# The square lattice Ising model on the rectangle III:
Hankel and Toeplitz determinants

Alfred Hucht  Faculty of Physics, University of Duisburg-Essen, 47048
Duisburg, Germany

(August 28, 2024)

###### Abstract

Based on the results obtained in [Hucht, J. Phys. A: Math. Theor. 50, 065201
(2017)], we show that the partition function of the anisotropic square lattice
Ising model on the $L\times M$ rectangle, with open boundary conditions in
both directions, is given by the determinant of a
$\nicefrac{{M}}{{2}}\times\nicefrac{{M}}{{2}}$ Hankel matrix, that
equivalently can be written as the Pfaffian of a skew-symmetric $M\times M$
Toeplitz matrix. The $M-1$ independent matrix elements of both matrices are
Fourier coefficients of a certain symbol function, which is given by the ratio
of two characteristic polynomials. These polynomials are associated to the
different directions of the system, encode the respective boundary conditions,
and are directly related through the symmetry of the considered Ising model
under exchange of the two directions. The results can be generalized to other
boundary conditions and are well suited for the analysis of finite-size
scaling functions in the critical scaling limit using Szegő’s theorem.

###### Contents

  1. 1 Introduction
    1. 1.1 The model
    2. 1.2 Starting point
    3. 1.3 The Onsager dispersion relation
  2. 2 Elliptic parametrization
    1. 2.1 Coupling constants parametrization
    2. 2.2 Eigenvalues parametrization
    3. 2.3 The Jacobi amplitudes $\omega$ and $\theta$
  3. 3 Results
    1. 3.1 Characteristic polynomials
    2. 3.2 Common eigenvectors
    3. 3.3 Hankelization
    4. 3.4 Contour integral representation
    5. 3.5 From Hankel to Toeplitz
  4. 4 Discussion
  5. A Useful elliptic identities
  6. B More characteristic polynomials
  7. C Some product identities
  8. D A block Hankel matrix identity

##  1 Introduction

The anisotropic two-dimensional Ising model [1] on the $L\times M$ square
lattice is one of the best investigated models in statistical mechanics. In
the thermodynamic limit $L,M\to\infty$, it has a continuous phase transition,
from a disordered high-temperature phase to an ordered low-temperature phase,
at a critical temperature $T_{\mathrm{c}}$. After the exact solution of the
periodic case by Onsager [2, 3], many authors have contributed to the
knowledge about this model under various aspects, such as different boundary
conditions (BCs) or surface effects [4, 5, 6].

Until some years ago, exact solutions for arbitrary temperatures were only
known for systems with periodic or antiperiodic boundary conditions in at
least one direction, as then a Fourier transform along this translationally
invariant direction could be used to diagonalize the problem in the
corresponding direction. The remaining direction could be handled afterwards
with a transfer matrix method, involving a $2\times 2$ transfer matrix, taking
into account arbitrary boundary conditions, or even line disorder [4].

This changed in 2016, when Baxter [7, 8] and Hucht [9, 10, 11] independently
presented exact results for the Ising model on the rectangle, with open
boundary conditions in both directions, expressing the partition function as
$M\times M$ and $M/2\times M/2$ determinants, respectively. While Baxter used
Kaufman’s spinor method [3] below $T_{\mathrm{c}}$ and focused on the
thermodynamic limit [7], where he calculated the corner contributions to the
free energy [12], Hucht utilized the dimer method of Kasteleyn and McCoy & Wu
[13, 14, 15, 4, 16], combined with Schur reductions and a block transfer
matrix formulation of Molinari [17], and derived closed expressions for finite
systems at arbitrary temperatures. The resulting transfer matrices directly
correspond to the spinor method matrices, connecting the so far unrelated
methods.

Near the critical temperature $T_{\mathrm{c}}$, the direction dependent bulk
correlation length of thermal fluctuations $\xi_{\infty}^{\delta}(T)$ diverges
according to
$\xi_{\infty}^{\delta}(T\,{>}\,T_{\mathrm{c}})\simeq\xi_{>}^{\delta}(T/T_{\mathrm{c}}-1)^{-\nu}$,
where $\delta=\,\leftrightarrow,\updownarrow$ denotes the directions
corresponding to $L$ and $M$, $\xi_{>}^{\delta}$ are interaction-dependent
metric factors, and the correlation length critical exponent is $\nu=1$ in the
two-dimensional Ising model111The relation “$\simeq$” denoted “asymptotically
equal” in the respective limit, and $\equiv$ denotes a definition.. If
$\xi_{\infty}^{\delta}(T)$ becomes of the order of the system size $L$ or $M$
in finite systems, interesting finite-size effects such as the critical
Casimir effect emerge, which describes an interaction of the system boundaries
mediated by long-range critical fluctuations [18, 19] in close analogy to the
quantum electrodynamical Casimir effect [20, 21]. These finite-size effects
can be described by universal finite-size scaling functions of the form
$\Theta(x,\rho)$, with temperature scaling variable
$x\equiv(T/T_{\mathrm{c}}-1)(M/\xi_{0}^{\updownarrow})^{1/\nu}$ and
generalized aspect ratio
$\rho\equiv(L/\xi_{0}^{\leftrightarrow})/(M/\xi_{0}^{\updownarrow})$, that
only depend on the bulk and surface universality classes of the model [22,
23], as well as on the BCs. They have been calculated exactly for many cases,
albeit mostly in strip geometry, where the aspect ratio $\rho$ of the system
goes to zero [24, 25, 26, 27, 28, 29, 30]. The theoretical results from exact
calculations, field theory and Monte Carlo simulations [29, 30, 27, 31, 32,
33, 34, 35, 36, 37, 38, 28, 39, 40, 41] were shown to be in excellent
agreement with experiments [42, 43, 44, 45, 46, 47, 48, 49] for various bulk
and surface universality classes.

Directly at the critical point, exact methods or conformal field theory [50,
51, 52, 53, 54] can be used to get exact expressions for the Casimir amplitude
$\Delta_{\mathrm{C}}(\rho)$ for arbitrary $\rho$. This has been done for
periodic [55, 56] as well as for open BCs [57, 58]. Using conformal maps,
these results can be used to investigate fluctuation-induced forces between
colloids in critical suspensions both in theory [59, 60, 61, 62, 63, 64, 65,
66, 67] and experiment [68, 69, 70, 71, 72, 73, 74, 75, 76]. At arbitrary
aspect ratios and temperatures, however, the finite-size scaling functions
must be derived from the exact solution of the finite system with the correct
BCs. For the Ising model, this has been done only in a few cases, namely for
the torus with periodic BCs in both directions [77, 78, 79], for the cylinder
with open or fixed at the boundaries in one direction [78, 80], and for the
open rectangle considered here [11].

In this work, the calculations of [9] are substantially simplified by (i) the
use of elliptic functions [5], (ii) a simplified normalization of the
eigenvectors of the relevant $M\times M$ transfer matrices, (iii) a
transformation to Hankel and Toeplitz matrices, and (iv) a representation of
the sums over transfer matrix eigenvalues through complex contour integrals,
having remarkably simple integrands. The two preceding publications [9] and
[11] will be denoted [I] and [II] in the following.

###  1.1 The model

Figure 1: The anisotropic square lattice Ising model on the $L\times M$
rectangle (black), together with its dual lattice (red). The transfer matrix
$\bm{\mathrm{\mathcal{T}}}$ (1.6) propagates as indicated.

This work focuses on the anisotropic square lattice Ising model on the
$L\times M$ open rectangle, shown in figure 1. Our aim is to calculate the
partition function

$Z=\operatorname{tr}\exp\left(\mathcal{K}^{\leftrightarrow}\sum_{\ell=1}^{L-1}\sum_{m=1}^{M}\sigma_{\ell,m}\sigma_{\ell+1,m}+\mathcal{K}^{\updownarrow}\sum_{\ell=1}^{L}\sum_{m=1}^{M-1}\sigma_{\ell,m}\sigma_{\ell,m+1}\right),$
(1.1)

with reduced couplings $\mathcal{K}^{\delta}=\beta J^{\delta}$ in direction
$\delta=\,\leftrightarrow,\updownarrow$, where the trace is over all $2^{LM}$
configurations of the $LM$ spins $\sigma_{\ell,m}=\pm 1$. We assume open BCs
both in horizontal ($L$) and in vertical ($M$) directions, and we assume even
$M$. The starting point of this work is the matrix product representation of
the well known Pfaffian representation by Kasteleyn, McCoy & Wu [14, 4],
derived in ref. [I, (29)] using a Schur reduction as well as a block transfer
matrix formula by Molinari [17]. We follow the notation of [I], with a few
clarifying minor modifications. Therefore, we define the dual

$a^{*}\equiv\frac{1-a}{1+a}$ (1.2)

of some quantity $a$, as well as the quite useful abbreviations222Subscripts
$a_{\pm}$ are consequently used in this manner, while superscripts $a^{\pm}$
may have different meanings.

$a_{\pm}\equiv\frac{a\pm a^{-1}}{2}$ (1.3)

introduced in (I.20), fulfilling $(a^{*})^{*}=a=a_{+}+a_{-}$. The reduced
couplings $\mathcal{K}^{\delta}$ are rewritten using the weights $z$ and $t$
according to

$z\equiv\tanh\mathcal{K}^{\leftrightarrow},\qquad
t\equiv(\tanh\mathcal{K}^{\updownarrow})^{*}=\mathrm{e}^{-2\mathcal{K}^{\updownarrow}}.$
(1.4)

The partition function $Z$ of the considered anisotropic Ising model with open
BCs in both directions is invariant under exchange of the two directions
$\leftrightarrow$ and $\updownarrow$, where both the system dimensions and the
coupling constants are exchanged according to the _swap transformation_

$\mathcal{S}:(L,M;\mathcal{K}^{\leftrightarrow},\mathcal{K}^{\updownarrow})\mapsto\mathcal{S}[(L,M;\mathcal{K}^{\leftrightarrow},\mathcal{K}^{\updownarrow})]=(M,L;\mathcal{K}^{\updownarrow},\mathcal{K}^{\leftrightarrow}),$
(1.5)

such that $Z=\mathcal{S}[Z]$. The swap transformation is a involution, as
$\mathcal{S}[\mathcal{S}[a]]=a$. We now summarize the required results from
reference [I].

###  1.2 Starting point

In equations (27-29) of [I] we showed that the square of the partition
function (1.1) is proportional to the determinant of a matrix product of the
form
$\langle\bm{\mathrm{e}}_{2}|\bm{\mathrm{\mathcal{T}}}_{2}^{L}|\bm{\mathrm{e}}_{2}\rangle$,
with real hyperbolic $2\times 2$ block transfer matrix
$\bm{\mathrm{\mathcal{T}}}_{\\!2}$ propagating in horizontal direction. In
this work, we will use a “Wick rotated” version of
$\bm{\mathrm{\mathcal{T}}}_{\\!2}$ according
to333$\bm{\mathrm{A}}\negmedspace^{\top}$ and $\bm{\mathrm{A}}^{\dagger}$
denote the transpose and the conjugate transpose of $\bm{\mathrm{A}}$,
respectively.
$\bm{\mathrm{\mathcal{T}}}\equiv\bm{\mathrm{\mathcal{I}}}\,\bm{\mathrm{\mathcal{T}}}_{\\!2}\,\bm{\mathrm{\mathcal{I}}}^{\dagger},{}$
with matrix $\bm{\mathrm{\mathcal{I}}}\equiv(\begin{smallmatrix}1&0\\\
0&\mathrm{i}\end{smallmatrix})\otimes\bm{\mathrm{1}}$, leading to the complex
orthogonal block transfer matrix

${}\bm{\mathrm{\mathcal{T}}}\equiv\begin{bmatrix}\bm{\mathrm{T}}_{+}&-\mathrm{i}\bm{\mathrm{T}}_{-}\\\
\,\mathrm{i}\bm{\mathrm{T}}_{-}&\bm{\mathrm{T}}_{+}\end{bmatrix},$ (1.6)

which fulfills
$\bm{\mathrm{\mathcal{T}}}^{-1}=\bm{\mathrm{\mathcal{T}}}^{\top}$. Note that
$\bm{\mathrm{\mathcal{T}}}$ is not unitary. The two real symmetric $M\times M$
matrices $\bm{\mathrm{T}}_{\pm}$ were given in (I.30) and can be simplified to

$\displaystyle\bm{\mathrm{T}}_{+}$
$\displaystyle=-\frac{t_{-}z_{-}}{2}{\,\,\underbrace{\\!\\!\begin{pmatrix}2{+}\frac{t^{*}}{z^{*}}&1\\\
1&2&\ddots\\\ &\ddots&\ddots&\ddots\\\ &&\ddots&2&1\\\
&&&1&2{+}t^{*}z^{*}\end{pmatrix}\\!\\!}_{{\textstyle\bm{\mathrm{C}}}}}\,\,+(t_{+}z_{+}+t_{-}z_{-})\bm{\mathrm{1}},$
(1.7a) $\displaystyle\bm{\mathrm{T}}_{-}$
$\displaystyle=-\frac{t_{-}z_{-}}{2}{\begin{pmatrix}&&&z^{*}&-\frac{1}{t^{*}}\\\
&&\iddots&-2{t^{*}}_{\\!+}&\frac{1}{z^{*}}\\\ &\iddots&\iddots&\iddots\\\
z^{*}&-2{t^{*}}_{\\!+}&\iddots\\\
-\frac{1}{t^{*}}&\frac{1}{z^{*}}\end{pmatrix}},$ (1.7b)

using the dual couplings $z^{*}$ and $t^{*}$ (the matrix $\bm{\mathrm{C}}$
will be needed later). They are related via the transfer matrix
$\bm{\mathrm{T}}=\bm{\mathrm{T}}_{+}+\bm{\mathrm{T}}_{-}$ according to (1.3),

$\bm{\mathrm{T}}_{\pm}=\tfrac{1}{2}\left(\bm{\mathrm{T}}\pm\bm{\mathrm{T}}^{-1}\right)\quad\Leftrightarrow\quad\bm{\mathrm{T}}^{\pm
1}=\bm{\mathrm{T}}_{+}\pm\bm{\mathrm{T}}_{-},$ (1.8)

from which directly follows that $\bm{\mathrm{\mathcal{T}}}$ can be block
diagonalized through a rotation by $\pi/4$,

$\bm{\mathrm{\mathcal{R}}}_{\frac{\pi}{4}}\bm{\mathrm{\mathcal{I}}}^{\dagger}\bm{\mathrm{\mathcal{T}}}\,\bm{\mathrm{\mathcal{I}}}\,\bm{\mathrm{\mathcal{R}}}_{-\frac{\pi}{4}}=\begin{bmatrix}\bm{\mathrm{T}}&\bm{\mathrm{0}}\\\
\bm{\mathrm{0}}&\bm{\mathrm{T}}^{-1}\end{bmatrix},$ (1.9)

with block rotation matrix
$\bm{\mathrm{\mathcal{R}}}_{\theta}\equiv(\begin{smallmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{smallmatrix})\otimes\bm{\mathrm{1}}$, for details
see chapter V in [I]. The open BCs in horizontal direction are represented
through the boundary state

$|\bm{\mathrm{e}}_{\mathrm{o}}\rangle\equiv\frac{1}{\sqrt{2}}|\begin{array}[]{cc}\bm{\mathrm{1}}&\mathrm{i}\bm{\mathrm{S}}\end{array}\rangle,\qquad\langle\bm{\mathrm{e}}_{\mathrm{o}}|\bm{\mathrm{e}}_{\mathrm{o}}\rangle=\bm{\mathrm{1}},$
(1.10)

where $|\bm{\mathrm{e}}_{\mathrm{o}}\rangle$ is a two element block vector,
with the $M\times M$ matrices

$\displaystyle\bm{\mathrm{1}}\equiv$
$\displaystyle\left(\begin{array}[]{ccc}1\\\ &\;\;\ddots\\\
&&1\end{array}\right),$ $\displaystyle\bm{\mathrm{S}}\equiv$
$\displaystyle\left(\begin{array}[]{ccc}&&1\\\ &\;\;\iddots\\\
1\end{array}\right).$ (1.17)

Together with the constant

$Z_{0}\equiv(1-z^{2})^{\frac{M}{2}}\left(\frac{2}{-z_{-}}\right)^{\negmedspace\frac{LM}{2}}\\!,$
(1.18)

the square of the partition function (1.1) reads

$Z^{2}=Z_{0}^{2}\det\langle\bm{\mathrm{e}}_{\mathrm{o}}|\bm{\mathrm{\mathcal{T}}}^{L}|\bm{\mathrm{e}}_{\mathrm{o}}\rangle,$
(1.19)

see also [7, (2.35)]. Defining the projectors

$\bm{\mathrm{S}}^{\pm}\equiv\frac{1}{2}\left(\bm{\mathrm{1}}\pm\bm{\mathrm{S}}\right),$
(1.20)

the argument of the determinant in (1.19) becomes

$\displaystyle\langle\bm{\mathrm{e}}_{\mathrm{o}}|\bm{\mathrm{\mathcal{T}}}^{L}|\bm{\mathrm{e}}_{\mathrm{o}}\rangle$
$\displaystyle=\langle\begin{array}[]{cc}\bm{\mathrm{S}}^{+}&\mathrm{i}\bm{\mathrm{S}}^{-}\end{array}|\begin{bmatrix}\bm{\mathrm{T}}{}^{L}&\bm{\mathrm{0}}\\\
\bm{\mathrm{0}}&\bm{\mathrm{T}}{}^{-L}\end{bmatrix}|\begin{array}[]{cc}\bm{\mathrm{S}}^{+}&\mathrm{i}\bm{\mathrm{S}}^{-}\end{array}\rangle$
(1.21c)
$\displaystyle=\bm{\mathrm{S}}^{+}\,\bm{\mathrm{T}}{}^{L}\,\bm{\mathrm{S}}^{+}+\bm{\mathrm{S}}^{-}\,\bm{\mathrm{T}}{}^{-L}\,\bm{\mathrm{S}}^{-}=\bm{\mathrm{M^{\top}}}\bm{\mathrm{M}},$
(1.21d)

with the matrix

$\bm{\mathrm{M}}\equiv\bm{\mathrm{x}}\big{(}\bm{\mathrm{T}}^{L/2}\,\bm{\mathrm{S}}^{+}+\bm{\mathrm{T}}^{-L/2}\,\bm{\mathrm{S}}^{-}\big{)},$
(1.22)

as $\bm{\mathrm{\bm{\mathrm{S}}^{+}\bm{\mathrm{S}}^{-}=\bm{\mathrm{0}}}}$. In
the following we will determine the eigenvalues
$\lambda_{\mu},\lambda_{\pm,\mu}$ and common eigenvectors
$\vec{x}_{\mu}=(\bm{\mathrm{x}})_{\mu}$ of $\bm{\mathrm{T}}$ and
$\bm{\mathrm{T}}_{\pm}$, which fulfill

$\displaystyle\bm{\mathrm{T}}\,\vec{x}_{\mu}$
$\displaystyle=\lambda_{\mu}\vec{x}_{\mu},$
$\displaystyle\bm{\mathrm{T}}_{\pm}\vec{x}_{\mu}$
$\displaystyle=\lambda_{\pm,\mu}\vec{x}_{\mu},$ (1.23)

where $\mu=1,\ldots,M$.

###  1.3 The Onsager dispersion relation

The next step in the calculation of the partition function (1.1) is the
determination of the eigenvalues $\lambda_{+,\mu}$ as zeroes of the
characteristic polynomial (CP) of the tridiagonal matrix $\bm{\mathrm{T}}_{+}$
from (1.7a), which will lead to the Onsager dispersion relation. As we will
discuss several characteristic polynomials (CPs) in the following, we first
define the CP $P_{\boldsymbol{a}}(x)$ of an arbitrary $M\times M$ matrix
$\bm{\mathrm{A}}$, with eigenvalues
$(\boldsymbol{a})_{\mu}=a_{\mu}\in\mathbb{C}$, to be

$P_{\boldsymbol{a}}(x)\equiv\det(x\bm{\mathrm{1}}-\bm{\mathrm{A}})=\prod_{\mu=1}^{M}(x-a_{\mu}),$
(1.24)

such that $P_{\boldsymbol{a}}(x)$ is a polynomial of degree $M$ in the
indeterminate $x\in\mathbb{C}$.

Using the well known recursion formula for tridiagonal matrices (see, e.g.,
[17]), in ref. [I] we derived444The sign change in the definition w.r.t. (I.9)
has no effect for even $M$.

$\displaystyle P_{\boldsymbol{\lambda_{+}}}(\lambda_{+})$
$\displaystyle=\det(\lambda_{+}\bm{\mathrm{1}}-\bm{\mathrm{T}}_{+})=\prod_{\mu=1}^{M}(\lambda_{+}-\lambda_{+,\mu})$
(1.25a) $\displaystyle=\left(\frac{t_{-}z_{-}}{2}\right)^{\negmedspace
M}\langle\begin{array}[]{cc}1/z^{*}&-t^{*}\end{array}|\,\bm{\mathrm{Q}}^{M}\,|\begin{array}[]{cc}z^{*}&t^{*}\end{array}\rangle,$
(1.25d)

where $\lambda_{+}\in\mathbb{C}$. We point out the obvious similarity between
equations (1.19) and (1.25d), which will become clearer later. The vertically
propagating $2\times 2$ transfer matrix

$\bm{\mathrm{Q}}\equiv\begin{pmatrix}2\frac{t_{+}z_{+}-\lambda_{+}}{t_{-}z_{-}}&-1\\\
1&0\end{pmatrix}=\begin{pmatrix}2\zeta_{+}&-1\\\
1&0\end{pmatrix}=\begin{pmatrix}2\cos\varphi\,&-1\\\ 1&0\end{pmatrix}$ (1.26)

has the eigenvalues $\zeta^{\pm 1}$ with modulus one555$\zeta^{\pm 1}$ were
denoted $q^{\pm}$ in (I.42)., such that the $n$-th power of $\bm{\mathrm{Q}}$
reads

$\bm{\mathrm{Q}}^{n}=\frac{1}{\sin\varphi}\begin{pmatrix}\sin([n+1]\varphi)&-\sin(n\varphi)\\\
\sin(n\varphi)&-\sin([n-1]\varphi)\end{pmatrix}.$ (1.27)

Here and in the following, we express the horizontal and vertical eigenvalues
$(\lambda,\zeta)$ through the introduced angles $(\gamma,\varphi)$ according
to

$\displaystyle\lambda$ $\displaystyle=\mathrm{e}^{\gamma},$
$\displaystyle\lambda_{+}$ $\displaystyle=\cosh\gamma,$
$\displaystyle\lambda_{-}$ $\displaystyle=\sinh\gamma,$ (1.28a)
$\displaystyle\zeta$ $\displaystyle=\mathrm{e}^{\mathrm{i}\varphi},$
$\displaystyle\zeta_{+}$ $\displaystyle=\cos\varphi,$ $\displaystyle\zeta_{-}$
$\displaystyle=\mathrm{i}\sin\varphi.$ (1.28b)

Equation (1.26) is the point in the calculation where the famous Onsager
dispersion relation

$\lambda_{+}+t_{-}z_{-}\zeta_{+}=t_{+}z_{+}\qquad\Leftrightarrow\qquad\cosh\gamma+t_{-}z_{-}\cos\varphi=t_{+}z_{+}$
(1.29)

between the eigenvalues $\lambda$ and $\zeta$ enters the stage, which plays
the key role in the exact solution of the square lattice Ising model [2]. It
relates the two “good” variables $\lambda$ and $\zeta$ for propagation in
$\leftrightarrow$ ($L$) and $\updownarrow$ ($M$) direction, respectively. As
pointed out by Baxter [5, chap. 15.10], a parametrization of relation (1.29)
using elliptic functions considerably simplifies the analysis. This
parametrization is introduced in the next chapter, and we return to the
characteristic polynomials in chapter 3.1.

##  2 Elliptic parametrization

The key idea behind the elliptic parametrization of the Onsager dispersion
relation (1.29) is (i) to substitute the coupling constants $(z,t)$ through
new constants $(k,\eta)$, where $k$ is temperature-like and $\eta$ encodes the
coupling anisotropy, and (ii) to introduce a complex variable $u$ that
simultaneously parametrizes the quantities $\zeta=\zeta(u)$ and
$\lambda=\lambda(u)$ in such a way that (1.29) is always fulfilled666Imagine
the simpler case of a circle $x^{2}+y^{2}=r^{2}$ being trigonometrically
parametrized, with $x(u)=r\cos u$, $y(u)=r\sin u$ and complex parameter $u$..
We recapitulate the parametrization of (1.29) through the Jacobi elliptic
functions $\operatorname{sn}(u,k)$, $\operatorname{cn}(u,k)$ and
$\operatorname{dn}(u,k)$ [81, 82, 83], with elliptic modulus $k$, by first
defining the Jacobi amplitude

$\phi\equiv\operatorname{am}(u,k)$ (2.1)

as the inverse function of the elliptic integral of the first kind

$u=F(\phi,k)\equiv\int_{0}^{\phi}(1-k^{2}\sin^{2}\theta)^{-1/2}\mathrm{d}\theta,$
(2.2)

for $\\{\phi,u,k\\}\in\mathbb{R}.$ Consequently, the Jacobi elliptic functions
are given by

$\operatorname{sn}(u,k)\equiv\sin\phi,\qquad\operatorname{cn}(u,k)\equiv\cos\phi,\qquad\operatorname{dn}(u,k)\equiv\frac{\partial\phi}{\partial
u}=\pm\sqrt{1-k^{2}\sin^{2}\phi},$ (2.3)

and fulfill the sum of squares identities

$\operatorname{sn}^{2}(u,k)+\operatorname{cn}^{2}(u,k)=k^{2}\operatorname{sn}^{2}(u,k)+\operatorname{dn}^{2}(u,k)=1.$
(2.4)

As common, we suppress the modulus $k$ if possible and write, e. g.,
$\operatorname{sn}u$ instead of $\operatorname{sn}(u,k)$. We follow Glashier’s
notation [83, (22.2.10)] and define all sixteen Jacobi elliptic functions

$\mathrm{pq}\,u\equiv\frac{\mathrm{pr}\,u}{\mathrm{qr}\,u}=\frac{1}{\mathrm{qp}\,u},\qquad\mathrm{where}\qquad\mathrm{p},\mathrm{q},\mathrm{r}\in\\{\mathrm{s},\mathrm{c},\mathrm{d},\mathrm{n}\\},$
(2.5)

including the four trivial ones,
$\operatorname{ss}u=\operatorname{cc}u=\operatorname{dd}u=\operatorname{nn}u=1$.

The Jacobi elliptic functions are double periodic and meromorphic in the
complex $u$-plane, that is they are analytic up to simple poles. The common
quarter-periodicity rectangle has the corners

$\\{u_{\mathrm{s}},u_{\mathrm{c}},u_{\mathrm{d}},u_{\mathrm{n}}\\}\equiv\\{0,K,K+\mathrm{i}K^{\prime},\mathrm{i}K^{\prime}\\},$
(2.6)

see figure 2, where we utilized the graphical interpretation from [83, (22.4)]
and associate the denomination
$\\{\mathrm{s},\mathrm{c},\mathrm{d},\mathrm{n}\\}$ with the four vertices of
the quarter-periodicity rectangle. Eventually, the quarter periods $K$ and
$K^{\prime}$ are the complete elliptic integrals of the first kind,

$\displaystyle K$ $\displaystyle\equiv F\big{(}\tfrac{\pi}{2},k\big{)},$
$\displaystyle K^{\prime}$ $\displaystyle\equiv
F\big{(}\tfrac{\pi}{2},k^{\prime}\big{)},$ (2.7)

cf. (2.2), with complementary modulus $k^{\prime}$ fulfilling $k^{2}+k^{\prime
2}=1$. The elliptic parametrization will lead to substantial simplifications
of the results from [I], as it firstly eliminates the sign ambiguities of the
square roots and secondly introduces certain functions of the parameter $u$
that substantially simplify the expressions.

###  2.1 Coupling constants parametrization

There are several possible ways to setup an elliptic parametrization of the
Onsager dispersion relation (1.29): one could set $k=\hat{k}\equiv
z_{-}/t_{-}$ such that $0\leq\hat{k}<1$ holds in the ordered phase. This
choice is usually used in the literature [84, 7]. However, we will argue that
many expressions become considerably simpler if we instead use the modulus

$k\equiv\frac{t_{-}}{z_{-}},$ (2.8)

being the reciprocal of $\hat{k}$. It obeys $0\leq k<1$ in the disordered
phase $T>T_{\mathrm{c}}$ and $k>1$ in the ordered phase $T<T_{\mathrm{c}}$,
similar to the reduced inverse temperature $\beta/\beta_{\mathrm{c}}$. Note
that $K$ becomes complex for $k>1$, and we can replace it with its real part,
$K\mapsto\Re(K)=K+\mathrm{i}K^{\prime}$, in the complex analysis in order to
keep an un-tilted quarter-periodicity rectangle.

The anisotropy parameter $\eta$ can be introduced in several ways, too. We
choose the definition

$\operatorname{sn}(2\eta)\equiv\frac{1}{\mathrm{i}t_{-}},$ (2.9)

such that $\eta$ is a purely imaginary point in the $u$-plane, with
$0\leq\Im(\eta)\leq K^{\prime}/2$. This leads to the identities

$\displaystyle t$
$\displaystyle=\frac{\operatorname{sn}\eta\operatorname{dn}\eta}{\mathrm{i}\operatorname{cn}\eta},$
$\displaystyle t_{+}$ $\displaystyle=\mathrm{i}\operatorname{cs}(2\eta),$
$\displaystyle t_{-}$
$\displaystyle=\frac{1}{\mathrm{i}}\operatorname{ns}(2\eta),$ (2.10a)
$\displaystyle z$
$\displaystyle=k\frac{\operatorname{sn}\eta\operatorname{cn}\eta}{\mathrm{i}\operatorname{dn}\eta},$
$\displaystyle z_{+}$
$\displaystyle=\frac{\mathrm{i}}{k}\operatorname{ds}(2\eta),$ $\displaystyle
z_{-}$ $\displaystyle=\frac{1}{\mathrm{i}k}\operatorname{ns}(2\eta).$ (2.10b)

For the dual couplings we find the corresponding expressions

$\displaystyle{t^{*}}_{\\!+}$
$\displaystyle=-\frac{t_{+}}{t_{-}}=\operatorname{cn}(2\eta),$
$\displaystyle{t^{*}}_{\\!-}$
$\displaystyle=\frac{1}{t_{-}}=\mathrm{i}\operatorname{sn}(2\eta),$ (2.11a)
$\displaystyle{z^{*}}_{\\!+}$
$\displaystyle=-\frac{z_{+}}{z_{-}}=\operatorname{dn}(2\eta),$
$\displaystyle{z^{*}}_{\\!-}$
$\displaystyle=\frac{1}{z_{-}}=\mathrm{i}k\operatorname{sn}(2\eta),$ (2.11b)

which implies that we can express $t$, $z$, $t^{*}$ and $z^{*}$ through the
Jacobi amplitude (2.1),

$\displaystyle t^{*}$
$\displaystyle=\mathrm{e}^{\mathrm{i}\operatorname{am}(2\eta)},$
$\displaystyle t$
$\displaystyle=-\mathrm{i}\tan\\!\left[{\textstyle\frac{1}{2}}\operatorname{am}(2\eta)\right],$
(2.12a) $\displaystyle z$
$\displaystyle=\mathrm{e}^{\mathrm{i}\operatorname{am}(2\tilde{\eta})},$
$\displaystyle z^{*}$
$\displaystyle=-\mathrm{i}\tan\\!\text{$\left[{\textstyle\frac{1}{2}}\operatorname{am}(2\tilde{\eta})\right]$},$
(2.12b)

where $\tilde{u}$ denotes the swap transform (1.5) of a point $u$ in the
complex $u$-plane,

$u\mapsto\tilde{u}\equiv\mathcal{S}(u)={\textstyle\frac{1}{2}}\mathrm{i}K^{\prime}-u.$
(2.13)

From (2.2) and the $t^{*}$ identity (2.12a) we conclude that

$2\eta=F(-\mathrm{i}\log t^{*},k)=\mathrm{i}K^{\prime}-F(-\mathrm{i}\log
z,k).$ (2.14)

Note that in the isotropic case $t^{*}=z$ the point $\eta$ lies symmetrically
at

$\eta_{\mathrm{iso}}\equiv{\textstyle\frac{1}{4}}\mathrm{i}K^{\prime}=\tilde{\eta}_{\mathrm{iso}},$
(2.15)

as $\mathcal{S}(\eta_{\mathrm{iso}})=\eta_{\mathrm{iso}}$. Therefore, the
transformation (2.13) is a point reflection of the complex $u$-plane at the
point $\eta_{\mathrm{iso}}$.

###  2.2 Eigenvalues parametrization

The relation of the eigenvalues $\lambda$ and $\zeta$ (1.28) to $u$ is also
ambiguous. We follow the literature [5, 84, 7] and define

$\frac{1}{\sqrt{\lambda\zeta^{\pm
1}}}=\mathrm{e}^{-\frac{\gamma\pm\mathrm{i}\varphi}{2}}\equiv\sqrt{k}\operatorname{sn}(u\mp\eta),$
(2.16)

such that

$\displaystyle\lambda$
$\displaystyle=\mathrm{e}^{\gamma}=\frac{1}{k\operatorname{sn}(u+\eta)\operatorname{sn}(u-\eta)},$
$\displaystyle\zeta$
$\displaystyle=\mathrm{e}^{\mathrm{i}\varphi}=\frac{\operatorname{sn}(u+\eta)}{\operatorname{sn}(u-\eta)}.$
(2.17)

The real and positive eigenvalues $\lambda_{\mu}$ correspond to values
$u_{\mu}$ on the real axis (for $\lambda_{\mu}>1$) as well as on the line
$\Im(u_{\mu})=\mathrm{i}K^{\prime}$ (for $\lambda_{\mu}<1$). With this
definition of $\lambda$, the constants $u_{\mathrm{p}}$ and
$\lambda_{\mathrm{p}}$ from (2.6) and (A.1) fulfill
$\lambda_{\mathrm{p}}=\lambda(u_{\mathrm{p}})$ and represent the upper and
lower bound of the spectrum of $\bm{\mathrm{T}}$ and $\bm{\mathrm{T}}^{-1}$
both above and below $T_{\mathrm{c}}$. In appendix A we give a subset of the
large number of identities which can be derived using elliptic functions
identities [81, 83]. Especially, we see from (A.11) that the eigenvalues
$\lambda$ and $\zeta$ are exchanged under the transformation $\mathcal{S}$
according to $\mathcal{S}[(\lambda,\zeta)]=(\zeta,\lambda)$.

Figure 2: Structure of the complex elliptic $u$-plane for $M=4$, paramagnetic
temperature $k=0.95$, and anisotropy $\eta=\frac{3}{4}\eta_{\mathrm{iso}}$.
The eigenvalues $\lambda_{\mu}$ translate to the points $u_{\mu}$ (red), while
the swap-transformed points $\tilde{u}_{\mu}$ (blue) represent the eigenvalues
$\zeta_{\mu}$ and lie point-symmetric w.r.t. the point $\eta_{\mathrm{iso}}$
(green) from (2.15). The corners (2.6) of the quarter-periodicity rectangle
are shown in black. The additional eigenvalues $\lambda_{\mu}^{-1}$ at
positions $\check{u}_{\mu}\equiv u_{\mu}+\mathrm{i}K^{\prime}$ (B.5) and the
corresponding eigenvalues $\zeta_{\mu}^{-1}$ are shown in light colors.

Besides the corners of the quarter-periodicity rectangle (2.6), and the
eigenvalues $u_{\mu}$, where the CP (1.25) is zero, there are four other
important points in the complex $u$-plane, namely the positions of the simple
poles and simple zeroes of $\lambda(u)$ and $\zeta(u)$. Because the Jacobi
$\operatorname{sn}u$ has a simple zero at $u=0$ and a simple pole at
$u=\mathrm{i}K^{\prime}$, we find

$\displaystyle u_{\lambda\to\infty,\zeta\to\infty}$ $\displaystyle=+\eta,$
$\displaystyle u_{\lambda\to 0,\zeta\to\infty}$
$\displaystyle=+\mathrm{i}K^{\prime}-\eta,$ (2.18a) $\displaystyle
u_{\lambda\to\infty,\zeta\to 0}$ $\displaystyle=-\eta,$ $\displaystyle
u_{\text{$\lambda\to$0,$\zeta\to$0}}$
$\displaystyle=-\mathrm{i}K^{\prime}+\eta.$ (2.18b)

These four points will be used later in the analysis of the complex structure
of the relevant integrals.

While $K$ and $K^{\prime}$ are the quarter periods of the Jacobi elliptic
functions (2.3), all relevant parameters and variables of the considered
system, such as $t,z,\lambda,\zeta$, can be written as meromorphic functions
of $2u$ and $2\eta$, see (2.12), (A.16), and (A.17). They are therefore double
periodic functions with quarter periods $K/2$ and $K^{\prime}/2$ and _half
periods_ $K$ and $K^{\prime}$, and can be depicted on the periodicity
rectangle going from $-K-\mathrm{i}K^{\prime}$ to $K+\mathrm{i}K^{\prime}$, as
shown in figure 2.

Finally, we remark that the CP zeroes show up in pairs
$(u_{\mu},-\bar{u}_{\mu})$, lying symmetric with respect to the imaginary
axis, see figure 2. The reason for this symmetry is the meromorphicity of the
Jacobi elliptic functions, and consequently of $\lambda(u)$ and $\zeta(u)$
from (2.17), in combination with the fact that both $\lambda$ and $\zeta$ are
real for purely imaginary $u$. Therefore, $\lambda$ and $\zeta$ transform
according to $f(u)=\overline{f(-\bar{u})}$ under conjugation. This symmetry is
sometimes called para-even in the literature, and $-\bar{u}$ is denoted the
para-conjugate of $u$ [85, 6.29]. We could restore the usual symmetry along
the real axis $f(u)=\overline{f(\bar{u})}$, valid for meromorphic functions
$f(u)$ that are real for real $u$, by rotating the complex plane by
$90^{\circ}$ using the Jacobi imaginary transform [83, §22.6(iv)]. This would
however break the simple relation to the Jacobi amplitude introduced in the
next section. Alternatively, one could also replace $u$ by $\mathrm{i}u$ in
all expressions as done in [5], but we will keep the present definition.

###  2.3 The Jacobi amplitudes $\omega$ and $\theta$

In the next chapter it turns out that the Jacobi amplitude $\omega$ of $2u$,
as well as its imaginary swap transform $\theta$,

$\displaystyle\omega$ $\displaystyle\equiv\operatorname{am}(2u),$
$\displaystyle\theta$
$\displaystyle\equiv\mathrm{i}\operatorname{am}(2\tilde{u}),$ (2.19)

with $\tilde{u}$ from (2.13), will play quite important roles in the
following777Note that Mathematica [86] can correctly handle the complex Jacobi
amplitude only since version 12.1.. They satisfy the identities

$\displaystyle\operatorname{sn}(2u)$ $\displaystyle=\sin\omega,$
$\displaystyle\operatorname{sn}(2\tilde{u})$
$\displaystyle=\frac{1}{\mathrm{i}}\sinh\theta=-\frac{1}{k}\operatorname{ns}(2u),$
(2.20a) $\displaystyle\operatorname{cn}(2u)$ $\displaystyle=\cos\omega,$
$\displaystyle\operatorname{cn}(2\tilde{u})$
$\displaystyle=\cosh\theta=\frac{\mathrm{i}}{k}\operatorname{ds}(2u),$ (2.20b)
$\displaystyle\operatorname{dn}(2u)$
$\displaystyle=-\coth\theta=\mathrm{i}\operatorname{cs}(2\tilde{u}),$
$\displaystyle\operatorname{dn}(2\tilde{u})$
$\displaystyle=\mathrm{i}\cot\omega=\mathrm{i}\operatorname{cs}(2u),$ (2.20c)

as well as

$\displaystyle\tan({\textstyle\frac{1}{2}}\omega)$
$\displaystyle=\frac{\operatorname{sn}u\operatorname{dn}u}{\operatorname{cn}u},$
$\displaystyle\mathrm{e}^{\pm\theta}$
$\displaystyle=\left(k\frac{\operatorname{sn}u\operatorname{cn}u}{\mathrm{i}\operatorname{dn}u}\right)^{\pm
1}\text{{}},$ (2.21)

and fulfill the symmetric relation

$\mathrm{i}k\sin\omega\,\sinh\theta=1.$ (2.22)

In summary, we have defined a double periodic complex manifold with the
topology of a torus as originally proposed by Baxter [5], to describe the two-
dimensional Ising model on the rectangle. This complex manifold will be called
the $u$-plane and is sketched in figure 2. The aspect ratio of the torus is
temperature-dependent and is encoded in the elliptic modulus $k$, while the
coupling anisotropy is described by a point $\eta$ on the torus. The $M$
eigenvalues $\lambda_{\mu}$ and $\zeta_{\mu}$ correspond to points $u_{\mu}$
and $\tilde{u}_{\mu}$, respectively, on the torus. With this set of
definitions, we now can return to the characteristic polynomial from section
1.3.

##  3 Results

###  3.1 Characteristic polynomials

The characteristic polynomial $P_{\boldsymbol{\lambda_{+}}}(\lambda_{+})$ of
the matrix $\bm{\mathrm{T}}_{+}$ was defined in chapter VI of [I]. It was used
to characterize the spectrum of $\bm{\mathrm{T}}_{+}$ and to derive the
finite-size scaling limit in [II]. For the transfer matrix
$\bm{\mathrm{T}}_{+}$ with eigenvalues $\lambda_{+,\mu}$, equation (1.25d) can
be simplified to

$P_{\boldsymbol{\lambda_{+}}}(\lambda_{+})=(1-{t^{*}}^{2})\left(\frac{t_{-}z_{-}}{2}\right)^{\negmedspace
M}\left[\cos(M\varphi)+\frac{t_{+}z_{-}\cos\varphi-
t_{-}z_{+}}{z_{-}\sin\varphi}\sin(M\varphi)\right],$ (3.1)

cf. (I.45), where the angle888For the correct determination of the sign of
$\varphi$ without elliptic parametrization, see chapter VI in [I]. $\varphi$
is given by the Onsager dispersion relation (1.29). Using the elliptic
parametrization from the last section, and especially the Jacobi amplitude
$\omega$ from (2.19), we find, using (2.10) and (A.17), the surprisingly
simple expressions

$\displaystyle P_{\boldsymbol{\lambda_{+}}}(\lambda_{+})$
$\displaystyle=(1-{t^{*}}^{2})\left(\frac{t_{-}z_{-}}{2}\right)^{\negmedspace
M}\big{[}\cos(M\varphi)-\operatorname{cs}(2u)\sin(M\varphi)\big{]}$ (3.2a)
$\displaystyle=(1-{t^{*}}^{2})\left(\frac{t_{-}z_{-}}{2}\right)^{\negmedspace
M}\frac{\sin(M\varphi-\omega)}{\sin(-\omega)},$ (3.2b)

as $\operatorname{cs}(2u)=\cot\omega$ by (2.20c). Here, $\lambda_{+}$ and
$\varphi$ depend on $u$ according to (2.17), (A.11a), or (A.16) and (A.17). As
a consequence, the eigenvalues $\lambda_{+,\mu}$ of $\bm{\mathrm{T}}_{+}$
fulfill the simple condition $M\varphi_{\mu}=\omega_{\mu}$. These
simplifications demonstrate the power of the elliptic parametrization in the
chosen form, and the introduced Jacobi amplitude $\omega$ (2.19) turns out to
be a phase shift in $\varphi$-space, describing the open BCs in $\updownarrow$
direction.

Due to the linear CP identity

$P_{c\boldsymbol{a}+b}(cx+b)=\prod_{\mu=1}^{M}(cx+b-ca_{\mu}-b)=c^{M}P_{\boldsymbol{a}}(x),$
(3.3)

we can rewrite (3.2b) in terms of the new variable

$\chi\equiv
2(\zeta_{+}+1)=4\cos^{2}({\textstyle\frac{1}{2}}\varphi)=\frac{2}{t_{-}z_{-}}(t_{+}z_{+}+t_{-}z_{-}-\lambda_{+}),{}$
(3.4)

eliminating the factor $(t_{-}z_{-}/2)^{M}$, and find the corresponding CP

$P_{\boldsymbol{\chi}}(\chi)=(1-{t^{*}}^{2})\,\frac{\sin(M\varphi-\omega)}{\sin(-\omega)}.$
(3.5)

Comparing (3.4) and (1.7a), we see that $P_{\boldsymbol{\chi}}(\chi)$ is the
characteristic polynomial of the matrix $\bm{\mathrm{C}}$, which therefore has
the eigenvalues $\chi_{\mu}$.

###  3.2 Common eigenvectors

With the help of the introduced elliptic parametrization, the matrix
$\bm{\mathrm{x}}$ of orthonormal common eigenvectors999The eigenvectors
$\vec{x}_{\mu}$ are row vectors in $\bm{\mathrm{x}}$, i. e.,
$\vec{x}_{\mu}=(\bm{\mathrm{x}})_{\mu}$. of $\bm{\mathrm{T}}_{\pm}$,
$\bm{\mathrm{T}}$ and $\bm{\mathrm{C}}$, defined in (1.23) and originally
given in (I.50), can be considerably simplified, too. Using the projectors
$\bm{\mathrm{S}}^{\pm}$ from (1.20), we can split $\bm{\mathrm{x}}$ into an
even part $\bm{\mathrm{x}}^{+}$ and an odd part $\bm{\mathrm{x}}^{-}$
according to
$\bm{\mathrm{x}}^{\pm}\equiv\bm{\mathrm{x}}\bm{\mathrm{S}}^{\pm}$, such that
$\bm{\mathrm{x}}=\bm{\mathrm{x}}^{+}+\bm{\mathrm{x}}^{-}$ and
$\bm{\mathrm{x}}^{\pm}=\pm\bm{\mathrm{x}}^{\pm}\bm{\mathrm{S}}$, i. e.,
$\bm{\mathrm{x}}^{+}$ ($\bm{\mathrm{x}}^{-}$) contains the symmetric (skew-
symmetric) parts of the eigenvectors. We get

$\displaystyle\bm{\mathrm{x}}^{+}$
$\displaystyle=\tfrac{1}{\sqrt{2}}\bm{\mathrm{D}}^{\frac{1}{2}}\left[\mathrm{e}^{-\frac{1}{2}(\theta_{\mu}-\psi_{\mu}+\frac{\mathrm{i}\pi}{2})}\,\frac{\cos({\textstyle\frac{m}{2}}\varphi_{\mu})}{\cos({\textstyle\frac{1}{2}}\varphi_{\mu})}\right]_{\mu=1,\,m\,\text{odd}}^{M},$
(3.6a) $\displaystyle\bm{\mathrm{x}}^{-}$
$\displaystyle=\tfrac{1}{\sqrt{2}}\bm{\mathrm{D}}^{\frac{1}{2}}\left[\mathrm{e}^{+\frac{1}{2}(\theta_{\mu}-\psi_{\mu}+\frac{\mathrm{i}\pi}{2})}\,\frac{\sin({\textstyle\frac{m}{2}}\varphi_{\mu})}{\sin({\textstyle\frac{1}{2}}\varphi_{\mu})}\right]_{\mu=1,\,m\,\text{odd}}^{M},$
(3.6b)

where $m$ runs over the odd integers between $-M$ and $M$. The angle $\psi$ is
defined through

$\mathrm{e}^{\frac{1}{2}(\theta-\psi)}\equiv\sqrt{\mathrm{i}z}\,\frac{\operatorname{cn}u}{\operatorname{cn}\eta},$
(3.7)

with $\theta$ from (2.19), cf. (2.21) and (A.12c), and fulfills

$\mathrm{e}^{-\psi}=-\mathrm{i}\zeta^{*}=-\tan({\textstyle\frac{1}{2}}\varphi),$
(3.8)

see also (C.3b). Using the CP $P_{\boldsymbol{\lambda_{+}}}(\lambda_{+})$ from
(3.2), or alternatively $P_{\boldsymbol{\chi}}(\chi)$ from (3.5), the diagonal
normalization matrix $\bm{\mathrm{D}}$ can be substantially simplified from
the expression given in (I.50), with the result

$\displaystyle(\bm{\mathrm{D}})_{\mu\mu}$
$\displaystyle=-\frac{t^{*}}{z_{-}}\left(\frac{t_{-}z_{-}}{2}\right)^{\negmedspace
M-1}\frac{1}{P_{\boldsymbol{\lambda_{+}}}^{\prime}(\lambda_{+,\mu})}$ (3.9a)
$\displaystyle=\frac{t^{*}}{z_{-}}\frac{1}{P_{\boldsymbol{\chi}}^{\prime}(\chi_{\mu})}.$
(3.9b)

Here we used a relation between the normalization of the eigenvectors of a
tridiagonal matrix and the derivative $P^{\prime}$ of its CP [87, chapter
7.9]. With these simplifications and the diagonal eigenvalue matrix
$(\bm{\mathrm{\Lambda}})_{\mu\mu}=\lambda_{\mu}$, we can rewrite (1.22) as

$\bm{\mathrm{M}}=\bm{\mathrm{\Lambda}}^{\negmedspace
L/2}\,\bm{\mathrm{x}}^{+}+\bm{\mathrm{\Lambda}}^{\negmedspace-L/2}\,\bm{\mathrm{x}}^{-},$
(3.10)

and the partition function (1.19) becomes $Z=Z_{0}\det\bm{\mathrm{M}}$.
Inserting the definition of $\bm{\mathrm{x}}^{\pm}$ into (3.10), we get the
result

$\hat{\bm{\mathrm{W}}}\equiv\frac{1}{\sqrt{2}}\left[\mathrm{e}^{\frac{1}{2}(L\gamma_{\mu}-\theta_{\mu}+\psi_{\mu}-\frac{\mathrm{i}\pi}{2})}\,\frac{\cos({\textstyle\frac{m}{2}}\varphi_{\mu})}{\cos({\textstyle\frac{1}{2}}\varphi_{\mu})}+\mathrm{e}^{-\frac{1}{2}(L\gamma_{\mu}-\theta_{\mu}+\psi_{\mu}-\frac{\mathrm{i}\pi}{2})}\,\frac{\sin(\frac{m}{2}\varphi_{\mu})}{\sin({\textstyle\frac{1}{2}}\varphi_{\mu})}\right]_{\mu=1,\,m\,\text{odd}}^{M},$
(3.11)

such that (1.21) becomes
$\bm{\mathrm{M}}^{\top}\bm{\mathrm{M}}=\hat{\bm{\mathrm{W}}}\vphantom{\bm{\mathrm{W}}}^{\top}\bm{\mathrm{D}}\hat{\bm{\mathrm{W}}}.$

As shown in [I], the matrix $\hat{\bm{\mathrm{W}}}$ is a generalized
Vandermonde matrix, such that a change of the base leaves the value of the
determinant invariant. Hence we can transform from the trigonometric base to
the simpler power base and get the corresponding matrix

$\bm{\mathrm{W}}\equiv\left[\mathrm{e}^{H(m)(L\gamma_{\mu}-\theta_{\mu}+\psi_{\mu})}\,\chi_{\mu}^{\frac{1}{2}(|m|-1)}\right]_{\mu=1,\,m\,\text{odd}}^{M},$
(3.12)

with the Heaviside step function $H$, where we have used the variable $\chi$
introduced in (3.4), and factored out the exponential for $m<0$ and moved it
to the diagonal matrix

${}\bm{\mathrm{F}}\equiv\left[\delta_{\mu\nu}\,\mathrm{e}^{-\frac{1}{2}(L\gamma_{\mu}-\theta_{\mu}+\psi_{\mu})}\right]_{\mu,\nu=1}^{M},$
(3.13)

such that $\det\hat{\bm{\mathrm{W}}}=\det\bm{\mathrm{F}}\det\bm{\mathrm{W}}$.
As $\det\bm{\mathrm{F}}=t^{-\frac{L}{2}}$ by (3.7), (C.1a) and (C.3b), the
resulting squared partition function now reads

$Z^{2}=Z_{0}^{2}\,t^{-L}\det(\bm{\mathrm{W}}^{\top}\bm{\mathrm{D}}\bm{\mathrm{W}}).$
(3.14)

Up to here, the calculation was similar to [I], with one major difference: the
occurrence of the terms $1/P_{\boldsymbol{\chi}}^{\prime}(\chi_{\mu})$ in the
diagonal matrix $\bm{\mathrm{D}}$ (3.9b) will enable us to use a relation
between characteristic polynomials, their associated Vandermonde matrices and
certain Hankel matrices.

###  3.3 Hankelization

The next significant simplification is obtained by utilizing a generalization
of the well known relation of the characteristic polynomial
$P_{\boldsymbol{x}}(x)$ and the related Vandermonde matrix

$\bm{\mathrm{V}}_{\\!\boldsymbol{x}}\equiv\left[\vphantom{A_{2}^{2}}x_{\mu}^{j}\right]_{\mu=1,\,j=0}^{M\hskip
6.02777ptM-1}$ (3.15)

to a certain Hankel matrix $\bm{\mathrm{H}}_{\boldsymbol{x}}$ [88, 89], also
known as ’Vandermonde factorization of a Hankel matrix’. Let

$P_{\boldsymbol{x}}(x)=\prod_{\mu=1}^{M}(x-x_{\mu})=\sum_{n=0}^{M}b_{n}x^{n},$
(3.16)

where $b_{M}=1$ by construction, as well as

$\displaystyle\bm{\mathrm{D}}_{\boldsymbol{x}}$
$\displaystyle\equiv\left[\frac{\delta_{\mu\nu}}{P_{\boldsymbol{x}}^{\prime}(x_{\mu})}\right]_{\mu,\nu=1}^{M},$
$\displaystyle\bm{\mathrm{H}}_{\boldsymbol{x}}^{-1}$
$\displaystyle\equiv\left[\vphantom{A_{2}^{2}}b_{i+j+1}\right]_{i,j=0}^{M-1},$
(3.17)

then

$\displaystyle\bm{\mathrm{V}}^{\top}_{\\!\boldsymbol{x}}\,\bm{\mathrm{D}}_{\boldsymbol{x}}\bm{\mathrm{V}}_{\\!\boldsymbol{x}}$
$\displaystyle=\bm{\mathrm{H}}_{\boldsymbol{x}},$
$\displaystyle{\det}^{2}\,\bm{\mathrm{V}}_{\\!\boldsymbol{x}}\det\bm{\mathrm{D}}_{\boldsymbol{x}}$
$\displaystyle=1.$ (3.18)

While $\bm{\mathrm{H}}_{\boldsymbol{x}}^{-1}$ is upper anti-triangular,
$\bm{\mathrm{H}}_{\boldsymbol{x}}$ itself is lower anti-triangular, e. g. for
$M=4$,

$\bm{\mathrm{H}}_{\boldsymbol{x}}^{-1}=\begin{pmatrix}b_{1}&b_{2}&b_{3}&1\\\
b_{2}&b_{3}&1&0\\\ b_{3}&1&0&0\\\
1&0&0&0\end{pmatrix}\quad\Rightarrow\quad\bm{\mathrm{H}}_{\boldsymbol{x}}=\begin{pmatrix}0&0&0&1\\\
0&0&1&\tilde{b}_{5}\\\ 0&1&\tilde{b}_{5}&\tilde{b}_{6}\\\
1&\tilde{b}_{5}&\tilde{b}_{6}&\tilde{b}_{7}\end{pmatrix},$ (3.19)

with101010As a side note, both $\tilde{b}_{n}=s_{(n-M)}(\boldsymbol{x})$ and
$b_{n}=(-1)^{n}s_{\boldsymbol{1}_{M-n}}(\boldsymbol{x})$ are Schur
polynomials.
$\tilde{b}_{n}=\sum_{\mu}x_{\mu}^{n-1}/P_{\boldsymbol{x}}^{\prime}(x_{\mu})$.

A direct computation (see appendix D) shows that a similar identity holds for
the generalized Vandermonde matrix $\bm{\mathrm{W}}$ (3.12) in conjunction
with $\bm{\mathrm{D}}$ from (3.9b), namely

$\bm{\mathrm{W}}^{\top}\bm{\mathrm{D}}\bm{\mathrm{W}}=\begin{bmatrix}\bm{\mathrm{0}}&\bm{\mathrm{S}}\hat{\bm{\mathrm{H}}}_{1}\\\
\hat{\bm{\mathrm{H}}}_{1}\bm{\mathrm{S}}&\hat{\bm{\mathrm{H}}}_{2}\end{bmatrix}$
(3.20)

with the $\nicefrac{{M}}{{2}}\times\nicefrac{{M}}{{2}}$ Hankel matrices

$\hat{\bm{\mathrm{H}}}_{\Delta}\equiv\left[\sum_{\mu=1}^{M}\frac{t^{*}}{z_{-}}\frac{\mathrm{e}^{\Delta(L\gamma_{\mu}-\theta_{\mu}+\psi_{\mu})}}{P_{\boldsymbol{\chi}}^{\prime}(\chi_{\mu})}\,\chi_{\mu}^{i+j}\right]_{i,j=0}^{\frac{M}{2}-1}.$
(3.21)

Note that (3.20) is block lower anti-triangular similar to
$\bm{\mathrm{H}}_{\boldsymbol{x}}$ in (3.19), as
$\hat{\bm{\mathrm{H}}}_{0}=\bm{\mathrm{0}}$. The important consequence of this
result is the generalized determinant identity

$\det(\bm{\mathrm{W}}^{\top}\bm{\mathrm{D}}\bm{\mathrm{W}})={\det}^{2}\,\bm{\mathrm{W}}\det\bm{\mathrm{D}}={\det}^{2}\,\hat{\bm{\mathrm{H}}}_{1},$
(3.22)

which leads to the surprising result that the Ising partition function can be
mapped to a Hankel determinant. Inserting the simplifications from above, and
defining $\bm{\mathrm{H}}\equiv 2\mathrm{i}z_{-}\hat{\bm{\mathrm{H}}}_{1},$ we
can draw the square root in (3.14) and get the compact exact expression for
the partition function of the square lattice Ising model on the rectangle,

$\displaystyle Z$ $\displaystyle=Z_{1}\det\bm{\mathrm{H}},$ $\displaystyle
Z_{1}$ $\displaystyle\equiv
t^{-\frac{L}{2}}z^{\frac{M}{2}}\left(-\frac{2}{z_{-}}\right)^{\negmedspace\frac{LM}{2}},$
(3.23a) where the $\nicefrac{{M}}{{2}}\times\nicefrac{{M}}{{2}}$ Hankel matrix
$\bm{\mathrm{H}}=\left[\vphantom{A_{2}^{2}}h_{i+j+1}\right]_{i,j=0}^{M/2-1}$
has the matrix elements
$h_{n}\equiv\sum_{\mu=1}^{M}\frac{2\mathrm{i}t^{*}\,\mathrm{e}^{L\gamma_{\mu}-\theta_{\mu}+\psi_{\mu}}}{P_{\boldsymbol{\chi}}^{\prime}(\chi_{\mu})}\,\chi_{\mu}^{n-1},$
(3.23b)

with $n=1,\ldots,M-1$. This expression represents a significant simplification
with respect to the result from [I]. However, in the next section we will
proceed further by rewriting the sum over $\mu$ as a complex contour integral,
inserting the known formula for $P_{\boldsymbol{\chi}}(\chi)$ from (3.5).

###  3.4 Contour integral representation

The matrix elements $h_{n}$ of the Hankel matrix $\bm{\mathrm{H}}$ (3.23b) can
be evaluated using complex contour integration, and the characteristic
polynomial $P_{\boldsymbol{\chi}}(\chi)$ plays a crucial role in this
calculation. In principle we use Cauchy’s residual theorem and calculate the
sum over $\mu$ in $h_{n}$ as a contour integral over a suitable contour $C$
around the points $u_{\mu}$ in the complex $u$-plane,

$\displaystyle{}h_{n}$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\oint_{C}\frac{2\mathrm{i}t^{*}\,\mathrm{e}^{L\gamma-\theta+\psi}}{P_{\boldsymbol{\chi}}^{\prime}(\chi)}\,\chi^{n-1}\,\frac{\partial\log
P_{\boldsymbol{\chi}}(\chi)}{\partial\chi}\frac{\partial\chi}{\partial
u}\,\mathrm{d}u$ (3.24a)
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\oint_{C}\frac{2\mathrm{i}t^{*}\,\mathrm{e}^{L\gamma-\theta+\psi}}{P_{\boldsymbol{\chi}}(\chi)}\,\chi^{n-1}\,\frac{\partial\chi}{\partial
u}\,\mathrm{d}u,$ (3.24b)

where, most importantly, the derivative $P_{\boldsymbol{\chi}}^{\prime}(\chi)$
cancels out.

While the CP $P_{\boldsymbol{\chi}}(\chi)$ has two sets of zeroes $u_{\mu}$
and $\check{u}_{\mu}$ corresponding to the eigenvalues $\lambda_{\mu}$ and
$\lambda_{\mu}^{-1}$, cf. (B.5), the contour $C$ only encloses the zeroes
$u_{\mu}$, see Figure 2. We can easily remove the additional zeroes
$\check{u}_{\mu}$ by employing a factorization analog to (B.2),

${}P_{\boldsymbol{\chi}}(\chi)=\frac{1-{t^{*}}^{2}}{2\mathrm{i}\sin\omega}\big{[}1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}\big{]}\big{[}1+\mathrm{e}^{-\mathrm{i}(M\varphi-\omega)}\big{]}.$
(3.25)

As the first (second) bracket vanishes at $u_{\mu}$ ($\check{u}_{\mu}$), we
can drop the additional zeroes $\check{u}_{\mu}$ by replacing the last term
with its value at the zeroes $u_{\mu}$, where $M\varphi-\omega=0$, to get

$h_{n}=\frac{1}{2\pi\mathrm{i}}\oint_{C}\frac{\mathrm{e}^{L\gamma-\theta+\psi}}{1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}}\,t_{-}\sin\omega\,\chi^{n-1}\,\frac{\partial\chi}{\partial
u}\,\mathrm{d}u.$ (3.26)

In the next step we use (3.8), (A.19b) and (A.20) to eliminate $\psi$ and
$\sin\omega$. Furthermore, we can move the zeroes of the numerator to the line
$\Im(u)=\frac{1}{2}K^{\prime}$ without changing the integral by adding one to
the numerator, such that the numerator is precisely the swap transform (1.5)
of the denominator,
$1-\mathrm{e}^{L\gamma-\theta}=\mathcal{S}[1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}]$,
and get the result

$h_{n}=\frac{1}{2\pi\mathrm{i}}\oint_{C}\frac{1-\mathrm{e}^{L\gamma-\theta}}{1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}}\,\chi^{n}\,\frac{\partial\gamma}{\partial
u}\,\mathrm{d}u.$ (3.27)

Due to the CP property (3.3) and the Vandermonde property of (3.11), the
determinant of (3.26) is invariant under a translation $\chi\mapsto\chi+c$.
This freedom is used in (3.27), as $\chi$ was defined in (3.4) in order to
obey $\chi=2\cot(\frac{1}{2}\varphi)\sin\varphi$.

Figure 3: Complex structure of the integrand (3.27) for $M=6$, $L=5$, $n=1$
and anisotropy $\eta=0.9\,\eta_{\mathrm{iso}}$, above ($k=0.6$, left) and
below ($k^{-1}=0.6$, right) the critical point. Yellow: $M$ zeroes of the
denominator, i. e., eigenvalues $\lambda_{\mu}$ of $\bm{\mathrm{T}}$, and
their para-conjugate. Pink: $L$ zeroes of the numerator and their para-
conjugate. Black: multiple zeroes/poles from (2.18a), with pole order
$\\{-4,1,7,2\\}$ from top to bottom in this case. Green: possible integration
paths. The contour lines are at constant modulus, with a dashed gray line at
1, and light (dark) gray lines at powers of 2 below (above) 1. The complex
phase is color coded, being {white, red, black, blue} at
$\\{1,\mathrm{i},-1,-\mathrm{i}\\}$, such that white turns to red (blue) at
zeroes (poles) under ccw. rotation. Note that $K$ becomes complex for $k>1$.

Figure 4: Complex structure of the integrand (3.27) from figure 3 (left),
mapped onto a torus. These pictures might (or might not) be helpful to better
understand the setup. Left: the upper circle is the real axis, with the odd
zeroes of the denominator (yellow), the outer circle has
$\Im(u)=-K^{\prime}/2$, with the even zeroes of the numerator (pink), and the
black point in front is at $-\eta$. The remaining zeroes are inside and at the
bottom. Right: The top circle is the imaginary axis $\Re(u)=0$, the front ring
is the real axis. This representation corresponds to the original directions
of the Ising model: the $\updownarrow$ direction goes along the $2M$
denominator zeroes (yellow) from front to back, the $\leftrightarrow$
direction goes along the $2L$ numerator zeroes (pink) from left to right, see
also table 1.

The resulting integrand is shown in figures 3. At the four points
$\\{u_{0,\infty},u_{\infty,\infty},u_{\infty,0},u_{0,0}\\}$ (2.18) (black
dots), the pole orders111111$n$-fold poles (zeroes) have pole order $n$
($-n$). $\\{n+1-M,n+1+L-M,n+1+L,n+1\\}$ are positive for certain
$n\in\\{1,\ldots,M-1\\}$. As the additional zeroes at $\check{u}_{\mu}$ are
removed, we can deform and simplify the integration contour $C$ to the four
straight lines (green), which pairwise enclose the CP zeroes (yellow), or
equivalently, the four points (2.18) (black dots). Due to the double periodic
complex plane, the integration paths at $\Re(u)=\pm K$ add up to zero. Note
that in the ordered phase below $T_{\mathrm{c}}$ the smallest real zero
$u_{1}$ reaches $\pm K$ and becomes complex, such that the integration contour
has to be modified as indicated. In figure 4 the complex torus is depicted.

Summarizing the last steps, we have found a compact representation for the
partition function of the anisotropic square lattice Ising model on the
rectangle as the determinant of a Hankel matrix
$\bm{\mathrm{H}}=\left[\vphantom{A_{2}^{2}}h_{i+j+1}\right]_{i,j=0}^{M/2-1}$,

$\displaystyle Z$ $\displaystyle=Z_{1}\det\bm{\mathrm{H}},$ $\displaystyle
h_{n}$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\oint_{C}\frac{1-\mathrm{e}^{L\gamma-\theta}}{1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}}\,\chi^{n}\,\frac{\partial\gamma}{\partial
u}\,\mathrm{d}u,$ (3.28)

with constant $Z_{1}$ from (3.23a). The $h_{n}$ are Fourier coefficients (in
the $\chi$ base) of a so-called symbol function (see below), which is given by
the ratio of two characteristic polynomials. These polynomials are associated
to the boundary conditions in the two directions. As the considered system is
invariant under an exchange of the two directions, the two CPs are directly
related by the symmetry (1.5) of the underlying square lattice Ising model.

Note that the $L$ zeroes $v_{\ell}$ of the numerator
$1-\mathrm{e}^{L\gamma-\theta}$ correspond to the $L$ eigenvalues of a
hypothetical $L\times L$ transfer matrix $\tilde{\bm{\mathrm{T}}}$ propagating
in vertical direction, which would have been used in an alternative rotated
setup. Only in the isotropic square system, where $L=M$ and
$\mathcal{K}^{\leftrightarrow}=\mathcal{K}^{\updownarrow}$, these zeroes
coincide with the swap-transformed zeroes,
$v_{\ell}=\mathcal{S}(u_{\mu})=\tilde{u}_{\mu}$. Unfortunately, we were not
able to simplify the integral (3.28) in this symmetric case.

###  3.5 From Hankel to Toeplitz

The Hankel matrix (3.27) can be further transformed using the identities
derived by Basor & Ehrhardt [90]. Using theorem 2.3, the symbols $a$ and $b$,
as well as the notation $T_{M}(a)$, $H_{M}(a)$ and $H_{M}[b]$ from [90], the
determinant $\det\bm{\mathrm{H}}$, with elements (3.27), corresponds to the
Hankel moment determinant $\det H_{M}[b]$ and can be transformed to the
symmetric Toeplitz plus Hankel determinant $\det(T_{M}(a)+H_{M}(a))$, with the
symbol

$a(\zeta)=\mathrm{e}^{-\psi}b(\zeta_{+})=\frac{1-\mathrm{e}^{L\gamma-\theta}}{1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}}\,\frac{\partial\gamma}{\mathrm{i}\partial\varphi},$
(3.29)

and with the Fourier components [90, (3)]

$\displaystyle a_{n}=a_{-n}$
$\displaystyle=\frac{1}{2\pi}\oint_{C}a(\mathrm{e}^{\mathrm{i}\varphi})\,\mathrm{e}^{-\mathrm{i}n\varphi}\,\frac{\partial\varphi}{\partial
u}\,\mathrm{d}u$ (3.30a)
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\oint_{C}\frac{1-\mathrm{e}^{L\gamma-\theta}}{1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}}\zeta^{-n}\,\frac{\partial\gamma}{\partial
u}\,\mathrm{d}u.$ (3.30b)

Note that (3.29) transforms to its reciprocal,
$\mathcal{S}[a(\zeta)]=1/a(\zeta)$, under exchange of the two directions. The
advantage of this representation is the occurrence of the simpler Fourier base
$\zeta^{-n}$, which eliminates the poles in the integrand stemming from
$\zeta\to\infty$ at the points $u_{\infty,\infty}=\eta$ and
$u_{0,\infty}=\mathrm{i}K^{\prime}-\eta$, provided $L<M$ (as $n\geq 0$). We
can therefore simplify the integration contour to a curve around the two
remaining poles at $u_{\infty,0}=-\eta$ and
$u_{0,0}=-\mathrm{i}K^{\prime}+\eta$. However, for arbitrary $L$ and $M$ the
pole order of the integrand (3.30b) at the four points (2.18) is
$\\{-n+2-M,-n+L-M,n+L,n\\}$ and can be positive in all four cases, such that
all four points must be enclosed by the contour.

Finally, from chapter 3 of [90] we borrow a clever transformation from the
symmetric Toeplitz plus Hankel representation to a skew-symmetric $M\times M$
Toeplitz matrix $\hat{\bm{\mathrm{T}}}$, with the result

$\displaystyle\hat{\bm{\mathrm{T}}}$
$\displaystyle=\left[\sum_{\mu=1}^{M}\frac{2\mathrm{i}t^{*}\,\mathrm{e}^{L\gamma_{\mu}-\theta_{\mu}}}{P_{\boldsymbol{\chi}}^{\prime}(\chi_{\mu})\sin\varphi_{\mu}}\cot(\tfrac{1}{2}\varphi_{\mu})\sin\\!\big{[}(i-j)\varphi_{\mu}\big{]}\right]_{i,j=0}^{M-1}$
(3.31a)
$\displaystyle=\left[\frac{1}{\pi\mathrm{i}}\oint_{C}\frac{1-\mathrm{e}^{L\gamma-\theta}}{1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}}\cot(\tfrac{1}{2}\varphi)\sin\\!\big{[}(i-j)\varphi\big{]}\,\frac{\partial\gamma}{\partial
u}\,\mathrm{d}u\right]_{i,j=0}^{M-1},$ (3.31b) such that
$\operatorname{Pf}\hat{\bm{\mathrm{T}}}=\det\bm{\mathrm{H}}$ and
$Z=Z_{1}\operatorname{Pf}\hat{\bm{\mathrm{T}}}.$ (3.31c)

As before, the integration contour $C$ can be freely deformed as long as it
enclosed all zeroes $u_{\mu}$ or, alternatively, the four points (2.18).

##  4 Discussion

In this work, we showed that the partition function of the anisotropic square
lattice Ising model on the $L\times M$ rectangle with open boundary conditions
in both directions is given by the determinant of a
$\nicefrac{{M}}{{2}}\times\nicefrac{{M}}{{2}}$ Hankel matrix
$\bm{\mathrm{H}}$, which equivalently can be written as the Pfaffian of a
skew-symmetric $M\times M$ Toeplitz matrix $\hat{\bm{\mathrm{T}}}$. The $M-1$
independent matrix elements of $\bm{\mathrm{H}}$ or $\hat{\bm{\mathrm{T}}}$
are Fourier coefficients of a symbol function (3.29), which is given by the
ratio of two characteristic polynomials. These polynomials are associated to
the different directions of the system, encode the respective boundary
conditions, and are directly related through the symmetry of the square
lattice Ising model under exchange of the two directions.

Table 1: Overview over quantities used in this work. The transformation $\mathcal{S}$ (1.5) maps between the two directions of the model. The right columns show the relation of our notation to the work of Baxter [7], where the transfer matrix propagated vertically.  direction |  $\leftrightarrow$ |  $\updownarrow$ |  $\updownarrow$ [7] |  $\leftrightarrow$ [7]
---|---|---|---|---
number of sites |  $L$ |  $M$ |  $M$ |  $N$
reduced coupling |  $\mathcal{K}^{\leftrightarrow}$ |  $\mathcal{K}^{\updownarrow}$ |  $H$ |  $H^{\prime}$
weights |  $(z,t)$ |  $(t^{*},z^{*})$ |  $(t^{*},u^{*})$ |  $(u,t)$
transfer matrix |  $\bm{\mathrm{\mathcal{T}}}$ |  $\bm{\mathrm{Q}}$ |  $\widehat{V}_{2}\widehat{V}_{1}$ |  $-$
eigenvalues |  $\lambda$ |  $\zeta$ |  $\lambda$ |  $z$
angles |  $\gamma$ |  $\mathrm{i}\varphi$ |  $-$ |  $-$
boundary angles |  $\theta$ |  $\mathrm{i}\omega$ |  $-$ |  $-$
characteristic polynomial |  $1-\mathrm{e}^{L\gamma-\theta}$ |  $1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}$ |  $-$ |  [7, (3.21)]
elliptic variables |  $(u,\eta)$ |  $(\tilde{u},\tilde{\eta})$ |  $(r,\nu/2)$ |  $(\bar{r},\bar{\nu}/2)$
elliptic modulus | $k$ | $k^{-1}$
$\Im(\text{zeroes})$ |  $(0,K^{\prime})$ |  $\pm K^{\prime}/2$ |  $-$ |  $-$

The considered anisotropic Ising model with open BCs in both directions is
invariant under exchange of the two directions $\leftrightarrow$ and
$\updownarrow$, such that both the system dimensions and the coupling
constants are exchanged according to
$L\stackrel{{\scriptstyle\mathcal{S}}}{{\leftrightarrow}}M$ and
$\mathcal{K}^{\leftrightarrow}\stackrel{{\scriptstyle\mathcal{S}}}{{\leftrightarrow}}\mathcal{K}^{\updownarrow}$,
see table 1. This _swap transformation_ $\mathcal{S}$ (1.5) leads to the
mapping of $z$ and $t$ to their respective duals,
$(z,t)\stackrel{{\scriptstyle\mathcal{S}}}{{\mapsto}}(t^{*},z^{*})$, as well
as to the exchange of the eigenvalues and angles according to
$\lambda\stackrel{{\scriptstyle\mathcal{S}}}{{\leftrightarrow}}\zeta$,
$\gamma\stackrel{{\scriptstyle\mathcal{S}}}{{\leftrightarrow}}\mathrm{i}\varphi$
and
$\theta\stackrel{{\scriptstyle\mathcal{S}}}{{\leftrightarrow}}\mathrm{i}\omega$.
In the complex $u$-plane, it corresponds to a point reflection
$(u,\eta)\stackrel{{\scriptstyle\mathcal{S}}}{{\mapsto}}(\tilde{u},\tilde{\eta})$
of the complex torus at the point
$\eta_{\mathrm{iso}}={\textstyle\frac{1}{4}}\mathrm{i}K^{\prime}$ from (2.15).
The elliptic modulus $k$, however, is invariant under this transformation.

This symmetry must not be confused with the related duality transformation
$\mathcal{D}:(z,t^{*})\mapsto\mathcal{D}[(z,t^{*})]=(t,z^{*})$ of the bulk
system, which maps a low-temperature system with spins $\sigma_{\ell,m}$ at
$k>1$ to a high-temperature system with the plaquette spins
$\tilde{\sigma}_{\ell,m}$ at $k\mapsto\tilde{k}=k^{-1}<1$, see figure 1. Note
that in systems with boundaries, this transform is known to change the
boundary conditions, e. g., from open to fixed.

We expect that our results can be extended to other boundary conditions by
using the corresponding characteristic polynomials, these generalizations are
left for future work.

From the Toeplitz determinant representation of the partition function, it is
straightforward to derive the (anisotropic) scaling limit $L,M\to\infty$,
$T\to T_{\mathrm{c}}$ at fixed scaling variables
$x\equiv(T/T_{\mathrm{c}}-1)(M/\xi_{0})^{1/\nu}$ and
$\rho\equiv(L/\xi_{0}^{\leftrightarrow})/(M/\xi_{0}^{\updownarrow})$ using
Szegő’s theorem [91, 92, 93]. This task will be addressed in a forthcoming
work.

###### Acknowledgements.

The author is grateful to Hendrik Oppenberg, Felix M. Schmidt and Wolfhard
Janke for helpful discussions and inspirations. Discussions with Jan
Büddefeld, Luca Cervellera and Nils Gluth are acknowledged. This work was
partially supported by the Deutsche Forschungsgemeinschaft through Grant HU
2303/1-1.

##  Appendix A Useful elliptic identities

In this appendix we will list identities arising from the elliptic
parametrization. We also simplify expressions given in chapter VI of [I].
Following McCoy & Wu [16] (who used $\alpha_{1}^{\pm 1},\alpha_{2}^{\pm 1}$)
and Iorgov & Lisovyy [84] (who used $\alpha^{\pm 1},\beta^{\pm 1}$), we
introduce abbreviations for products and ratios of $z$ and $t$, however we
again utilize Glashier’s notation (2.6) and define the constants

${\lambda_{\mathrm{n}}}\equiv
tz,\qquad{\lambda_{\mathrm{s}}}\equiv\frac{1}{tz},\qquad{\lambda_{\mathrm{c}}}\equiv\frac{t}{z},\qquad{\lambda_{\mathrm{d}}}\equiv\frac{z}{t},$
(A.1)

with
${\lambda_{\mathrm{n}}}{\lambda_{\mathrm{s}}}={\lambda_{\mathrm{c}}}{\lambda_{\mathrm{d}}}=1$.
This leads to the identities

$\displaystyle\operatorname{sn}^{2}\eta$
$\displaystyle=-{\lambda_{\mathrm{n}}}k^{-1},$
$\displaystyle\operatorname{cn}^{2}\eta$
$\displaystyle=\frac{{\lambda_{\mathrm{n}}}{\lambda_{\mathrm{n}}}_{,-}}{tt_{-}}=1+{\lambda_{\mathrm{n}}}k^{-1},$
$\displaystyle\operatorname{dn}^{2}\eta$
$\displaystyle=\frac{{\lambda_{\mathrm{n}}}{\lambda_{\mathrm{n}}}_{,-}}{zz_{-}}=1+{\lambda_{\mathrm{n}}}k.$
(A.2)

Furthermore, We list relations to the primary reduced couplings
$\mathcal{K}^{\delta}$ from (1.1),

$\displaystyle t$ $\displaystyle=\exp(-2\mathcal{K}^{\updownarrow}),$
$\displaystyle t_{+}$ $\displaystyle=\cosh(2\mathcal{K}^{\updownarrow}),$
$\displaystyle t_{-}$ $\displaystyle=-\sinh(2\mathcal{K}^{\updownarrow}),$
(A.3a) $\displaystyle z$ $\displaystyle=\tanh\mathcal{K}^{\leftrightarrow},$
$\displaystyle z_{+}$ $\displaystyle=\coth(2\mathcal{K}^{\leftrightarrow}),$
$\displaystyle z_{-}$
$\displaystyle=-\operatorname{csch}(2\mathcal{K}^{\leftrightarrow}),$ (A.3b)
whereas for the dual couplings $z^{*},t^{*}$ the directions $\leftrightarrow$
and $\updownarrow$ are to be exchanged, $\displaystyle z^{*}$
$\displaystyle=\exp(-2\mathcal{K}^{\leftrightarrow}),$
$\displaystyle{z^{*}}_{\\!+}$
$\displaystyle=\cosh(2\mathcal{K}^{\leftrightarrow}),$
$\displaystyle{z^{*}}_{\\!-}$
$\displaystyle=-\sinh(2\mathcal{K}^{\leftrightarrow}),$ (A.3c) $\displaystyle
t^{*}$ $\displaystyle=\tanh\mathcal{K}^{\updownarrow},$
$\displaystyle{t^{*}}_{\\!+}$
$\displaystyle=\coth(2\mathcal{K}^{\updownarrow}),$
$\displaystyle{t^{*}}_{\\!-}$
$\displaystyle=-\operatorname{csch}(2\mathcal{K}^{\updownarrow}).$ (A.3d)

Defining the dual primary reduced couplings $\tilde{\mathcal{K}}^{\delta}$ via

$\exp(-2\tilde{\mathcal{K}}^{\delta})=\tanh\mathcal{K}^{\delta},$ (A.4)

we conclude from (2.12) that the following simple relations hold between
$\tilde{\mathcal{K}}^{\delta}$ and $\eta$,

$\displaystyle 2\mathrm{i}\tilde{\mathcal{K}}^{\updownarrow}$
$\displaystyle=\operatorname{am}(2\eta),$ $\displaystyle
2\mathrm{i}\tilde{\mathcal{K}}^{\leftrightarrow}$
$\displaystyle=\operatorname{am}(2\tilde{\eta}),$ (A.5)

such that the Jacobi amplitude (2.1) represents a direct connection between
the physical reduced couplings $\mathcal{K}^{\delta}$ and the parameter
$\eta$, and (2.14) can be written using the elliptic Integral of the first
kind, cf. (2.2),

$2\eta=F(2\mathrm{i}\tilde{\mathcal{K}}^{\updownarrow},k)=\mathrm{i}K^{\prime}-F(2\mathrm{i}\tilde{\mathcal{K}}^{\leftrightarrow},k).$
(A.6)

We now turn to the eigenvalues $\lambda$ and $\zeta$. Defining the
abbreviation

$Q(u,\eta)\equiv\sqrt{{\lambda_{\mathrm{n}}}-\lambda}=\sqrt{\frac{(k\operatorname{sn}^{2}\eta)^{2}-1}{k\operatorname{sn}^{2}u-k\operatorname{sn}^{2}\eta}},$
(A.7)

we can express the four roots from chapter VI of [I] as meromorphic functions
of $u$, eliminating the ambiguous signs of the square roots,

$\displaystyle\sqrt{{\lambda_{\mathrm{n}}}-\lambda}$
$\displaystyle=Q(u,\eta)\frac{\operatorname{nn}u}{\operatorname{nn}\eta},$
$\displaystyle\sqrt{{\lambda_{\mathrm{s}}}-\lambda}$
$\displaystyle=Q(u,\eta)\frac{\operatorname{sn}u}{\operatorname{sn}\eta},$
(A.8a) $\displaystyle\sqrt{{\lambda_{\mathrm{c}}}-\lambda}$
$\displaystyle=Q(u,\eta)\frac{\operatorname{cn}u}{\operatorname{cn}\eta},$
$\displaystyle\sqrt{{\lambda_{\mathrm{d}}}-\lambda}$
$\displaystyle=Q(u,\eta)\frac{\operatorname{dn}u}{\operatorname{dn}\eta}.$
(A.8b)

Note that we have used the trivial elliptic function $\operatorname{nn}u\equiv
1$ in order to illustrate the systematics. Using (2.13) we can derive analog
expressions for the $\zeta$s,

$\displaystyle\sqrt{\zeta_{\mathrm{n}}-\zeta}$
$\displaystyle=Q(\tilde{u},\tilde{\eta})\frac{\operatorname{nn}\tilde{u}}{\operatorname{nn}\tilde{\eta}}{},$
$\displaystyle\sqrt{\zeta_{\mathrm{s}}-\zeta}$
$\displaystyle=Q(\tilde{u},\tilde{\eta})\frac{\operatorname{sn}\tilde{u}}{\operatorname{sn}\tilde{\eta}},$
(A.9a) $\displaystyle\sqrt{\zeta_{\mathrm{c}}-\zeta}$
$\displaystyle=Q(\tilde{u},\tilde{\eta})\frac{\operatorname{cn}\tilde{u}}{\operatorname{cn}\tilde{\eta}},$
$\displaystyle\sqrt{\zeta_{\mathrm{d}}-\zeta}$
$\displaystyle=Q(\tilde{u},\tilde{\eta})\frac{\operatorname{dn}\tilde{u}}{\operatorname{dn}\tilde{\eta}},$
(A.9b)

where we defined

$\zeta_{\mathrm{n}}\equiv
z^{*}t^{*},\qquad\zeta_{\mathrm{s}}\equiv\frac{1}{z^{*}t^{*}},\qquad\zeta_{\mathrm{c}}\equiv\frac{z^{*}}{t^{*}},\qquad\zeta_{\mathrm{d}}\equiv\frac{t^{*}}{z^{*}},$
(A.10)

in analogy to (A.1). From (A.8a) and (A.9a) we further derive

$\displaystyle\lambda$
$\displaystyle=\frac{1-k^{2}\operatorname{sn}^{2}u\operatorname{sn}^{2}\eta}{k(\operatorname{sn}^{2}u-\operatorname{sn}^{2}\eta)},$
$\displaystyle\zeta$
$\displaystyle=\frac{1-k^{2}\operatorname{sn}^{2}\tilde{u}\operatorname{sn}^{2}\tilde{\eta}}{k(\operatorname{sn}^{2}\tilde{u}-\operatorname{sn}^{2}\tilde{\eta})},$
(A.11a) which can be expressed using (1.2), $\displaystyle\lambda^{*}$
$\displaystyle=-\frac{(k\operatorname{sn}^{2}u)^{*}}{(k\operatorname{sn}^{2}\eta)^{*}},$
$\displaystyle\zeta^{*}$
$\displaystyle=-\frac{(k\operatorname{sn}^{2}\tilde{u})^{*}}{(k\operatorname{sn}^{2}\tilde{\eta})^{*}}.$
(A.11b)

Inserting the elliptic expressions (A.8) into (I.46) we find

$\displaystyle\sin\frac{\varphi}{2}$
$\displaystyle=-\frac{\sqrt{{\lambda_{\mathrm{c}}}-\lambda}\sqrt{{\lambda_{\mathrm{d}}}-\lambda}}{2\sqrt{\lambda
t_{-}z_{-}}}$ $\displaystyle=-\frac{Q^{2}(u,\eta)}{2\sqrt{\lambda
t_{-}z_{-}}}\frac{\operatorname{cn}u}{\operatorname{cn}\eta}\frac{\operatorname{dn}u}{\operatorname{dn}\eta},$
(A.12a) $\displaystyle\cos\frac{\varphi}{2}$
$\displaystyle=\frac{\sqrt{{\lambda_{\mathrm{n}}}-\lambda}\sqrt{{\lambda_{\mathrm{s}}}-\lambda}}{2\mathrm{i}\sqrt{\lambda
t_{-}z_{-}}}$ $\displaystyle=\frac{Q^{2}(u,\eta)}{2\mathrm{i}\sqrt{\lambda
t_{-}z_{-}}}\frac{\operatorname{nn}u}{\operatorname{nn}\eta}\frac{\operatorname{sn}u}{\operatorname{sn}\eta},$
(A.12b) $\displaystyle\tan\frac{\varphi}{2}$
$\displaystyle=\frac{1}{\mathrm{i}}\frac{\sqrt{{\lambda_{\mathrm{c}}}-\lambda}\sqrt{{\lambda_{\mathrm{d}}}-\lambda}}{\sqrt{{\lambda_{\mathrm{n}}}-\lambda}\sqrt{{\lambda_{\mathrm{s}}}-\lambda}}$
$\displaystyle=\frac{1}{\mathrm{i}}\frac{\operatorname{nn}\eta}{\operatorname{nn}u}\frac{\operatorname{sn}\eta}{\operatorname{sn}u}\frac{\operatorname{cn}u}{\operatorname{cn}\eta}\frac{\operatorname{dn}u}{\operatorname{dn}\eta},$
(A.12c)

while (I.47) at the eigenvalues $\lambda_{\mu}$ become

$\displaystyle\pm\sin\frac{M\varphi_{\mu}}{2}$
$\displaystyle=\sqrt{t}\frac{\sqrt{{\lambda_{\mathrm{s}}}-\lambda_{\mu}}\sqrt{{\lambda_{\mathrm{d}}}-\lambda_{\mu}}}{2\sqrt{\lambda_{\mu}t_{-}\lambda_{\mu,-}}}$
$\displaystyle=\frac{\sqrt{t}}{2}\frac{Q^{2}(u_{\mu},\eta)}{\sqrt{\lambda_{\mu}t_{-}\lambda_{\mu,-}}}\frac{\operatorname{sn}u_{\mu}}{\operatorname{sn}\eta}\frac{\operatorname{dn}u_{\mu}}{\operatorname{dn}\eta},$
(A.13a) $\displaystyle\pm\cos\frac{M\varphi_{\mu}}{2}$
$\displaystyle=\frac{1}{\sqrt{t}}\frac{\sqrt{{\lambda_{\mathrm{n}}}-\lambda_{\mu}}\sqrt{{\lambda_{\mathrm{c}}}-\lambda_{\mu}}}{2\mathrm{i}\sqrt{\lambda_{\mu}t_{-}\lambda_{\mu,-}}}$
$\displaystyle=\frac{1}{2\mathrm{i}\sqrt{t}}\frac{Q^{2}(u_{\mu},\eta)}{\sqrt{\lambda_{\mu}t_{-}\lambda_{\mu,-}}}\frac{\operatorname{nn}u_{\mu}}{\operatorname{nn}\eta}\frac{\operatorname{cn}u_{\mu}}{\operatorname{cn}\eta},$
(A.13b) $\displaystyle\tan\frac{M\varphi_{\mu}}{2}$
$\displaystyle=\mathrm{i}t\frac{\sqrt{{\lambda_{\mathrm{s}}}-\lambda_{\mu}}\sqrt{{\lambda_{\mathrm{d}}}-\lambda_{\mu}}}{\sqrt{{\lambda_{\mathrm{n}}}-\lambda_{\mu}}\sqrt{{\lambda_{\mathrm{c}}}-\lambda_{\mu}}}$
$\displaystyle=\frac{\operatorname{sn}u_{\mu}\operatorname{dn}u_{\mu}}{\operatorname{nn}u_{\mu}\operatorname{cn}u_{\mu}}.$
(A.13c)

We additionally list the identities

$\displaystyle t_{-}z_{-}\mathrm{i}\sin\varphi$
$\displaystyle=-\frac{1}{2\lambda}\sqrt{{\lambda_{\mathrm{n}}}-\lambda}\sqrt{{\lambda_{\mathrm{s}}}-\lambda}\sqrt{{\lambda_{\mathrm{c}}}-\lambda}\sqrt{{\lambda_{\mathrm{d}}}-\lambda}$
(A.14a)
$\displaystyle=\sqrt{{\lambda_{\mathrm{s}}}_{,+}-\lambda_{+}}\sqrt{{\lambda_{\mathrm{d}}}_{,+}-\lambda_{+}}$
(A.14b)
$\displaystyle=-\frac{Q^{4}(u,\eta)}{2\lambda}\frac{\operatorname{nn}u}{\operatorname{nn}\eta}\frac{\operatorname{sn}u}{\operatorname{sn}\eta}\frac{\operatorname{cn}u}{\operatorname{cn}\eta}\frac{\operatorname{dn}u}{\operatorname{dn}\eta}.$
(A.14c) as well as
$\displaystyle\sqrt{2t_{-}z_{-}}\,\mathrm{i}\sin\frac{\varphi}{2}$
$\displaystyle=\sqrt{{\lambda_{\mathrm{d}}}_{,+}-\lambda_{+}},$
$\displaystyle\sqrt{2t_{-}z_{-}}\cos\frac{\varphi}{2}$
$\displaystyle=\sqrt{{\lambda_{\mathrm{s}}}_{,+}-\lambda_{+}}.$ (A.14d)

Using the addition theorem [81, (2.4.22)]

$k\operatorname{sn}u\,\operatorname{sn}v=k\frac{\operatorname{cn}(u-v)-\operatorname{cn}(u+v)}{\operatorname{dn}(u-v)+\operatorname{dn}(u+v)}=\frac{1}{k}\frac{\operatorname{dn}(u-v)-\operatorname{dn}(u+v)}{\operatorname{cn}(u-v)+\operatorname{cn}(u+v)}$
(A.15)

we derive the important identities

$\displaystyle\lambda=\mathrm{e}^{\gamma}$
$\displaystyle=-k\frac{\operatorname{cn}(2u)+\operatorname{cn}(2\eta)}{\operatorname{dn}(2u)-\operatorname{dn}(2\eta)}=-\frac{1}{k}\frac{\operatorname{dn}(2u)+\operatorname{dn}(2\eta)}{\operatorname{cn}(2u)-\operatorname{cn}(2\eta)},$
(A.16a) $\displaystyle\lambda_{+}=\cosh\gamma$
$\displaystyle=-k\frac{\operatorname{cn}(2u)\operatorname{dn}(2u)+\operatorname{cn}(2\eta)\operatorname{dn}(2\eta)}{\operatorname{dn}^{2}(2u)-\operatorname{dn}^{2}(2\eta)},$
(A.16b) $\displaystyle\lambda_{-}=\sinh\gamma$
$\displaystyle=-k\frac{\operatorname{cn}(2u)\operatorname{dn}(2\eta)+\operatorname{cn}(2\eta)\operatorname{dn}(2u)}{\operatorname{dn}^{2}(2u)-\operatorname{dn}^{2}(2\eta)},$
(A.16c)

and, by the swap transformation (2.13),

$\displaystyle\zeta=\mathrm{e}^{\mathrm{i}\varphi}$
$\displaystyle=-\frac{\operatorname{ds}(2u)+\operatorname{ds}(2\eta)}{\operatorname{cs}(2u)-\operatorname{cs}(2\eta)}=-\frac{\operatorname{cs}(2u)+\operatorname{cs}(2\eta)}{\operatorname{ds}(2u)-\operatorname{ds}(2\eta)},$
(A.17a) $\displaystyle\zeta_{+}=\cos\varphi$
$\displaystyle=-\frac{\operatorname{ds}(2u)\operatorname{cs}(2u)+\operatorname{ds}(2\eta)\operatorname{cs}(2\eta)}{\operatorname{cs}^{2}(2u)-\operatorname{cs}^{2}(2\eta)},$
(A.17b) $\displaystyle\zeta_{-}=\mathrm{i}\sin\varphi$
$\displaystyle=-\frac{\operatorname{ds}(2u)\operatorname{cs}(2\eta)+\operatorname{ds}(2\eta)\operatorname{cs}(2u)}{\operatorname{cs}^{2}(2u)-\operatorname{cs}^{2}(2\eta)},$
(A.17c)

which express the eigenvalues $\lambda$ and $\zeta$ as functions of $2u$.

We now turn to derivatives. From (2.16) and

$\frac{\partial}{\partial
u}\log\operatorname{sn}(u\pm\eta)=\frac{\operatorname{cn}(u\pm\eta)\operatorname{dn}(u\pm\eta)}{\operatorname{sn}(u\pm\eta)}=k\lambda_{-}[\operatorname{sn}(2u)\mp\operatorname{sn}(2\eta)]$
(A.18)

we see that the derivatives of $\varphi$ and $\gamma$ w.r.t. $u$ become

$\displaystyle\frac{1}{2}\frac{\partial\varphi}{\partial u}$
$\displaystyle=\,\,\mathrm{i}k\operatorname{sn}(2\eta)\,\lambda_{-}$
$\displaystyle=\frac{1}{z_{-}}\sinh\gamma$
$\displaystyle=-\frac{\sin\varphi}{\sin\omega},$ (A.19a)
$\displaystyle\frac{1}{2}\frac{\partial\gamma}{\partial u}$
$\displaystyle=-k\operatorname{sn}(2u)\,\lambda_{-}$
$\displaystyle=t_{-}\sin\varphi$
$\displaystyle=\mathrm{i}\frac{\sinh\gamma}{\sinh\theta},$ (A.19b)

from which other identities, such as

$\frac{\partial\gamma}{\partial\varphi}=-t_{-}\sin\omega=t_{-}z_{-}\frac{\sin\varphi}{\sinh\gamma},\qquad\frac{\partial\chi}{\partial
u}=\frac{\chi^{2}-4}{\sin\omega},$ (A.20)

are easily calculated.

In the ordered phase where $k>1$, the angle $\varphi_{1}$ becomes complex [I,
chap. 6], leading to a complex value of $u_{1}$. The correct mapping from the
eigenvalues $\lambda_{\mu}$ to the elliptic variable $u_{\mu}$, respecting
this behavior and being valid at arbitrary temperatures, can be expressed
using the inverse Jacobi dn, see (A.8),

$u_{\mu}=\operatorname{dn}^{(-1)}\negmedspace\left[\operatorname{dn}\eta\,\frac{\sqrt{{\lambda_{\mathrm{d}}}-\lambda_{\mu}}}{\sqrt{{\lambda_{\mathrm{n}}}-\lambda_{\mu}}}\right].$
(A.21)

While it is tempting to utilize the simpler relation (2.19)

$u_{\mu}=\frac{1}{2}F(M\varphi_{\mu},k),$ (A.22)

it won’t give correct results for even $\mu$ and below $T_{\mathrm{c}}$,
because the elliptic integral $F$ does not have the correct branch cuts for
these cases.

##  Appendix B More characteristic polynomials

In [II] we used the finite-size scaling limit of
$P_{\boldsymbol{\lambda_{+}}}$ to locate the zeroes in the complex plane and
to perform the corresponding Cauchy integrals. We had to distinguish between
even and odd zeroes and defined an alternating counting polynomial from
$P_{\boldsymbol{\lambda_{+}}}$. Now we will demonstrate that it is much easier
to analyze the complex structure of the system by using the CP of the transfer
matrix $\bm{\mathrm{T}}$ instead of $\bm{\mathrm{T}}_{+}$. While it is
possible but cumbersome to derive the CP of $\bm{\mathrm{T}}$, with
eigenvalues $\lambda_{\mu}$,

$P_{\boldsymbol{\lambda}}(\lambda)\equiv\det(\lambda\bm{\mathrm{1}}-\bm{\mathrm{T}})=\prod_{\mu=1}^{M}(\lambda-\lambda_{\mu})$
(B.1)

from scratch analogously to (1.25a) [I], we instead proceed in a much simpler
way and derive it directly from (3.2): Using (1.3), we first factorize the
right hand side of (1.25a),

$P_{\boldsymbol{\lambda_{+}}}(\lambda_{+})=\prod_{\mu=1}^{M}(\lambda_{+}-\lambda_{\mu,+})=\prod_{\mu=1}^{M}\frac{(\lambda-\lambda_{\mu})(\lambda^{-1}-\lambda_{\mu})}{2(0-\lambda_{\mu})}=\frac{1}{2^{M}t}P_{\boldsymbol{\lambda}}(\lambda)P_{\boldsymbol{\lambda}}(\lambda^{-1}),$
(B.2)

as $\lambda_{\mu,+}=\lambda_{+.\mu}$ and, cf. (I.35),

$P_{\boldsymbol{\lambda}}(0)=\det(-\bm{\mathrm{T}})=\det\bm{\mathrm{T}}=t.$
(B.3)

Employing the trigonometric factorization identity

$\frac{\sin\left(M\varphi-\omega\right)}{\sin(-\omega)}=\frac{\sin\left({\textstyle\frac{1}{2}}[M\varphi-\omega]\right)}{\sin\left(-{\textstyle\frac{1}{2}}\omega\right)}\frac{\cos\left({\textstyle\frac{1}{2}}[M\varphi-\omega]\right)}{\cos\left(-{\textstyle\frac{1}{2}}\omega\right)}$
(B.4)

as well as the identities

$\lambda(\check{u})\lambda(u)=1,\quad\varphi(\check{u})=-\varphi(u),\quad\omega(\check{u})+\omega(u)=\pi,\quad\cot[{\textstyle\frac{1}{2}}\omega(\check{u})]=\tan[{\textstyle\frac{1}{2}}\omega(u)],$
(B.5)

with inversion transform $u\mapsto\check{u}\equiv u+\mathrm{i}K^{\prime}$, we
see that $P_{\boldsymbol{\lambda}}(\lambda)$ and
$P_{\boldsymbol{\lambda}}(\lambda^{-1})$ are given by the remarkably simple
formulas

$\displaystyle P_{\boldsymbol{\lambda}}(\lambda)$
$\displaystyle=(1-t^{*})\left(-t_{-}z_{-}\lambda\right)^{\frac{M}{2}}\left[\cos\left({\textstyle\frac{1}{2}}M\varphi\right)-\cot\left({\textstyle\frac{1}{2}}\omega\right)\sin\left({\textstyle\frac{1}{2}}M\varphi\right)\right]$
(B.6a)
$\displaystyle=(1-t^{*})\left(-t_{-}z_{-}\lambda\right)^{\frac{M}{2}}\frac{\sin\left({\textstyle\frac{1}{2}}[M\varphi-\omega]\right)}{\sin\left(-{\textstyle\frac{1}{2}}\omega\right)}$
(B.6b)
$\displaystyle=(1-t^{*})\left(-\frac{t_{-}z_{-}\lambda}{\zeta}\right)^{\negmedspace\frac{M}{2}}\frac{1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}}{1-\mathrm{e}^{-\mathrm{i}\omega}},$
(B.6c) $\displaystyle P_{\boldsymbol{\lambda}}(\lambda^{-1})$
$\displaystyle=(1-t^{*})\left(-t_{-}z_{-}\lambda^{-1}\right)^{\frac{M}{2}}\left[\cos\left({\textstyle\frac{1}{2}}M\varphi\right)+\tan\left({\textstyle\frac{1}{2}}\omega\right)\sin\left({\textstyle\frac{1}{2}}M\varphi\right)\right]$
(B.6d)
$\displaystyle=(1-t^{*})\left(-t_{-}z_{-}\lambda^{-1}\right)^{\frac{M}{2}}\frac{\cos\left({\textstyle\frac{1}{2}}[M\varphi-\omega]\right)}{\cos\left(-{\textstyle\frac{1}{2}}\omega\right)}$
(B.6e)
$\displaystyle=(1-t^{*})\left(-\frac{t_{-}z_{-}}{\lambda\zeta}\right)^{\negmedspace\frac{M}{2}}\frac{1+\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}}{1+\mathrm{e}^{-\mathrm{i}\omega}},$
(B.6f)

The additional factor $(-\lambda)^{M/2}$ in
$P_{\boldsymbol{\lambda}}(\lambda)$ follows from (B.3) in the known limits
$\lambda\to\\{0,\infty\\}$, see (2.18). Utilizing a factorization similar to
(B.2), the CP of $\bm{\mathrm{T}}_{-}$ can also be derived,

$P_{\boldsymbol{\lambda_{-}}}(\lambda_{-})=\prod_{\mu=1}^{M}(\lambda_{-}-\lambda_{\mu,-})=\prod_{\mu=1}^{M}\frac{(\lambda-\lambda_{\mu})(-\lambda^{-1}-\lambda_{\mu})}{2(0-\lambda_{\mu})}=\frac{1}{2^{M}t}P_{\boldsymbol{\lambda}}(\lambda)P_{\boldsymbol{\lambda}}(-\lambda^{-1}).$
(B.7)

Finally, from (3.25) and Liouville’s theorem [5, 15.3] we derived the CPs

$\displaystyle P_{\boldsymbol{\zeta}}(\zeta)$
$\displaystyle=(1-t^{*})\frac{1-\mathrm{e}^{\mathrm{i}(M\varphi-\omega)}}{1-\mathrm{e}^{-\mathrm{i}\omega}}\prod_{\mu=1}^{M}\frac{\operatorname{sn}(\eta+u_{\mu})}{\operatorname{sn}(u+u_{\mu})},$
(B.8) $\displaystyle P_{\boldsymbol{\zeta}}(\zeta^{-1})$
$\displaystyle=(1-t^{*})\frac{1+\mathrm{e}^{-\mathrm{i}(M\varphi-\omega)}}{1+\mathrm{e}^{\mathrm{i}\omega}}\prod_{\mu=1}^{M}k\operatorname{sn}(\eta+u_{\mu})\operatorname{sn}(u+u_{\mu}).$
(B.9)

##  Appendix C Some product identities

Using the CPs (3.2), (B.6), (B.7), we have the following identities (remember
that $M$ is even): the determinants are given by

$\displaystyle\det\bm{\mathrm{T}}_{\hphantom{-}}$
$\displaystyle=\prod_{\mu=1}^{M}\lambda_{\mu\hphantom{,-}}=P_{\boldsymbol{\lambda}}(0)_{\hphantom{-}}=t\,,$
(C.1a) $\displaystyle\det\bm{\mathrm{T}}_{-}$
$\displaystyle=\prod_{\mu=1}^{M}\lambda_{\mu,-}=P_{\boldsymbol{\lambda_{-}}}(0)=(1-{t^{*}}^{2})\left(\frac{\mathrm{i}z_{-}}{1-{t^{*}}^{2}}\right)^{\negmedspace
M}\negmedspace,$ (C.1b) $\displaystyle\det\bm{\mathrm{T}}_{+}$
$\displaystyle=\prod_{\mu=1}^{M}\lambda_{\mu,+}=P_{\boldsymbol{\lambda_{+}}}(0)=(1-{t^{*}}^{2})\left(\frac{t_{-}z_{-}}{2}\right)^{\negmedspace
M}\frac{\sin(M\varphi_{\lambda_{+}\to 0}-\omega_{\lambda_{+}\to
0})}{\sin(-\omega_{\lambda_{+}\to 0})},$ (C.1c)

with (2.18) and

$u_{\lambda_{+}\to
0}=\operatorname{sn}^{(-1)}\sqrt{\frac{(\mathrm{i}{\lambda_{\mathrm{n}}})^{*}}{\mathrm{i}k}},\qquad
u_{\lambda_{-}\to 0}=K+\tfrac{1}{2}\mathrm{i}K^{\prime}.$ (C.2)

Furthermore, we have the following product identities for the Jacobi elliptic
functions

$\displaystyle\prod_{\mu=1}^{M}\sqrt{-tz}\,\frac{\operatorname{sn}u_{\mu}}{\operatorname{sn}\eta}$
$\displaystyle=\sqrt{1-M{\lambda_{\mathrm{s}}}_{,-}z_{-}^{-1}}\,,$ (C.3a)
$\displaystyle\prod_{\mu=1}^{M}\sqrt{\mathrm{i}z}\,\frac{\operatorname{cn}u_{\mu}}{\operatorname{cn}\eta}$
$\displaystyle=1\,,$ (C.3b)
$\displaystyle\prod_{\mu=1}^{M}\sqrt{\mathrm{i}t}\,\frac{\operatorname{dn}u_{\mu}}{\operatorname{dn}\eta}$
$\displaystyle=\sqrt{1+M{\lambda_{\mathrm{d}}}_{,-}z_{-}^{-1}}\,.$ (C.3c)

For products over $\lambda_{\mu}$ we find the identities

$\displaystyle\prod_{\mu=1}^{M}({\lambda_{\mathrm{n}}}-\lambda_{\mu})$
$\displaystyle=P_{\boldsymbol{\lambda}}({\lambda_{\mathrm{n}}})=(1-t^{*})(t_{-}z_{-}{\lambda_{\mathrm{n}}})^{\frac{M}{2}},$
(C.4a) $\displaystyle\prod_{\mu=1}^{M}({\lambda_{\mathrm{c}}}-\lambda_{\mu})$
$\displaystyle=P_{\boldsymbol{\lambda}}({\lambda_{\mathrm{c}}})=(1-t^{*})(-t_{-}z_{-}{\lambda_{\mathrm{c}}})^{\frac{M}{2}},$
(C.4b) $\displaystyle\prod_{\mu=1}^{M}({\lambda_{\mathrm{s}}}-\lambda_{\mu})$
$\displaystyle=P_{\boldsymbol{\lambda}}({\lambda_{\mathrm{s}}})=(1-t^{*})(t_{-}z_{-}{\lambda_{\mathrm{s}}})^{\frac{M}{2}}\left(1-M{\lambda_{\mathrm{s}}}_{,-}z_{-}^{-1}\right),$
(C.4c) $\displaystyle\prod_{\mu=1}^{M}({\lambda_{\mathrm{d}}}-\lambda_{\mu})$
$\displaystyle=P_{\boldsymbol{\lambda}}({\lambda_{\mathrm{d}}})=(1-t^{*})(-t_{-}z_{-}{\lambda_{\mathrm{d}}})^{\frac{M}{2}}\left(1+M{\lambda_{\mathrm{d}}}_{,-}z_{-}^{-1}\right),$
(C.4d)

and for products over $\zeta_{\mu}$ we derive

$\displaystyle\prod_{\mu=1}^{M}\sin\frac{\varphi_{\mu}}{2}$
$\displaystyle=\frac{1-t^{*}}{(2\mathrm{i})^{M}\sqrt{t}}\sqrt{1+M{\lambda_{\mathrm{d}}}_{,-}z_{-}^{-1}}\,,$
(C.5a) $\displaystyle\prod_{\mu=1}^{M}\cos\frac{\varphi_{\mu}}{2}$
$\displaystyle=\frac{1-t^{*}}{2^{M}\sqrt{t}}\sqrt{1-M{\lambda_{\mathrm{s}}}_{,-}z_{-}^{-1}}\,,$
(C.5b) $\displaystyle\prod_{\mu=1}^{M}\tan\frac{\varphi_{\mu}}{2}$
$\displaystyle=(-1)^{M/2}\,\frac{\sqrt{1+M{\lambda_{\mathrm{d}}}_{,-}z_{-}^{-1}}}{\sqrt{1-M{\lambda_{\mathrm{s}}}_{,-}z_{-}^{-1}}}=\prod_{\mu=1}^{M}\mathrm{e}^{-\theta_{\mu}}=\prod_{\mu=1}^{M}\mathrm{e}^{-\psi_{\mu}}\,.$
(C.5c)

Finally, from the factorization

$(t_{-}z_{-}\mathrm{i}\sin\varphi)^{2}=(t_{+}z_{+}-\lambda_{+})^{2}-t_{-}^{2}z_{-}^{2}=({\lambda_{\mathrm{s}}}_{,+}-\lambda_{+})({\lambda_{\mathrm{d}}}_{,+}-\lambda_{+}),$
(C.6)

as ${\lambda_{\mathrm{s}}}_{,+}=t_{+}z_{+}+t_{-}z_{-}$ and
${\lambda_{\mathrm{d}}}_{,+}=t_{+}z_{+}-t_{-}z_{-}$, we deduce the closed form
expression

$\displaystyle\prod_{\mu=1}^{M}(t_{-}z_{-}\mathrm{i}\sin\varphi_{\mu})^{2}$
$\displaystyle=\prod_{\mu=1}^{M}({\lambda_{\mathrm{s}}}_{,+}-\lambda_{\mu,+})({\lambda_{\mathrm{d}}}_{,+}-\lambda_{\mu,+})=P_{\boldsymbol{\lambda_{+}}}({\lambda_{\mathrm{s}}}_{,+})P_{\boldsymbol{\lambda_{+}}}({\lambda_{\mathrm{d}}}_{,+})$
$\displaystyle=(1-t^{*2})^{2}\left(\frac{t_{-}z_{-}}{2}\right)^{\negmedspace
2M}\left(1-M\frac{{\lambda_{\mathrm{s}}}_{,-}}{z_{-}}\right)\left(1+M\frac{{\lambda_{\mathrm{d}}}_{,-}}{z_{-}}\right).$
(C.7)

##  Appendix D A block Hankel matrix identity

Let $\bm{\mathrm{\mathcal{V}}}_{\boldsymbol{g},\boldsymbol{x}}$ be the
generalized $1\times B$ block Vandermonde matrix with $M\times N$ blocks

$\bm{\mathrm{\mathcal{V}}}_{\boldsymbol{g},\boldsymbol{x}}\equiv\left[\vphantom{A_{2}^{2}}g_{\mu}^{b}\,x_{\mu}^{n}\right]_{b=0;\,\,\mu=1,\,n=0}^{B-1\,M\hskip
6.02777ptN-1}=\left[\vphantom{A_{2}^{2}}\bm{\mathrm{G}}^{b}\,\bm{\mathrm{V}}_{\\!\boldsymbol{x}}\right]_{b=0}^{B-1},$
(D.1)

with

$\bm{\mathrm{V}}_{\\!\boldsymbol{x}}\equiv\left[\vphantom{A_{2}^{2}}x_{\mu}^{n}\right]_{\mu=1,\,n=0}^{M\hskip
6.02777ptN-1},\qquad\bm{\mathrm{G}}\equiv\left[\vphantom{A_{2}^{2}}\delta_{\mu\nu}\,g_{\mu}\right]_{\mu,\nu=1}^{M}.$
(D.2)

As example, for $M=4$, $B=3$ and $N=3$ we have

$\bm{\mathrm{\mathcal{V}}}_{\boldsymbol{g},\boldsymbol{x}}=\left[\begin{array}[]{ccc|ccc|ccc}1&x_{1}&x_{1}^{2}&g_{1}&g_{1}x_{1}&g_{1}x_{1}^{2}&g_{1}^{2}&g_{1}^{2}x_{1}&g_{1}^{2}x_{1}^{2}\\\
1&x_{2}&x_{2}^{2}&g_{2}&g_{2}x_{2}&g_{2}x_{2}^{2}&g_{2}^{2}&g_{2}^{2}x_{2}&g_{2}^{2}x_{2}^{2}\\\
1&x_{3}&x_{3}^{2}&g_{3}&g_{3}x_{3}&g_{3}x_{3}^{2}&g_{3}^{2}&g_{3}^{2}x_{3}&g_{3}^{2}x_{3}^{2}\\\
1&x_{4}&x_{4}^{2}&g_{4}&g_{4}x_{4}&g_{4}x_{4}^{2}&g_{4}^{2}&g_{4}^{2}x_{4}&g_{4}^{2}x_{4}^{2}\end{array}\right].$
(D.3)

Furthermore, let $\bm{\mathrm{D}}$ be an arbitrary $M\times M$ diagonal
matrix. Then, the $B\times B$ block Hankel matrix with $N\times N$ blocks

$\bm{\mathrm{\mathcal{H}}}_{\boldsymbol{g},\boldsymbol{x}}\equiv\left[\sum_{\mu=1}^{M}d_{\mu}g_{\mu}^{a+b}\,x_{\mu}^{m+n}\right]_{a,b=0;\,m,n=0}^{B-1\;\;\;N-1}=\left[\bm{\mathrm{V}}^{\top}_{\\!\boldsymbol{x}}\,\bm{\mathrm{D}}\,\bm{\mathrm{G}}^{a+b}\,\bm{\mathrm{V}}_{\\!\boldsymbol{x}}\right]_{a,b=0}^{B-1}$
(D.4)

trivially fulfills the identity

$\bm{\mathrm{\mathcal{H}}}_{\boldsymbol{g},\boldsymbol{x}}=\bm{\mathrm{\mathcal{V}}}^{\top}_{\\!\boldsymbol{g},\boldsymbol{x}}\bm{\mathrm{D}}\bm{\mathrm{\mathcal{V}}}_{\\!\boldsymbol{g},\boldsymbol{x}}.$
(D.5)

Note that the upper left block
$(\bm{\mathrm{\mathcal{H}}}_{\boldsymbol{g},\boldsymbol{x}})_{0,0}=\bm{\mathrm{V}}^{\top}_{\\!\boldsymbol{x}}\,\bm{\mathrm{D}}\,\bm{\mathrm{V}}_{\\!\boldsymbol{x}}$
is free of $g_{\mu}$ and is therefore a usual Vandermonde product in
$x_{\mu}$. Consequently, if the matrix $\bm{\mathrm{D}}$ is set to the
diagonal matrix $\bm{\mathrm{D}}_{\boldsymbol{x}}$ with the reciprocal first
derivatives of the characteristic polynomial $P_{\boldsymbol{x}}(x)$ from
(3.17),

$\bm{\mathrm{D}}_{\boldsymbol{x}}\equiv\left[\frac{\delta_{\mu\nu}}{P_{\boldsymbol{x}}^{\prime}(x_{\mu})}\right]_{\mu,\nu=1}^{M},$
(D.6)

and if additionally $M\geq 2N$, then
$(\bm{\mathrm{H}}_{\boldsymbol{g},\boldsymbol{x}})_{0,0}$ vanishes
identically. For $B=2$ and $N=M/2$, this leads to equation (3.20).

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