$\Bigg{(}\frac{1+({\rm i}\zeta_{j})^{r}}{1-({\rm
i}\zeta_{j})^{r}}\Bigg{)}^{\frac{N}{r}}=-\mbox{e}^{{\rm i}\pi(2{\tt
k}+S^{z})}\ .$ (D.1)

In sec. 2.1 we described the scaling limit of the eigenvalue of the $Q$ \-
operator for the ground state of the spin chain. Formula (2.32) remains valid
in the case of any low energy Bethe state provided $n>r$. However, at $n=r$ it
should be modified as

$\operatorname*{slim}_{N\to\infty}\bigg{(}\frac{N}{2r}\bigg{)}^{\frac{1}{2r}(-1)^{A}E^{r}}\,A_{+}\Big{(}\big{(}2N/\pi\big{)}^{-\frac{1}{r}}\,E\Big{)}=D_{+}(E)\qquad\qquad\qquad\qquad(n=r)\,,$
(D.2)

where recall that the symbol “$\operatorname*{slim}$” is there as a reminder
that the formula holds true only for the low energy states. The necessity of
the $N$ \- dependent factor can be illustrated on the ground state eigenvalue.
The scaling limit of the Bethe roots for the ground state, where $S^{z}=0$,
yields

$\displaystyle E^{(a)}_{m}=\mbox{e}^{\frac{{\rm i}\pi}{r}(2a-\frac{r-1}{2})}\
\big{(}(2m_{a}-1+2{\tt k})\,r\big{)}^{\frac{1}{r}}\ \ \ \ \ \qquad\qquad\
(a=1,\ldots,r)$ (D.3)

with

$m_{a}=m=1,2,\ldots\ .$ (D.4)

Then, it is easy to see that the limit in the l.h.s. of (D.2) exists and leads
to

$D_{+}^{({\rm vac})}=\frac{\Gamma(\frac{1}{2}+{\tt
k})}{\Gamma\big{(}\frac{1}{2}+{\tt k}+\epsilon^{r}\big{)}}\ ,$ (D.5)

where we use the short-cut notation

$\displaystyle\epsilon=\mbox{e}^{-\frac{{\rm
i}\pi}{2r}(r+1)}\,(2r)^{-\frac{1}{r}}\ E\ .$ (D.6)

Staying in the sector $S^{z}=0$, we now consider the scaling limit of
$A_{+}(\zeta)$ for a low energy excited Bethe state. The chiral state
$|\alpha\rangle$ appearing in eq. (2.108) can be characterized by the location
of a finite number of “holes” at $n^{-}_{j,a}$ in the vacuum distribution of
the integers $m_{a}$ given by (D.4) as well as “particles” $n^{+}_{j,a}$. The
positive integers $n^{\pm}_{j,a}$ are ordered as

$\displaystyle 1\leq
n_{1,a}^{\pm}<n_{2,a}^{\pm}<\ldots<n_{M^{\pm}_{a},a}^{\pm}\ .$ (D.7)

Note that for the particles, ${\tt k}-n^{+}_{j,a}+\frac{1}{2}$ is a negative
number and we take its $r$-th root to be $\mbox{e}^{\frac{{\rm i}\pi}{r}}\
(n^{+}_{j,a}-{\tt k}-\frac{1}{2})^{\frac{1}{r}}$ so that the corresponding
$E_{m}^{(a)}$ lies on the ray $\arg(E_{m}^{(a)})=\frac{\pi}{2r}\,(4a-r-1)$
(${\rm mod}\ 2\pi$). Then the scaling limit of $A_{+}(\zeta)$ yields

$\displaystyle D_{+}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\frac{1}{2}+{\tt k})}{\Gamma\big{(}\frac{1}{2}+{\tt
k}+\epsilon^{r}\big{)}}\ \frac{\prod_{a=1}^{r}\prod_{j=1}^{M^{-}_{a}}\
\mbox{e}^{\frac{2\pi{\rm i}}{r}a}\big{(}n_{j,a}^{-}-\frac{1}{2}+{\tt
k}\big{)}^{\frac{1}{r}}}{\prod_{a=1}^{r}\prod_{j=1}^{M^{+}_{a}}\mbox{e}^{\frac{\pi{\rm
i}}{r}(2a+1)}\big{(}n_{j,a}^{+}-\frac{1}{2}-{\tt k}\big{)}^{\frac{1}{r}}}$
(D.8) $\displaystyle\times$
$\displaystyle\frac{\prod_{a=1}^{r}\prod_{j=1}^{M^{+}_{a}}\
\Big{(}\mbox{e}^{\frac{\pi{\rm
i}}{r}(2a+1)}\big{(}n_{j,a}^{+}-\frac{1}{2}-{\tt
k}\big{)}^{\frac{1}{r}}+\epsilon\Big{)}}{\prod_{a=1}^{r}\prod_{j=1}^{M^{-}_{a}}\
\Big{(}\mbox{e}^{\frac{2\pi{\rm i}}{r}a}\big{(}n_{j,a}^{-}-\frac{1}{2}+{\tt
k}\big{)}^{\frac{1}{r}}+\epsilon\Big{)}}\ .$

As $\epsilon\to+\infty$, $D_{+}$ develops the asymptotic behaviour

$\displaystyle\log D_{+}\asymp\log({\mathfrak{C}_{+}})-\epsilon^{r}\
\big{(}r\log(\epsilon)-1\big{)}-\big{(}{\tt M}+r{\tt
k}\big{)}\,\log(\epsilon)+\sum_{s=1}^{\infty}\frac{(-1)^{s}}{s}\ D_{+,s}\
\epsilon^{-s}\,.$ (D.9)

Here ${\tt M}$ stands for the difference between the total number of holes and
particles,

$\displaystyle{\tt M}=\sum_{a=1}^{r}\big{(}M_{a}^{-}-M_{a}^{+}\big{)}\,,$
(D.10)

while $\mathfrak{C}_{+}$ is given by

$\displaystyle{\mathfrak{C}_{+}}=\frac{\Gamma(\frac{1}{2}+{\tt
k})}{\sqrt{2\pi}}\frac{\prod_{a=1}^{r}\prod_{j=1}^{M^{-}_{a}}\
\mbox{e}^{\frac{2\pi{\rm i}}{r}a}\big{(}n_{j,a}^{-}-\frac{1}{2}+{\tt
k}\big{)}^{\frac{1}{r}}}{\prod_{a=1}^{r}\prod_{j=1}^{M^{+}_{a}}\mbox{e}^{\frac{\pi{\rm
i}}{r}(2a+1)}\big{(}n_{j,a}^{+}-\frac{1}{2}-{\tt k}\big{)}^{\frac{1}{r}}}\ .$
(D.11)

The coefficients $D_{+,s}$ entering into the sum read as

$\displaystyle D_{+,s}$ $\displaystyle=$ $\displaystyle r\delta_{s,0\,({\rm
mod}\,r)}\ \frac{B_{\frac{s}{r}+1}(\frac{1}{2}+{\tt k})}{\frac{s}{r}+1}$
$\displaystyle+$
$\displaystyle\sum_{a=1}^{r}\Bigg{(}\,\sum_{j=1}^{M_{a}^{-}}\mbox{e}^{\frac{{\rm
i}\pi}{r}\,2as}\,\Big{(}n_{j,a}^{-}-\tfrac{1}{2}+{\tt k}\Big{)}^{\frac{s}{r}}\
-\sum_{j=1}^{M_{a}^{+}}\mbox{e}^{\frac{{\rm
i}\pi}{r}(2a+1)s}\,\Big{(}n_{j,a}^{+}-\tfrac{1}{2}-{\tt
k}\Big{)}^{\frac{s}{r}}\,\Bigg{)}\,,$

where $B_{m+1}(z)$ are the Bernoulli polynomials.

Within the usual ODE/IQFT interpretation [5], the coefficients $D_{+,s}$ in
the expansion (D.9) would be the eigenvalues of certain integrals of motion.
In particular, $D_{+,r}=I_{1}$ — the eigenvalue of the first local integral of
motion,

$\hat{I}_{1}=\int_{0}^{2\pi}\frac{{\rm d}x}{2\pi}\,T\,,$ (D.13)

which is the chiral component of the CFT Hamiltonian (2.107). Formula (D),
specialized to $s=r$ yields

$I_{1}=\frac{r}{2}\ \bigg{(}{\tt k}+\frac{{\tt M}}{r}\bigg{)}^{2}-\frac{{\tt
M}^{2}}{2r}-\frac{r}{24}+\sum_{a=1}^{r}\bigg{(}\sum_{j=1}^{M_{a}^{-}}\big{(}n_{j,a}^{-}-\tfrac{1}{2}\big{)}+\sum_{j=1}^{M_{a}^{+}}\big{(}n_{j,a}^{+}-\tfrac{1}{2}\big{)}\,\bigg{)}\
.$ (D.14)

Let’s write the integer ${\tt M}$ as

$\displaystyle{\tt M}={\tt m}+r{\tt w}$ (D.15)

with

${\tt w}=0,\pm 1,\pm 2,\ldots\,,\qquad\qquad{\tt m}=0,1,2,\ldots,r-1\,.$
(D.16)

Then the state with the lowest value of $I_{1}$ with given ${\tt M}\geq 0$
would have all $M_{a}^{+}=0$, while ${\tt m}$ members of the set
$\\{M_{a}^{-}\\}_{a=1}^{r}$ would be equal to ${\tt w}+1$ and the rest would
coincide with ${\tt w}$. The location of the holes is described by
$n_{j,a}^{-}=j$ for $j\leq M_{a}^{-}$. For ${\tt M}<0$ all $M_{a}^{-}=0$,
while the set $\\{M_{a}^{+}\\}_{a=1}^{r}$ is comprised of $r-{\tt m}$ integers
$|{\tt w}|$ and ${\tt m}$ integers $|{\tt w}+1|$. Then one can re-write
$I_{1}$ in the form which is applicable for any ${\tt M}\in\mathbb{Z}$,

$I_{1}=\frac{r}{2}\ \bigg{(}{\tt k}+{\tt w}+\frac{\tt
m}{r}\bigg{)}^{2}+\frac{{\tt m}\,(r-{\tt m})}{2r}-\frac{r}{24}+{\tt L}\,,$
(D.17)

where ${\tt L}$ is a non-negative integer given by

${\tt
L}=\sum_{a=1}^{r}\bigg{(}\sum_{j=1}^{M_{a}^{-}}\big{(}n_{j,a}^{-}-\tfrac{1}{2}\big{)}+\sum_{j=1}^{M_{a}^{+}}\big{(}n_{j,a}^{+}-\tfrac{1}{2}\big{)}\,\bigg{)}-{\tt
m}\sum_{j=1}^{|{\tt w}+1|}\,(j-\tfrac{1}{2})-(r-{\tt m})\sum_{j=1}^{|{\tt
w}|}\,(j-\tfrac{1}{2})\ .$ (D.18)

Note that the states with ${\tt L}=0$, having the lowest value of $I_{1}$, are
$\frac{r!}{{\tt m}!(r-{\tt m})!}$ times degenerate. Analogous computations can
be repeated for the chiral state $|\bar{\alpha}\rangle$. One should keep in
mind that in the sector with fixed $S^{z}$, the total number of (left and
right) holes should coincide with the total number of particles, i.e.,

$\sum_{a=1}^{r}\big{(}M^{-}_{a}+\bar{M}^{-}_{a}\big{)}=\sum_{a=1}^{r}\big{(}M^{+}_{a}+\bar{M}^{+}_{a}\big{)}\,.$
(D.19)

In the above considerations, we focused on the sector $S^{z}=0$ to make the
discussion as simple as possible. This can be extended to the case of $S^{z}$
being any (half-)integer. The final expressions for $I_{1}$ and $\bar{I}_{1}$
are given by formulae (2.125)-(2.127) where one should set $n=r$ therein.
Then, the eigenvalues of the Hamiltonian $\hat{H}_{\rm CFT}$ (2.106) coincide
with $I_{1}+\bar{I}_{1}$, while for the $r$ \- site lattice translation
operator they are described by (2.128). This result motivated the conjecture
contained in sec.2.5.1. The latter was verified numerically on some specific
examples away from the free fermion point.

##  Appendix E Scaling limit for $r$, $A$ even and
$(\mu,j)=(\frac{r}{2},\frac{A}{2})$

Here we present the numerical scheme mentioned in sec. 5.1 which allows one to
confirm that the scaling limit of the Bethe roots with the single non-trivial
RG invariant taken as in (5.4) is described by the differential equation (5.1)
with $\mu=\frac{r}{2}$.

As was explained in sec. 5.2, in the case $\mu=\frac{r}{2}$ the ground state
Bethe roots, where the inhomogeneities obey the $r$ \- site periodicity
condition $\eta_{J+r}=\eta_{J}$, are simply related to those for the spin
chain with $\tilde{\eta}_{J+2}=\tilde{\eta}_{J}$. The parameters of the
reduced system $(\tilde{N},\tilde{q},\tilde{\eta}_{\ell})$ are expressed in
terms of the original one $(N,q,\eta_{\ell})$ as in eqs. (5.23) and (5.26)
with $\sigma=\frac{r}{2}$. In particular,

$\tilde{N}=2N/r\,,\qquad\qquad\tilde{q}=\exp\Big{(}{\frac{{\rm
i}\pi}{\tilde{n}+2}}\Big{)}\,,\qquad\qquad{\rm
where}\qquad\qquad\tilde{n}=2n/r\,.$ (E.1)

From the definition of the RG invariant (5.4) and the normalization condition
(2.34) one has

$\displaystyle\frac{1}{\tilde{\eta}_{1}}+\frac{1}{\tilde{\eta}_{2}}$
$\displaystyle=$ $\displaystyle(-1)^{\frac{r}{2}-1}\
{\mathfrak{a}}_{\frac{r}{2}}\
\bigg{(}\frac{4\tilde{N_{0}}}{\tilde{N}}\bigg{)}\
\log\bigg{(}\frac{r\tilde{N}}{4\tilde{N_{0}}}\bigg{)}\,$
$\displaystyle\frac{1}{\tilde{\eta}_{1}^{2}}+\frac{1}{\tilde{\eta}_{2}^{2}}$
$\displaystyle=$ $\displaystyle-2\,,$ (E.2)

where $\tilde{N}_{0}$ is given by formula (1.17) with $r$ and $n$ replaced by
$2$ and $\tilde{n}$, respectively. In what follows it would be convenient to
use the parameter $\alpha$, defined as

$\alpha=\arg(\tilde{\eta}_{1})=-\arg(\tilde{\eta}_{2})\,,\qquad\qquad\qquad\qquad\alpha\in(0,\pi)\,.$
(E.3)

Then the system of equations (E) determining the inhomogeneities implies

$\alpha=\frac{\pi}{2}+(-1)^{\frac{r}{2}}\ {\mathfrak{a}}_{\frac{r}{2}}\
\bigg{(}\frac{2\tilde{N_{0}}}{\tilde{N}}\bigg{)}\
\log\bigg{(}\frac{r\tilde{N}}{4\tilde{N_{0}}}\bigg{)}\,\
+O\big{(}(\log\tilde{N})^{3}/\tilde{N}^{3}\big{)}\ .$ (E.4)

The scaling limit of the 2 - site periodic spin chain in the regime with
$\arg(\tilde{q})\in(0,\frac{\pi}{2})$ has attracted a lot of attention in the
literature, see [35, 23, 37, 38, 24] and the more recent papers [20, 39]. The
interest originated following the work of Jacobsen and Saleur [35], which was
devoted to the case when the parameter $\alpha$ in (E.3) is equal to
$\frac{\pi}{2}$. In the subsequent papers [23] and [37], the energy spectrum
for — in the terminology used in sec. 7 — primary Bethe states was studied.
These were originally defined in [23] using the logarithmic form of the Bethe
Ansatz equations. The corresponding Bethe roots turn out to be all real and
the states can be characterized by the integer ${\tt m}$, which coincides with
the difference between the number of positive $(\tilde{\zeta}_{m}>0)$ and
negative $(\tilde{\zeta}_{m}<0)$ roots. In ref.[37] an important relation was
found between ${\tt m}$ and the quantity

$s=\frac{\tilde{n}}{2\pi}\log\big{(}{\cal K}^{(2)}\,/\,{\cal
K}^{(1)}\big{)}\,,$ (E.5)

which is expressed in terms of the eigenvalues of the quasi-shift operators

${\cal K}^{(\ell)}=\mbox{e}^{{\rm i}\pi{\tt
k}}\,{\tilde{q}}^{-\frac{\tilde{N}}{2}}\,\prod_{m=1}^{\frac{\tilde{N}}{2}-\tilde{S}^{z}}\,\frac{\tilde{\zeta}_{m}+\tilde{\eta}_{\ell}\,\tilde{q}^{+1}}{\tilde{\zeta}_{m}+\tilde{\eta}_{\ell}\,\tilde{q}^{-1}}\qquad\qquad\qquad(\ell=1,2)\,.$
(E.6)

In the case of the primary Bethe states there is a natural way of assigning
them a dependence on the lattice size so that $s$ becomes a function of
$\tilde{N}$. Then it was observed from the numerical data that, with a
properly chosen branch of the logarithm in the definition (E.5), the leading
large - $\tilde{N}$ behaviour of $s$ for fixed ${\tt m}$ is given by

$s\sim\frac{\pi{\tt
m}}{2\log(\tilde{N})}\qquad\qquad\qquad(\tilde{N}\to\infty)\ .$ (E.7)

Moreover, the so-called “quantization condition” was proposed that provides a
refinement to this leading asymptotic. The more precise version of the
relation, which is also applicable to the case of quasi-periodic boundary
conditions, appeared in ref.[20] and reads explicitly as

$4s\log\bigg{(}\frac{\tilde{N}}{2\tilde{N}_{0}}\bigg{)}-\delta(s)-2\pi\,{\tt
m}=O\big{(}(\log\tilde{N})^{-\infty}\big{)}\ ,$ (E.8)

where

$\displaystyle\delta(s)=-2{\rm i}\,\log\Bigg{(}2^{-\frac{2{\rm
i}s}{\tilde{n}}(\tilde{n}+2)}\,\frac{\Gamma(\frac{1}{2}+\tilde{p}+\frac{{\rm
i}s}{2})}{\Gamma(\frac{1}{2}+\tilde{p}-\frac{{\rm
i}s}{2})}\frac{\Gamma(\frac{1}{2}+\bar{\tilde{p}}+\frac{{\rm
i}s}{2})}{\Gamma(\frac{1}{2}+\bar{\tilde{p}}-\frac{{\rm i}s}{2})}\Bigg{)}$
(E.9)

and

$\tilde{p}=\frac{\tilde{S}^{z}}{2}+\frac{\tilde{n}+2}{2}\ \big{(}{\tt k}+{\tt
w}\big{)}\,,\qquad\qquad\bar{\tilde{p}}=\frac{\tilde{S}^{z}}{2}-\frac{\tilde{n}+2}{2}\
\big{(}{\tt k}+{\tt w}\big{)}\ .$ (E.10)

The quantization condition holds true up to power law corrections in
$\tilde{N}$, as indicated by the r.h.s. of (E.8). For given ${\tt m}$, it
gives a remarkably accurate description for the value of $s$ at large
$\tilde{N}$ [44]. It follows that $s(\tilde{N})$ tends to zero as
$\tilde{N}\to\infty$ if the integer ${\tt m}$ is left unchanged. A non-trivial
scaling limit can be achieved by increasing $|{\tt m}|$ simultaneously with
$\tilde{N}$ in such a way that $s$ is kept fixed, i.e., treated as the RG
invariant. Then (E.8) is interpreted as an equation specifying the $\tilde{N}$
\- dependence of ${\tt m}$. This way, the states would be characterized by
$s$, which may take any real value. The corresponding eigenvalue of the CFT
Hamiltonian (2.106) is given by (7.20) with $r=2$. In the work [20] it was
proposed that the differential equation

$\Big{[}-\partial^{2}_{\tilde{y}}+\tilde{p}^{2}+\mbox{e}^{(\tilde{n}+2)\tilde{y}}-\tilde{E}^{2}\,\mbox{e}^{2\tilde{y}}-s\,\tilde{E}\,\mbox{e}^{\tilde{y}}\Big{]}\psi=0$
(E.11)

describes the scaled Bethe roots for the primary Bethe states when the scaling
limit is taken with $s$ being treated as an RG invariant.

The quantization condition was originally found for the case when $\alpha$
from (E.3) is equal to $\frac{\pi}{2}$. It was generalized by Frahm and Seel
in ref.[24] to $\alpha\in(\gamma,\pi-\gamma)$ being a fixed number independent
of the lattice size. For our purposes, it is sufficient to use the results of
that paper specialized to the ground state, i.e., the state of the spin chain
Hamiltonian with the lowest possible energy. The important difference between
$\alpha=\frac{\pi}{2}$ and $\alpha\neq\frac{\pi}{2}$ is that the value of $s$
defined as in (E.5) for the ground state in the former case vanishes, while
for the latter it is no longer zero. Moreover, as $\tilde{N}\to\infty$,

$s_{\scriptscriptstyle\rm GS}(\tilde{N})=\tilde{N}b_{\infty}+O(1)\ ,$ (E.12)

where $s_{\scriptscriptstyle\rm GS}$ denotes the value of $s$ computed for the
ground state and

$\displaystyle b_{\infty}=-\frac{n}{\pi}\int_{0}^{\infty}\frac{\mbox{d}t}{t}\
\frac{\sinh(t)\,\sinh\big{(}(\tilde{n}+1)t\big{)}}{\sinh({\tilde{n}}t)\,\sinh\big{(}(\tilde{n}+2)t\big{)}}\,\sinh\big{(}(1-\tfrac{2\alpha}{\pi})\,(\tilde{n}+2)t\big{)}\
.$ (E.13)

The remainder term in (E.12) admits a more accurate description. Namely,
denoting it as

$s_{\alpha}\equiv s_{\scriptscriptstyle\rm GS}(\tilde{N})-\tilde{N}b_{\infty}\
,$ (E.14)

the following quantization condition holds true

$4s_{\alpha}\,\log\bigg{(}\frac{\tilde{N}}{2\tilde{N}_{0}}\bigg{)}-\delta_{\tt
k}(s_{\alpha})=2\pi{\tt
m}_{\alpha}+\frac{\tilde{n}+2}{\tilde{n}}\,(\pi-2\alpha)\,\tilde{N}+O\big{(}(\log\tilde{N})^{-\infty}\big{)}\,.$
(E.15)

Here $\delta_{\tt k}(s)$ is given by (E.9) with
${\tilde{p}}=-\bar{{\tilde{p}}}=\frac{1}{2}(\tilde{n}+2)\,{\tt k}$ and ${\tt
m}_{\alpha}$ is the nearest integer to
$-\frac{\tilde{n}+2}{2\pi\tilde{n}}\,(\pi-2\alpha)\,\tilde{N}$. Despite the
similarity of the above with (E.8) one should keep in mind that the latter
relation gives the $\tilde{N}$ dependence of $s$ (E.5) for the primary Bethe
states, while (E.15) describes how $s_{\alpha}$, defined in (E.14) depends on
$\tilde{N}$ for the ground state of the spin chain only.

In the relation (E.15) the parameter $\alpha$ is assumed to be fixed.
Nevertheless, it inspired us to introduce the formal variable $s$ as the
solution of the equation

$4s\,\log\bigg{(}\frac{\tilde{N}}{2\tilde{N}_{0}}\bigg{)}-\delta_{\tt
k}(s)=(-1)^{\frac{r}{2}-1}\ \mathfrak{a}_{\frac{r}{2}}\
\frac{4\,(\tilde{n}+2)}{\tilde{n}}\,\tilde{N}_{0}\,\log\bigg{(}\frac{r\tilde{N}}{4\tilde{N}_{0}}\bigg{)}\
,$ (E.16)

which is obtained from (E.15) by setting ${\tt m}_{\alpha}=0$; substituting
$\alpha$ for the $\tilde{N}$ dependent expression in the r.h.s. of (E.4); and
ignoring all correction terms. In the original scheme for performing the
scaling limit $\mathfrak{a}_{\frac{r}{2}}$ was treated as the RG invariant. We
now switch the schemes and suppose that $s$ does not depend on the number of
lattice sites, whereas the $\tilde{N}$ dependence of
$\mathfrak{a}_{\frac{r}{2}}$ is dictated by (E.16). It turns out that in the
new scheme the sums over the Bethe roots

$\bigg{(}\frac{2\tilde{N}_{0}}{\tilde{N}}\bigg{)}^{\frac{\tilde{n}j}{\tilde{n}+2}}\
\frac{1}{j}\ \sum_{m=1}^{\tilde{N}/2}(\tilde{\zeta}_{m})^{-j}$

tend to their limiting values with corrections that decay as a power law in
$\tilde{N}$. In contrast, if $\mathfrak{a}_{\frac{r}{2}}$ were to be kept
fixed, the corrections would decay considerably more slowly — as the inverse
power of $\log(\tilde{N})$. Thus, the new scheme allows one to observe the
scaling behaviour for $\tilde{N}=\frac{2N}{r}\sim 1000$. On the other hand the
definition (E.16) implies

$s+(-1)^{\frac{r}{2}}\,\mathfrak{a}_{\frac{r}{2}}\,\frac{\tilde{n}+2}{\tilde{n}}\,\tilde{N}_{0}\,=O\big{(}1/\log(\tilde{N})\big{)}\
.$ (E.17)

Hence, the change of scheme will not affect the scaling limit, but rather, the
rate at which it is achieved as $\tilde{N}\to\infty$. In this regard, one may
note that the choice of the constant $rN_{0}$ in the logarithm in (5.4) for
the RG invariant $\mathfrak{a}_{\frac{r}{2}}$ is not essential: the constant
can be replaced by any positive number.

This way, we were able to establish that the scaling limit of the Bethe roots
is described by the ODE (E.11). Taking into account the relations (E.1)
between the parameters of the original and reduced systems and performing the
change of variables

$\tilde{y}=\frac{r}{2}\,{y}-\frac{r}{n+r}\,\log\Big{(}\frac{r}{2}\Big{)}$
(E.18)

and

$\displaystyle\tilde{p}$ $\displaystyle=$
$\displaystyle\frac{2}{r}\,p\,,\qquad\qquad\qquad\tilde{E}=(-1)^{\frac{A}{2}}\,\Big{(}\frac{2}{r}\Big{)}^{\frac{n}{n+r}}\,E^{\frac{r}{2}}$
(E.19a) $\displaystyle s$ $\displaystyle=$
$\displaystyle(-1)^{\frac{A}{2}}\,\frac{2c}{r}$ (E.19b)

one arrives at the differential equation (5.1) with $\mu=\frac{r}{2}$. The
$\mathfrak{a}_{\frac{r}{2}}$ — $c$ relation (5.5) follows by combining (E.19b)
with (E.17).

##  Appendix F Supplementary tables

Here we give the specification of the RG invariants $\mathfrak{a}_{s}$ (1.22),
such that the scaled Bethe roots (1.16) are described by the ODE

$\displaystyle\bigg{[}-\partial_{y}^{2}+p^{2}+\mbox{e}^{(n+r)y}-(-1)^{A}\,E^{r}\
\mbox{e}^{ry}-c\,E^{\mu}\ \mbox{e}^{\big{(}(A\mu-
rj)\,\frac{n+r}{r}+\mu\big{)}y}\,\bigg{]}\,\psi=0$ (F.1)

with $(\mu,j)$ taken from the set (1.24).

For $r=3,4,\ldots,10$, $A=1,2,\ldots,r-2$ and all possible pairs $(\mu,j)$,
presented in the tables are the values of the integer $s$ corresponding to the
minimal non-trivial set of RG invariants along with their exponents $d_{s}$.
All other sums $\sum_{\ell=1}^{r}(\eta_{\ell})^{-s}$ can be taken to be zero.
Also, the normalization condition

$\frac{1}{r}\,\sum_{\ell=1}^{r}(\eta_{\ell})^{-r}=(-1)^{r-1}$ (F.2)

is being assumed. The values of the non-vanishing RG invariants are expressed
in terms of the coefficient $c$ of the ODE as in formulae (4.22) - (4.26).
Then, the scaling limit of the ground state Bethe roots is described by the
differential equation for any $n>n_{\rm min}$, where $n_{\rm min}$ is listed
in the last column of the tables.

| $A$ | $(\mu,j)$ | $s$ | $d_{s}$ | $n_{\rm min}$
---|---|---|---|---|---
$r=3$ | $\vspace{-0.0cm}1$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{2}{3}$ | $0$
$r=4$ | $\vspace{-0.0cm}1$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=1$ | $0$
| $\vspace{-0.0cm}2$ | $(1,0)$ | $\\{1,2\\}$ | $d_{1}=\frac{1}{2},\,d_{2}=1$ | $0$
$r=5$ | $\vspace{-0.2cm}1$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{6}{5}$ | $\frac{5}{2}$
| $(2,0)$ | $\\{2,4,1\\}$ | $d_{2}=\frac{2}{5},\,d_{4}=\frac{4}{5},\,d_{1}=\frac{6}{5}$ | $\frac{5}{2}$
$\vspace{-0.2cm}2$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{4}{5}$ | 0
| $(3,1)$ | $\\{3,1\\}$ | $d_{3}=\frac{2}{5},\,d_{1}=\frac{4}{5}$ | 0
| $\vspace{-0.2cm}3$ | $(1,0)$ | $\\{1,2\\}$ | $d_{1}=\frac{2}{5},\,d_{2}=\frac{4}{5}$ | 0
|  | $(2,1)$ | $\\{2\\}$ | $d_{2}=\frac{4}{5}$ | 0
$r=6$ | $\vspace{-0.2cm}1$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{4}{3}$ | $6$
| $(2,0)$ | $\\{2\\}$ | $d_{2}=\frac{2}{3}$ | $0$
$\vspace{-0.0cm}2$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=1$ | 0
$\vspace{-0.0cm}3$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{2}{3}$ | 0
| $\vspace{-0.2cm}4$ | $(1,0)$ | $\\{1,2,3\\}$ | $d_{1}=\frac{1}{3},\,d_{2}=\frac{2}{3},\,d_{3}=1$ | 0
|  | $(2,1)$ | $\\{2\\}$ | $d_{2}=\frac{2}{3}$ | 0
$r=7$ | $\vspace{0.1cm}1$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{10}{7}$ | $\frac{21}{2}$
$(2,0)$ | $\\{2\\}$ | $d_{2}=\frac{6}{7}$ | $0$
| $(3,0)$ | $\\{3,6,2,5,1\\}$ | $d_{3}=\frac{2}{7},\,d_{6}=\frac{4}{7},\,d_{2}=\frac{6}{7},\,d_{5}=\frac{8}{7},\,d_{1}=\frac{10}{7}$ | $\frac{21}{2}$
$\vspace{0.1cm}2$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{8}{7}$ | $\frac{7}{4}$
$(2,0)$ | $\\{2,4,6,1\\}$ | $d_{2}=\frac{2}{7},\,d_{4}=\frac{4}{7},\,d_{6}=\frac{6}{7},\,d_{1}=\frac{8}{7}$ | $\frac{7}{4}$
| $(4,1)$ | $\\{4,1\\}$ | $d_{4}=\frac{4}{7},\,d_{1}=\frac{8}{7}$ | $\frac{7}{4}$
$\vspace{0.1cm}3$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{6}{7}$ | $0$
$(3,1)$ | $\\{3\\}$ | $d_{3}=\frac{4}{7}$ | $0$
| $(5,2)$ | $\\{5,3,1\textasciiacute\\}$ | $d_{5}=\frac{2}{7},\,d_{3}=\frac{4}{7},\,d_{1}=\frac{6}{7}$ | $0$
$\vspace{0.1cm}4$ | $(1,0)$ | $\\{1,2\\}$ | $d_{1}=\frac{4}{7},\,d_{2}=\frac{8}{7}$ | $\frac{7}{2}$
| $(2,1)$ | $\\{2\\}$ | $d_{2}=\frac{8}{7}$ | $\frac{7}{2}$
|  | $(4,2)$ | $\\{4,1,5,2\\}$ | $d_{4}=\frac{2}{7},\,d_{1}=\frac{4}{7},\,d_{5}=\frac{6}{7},\,d_{2}=\frac{8}{7}$ | $\frac{7}{2}$
| $\vspace{0.1cm}5$ | $(1,0)$ | $\\{1,2,3\\}$ | $d_{1}=\frac{2}{7},\,d_{2}=\frac{4}{7},\,d_{3}=\frac{6}{7}$ | $0$
| $(2,1)$ | $\\{2\\}$ | $d_{2}=\frac{4}{7}$ | $0$
|  | $(3,2)$ | $\\{3\\}$ | $d_{3}=\frac{6}{7}$ | $0$

| $A$ | $(\mu,j)$ | $s$ | $d_{s}$ | $n_{\rm min}$
---|---|---|---|---|---
$r=8$ | $\vspace{0.1cm}1$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{3}{2}$ | $16$
$(2,0)$ | $\\{2\\}$ | $d_{2}=1$ | $0$
| $(3,0)$ | $\\{3,6,1\\}$ | $d_{3}=\frac{1}{2},\,d_{6}=1,\,d_{1}=\frac{3}{2}$ | $16$
$\vspace{0.1cm}2$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{5}{4}$ | $4$
$(2,0)$ | $\\{2,4\\}$ | $d_{2}=\frac{1}{2},\,d_{4}=1$ | $0$
| $(5,1)$ | $\\{5,2,7,4\\}$ | $d_{5}=\frac{1}{4},\,d_{2}=\frac{1}{2},\,d_{7}=\frac{3}{4},\,d_{4}=1$ | $0$
$\vspace{0.1cm}3$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=1$ | $0$
| $(3,1)$ | $\\{3\\}$ | $d_{3}=1$ | $0$
$\vspace{0.1cm}4$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{3}{4}$ | 0
| $(3,1)$ | $\\{3,6,1,4\\}$ | $d_{3}=\frac{1}{4},\,d_{6}=\frac{1}{2},\,d_{1}=\frac{3}{4},\,d_{4}=1$ | $0$
$\vspace{0.1cm}5$ | $(1,0)$ | $\\{1,2\\}$ | $d_{1}=\frac{1}{2},\,d_{2}=1$ | $0$
| $(2,1)$ | $\\{2\\}$ | $d_{2}=1$ | $0$
|  | $(5,3)$ | $\\{5,2\\}$ | $d_{5}=\frac{1}{2},\,d_{2}=1$ | $0$
| $\vspace{-0.1cm}6$ | $(1,0)$ | $\\{1,2,3,4\\}$ | $d_{1}=\frac{1}{4},\,d_{2}=\frac{1}{2},\,d_{3}=\frac{3}{4},\,d_{4}=1$ | $0$
| $(2,1)$ | $\\{2,4\\}$ | $d_{2}=\frac{1}{2},\,d_{4}=1$ | $0$
|  | $(3,2)$ | $\\{3\\}$ | $d_{3}=\frac{3}{4}$ | $0$

| $A$ | $(\mu,j)$ | $s$ | $d_{s}$ | $n_{\rm min}$
---|---|---|---|---|---
$r=9$ | $\vspace{-0.1cm}1$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{14}{9}$ | $\frac{45}{2}$
$(2,0)$ | $\\{2\\}$ | $d_{2}=\frac{10}{9}$ | $\frac{9}{4}$
$(3,0)$ | $\\{3\\}$ | $d_{3}=\frac{2}{3}$ | $0$
| $(4,0)$ | $\\{4,8,3,7,2,6,1\\}$ | $d_{4}=\frac{2}{9},\,d_{8}=\frac{4}{9},\,d_{3}=\frac{2}{3},\,d_{7}=\frac{8}{9},$ | $\frac{45}{2}$
|  |  | $d_{2}=\frac{10}{9},\,d_{6}=\frac{12}{9},\,\,d_{1}=\frac{14}{9}$ |
$\vspace{0.125cm}2$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{4}{3}$ | $\frac{27}{4}$
$(2,0)$ | $\\{2\\}$ | $d_{2}=\frac{2}{3}$ | $0$
| $(5,1)$ | $\\{5,1\\}$ | $d_{5}=\frac{2}{3},\,d_{1}=\frac{4}{3}$ | $\frac{27}{4}$
$\vspace{-0.1cm}3$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{10}{9}$ | $\frac{3}{2}$
$(2,0)$ | $\\{2,4,6,8,1\\}$ | $d_{2}=\frac{2}{9},\,d_{4}=\frac{4}{9},\,d_{6}=\frac{2}{3},\,d_{8}=\frac{8}{9},\,d_{1}=\frac{10}{9}$ | $\frac{3}{2}$
| $(4,1)$ | $\\{4,8,3\\}$ | $d_{4}=\frac{4}{9},\,d_{8}=\frac{8}{9}$ | 0
$\vspace{-0.1cm}4$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{8}{9}$ | 0
$(3,1)$ | $\\{3\\}$ | $d_{3}=\frac{2}{3}$ | $0$
$(5,2)$ | $\\{5,1\\}$ | $d_{5}=\frac{4}{9},\,d_{1}=\frac{8}{9}$ | $0$
| $(7,3)$ | $\\{7,5,3,1\\}$ | $d_{7}=\frac{2}{9},\,d_{5}=\frac{4}{9},\,d_{3}=\frac{2}{3},\,d_{1}=\frac{8}{9}$ | $0$
| $\vspace{0.1cm}5$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{2}{3}$ | $0$
| $(2,1)$ | $\\{2\\}$ | $d_{2}=\frac{4}{3}$ | $\frac{27}{2}$
|  | $(4,2)$ | $\\{4\\}$ | $d_{4}=\frac{2}{3}$ | $0$
| $\vspace{0.115cm}6$ | $(1,0)$ | $\\{1,2\\}$ | $d_{1}=\frac{4}{9},\,d_{2}=\frac{8}{9}$ | $0$
| $(2,1)$ | $\\{2\\}$ | $d_{2}=\frac{8}{9}$ | $0$
|  | $(5,3)$ | $\\{5,1,6,2\\}$ | $d_{5}=\frac{2}{9},\,d_{1}=\frac{4}{9},\,d_{6}=\frac{2}{3},\,d_{2}=\frac{8}{9}$ | $0$
| $\vspace{-0.0cm}7$ | $(1,0)$ | $\\{1,2,3,4\\}$ | $d_{1}=\frac{2}{9},\,d_{2}=\frac{4}{9},\,d_{3}=\frac{2}{3},\,d_{4}=\frac{8}{9}$ | $0$
| $(2,1)$ | $\\{2,4\\}$ | $d_{2}=\frac{4}{9},\,d_{4}=\frac{8}{9}$ | $0$
| $(3,2)$ | $\\{3\\}$ | $d_{3}=\frac{2}{3}$ | $0$
|  | $(4,3)$ | $\\{4\\}$ | $d_{4}=\frac{8}{9}$ | $0$

| $A$ | $(\mu,j)$ | $s$ | $d_{s}$ | $n_{\rm min}$
---|---|---|---|---|---
$r=10$ | $\vspace{-0.1cm}1$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{8}{5}$ | $30$
$(2,0)$ | $\\{2\\}$ | $d_{2}=\frac{6}{5}$ | $5$
$(3,0)$ | $\\{3\\}$ | $d_{3}=\frac{4}{5}$ | $0$
| $(4,0)$ | $\\{4,8,2\\}$ | $d_{4}=\frac{2}{5},\,d_{8}=\frac{4}{5},\,d_{2}=\frac{6}{5}$ | $5$
$\vspace{-0.1cm}2$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{7}{5}$ | $10$
$(2,0)$ | $\\{2\\}$ | $d_{2}=\frac{4}{5}$ | $0$
$(3,0)$ | $\\{3,6,9,2,5,1\\}$ | $d_{3}=\frac{1}{5},\,d_{6}=\frac{2}{5},\,d_{9}=\frac{3}{5},\,d_{2}=\frac{4}{5},\,d_{5}=1,\,d_{1}=\frac{7}{5}$ | $10$
| $(6,1)$ | $\\{6,2\\}$ | $d_{6}=\frac{2}{5},\,d_{2}=\frac{4}{5}$ | $0$
$\vspace{-0.1cm}3$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{6}{5}$ | $\frac{10}{3}$
$(2,0)$ | $\\{2,4\\}$ | $d_{2}=\frac{2}{5},\,d_{4}=\frac{4}{5}$ | $0$
$(4,1)$ | $\\{4\\}$ | $d_{4}=\frac{4}{5}$ | 0
| $(7,2)$ | $\\{7,4,1\\}$ | $d_{7}=\frac{2}{5},\,d_{4}=\frac{4}{5},\,d_{1}=\frac{6}{5}$ | $\frac{10}{3}$
$\vspace{-0.2cm}4$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=1$ | $0$
| $(3,1)$ | $\\{3\\}$ | $d_{3}=1$ | $0$
$\vspace{-0.2cm}5$ | $(1,0)$ | $\\{1\\}$ | $d_{1}=\frac{4}{5}$ | $0$
| $(3,1)$ | $\\{3,6\\}$ | $d_{3}=\frac{2}{5},\,d_{6}=\frac{4}{5}$ | $0$
$\vspace{-0.0cm}6$ | $(1,0)$ | $\\{1,2\\}$ | $d_{1}=\frac{3}{5},\,d_{2}=\frac{6}{5}$ | $5$
$(2,1)$ | $\\{2\\}$ | $d_{2}=\frac{6}{5}$ | $5$
| $(4,2)$ | $\\{4,8,2\\}$ | $d_{4}=\frac{2}{5},\,d_{8}=\frac{4}{5},\,d_{2}=\frac{6}{5}$ | $5$
|  | $(7,4)$ | $\\{7,4,1,8,5\\}$ | $d_{7}=\frac{1}{5},\,d_{4}=\frac{2}{5},\,d_{1}=\frac{3}{5},\,d_{8}=\frac{4}{5},\,d_{5}=1$ | $0$
| $\vspace{-0.0cm}7$ | $(1,0)$ | $\\{1,2\\}$ | $d_{1}=\frac{2}{5},\,d_{2}=\frac{4}{5}$ | $0$
| $(2,1)$ | $\\{2\\}$ | $d_{2}=\frac{4}{5}$ | $0$
| $(3,2)$ | $\\{3\\}$ | $d_{3}=\frac{6}{5}$ | $10$
|  | $(6,4)$ | $\\{6,2\\}$ | $d_{6}=\frac{2}{5},\,d_{2}=\frac{4}{5}$ | $0$
| $\vspace{-0.05cm}8$ | $(1,0)$ | $\\{1,2,3,4,5\\}$ | $d_{1}=\frac{1}{5},\,d_{2}=\frac{2}{5},\,d_{3}=\frac{3}{5},\,d_{4}=\frac{4}{5},\,d_{5}=1$ | $0$
| $(2,1)$ | $\\{2,4\\}$ | $d_{2}=\frac{2}{5},\,d_{4}=\frac{4}{5}$ | $0$
| $(3,2)$ | $\\{3\\}$ | $d_{3}=\frac{3}{5}$ | $0$
|  | $(4,3)$ | $\\{4\\}$ | $d_{4}=\frac{4}{5}$ | $0$

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