How Coordinate Bethe Ansatz Works for Inozemtsev Model

Abstract

Three decades ago, Inozemtsev discovered an isotropic long-range spin chain with elliptic pair potential that interpolates between the Heisenberg and Haldane–Shastry spin chains while admitting an exact solution throughout, based on a connection with the elliptic quantum Calogero–Sutherland model. Though Inozemtsev’s spin chain is widely believed to be quantum integrable, the underlying algebraic reason for its exact solvability is not yet well understood. As a step in this direction we refine Inozemtsev’s ‘extended coordinate Bethe ansatz’ and clarify various aspects of the model’s exact spectrum and its limits. We identify quasimomenta in terms of which the M-particle energy is close to being (functionally) additive, as one would expect from the limiting models. This moreover makes it possible to rewrite the energy and Bethe-ansatz equations on the elliptic curve, turning the spectral problem into a rational problem as might be expected for an isotropic spin chain. We treat the $M=2$ particle sector and its limits in detail. We identify an S-matrix that is independent of positions despite the more complicated form of the extended coordinate Bethe ansatz. We show that the Bethe-ansatz equations reduce to those of Heisenberg in one limit and give rise to the ‘motifs’ of Haldane–Shastry in the other limit. We show that, as the interpolation parameter changes, the ‘scattering states’ from Heisenberg become Yangian highest-weight states for Haldane–Shastry, while bound states become (${{\mathfrak {sl}}}_2$-highest weight versions of) affine descendants of the magnons from $M=1$. We are able to treat this at the level of the wave function and quasimomenta. For bound states we find an equation that, for given Bethe integers, relates the ‘critical’ values of the spin-chain length and the interpolation parameter for which the two complex quasimomenta collide; it reduces to the known equation for the ‘critical length’ in the limit of the Heisenberg spin chain. We also elaborate on Inozemtsev’s proof of the completeness for $M=2$ by passing to the elliptic curve. Our review of the two-particle sectors of the Heisenberg and Haldane–Shastry spin chains may be of independent interest.

Appendices

Appendix A. Notation Compared with the Literature

In this work we have made some changes with respect to Inozemtsev’s extended Bethe ansatz, see Sect. 3.2. To facilitate comparison with the literature we have summarised the notation used for important quantities here and in the literature in Table 4. When applicable the relation to our notation is indicated in square brackets; for example, the ${\varvec{p}}$ of [Ino95] coincides with our $\widetilde{{\varvec{p}}} - {\varvec{\varphi }}$.

Table 4 Comparison of notations used in this work and Inozemtsev’s papers

Appendix B. Elliptic Functions

In this work we make use of six sets of Weierstrass elliptic functions associated to six different lattices depending on the spin chain length $L \in {\mathbb {N}}$ and the interpolation parameter $\kappa \in {\mathbb {R}}_+$ setting $\omega = \mathrm {i}\pi /\kappa$. The six lattices can be divided into three ‘coordinate lattices’ along with three (reciprocal) ‘momentum lattices’ that differ by a rescaling by $-2\pi /\omega$ (see Sect. 3.4). The Hamiltonian and wave function are defined on the coordinate lattice ${\mathbb {L}}$ with periods $(\omega _1,\omega _2) = (L,\omega )$. The elliptic sums discussed in Appendix C are naturally defined on the lattice $\bar{\,\,{\mathbb {L}}}$ with periods $({\bar{\omega }}_1,{\bar{\omega }}_2) = (1,\omega )$, whereas the coordinate-version $\gamma$ of the scattering phase $\varphi$ in e.g. (4.4) behaves most naturally on the lattice $\hat{\,\, {\mathbb {L}}}$ with periods $({\hat{\omega }}_1,{\hat{\omega }}_2) = ( L, L\,\omega )$, which appears in Appendix D.3–D.4. The momentum analogues are decorated with a ‘${}^\vee$’. The lattices ${\mathbb {L}}^{\!\vee }$ with periods $(\omega _1^\vee ,\omega _2^\vee ) \,{:}{=}\,\, (2\pi , -2\pi L /\omega )$ and $\bar{\,\,{\mathbb {L}}}^{\!\vee }$ with $({\bar{\omega }}_1^\vee ,{\bar{\omega }}_2^\vee ) \,{:}{=}\,\, (2\pi , -2\pi /\omega )$ are discussed at length in Sect. 3.4, providing the natural setting for expressions involving the quasimomenta. The lattice $\hat{\,\,{\mathbb {L}}}^{\vee }$ with periods $(\hat{\omega }^{\vee }_1,\hat{\omega }^{\vee }_2) \,{:}{=}\,\, (2\pi L, -2\pi L/\omega )$ is the natural choice for the scattering phase $\varphi$ and therefore the lattice that defines the elliptic curve on which the spectral problem gets rationalised in Sect. 4.5.

Now we turn to the associated elliptic functions. Useful references are [OOL+, AS48, WW02].

1.1 B.1. Weierstrass functions

The Weierstrass elliptic function with periods $(\omega _1,\omega _2)$ is

$$\begin{aligned} \wp (z) = \frac{1}{z^2} + \sum _{j,k \in {\mathbb {Z}}} {}^{\!\!\!^{\scriptstyle *} }\left( \frac{1}{(z+j \, \omega _1 + k\, \omega _2 )^2} - \frac{1}{(j\, \omega _1 + k\, \omega _2)^2}\right) , \end{aligned}$$

(B.1)

where the asterisk indicates that the term with $j=k=0$ is to be omitted from the sum. This (even) function is related to the other two (odd) Weierstrass functions by

$$\begin{aligned} \wp (z) = -\zeta '(z) \, , \qquad \qquad \zeta (z) = (\log \sigma (z))' = \frac{\sigma '(z)}{\sigma (z)} \, . \end{aligned}$$

(B.2)

These two functions are doubly quasiperiodic with quasiperiods $(\omega _1,\omega _2)$. Denote (twice) the values of $\zeta$ at half its quasiperiods by

$$\begin{aligned} \eta _b \,{:}{=}\,\, 2\,\zeta (\omega _1/2) \, , \qquad b=1,2\, . \end{aligned}$$

(B.3a)

These constants obey the Legendre relation

$$\begin{aligned} \omega _2 \, \eta _1 - \omega _1 \, \eta _2 = 2 \pi \mathrm {i}\, . \end{aligned}$$

(B.3b)

Then the (arithmetic) quasiperiodicity of $\zeta$ reads $\zeta (z+\omega _b)=\zeta (z) + \eta _b$ for $b=1,2$, while $\sigma$ is doubly (geometric) quasiperiodic, $\sigma (z+\omega _b) = {-}\mathrm {e}^{(z+\omega _b/2)\,\eta _b }\,\sigma (z)$.

We make frequent use of the functions

$$\begin{aligned} \rho _b(z) \,{:}{=}\,\, \zeta (z) - \frac{\eta _b}{\omega _b} \, z \, , \qquad b=1,2\,. \end{aligned}$$

(B.4)

These functions are odd, $\rho _b(-z) = -\rho _b(z)$, periodic in one direction, $\rho _b(z+\omega _b) = \rho _b(z)$, and (arithmetically) quasiperiodic in the other, $\rho _b(z+\omega _a) = \rho _b(z) + (-1)^a\, 2\pi \mathrm {i}/\omega _a$ for $a\ne b$. Due to the Legendre relation they differ by $\rho _2(z) - \rho _1(z) = (2\pi \mathrm {i})/(L\,\omega )\,z$.

We will need the doubling formulae, valid as long as $u,v,u+v \notin {\mathbb {L}}$:

$$\begin{aligned} \wp (u+v) = \mathrm {P}(u,v)^2 -\wp (u) - \wp (v) \, , \qquad \rho _b(u+v) = \mathrm {P}(u,v) + \rho _b(u) + \rho _b(v) \, ,\nonumber \\ \end{aligned}$$

(B.5a)

where, to make the structure of the equations more clear, we write

$$\begin{aligned} \mathrm {P}(u,v) \,{:}{=}\,\, \frac{1}{2} \frac{\wp '(u) - \wp '(v)}{\wp (u) - \wp (v)} \, . \end{aligned}$$

(B.5b)

1.2 B.2. Jacobi theta functions

The odd Jacobi theta function $\vartheta$ is defined as

$$\begin{aligned} \vartheta (z | \tau ) \,{:}{=}\,\, 2 \sum _{n=0}^\infty (-1)^n \, q^{(n+1/2)^2}\sin ( (2n+1)\,z)\, , \qquad q = \mathrm {e}^{\mathrm {i}\pi \tau }\, \end{aligned}$$

(B.6)

It is double quasiperiodic, $\vartheta (z +\pi | \tau ) = -\vartheta (z | \tau )$ and $\vartheta (z + \pi \,\tau | \tau ) = - q^{-1} \, \mathrm {e}^{-2\mathrm {i}z} \, \vartheta (z | \tau )$, and entire with a simple zero at the origin and its lattice shifts. We will use two variants of this function, which we denote by

$$\begin{aligned} \theta _1(z) \,{:}{=}\,\, \vartheta \bigl (\pi \,z/L\bigm |\omega /L\bigr ) \, , \qquad \theta _2(z)\,{:}{=}\,\, \vartheta \bigl (\pi \,z/\omega \bigm |{-L}/\omega \bigr ) \, , \end{aligned}$$

(B.7)

and are related by $\theta _1(z) = \mathrm {i}\, (\mathrm {i}\, L/\omega )^{1/2} \, \mathrm {e}^{-\mathrm {i}\pi \,z^2/(L\omega )} \, \theta _2(z) = \mathrm {i}\, (L \, \kappa /\pi )^{1/2} \, \mathrm {e}^{-\kappa \, z^2/L} \, \theta _2(z)$. These functions inherit the double quasiperiodicy in the form $\theta _b(z+\omega _b) = -\theta _b(z)$ along with $\theta _1(z+\omega ) = -\mathrm {e}^{-2\, \pi \mathrm {i}z/L -\pi \mathrm {i}\, \omega /L} \theta _1(z)$ and $\theta _2(z+L) = -\mathrm {e}^{2\, \pi \mathrm {i}z/\omega +\pi \mathrm {i}\, L/\omega } \theta _2(z)$. In particular these functions are closely related to Weierstrass $\sigma$. We have

$$\begin{aligned} \sigma (z)= & {} \exp \biggl (\frac{\eta _b \, z^2}{2\,\omega _b}\biggr ) \, \frac{\theta _b(z)}{\theta _b'(0)} \, , \qquad \zeta (z) = \bigl (\log \theta _b(z)\bigr )' + \frac{\eta _b}{\omega _b}\,z \, , \qquad \nonumber \\ \wp (z)= & {} -\bigl (\log \theta _b(z)\bigr )'' - \frac{\eta _b}{\omega _b} \, . \end{aligned}$$

(B.8)

Notice that (B.4) naturally arises as

$$\begin{aligned} \rho _b(z) = \bigl (\log \theta _b(z)\bigr )' = \frac{\theta '_b(z)}{\theta _b(z)} \, . \end{aligned}$$

(B.9a)

This is a close cousin of the Jacobi zeta function. Its derivative

$$\begin{aligned} \rho '_b(z) = -\wp (z)- \frac{\eta _b}{\omega _b} \end{aligned}$$

(B.9b)

appears in Sect. 2.2 as the pair potential (for $b=2$). That is, the Weierstrass elliptic functions $\wp : \zeta : \sigma$ have Jacobi counterparts $V_{\text {s}} : \rho _2 : \theta _2$ (along with a version for $b=1$). Let us stress once more that, despite their subscripts, both of (B.7) are versions of the odd Jacobi theta function (B.6).

Appendix C. Inozemtsev’s Trick for Computing Elliptic Sums

To compute the sums that appear in various places of the analysis of Inozemtsev’s spin chain we often use the fact that two elliptic functions with the same pole structure must be equal up to a constant, as stipulated by Louiville’s theorem. This allows us to find simple explicit quasiperiodic functions for complicated object of which we only know its poles and quasiperiodicity properties. This trick was used repeatedly by Inozemtsev [Ino90, Ino92, Ino95].

Consider an analytic function W that is doubly (quasi)periodic on the lattice $\bar{\,\,{\mathbb {L}}} = {\mathbb {Z}} \oplus \omega \, {\mathbb {Z}}$,

$$\begin{aligned} W(z+1) = W(z) \, , \quad W(z+\omega ) = \mathrm {e}^{\mathrm {i}\,\delta } \, W(z)\, , \end{aligned}$$

(C.1)

where we assume $\delta \ne 0$ to ensure W is not doubly periodic, and whose only pole in the fundamental parallelogram occurs at $z=0$, with Laurent expansion

$$\begin{aligned} W(z) = \frac{c_{-2}}{z^2} + \frac{c_{-1}}{z} + c_0 + O(z) \, . \end{aligned}$$

(C.2)

For us the real periodicity of W is typically true only on-shell, i.e. by virtue of the Bethe-ansatz equations (3.18). In order to rewrite W in terms of known elliptic functions we first engineer an elliptic function with first and second order poles at $z=0$. This goes as follows. The second-order pole is taken care of by ${\bar{\wp }}(z)$, which is doubly periodic. For the first-order pole we use ${\bar{\zeta }}(z)$, which has the right expansion, but does not have the correct double quasiperiodicity properties. To remedy this we subtract another zeta function with argument shifted by some $s \ne 0$ to be determined:

$$\begin{aligned} {\bar{\zeta }}(z)-{\bar{\zeta }}(z+s) \, . \end{aligned}$$

So far our ansatz therefore is

$$\begin{aligned} A\,{\bar{\wp }}(z) + B\,\Bigl ({\bar{\zeta }}(z)-{\bar{\zeta }}(z+s)\Bigr ) \, , \end{aligned}$$

(C.3)

which has the right double quasiperiodicity, but has a spurious pole at $z=-s$. To remove this spurious pole we multiply the ansatz by ${\bar{\sigma }}(z+s)$, and divide by ${\bar{\sigma }}(z-s)$ to compensate for the z-dependent behaviour under $z \mapsto z+\omega _b$,

$$\begin{aligned} \frac{{\bar{\sigma }}(z+s)}{{\bar{\sigma }}(z-s)} \, \biggl ( A\,{\bar{\wp }}(z) + B\,\Bigl ({\bar{\zeta }}(z)-{\bar{\zeta }}(z+s)\Bigr ) \biggr ). \end{aligned}$$

However, this prefactor introduces another spurious pole at $z=s$; to compensate for this we add a zero at that point by subtracting the value of (C.3) at $z=s$:

$$\begin{aligned} \frac{{\bar{\sigma }}(z+s)}{{\bar{\sigma }}(z-s)}\, \biggl (A\,\Bigl ({\bar{\wp }}(z)-{\bar{\wp }}(s)\Bigr ) + B\,\Bigl ({\bar{\zeta }}(z)-{\bar{\zeta }}(z+s)-{\bar{\zeta }}(s)+{\bar{\zeta }}(2s)\Bigr ) \biggr ). \end{aligned}$$

To finish we note that the first factor has two quasiperiods, but only one quasiperiodicity parameter to tune. Multiplying the ansatz with the simplest doubly quasiperiodic, $\mathrm {e}^{q \, z}$, we arrive at the Weierstrass form of our function W:

$$\begin{aligned} \mathrm {e}^{q \, z} \, \frac{{\bar{\sigma }}(z+s)}{{\bar{\sigma }}(z-s)} \, \biggl (A \, \Bigl ({\bar{\wp }}(z)-{\bar{\wp }}(s)\Bigr ) + B\, \Bigl ({\bar{\zeta }}(z)-{\bar{\zeta }}(z+s)-{\bar{\zeta }}(s)+{\bar{\zeta }}(2s)\Bigr )\Bigr ) \, . \end{aligned}$$

(C.4)

This is the general form of a doubly quasiperiodic function with second and first-order poles at the origin. The parameters q and s are fixed by (C.1), yielding the system of linear equations

$$\begin{aligned} q+ 2\, {\bar{\eta }}_1 \, s = 0 \, , \qquad q \, \omega +2 \, {\bar{\eta }}_2 \, s = \mathrm {i}\,\delta \, , \end{aligned}$$

with unique solution

$$\begin{aligned} s= -\frac{\delta }{4 \pi } \, , \qquad q = \frac{{\bar{\eta }}_1}{2\pi } \, \delta \, . \end{aligned}$$

(C.5)

The final step is to determine the residues A, B. To this end we expand (C.4) at $z=0$,

$$\begin{aligned} -\frac{A}{z^2} + \frac{\bigl (q + 2\,{\bar{\zeta }}(s)\bigr ) A - B}{z} + O(z^0) \, , \end{aligned}$$

which matches (C.2) provided

$$\begin{aligned} A = -c_{-2} \, , \qquad B = c_{-2} \, \bigl (q + 2\,{\bar{\zeta }}(s)\bigr ) - c_{-1} \, . \end{aligned}$$

(C.6)

Putting everything together we conclude that the function W can be written as

$$\begin{aligned} \begin{aligned} W(z)&= \mathrm {e}^{q \, z} \, \frac{{\bar{\sigma }}(z+s)}{{\bar{\sigma }}(z-s)} \, \biggl ( {-}c_{-2} \,\Bigl ({\bar{\wp }}(z)-{\bar{\wp }}(s)\Bigr ) \\&\quad + \Bigl (c_{-2}\,\bigl (q + 2{\bar{\zeta }}(s) \bigr ) - c_{-1}\Bigr ) \,\Bigl ({\bar{\zeta }}(z)-{\bar{\zeta }}(z+s)-{\bar{\zeta }}(s)+{\bar{\zeta }}(2s)\Bigr )\bigg ). \end{aligned} \end{aligned}$$

(C.7)

supplemented by (C.5).

Note that we did not discuss the possibility that the above expression for W is off by a constant, since the quasiperiodicity (rather than periodicity) demands that such a constant is zero. This means in particular that we can express $c_0$, the zeroth order term in the Laurent expansion (C.2) of W in terms of the other expansion coefficients along with s and q:

$$\begin{aligned} c_0 = -c_{-2}\left( {\bar{\wp }}(s) +\tfrac{1}{2}q^2-2\,{\bar{\zeta }}(s)^2+q\,{\bar{\zeta }}(2s)+2\,{\bar{\zeta }}(s)\,{\bar{\zeta }}(2s) \right) +c_{-1}\bigl (q+{\bar{\zeta }}(2s)\bigr ).\nonumber \\ \end{aligned}$$

(C.8)

This will prove to be very useful for the computation of various sums, which we will now discuss in turn.

1.1 C.1. Dispersion

Since there are some subtleties in the evaluation of the sums in (3.6) needed to find the dispersion relation we include a detailed computation here. This also serves as a warm-up for the general-M case in Appendix C.3. We wish to compute the sums

$$\begin{aligned} \sum _{j=1}^{L-1} \wp (j) \, , \qquad \sum _{j=1}^{L-1} \mathrm {e}^{\mathrm {i}\, p \, j} \, \wp (j) \, . \end{aligned}$$

We will start with the p-dependent sum, which is a variant of a sum computed in [Ino90]. Following Inozemtsev let us consider the function

$$\begin{aligned} W(z) \,{:}{=}\,\, \sum _{j \in {\mathbb {Z}}_L} \mathrm {e}^{\mathrm {i}\, p \, (j+z)} \, \wp (j+z) \, . \end{aligned}$$

(C.9)

The presence of the exponential can be exploited: it makes W doubly quasiperiodic,

$$\begin{aligned} W(z+1) = W(z) \, , \qquad W(z+\omega ) = \mathrm {e}^{\mathrm {i}\, p_m \, \omega } \, W_m(z) \, , \end{aligned}$$

(C.10)

iff $\mathrm {e}^{\mathrm {i}\, L \, p} = 1$, which is required by the periodicity of the one-particle wave function anyway. Therefore W can be expressed in the Weierstrass form (C.7) with the choices (C.5) with $\delta = p \, \omega$ for the two parameters s and q:

$$\begin{aligned} s= -\frac{p \, \omega }{4 \pi }, \qquad q = \frac{{\bar{\eta }}_1}{2\pi } \, p \, \omega , \end{aligned}$$

(C.11)

including the now important restriction that we must have $s\ne 0$ to ensure W is only quasiperiodic and not periodic. Since W has only a second order pole at zero with residue $c_{-2} = 1$ we can use (C.8) to find that

$$\begin{aligned} -c_0 = {\bar{\wp }}(s) + \tfrac{1}{2} \, q^2 - 2\, {\bar{\zeta }}(s)^2 + q \, {\bar{\zeta }}(2s) + 2\, {\bar{\zeta }}(s) \, {\bar{\zeta }}(2s) \, , \end{aligned}$$

(C.12)

which by computing the zeroth-order term of the sum directly is seen to equal the sum

$$\begin{aligned} \sum _{j=1}^{L-1} \wp (j) \, \mathrm {e}^{\mathrm {i}\, p \, j} \end{aligned}$$

we are after. Plugging in s and q and using the doubling formulae for ${\bar{\wp }}$ and ${\bar{\zeta }}$ we conclude

$$\begin{aligned} \sum _{j=1}^{L-1} \wp (j) \, \mathrm {e}^{\mathrm {i}\, p \, j} = \frac{1}{2} {\bar{\wp }}\Bigl (\frac{\omega p}{2\,\pi }\Bigr ) -\frac{1}{2} \left( {\bar{\zeta }}\Bigl (\frac{\omega p}{2\,\pi }\Bigr ) - \frac{{\bar{\eta }}_2}{\omega } \frac{\omega p}{2\,\pi } \right) ^{\!2} \qquad \text {provided} \quad p \in \frac{2\pi }{L} \,({\mathbb {Z}}_L {\setminus } \{0\} ) \, . \end{aligned}$$

(C.13)

This equality only holds when $p = 2\pi I/L$ with $I \in {\mathbb {Z}}_L$ quantised by the periodicity condition,³⁹ and moreover $s\ne 0$, which implies $I \ne 0 \text { mod } L$.

It remains to compute the sum with $p=0$, which we also use to get the elliptic M-particle difference equation (3.13).⁴⁰ To this end we analyse the function

$$\begin{aligned} W(z) = \sum _{j=0}^{L-1} \wp (z+j), \end{aligned}$$

(C.14)

which is obviously doubly periodic with periods $(1,\omega )$ and has a double pole at the origin. This means that

$$\begin{aligned} W(z) = {\bar{\wp }}(z) +C, \end{aligned}$$

with C a constant. By performing the Laurent expansion we see that C equals the sum we are after. We can find an expression for C by analysing $W(z) - {\bar{\wp }}(z)$: as long as we pick $\tau$ such that we do not pass any poles we have

$$\begin{aligned} \begin{aligned} C \, \omega&= \int _{\tau -\omega /2}^{\tau +\omega /2} \bigl ( W(z) - {\bar{\wp }}(z) \bigr ) \mathrm {d}z \\&= {\bar{\zeta }}\Bigl (\tau + \frac{\omega }{2}\Bigr ) - {\bar{\zeta }}\Bigl (\tau - \frac{\omega }{2}\Bigr ) - \sum _{j=0}^{L-1} \left( {\bar{\zeta }}\Bigl (\tau + j + \frac{\omega }{2}\Bigr ) - {\bar{\zeta }}\Bigl (\tau + j - \frac{\omega }{2}\Bigr ) \right) . \end{aligned} \end{aligned}$$

Differentiation with respect to $\tau$ shows that this expression, and in fact each pair of $\zeta$s, is constant. This allows us to choose $\tau$ as we like. Sending $\tau \rightarrow 0$ (or to $-n$ for the summands) we find

$$\begin{aligned} C \, \omega = 2 \left( {\bar{\zeta }}(\omega /2) - L \, \zeta (\omega /2)\right) \, . \end{aligned}$$

We conclude that

$$\begin{aligned} \sum _{j=1}^{L-1} \wp (j) = \frac{{\bar{\eta }}_2 - L \, \eta _2}{\omega } = {\bar{\eta }}_1 - \eta _1 \, . \end{aligned}$$

(C.15)

1.2 C.2. Highest-weight sum

In order to prove the highest-weight property of the $M=2$ eigenfunctions we need to compute the sum

$$\begin{aligned} \sum _{n=1}^{L-1} \! \mathrm {e}^{\mathrm {i}\, p_1\,n} \, \chi _2(n,\gamma )\, . \end{aligned}$$

(C.16)

To compute it we first introduce the function

$$\begin{aligned} W(z) = \sum _{n'= 0 }^{L-1} \mathrm {e}^{\mathrm {i}p_1 (n'+z)} \chi _2(n'+z,\gamma )\, , \end{aligned}$$

(C.17)

which, due to the periodicity properties of $\chi _2$ has the following periodicity properties:

$$\begin{aligned} W(z+1) = W(z)\, , \quad W(z+\omega ) = \mathrm {e}^{\mathrm {i}p_1 \omega } W(z)\, , \end{aligned}$$

(C.18)

where the first equation is only true on-shell, i.e. as long as $p_1$ and $\gamma$ satisfy (4.12). Further inspection shows that W has only one pole in the fundamental parallelogram, of first order. This means that we can again postulate that W is of the form (C.7), where we can now choose $w_ {-2} = 0$ and find the parameters q and s from the quasiperiods (for which we must assume that $p_1 \ne 0$). To continue we first note that at $z=0$ W expands as

$$\begin{aligned} W(z) = \frac{1}{z} + \mathrm {i}p_1 + \rho _2(\gamma ) +\sum _{n' (\ne 0) }^{L-1} \mathrm {e}^{\mathrm {i}p_1 n'} \chi _2(n',\gamma )+O(z)\, . \end{aligned}$$

(C.19)

Using the relation between the Laurent coefficients given in (C.8) we straightforwardly find

$$\begin{aligned} \begin{aligned} \sum _{n' (\ne 0) }^{L-1} \mathrm {e}^{\mathrm {i}p_1 n'} \chi _2(n',\gamma ) = - \rho _2(\gamma ) -\mathrm {i}p_1 -{\bar{\rho }}_1\left( \frac{\omega \,p_1}{2\pi }\right) = -\rho _2(\gamma )-{\bar{\rho }}_2\left( \frac{\omega \,p_1}{2\pi }\right) \, .\nonumber \end{aligned}\\ {} \end{aligned}$$

(C.20)

1.3 C.3. M-particle sums

Recall the short-hand ${\varvec{r}} \,{:}{=}\,\, {\varvec{p}} - \tilde{{\varvec{p}}}$. We wish to compute the sum

$$\begin{aligned} \Sigma _m({\varvec{n}}) \,{:}{=}\,\, \sum _{j (\notin {\varvec{n}})}^L \! \wp (n_{m}-j) \, [{\widetilde{\Psi }}_{\tilde{{\varvec{p}}}}({\varvec{n}}) \, \mathrm {e}^{\mathrm {i}{\varvec{r}}\cdot {\varvec{n}}}]_{n_m \mapsto j} \, \end{aligned}$$

(C.21)

defined in (3.21). Finding an explicit expression for this sum is the main task in order to analyse the difference equation and follows the strategy of Inozemtsev [Ino95].

We view ${\varvec{n}}$ as fixed and introduce the function

$$\begin{aligned} W_m(z) \,{:}{=}\,\, \sum _{n' \in {\mathbb {Z}}_L} \wp (n_{m}- n' -z) \, {\widetilde{\Psi }}_{{\varvec{r}}}(n_{1},\dots \,,n' +z,\dots \,, n_{M}), \end{aligned}$$

(C.22)

which obeys the periodicity conditions

$$\begin{aligned} W_m(z+1) = W_m(z), \quad W_m(z+\omega ) = \mathrm {e}^{\mathrm {i}p_m \omega } \, W_m(z), \end{aligned}$$

(C.23)

using the Bethe-ansatz equations (3.18). Its only pole in the fundamental parallellogram occurs at zero with Laurent expansion matching (C.2). Therefore $W_m$ can be expressed as (C.7) with q and s determined by (C.5) with $\delta = p_m \, \omega$. Because of this we know that we can express the constant term $c_0$ of $W_m$ as in (C.8), which we will here sometimes abbreviate as $c_0 = Y_m \, c_{-2} + X_m \, c_{-1}$. On the other hand we can directly evaluate the Laurent coefficients $c_{-2}$, $c_{-1}$ and $c_0$ by performing the relevant residue integrals, see below. Recall that ${\widetilde{\Phi }}_{{\tilde{p}}}({\varvec{n}})$ denotes the numerator from (3.16) and abbreviate ${\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{p}}}({\varvec{n}}) \,{:}{=}\,\, {\widetilde{\Phi }}_{{\tilde{p}}}({\varvec{n}}) \, \mathrm {e}^{\mathrm {i}{\varvec{r}} \cdot {\varvec{n}}}$. Then we claim that

$$\begin{aligned} c_0= & {} {} \Sigma _m({\varvec{n}}) + \frac{1}{2} \, \partial ^2_{n_m} {\widetilde{\Psi }}_{{\varvec{r}}}({\varvec{n}}) \nonumber \\&+ \frac{ 1}{\Delta ({\varvec{n}})} \!\sum _{m' (\ne m)}^M \!\!\! \Pi _{mm'}({\varvec{n}}) \Bigg (\! \biggl (\frac{\wp '(n_{m'} - n_{m})}{\wp (n_{m'} - n_{m})} + \!\! \sum _{m''(\ne m,m')}^{M} \!\!\!\!\!\!\! \zeta (n_{m''} - n_{m'}) \biggr ) \, \bigl [ {\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{r}}}({\varvec{n}})\bigr ]_{n_m=n_{m'}} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad \qquad + \bigl [ \partial _{n_m} {\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{r}}}({\varvec{n}})\bigr ]_{n_{m} = n_{m'}} \Bigg ), \end{aligned}$$

(C.24)

where

$$\begin{aligned} \Pi _{m m'}({\varvec{n}}) \,{:}{=}\,\, \wp (n_m-n_{m'}) \, \sigma (n_{m}-n_{m'})\prod _{m''(\ne m,m')}^M \frac{\sigma (n_{m''} - n_{m})}{\sigma (n_{m''} - n_{m'})} \, . \end{aligned}$$

(C.25)

By equating this expression for $c_0$ with the expression (C.8) containing $c_{-2}$ and $c_{-1}$ we find an expression for $\Sigma _m({\varvec{n}})$, which can be plugged into the M-particle difference equation (3.20).

Using the definition (C.22) we can compute the Laurent coefficients of $W_m$ directly.

Computing $c_{-2}$. To find $c_{-2}$ we only need to consider the second-order poles in the sum, which is relatively easy. Since we assumed that the summands in the ansatz have no second-order poles they can only occur in the $\wp$ function and restricted to the $(1,\omega )$ lattice the only relevant one occurs when $n'=n_m$. Thus we find

$$\begin{aligned} c_{-2} = {\widetilde{\Psi }}_{{\tilde{{\varvec{p}}}},{\varvec{r}}}({\varvec{n}}). \end{aligned}$$

(C.26)

Computing $c_{-1}$. Things already get a little more interesting when we consider the first order poles, as there are many more of those. Nevertheless, let us start noting that there is also a contribution coming from the second-order pole at $n'=n_m$, yielding $\partial _{n_m} {\widetilde{\Psi }}_{{\tilde{{\varvec{p}}}},{\varvec{r}}}$. Further contributions from this pole combined with possible zeroes that compensate are impossible, since such a zero would mean that the entire eigenfunction is zero. Therefore all further contributions come from the first order poles in the eigenfunctions in the summands with $n'\ne n_{m}$. So let us set $n'=n_{m'}$ with $m'\ne m$, so that the summand reads

$$\begin{aligned} \wp (n_m-n_{m'}-z) \, \frac{{\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{r}}}(n_1,\ldots ,n_{m'}+z,\ldots ,n_{M})}{\Delta (n_1,\ldots ,n_{m'}+z,\ldots ,n_{M})} \, . \end{aligned}$$

As the pole comes from $1/\Delta$ we can set $z=0$ everywhere else in the expression, yielding

$$\begin{aligned} \begin{aligned}&\wp (n_m-n_{m'}) \, {\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{r}}}(n_1,\ldots ,n_{m'},\ldots ,n_{M}) \\&\quad \times \Biggl (\, \prod _{\begin{array}{c} m''< m''' \\ (\ne m) \end{array}}^M \!\!\!\! \sigma (n_{m''} - n_{m'''}) \prod _{m''=1}^{m-1} \!\! \sigma (n_{m''} - n_{m'}-z) \prod _{m'''(>m)}^M \!\!\!\!\! \sigma (n_{m'}+z - n_{m'''}) \Biggr )^{\!\! -1}. \end{aligned} \end{aligned}$$

If $m'<m$ the residue is

$$\begin{aligned} \begin{aligned}&{-}\wp (n_m-n_{m'}) \, {\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{r}}}(n_1,\ldots ,n_{m'},\ldots ,n_{M}) \\&\qquad \times \Biggl (\, \prod _{\begin{array}{c} m''< m''' \\ (\ne m) \end{array}}^M \!\!\!\! \sigma (n_{m''} - n_{m'''}) \prod _{m'' (\ne m')}^{m-1} \!\! \sigma (n_{m''} - n_{m'}) \prod _{n'''(>m)}^M \!\! \sigma (n_{m'} - n_{m'''}) \Biggr )^{\!\! -1} \\&\quad = {-}\wp (n_m-n_{m'}) \, {\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{r}}}(n_1,\ldots ,n_{m'},\ldots ,n_{M}) \\&\qquad \times \Biggl (\,\prod _{m''< m''' }^M \!\!\!\! \sigma (n_{m''} - n_{m'''}) \prod _{m'' (\ne m')}^{m-1} \frac{\sigma (n_{m''} - n_{m'})}{\sigma (n_{m''} - n_{m})} \prod _{m'''(>m)}^M \frac{\sigma (n_{m'} - n_{m'''})}{\sigma (n_{m} - n_{m'''})} \Biggr )^{\!\! -1} \sigma (n_{m'}-n_{m}) \, , \end{aligned} \end{aligned}$$

where we noted the minus sign explicitly coming from the minus in front of the z that gives the pole and introduced extra factors of $\sigma$ to simplify the products. Namely, we see that the first product is basically $\Delta (n_1,\ldots ,n_M)$. So we find

$$\begin{aligned} \wp (n_m-n_{m'}) \, \frac{ {\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{r}}} (n_1,\ldots ,n_{m'},\ldots ,n_{M}) }{ \Delta (n_1,\ldots ,n_M) } \, \sigma (n_{m}-n_{m'})\prod _{m''(\ne m,m')}^M \frac{\sigma (n_{m''} - n_{m})}{\sigma (n_{m''} - n_{m'})} \, . \end{aligned}$$

Repeating the same argument for $m'>m$ one finds the same result, the lack of a minus sign being compensated for by the oddness of the separate $\sigma$ function. This expression is the same as that obtained by Inozemtsev in [Ino95] up to a factor $\wp (n_m-n_{m'})$. So we finally have

$$\begin{aligned} \begin{aligned} c_{-1}&= \partial _{n_m} {\widetilde{\Psi }}_{{\tilde{{\varvec{p}}}},{\varvec{r}}} \, + \, \frac{1}{\Delta ({\varvec{n}})} \! \sum _{m'(\ne m)}^M \!\!\! {\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{r}}}(n_1,\ldots ,n_{m'},\ldots ,n_{M}) \, \Pi _{m m'}({\varvec{n}}) \, , \end{aligned} \end{aligned}$$

(C.27)

where we define⁴¹

$$\begin{aligned} \Pi _{m m'}({\varvec{n}}) \,{:}{=}\,\, \wp (n_m-n_{m'}) \, \sigma (n_{m}-n_{m'})\prod _{m'' (\ne m,m')}^M \frac{\sigma (n_{m''} - n_{m})}{\sigma (n_{m''} - n_{m'})} \, . \end{aligned}$$

(C.28)

Computing $c_0$. Of course now comes the most important coefficient, the one that contains the sum we are actually after. It has three contributions: the direct contribution from setting $z=0$ is exactly $\Sigma _m({\varvec{n}})$. The contribution from the second order pole is also straightforward, as it is simply $\partial _{n_m}^2 {\widetilde{\Psi }}_{{\varvec{r}}}/2$. The contribution coming from the first order poles at $n'=n_{m'}$ is the most complicated, as it requires us to differentiate

$$\begin{aligned} \begin{aligned}&\wp (n_m-n_{m'}-z) \, {\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{r}}}(n_1,\ldots ,n_{m'}+z,\ldots ,n_{M}) \\&\quad \times \Biggl (\, \prod _{\begin{array}{c} m''< m''' \\ (\ne m) \end{array}}^M \!\!\!\! \sigma (n_{m''} - n_{m'''}) \prod _{m''=1}^{m-1} \!\! \sigma (n_{m''} - n_{m'}-z) \prod _{m'''(>m)}^M \!\! \sigma (n_{m'}+z - n_{m'''}) \Biggr )^{\!\! -1} \end{aligned} \end{aligned}$$

with respect to z before setting it to zero. Doing this yields

$$\begin{aligned} \frac{\Pi _{mm'}({\varvec{n}})}{\Delta ({\varvec{n}})} \Biggl (\, \biggl (\, \frac{\wp '(n_{m'} - n_{m})}{\wp (n_{m'} - n_{m})} +\sum _{m''(\ne m,m')}^{M} \!\!\!\!\!\!\! \zeta (n_{m''} - n_{m'}) \biggr ) F_{{\varvec{r}}}({\varvec{n}})\big |_{n_m=n_{m'}} +\partial _{n_m} {\widetilde{\Phi }}_{\tilde{{\varvec{p}}},{\varvec{r}}}({\varvec{n}})\big |_{n_{m} = n_{m'}} \Biggr ) \, . \end{aligned}$$

In this way we obtain (C.24).

Appendix D. Further Proofs

1.1 D.1. A simplification

Before we can plug (C.24) into the M-particle difference equation (3.20) let us consider the action of the permutation $w \in S_M$ on the coordinates in $\Sigma ({\varvec{n}}_w)$. In the first line of (C.24) we can simply replace ${\varvec{n}} \rightarrow {\varvec{n}}_w$, while in the second line the antisymmetry of the Vandermonde factor, $\Delta ({\varvec{n}}_w) = \mathrm {sgn}(w) \, \Delta ({\varvec{n}})$, yields a total antisymmetrisation. This allows the summands in that line to be simplified in pairs (whence the factor 1/2 in front) with permutations differing by the transposition $(m,m') \in S_M$. Indeed, define

$$\begin{aligned} Z_{mm'}({\varvec{n}})&{:}{=}&\Pi _{mm'}({\varvec{n}}) \Bigg ( \biggl ( \frac{\wp '(n_{m'} - n_{m})}{\wp (n_{m'} - n_{m})} -X_m +\sum _{\alpha (\ne m,m')}^{M} \zeta (n_{\alpha } - n_{m'}) \biggr ) {\widetilde{\Phi }}_{\tilde{{\varvec{p}}}}({\varvec{n}}) \big |_{n_m=n_{m'}} \nonumber \\&\quad \qquad \qquad \qquad +\partial _{n_m} {\widetilde{\Phi }}_{\tilde{{\varvec{p}}}}({\varvec{n}}) \, \mathrm {e}^{\mathrm {i}{\varvec{r}}\cdot {\varvec{n}}} \big |_{n_{m} = n_{m'}} \Bigg ). \end{aligned}$$

(D.1)

Then it follows that

$$\begin{aligned} \frac{1}{\Delta ({\varvec{n}})} \sum _{w\in S_M} \!\! \mathrm {sgn}(w) \!\! \sum _{m\ne m'}^M \!\! Z_{mm'}({\tilde{n}}) = \frac{1}{2\,\Delta ({\varvec{n}})}\sum _{w\in S_M} \!\! \mathrm {sgn}(w) \! \sum _{m\ne m'}^M \! \left( Z_{mm'}({\tilde{n}}) - Z_{m'm}({\tilde{n}}_{\tau })\right) ,\nonumber \!\!\!\!\!\!\!\!\\ \end{aligned}$$

(D.2)

where $\tau \,{:}{=}\,\, (m m')$ is a transposition. By noting that

$$\begin{aligned} \begin{aligned} \Pi _{mm'}({\varvec{n}})&= \Pi _{m'm}({\varvec{n}}_{\tau }) \\ {\widetilde{\Phi }}_{\tilde{{\varvec{p}}}}({\varvec{n}}) \, \mathrm {e}^{\mathrm {i}{\varvec{r}}\cdot {\varvec{n}}} \big |_{n_m = n_{m'}}&= {\widetilde{\Phi }}_{\tilde{{\varvec{p}}}}({\varvec{n}}_\tau ) \, \mathrm {e}^{\mathrm {i}{\varvec{r}}\cdot {\varvec{n}}_\tau } \big |_{n_{\tau (m')} = n_{\tau (m)}}, \\ \frac{\wp '(n_{m'} - n_{m})}{\wp (n_{m'} - n_{m})}&= \frac{\wp '(n_{\tau (m)} - n_{\tau (m')})}{\wp (n_{\tau (m)} - n_{\tau (m')})}, \\ \sum _{m''(\ne m,m')}^{M} \!\!\!\!\!\!\!\! \zeta (n_{m''} - n_{m'})&= \sum _{m''(\ne m,m')}^{M} \!\!\!\!\!\!\!\! \zeta (n_{\tau (m'')} - n_{\tau (m)}), \end{aligned} \end{aligned}$$

(D.3)

we can drastically simplify

$$\begin{aligned} Z_{mm'}({\varvec{n}}) - Z_{m'm}({\varvec{n}}_{\tau }) = \Pi _{mm'}({\varvec{n}})\left( (\partial _{n_m}-X_m) - (\partial _{n_{m'}}-X_{m'}) \right) {\widetilde{\Phi }}_{\tilde{{\varvec{p}}}}({\varvec{n}}) \, \mathrm {e}^{\mathrm {i}{\varvec{r}}\cdot {\varvec{n}}} \, \big |_{n_{m} = n_{m'}}.\nonumber \\ \end{aligned}$$

(D.4)

Plugging (C.24) with ${\varvec{n}} \rightarrow {\varvec{n}}_w$ into the M-particle difference equation (3.20) yields a left-hand side containing the sum (D.2). Using (D.4) we obtain (3.22).

1.2 D.2. Derivatives of the numerator

By assuming that ${\widetilde{\Psi }}_{\tilde{{\varvec{p}}}}$ is an eigenfunction of the eCS Hamiltonian at $\beta =2$ we can deduce further information about its numerator ${\widetilde{\Phi }}_{\tilde{{\varvec{p}}}} = \Delta \, {\widetilde{\Psi }}_{\tilde{{\varvec{p}}}}$, see (3.16), from the eCS eigenvalue equation. Carefully cancelling the denominator $\Delta$ we find

$$\begin{aligned} \sum _{m=1}^M \partial ^2_{x_m}&{\widetilde{\Phi }}_{\tilde{{\varvec{p}}}}({\varvec{x}}) + \left( 2 \, {\widetilde{E}} - \sum _{m\ne m'}^M \!\! \wp (x_m-x_{m'}) + \sum _{m=1}^M \biggl (\, \sum _{m'(\ne m)}^M \!\!\!\! \zeta (x_m-x_{m'}) \biggr )^{\!\! 2} \ \right) \, {\widetilde{\Phi }}_{\tilde{{\varvec{p}}}}({\varvec{x}}) \\&\quad = \sum _{m\ne m'}^M \!\!\zeta (x_m -x_{m'}) \, (\partial _{m} - \partial _{m'}) \, {\widetilde{\Phi }}_{\tilde{{\varvec{p}}}}({\varvec{x}}) \, . \end{aligned}$$

This looks slightly different than equation (62b) in [Ino03] but luckily we can draw the same conclusions: the left-hand side of this equation is analytic, as every pole from $\wp$ gets cancelled by one from the square of $\zeta$ functions. Since the right-hand side has $\zeta$ functions as well (and therefore poles), we see that it must be true that

$$\begin{aligned} (\partial _{m} - \partial _{m'}) \, {\widetilde{\Phi }}_{\tilde{{\varvec{p}}}}({\varvec{x}}) \big |_{x_{m} = x_{m'}} = 0 \end{aligned}$$

to ensure the analyticity of the right-hand side.

1.3 D.3. Trivial symmetry map

In this appendix we try to give some more explanation to one of the central tools Inozemtsev used in [Ino93] to perform his version of the counting of the $M=2$ solutions as we have done in Sect. 4.5.4, because it turns out to be a nice piece of Weierstrass analysis that can be applicable in other situations. In order to facilitate comparison with Inozemtsev’s work we will work on the lattice $\hat{\,\,{\mathbb {L}}}$ with periods $(L, L\,\omega )$, which is the coordinate version of the lattice $\hat{\, {\mathbb {L}}}^{\vee }$ defined in (4.68).

Let us first define the map ${\hat{R}}_x$ on the fundamental parallelogram of $\hat{\,\,{\mathbb {L}}}$ by requiring it to satisfy $x_{{\hat{R}}_x(\gamma )} = x_{\gamma }$, i.e. $\gamma$ and ${\hat{R}}_x(\gamma )$ belong to the same solution of (4.78). Here $\gamma$ relates to $\varphi$ as in (4.10). Note that there is a unique $\gamma ' \ne \gamma$ inside the fundamental parallelogram with the same image under $\wp$ since the latter is a second-order elliptic function, implying the map ${\hat{R}}_x$ is indeed well-defined.

More precisely, let $\gamma$ be in the fundamental parallelogram and

$$\begin{aligned} x_{\gamma } = \wp (\gamma -I_\text {tot} \, \omega /2) \in {\mathbb {C}}\, . \end{aligned}$$

(D.5)

We can find this $\gamma '$ by congruence:

$$\begin{aligned} x_{\gamma } = \wp \bigl (-(\gamma - \omega \, I_\text {tot}/2)\bigr ) = \wp ((-\gamma + \omega I_\text {tot} ) - I_\text {tot} \omega /2 ) \, , \end{aligned}$$

(D.6)

which tells us that the second solution is

$$\begin{aligned} \gamma ' = -\gamma + \omega \, I_\text {tot} \, \ \mathrm {mod }\,\,\!\!\hat{\,\,{\mathbb {L}}} \, , \end{aligned}$$

(D.7)

i.e. we can define the map ${\hat{R}}_x$ by

$$\begin{aligned} {\hat{R}}_x(\gamma ) \,{:}{=}\,\, (-\gamma + I_\text {tot} \, \omega )\, \mathrm {mod }\,\, \!\!\hat{\,\,{\mathbb {L}}}\, . \end{aligned}$$

(D.8)

It is not so hard to work out what the modulo means in practice and derive a formula for ${\hat{R}}_x(\gamma )$. By assumption $0\le {\mathrm {Re}}\gamma < L$ and $0\le {\mathrm {Im}}\,\gamma \le L |\omega |$. First we consider the real part of $\gamma '$. If $\gamma$ is purely imaginary then so is $\gamma '$ and we do not have to adjust the real part, while if ${\mathrm {Re}}\gamma$ is positive then ${\mathrm {Re}}(I_\text {tot} \, \omega -\gamma )<0$ so we should add L to land in the fundamental parallelogram. In total, then, the real shift can be written as $L \, \mathrm {sgn}({\mathrm {Re}}\gamma )$, with the convention that $\mathrm {sgn}\,0 = 0$. To determine the imaginary shift there are three cases to consider. If $I_\text {tot} \, {\mathrm {Im}}\omega - {\mathrm {Im}}\gamma$ is positive we do not have to do anything, while if it is negative we can add $L\,\omega$ to land in the fundamental parallelogram, and when

$$\begin{aligned} I_\text {tot} \, {\mathrm {Im}}\omega - {\mathrm {Im}}\gamma =0 \end{aligned}$$

(D.9)

then $\gamma ' = -{\mathrm {Re}}\gamma$ is real. This means only the real shift by L is necessary in order for the solution to be in the fundamental parallelogram. We therefore find that

$$\begin{aligned} \gamma '= {\hat{R}}_x(\gamma ) = I_\text {tot} \, \omega - \gamma + L \, \mathrm {sgn}({\mathrm {Re}}\gamma ) + L \, \omega \, \theta (I_\text {tot}\, |\omega | -{\mathrm {Im}}\gamma )\, , \end{aligned}$$

(D.10)

where $\theta$ is the Heaviside step function satisfying $\theta (0) = 0$.

This expression looks a lot like the equation for $\gamma _0'$ below (14) in [Ino93], with the only difference sitting in the final term. Before exploring this difference in more details, let us first note that the map ${\hat{R}}_x$ has a nice geometric meaning. Indeed, $\gamma ' = {\hat{R}}_x(\gamma )$ is the unique second solution to $x_{\gamma '} = x_\gamma$, which has $y_{{\hat{R}}_x(\gamma )} = -y_{\gamma }$. Therefore ${\hat{R}}_x$ maps a point $(x_{\gamma }, y_{\gamma })$ on the elliptic curve to the point $(x_{\gamma }, - y_{\gamma })$: it is nothing but a reflection in the x-axis.

To investigate the difference between ${\hat{R}}_x$ and the expression for $\gamma _0'$ in [Ino93], let us first note that on the set of solutions to the constraints the maps are nearly identical, with the only difference occurring when (D.9) holds. However, solutions like this are extremely rare: consider such a solution $\gamma = \gamma _\text {r} + I_\text {tot} \, \omega$ where $\gamma _\text {r} \in [0,L]$. Plugging in this form into the constraint yields

$$\begin{aligned} 2 \, \rho _2(\gamma _\text {r} ) - {\bar{\rho }}_2\biggl (\frac{\gamma _\text {r}}{L}\biggr ) - {\bar{\rho }}_2\biggl (\frac{\gamma _\text {r}+I_\text {tot} \, \omega }{L}\biggr ) = 0\, , \end{aligned}$$

(D.11)

where we used the periodicity of $\rho _2$ to omit the shift by $I_\text {tot} \, \omega$ in the first term. Now note that $\rho _2$ and ${\bar{\rho }}_2$ are analytic functions away from poles (i.e. meromorphic) and that there is only one term here that will generate a non-trivial imaginary part as long as $I_\text {tot} \not \cong 0$ mod L/2.⁴²

Thus this can happen only if $I_\text {tot} = 0,L/2$, but $I_\text {tot} = 0$ yields the trivial solution $\gamma = L/2$. Choosing $I_\text {tot} = L/2$ yields a unique non-trivial solution with $\gamma _\text {r} = L/2$, i.e. $\gamma = L/2 + L\omega /2$. This is not a trivial root: the two-particle wave function is nonzero for this $\gamma$. Now we see that ${\hat{R}}_x(L/2 + L\omega /2) = L/2$, while $\gamma '_0 = L/2 + L\omega /2$. So both expressions send this solution to a solution in the fundamental domain, but in the case discussed in [Ino93] we have $\gamma '_0 = \gamma _0$.

Let us finally note that this analysis can be performed on any of the lattices. For our purposes the lattice $\hat{\,{\mathbb {L}}}^{\vee }$, in terms of $\varphi$, is most convenient. Following the preceding steps one readily finds that on that lattice the action of the reflection $\hat{R}^{\vee }_x$ is given by

$$\begin{aligned} \hat{R}^{\vee }_x(\varphi ) \,{:}{=}\,\, -\varphi -2\pi \,I_\text {tot} + \frac{2\pi L}{\omega } \,\mathrm {sgn}({\mathrm {Im}}\varphi ) + \, 2\pi L \, \theta (2\pi \, I_\text {tot} + {\mathrm {Re}}\varphi ) \, . \end{aligned}$$

(D.12)

1.4 D.4. Trivial roots are nonphysical

In Sect. 4 we simplified (4.75) by discarding the four simple explicit solutions (4.24). Let us show that this is allowed by analysing these trivial solutions. First, all the trivial solutions of (4.75) can be identified with the solutions to⁴³

$$\begin{aligned} {\hat{R}}_x(\gamma ) = \gamma \, , \end{aligned}$$

(D.13)

which on the elliptic curve is easy to see: checking that all our trivial solutions satisfy the above equation is straightforward, so let us consider the converse. If $x_{{\hat{R}}_x(\gamma )} = x_{\gamma } \in {\mathbb {C}}$, then we have $y_{{\hat{R}}_x(\gamma )} = -y_{\gamma }$, so in particular if ${\hat{R}}_x(\gamma ) = \gamma$ we find $y_{\gamma } = -y_{\gamma }$, i.e. $y_{\gamma } = 0$. This yields the last three trivial solutions listed in (4.24). The fourth trivial root at $\gamma = I_\text {tot} \, \omega /2$ is slightly more subtle: for this value $x_\gamma$ and $y_\gamma$ diverge, i.e. $x_\gamma \notin {\mathbb {C}}$. This implies we cannot follow the approach above as there is only one point for which $x_\gamma$ diverges in the fundamental parallelogram. Moreover, we technically have not proven the validity of (D.8) for such $\gamma$. Nevertheless, by the previous we see that if $x_{{\hat{R}}_x(\gamma )} = x_\gamma \notin {\mathbb {C}}$ it must be that $\gamma$ lies at a pole, which can only be $\gamma = I_\text {tot} \, \omega /2$. Thus we see that the set of trivial solutions is in one-to-one correspondence with the solutions to (D.13).

Now we would like to furthermore demonstrate that the two-particle wave function corresponding to a trivial root vanishes. On the lattice $\hat{\,{\mathbb {L}}}^{\vee }$ the roots in question are

$$\begin{aligned} \varphi = -I_\text {tot} \, \pi \, , \quad \varphi = (L-I_\text {tot}) \, \pi \, , \quad \varphi = -I_\text {tot} \, \pi + \mathrm {i}\,L\,\kappa \, , \quad \varphi = (L-I_\text {tot})\, \pi + \mathrm {i}\,L\,\kappa \, , \end{aligned}$$

which on the corresponding coordinate lattice $\hat{\,\,{\mathbb {L}}}$ yields

$$\begin{aligned} \gamma = \frac{I_\text {tot} \, \omega }{2} \, ,\quad \gamma = \frac{I_\text {tot} \, \omega + L}{2}, \quad \gamma = \frac{(I_\text {tot}+L) \, \omega }{2} \, , \quad \gamma = \frac{(I_\text {tot}+L) \, \omega +L }{2} \, .\nonumber \\ \end{aligned}$$

(D.14)

The $M=2$ wave function can be written in terms of $\gamma$ as

$$\begin{aligned} \Psi\sim & {} \frac{2}{\sigma (\gamma )} \left( \exp \left( \frac{2\pi \mathrm {i}\, I_\text {tot} \, n_1 + 2 \, \eta _1 \gamma \, \delta n }{L}\right) \sigma (\gamma -\delta n) +\exp \left( \frac{2\pi \mathrm {i}\, I_\text {tot} \,n_2}{L}\right) \sigma (-\gamma -\delta n) \right) ,\nonumber \\ \end{aligned}$$

(D.15)

where we have omitted irrelevant regular non-vanishing factors. The overall factor is finite as long as $\gamma$ does not lie on the lattice. The mechanism for vanishing is the same for all four roots: we check that at these roots the two occurring sigma functions are the same by applying the quasi-periodicity rule we defined just below (B.3b) as often as necessary. The easiest one is the root $\gamma = I_\text {tot} \, \omega /2$: by applying the quasi-periodicity rule $I_\text {tot}$ times we obtain

$$\begin{aligned} \sigma \left( \frac{I_\text {tot} \, \omega }{2} -\delta n\right) = (-1)^{I_\text {tot}} \, \mathrm {e}^{-\delta n \, I_\text {tot} \, \eta _2} \, \sigma \left( -\frac{I_\text {tot} \, \omega }{2} -\delta n\right) , \end{aligned}$$

(D.16)

thus we can rewrite the first factor in the brackets of (D.15) as

$$\begin{aligned} \begin{aligned}&(-1)^{I_\text {tot}} \exp \biggl (\frac{2\pi \mathrm {i}\, n_1 + 2\gamma \, \eta _1 \delta n }{L} -\delta n \, I_\text {tot} \, \eta _2 \biggr ) \, \sigma \biggl ( -\frac{I_\text {tot} \omega }{2} -\delta n \biggr ) \\&\quad = (-1)^{I_\text {tot}} \exp \biggl (\frac{2\pi \mathrm {i}\, I_\text {tot} \, n_2}{L} \biggr ) \, \sigma \biggl ( -\frac{I_\text {tot} \, \omega }{2} -\delta n\biggr ). \end{aligned} \end{aligned}$$

This directly implies that the wave function vanishes as long as $I_\text {tot}$ is odd. Note that the possible pole occurring due the first factor in (D.15) is avoided due to this requirement on $I_\text {tot}$ as well.

Performing a similar analysis on $\gamma = (I_\text {tot}\, \omega + L)/2$ shows that we can rewrite the first term in the brackets of (D.15) as

$$\begin{aligned} {-}\! \exp \biggl (\frac{2\pi \mathrm {i}\, I_\text {tot} \, n_2}{L} \biggr ) \, \sigma \biggl ( -\frac{I_\text {tot} \, \omega }{2}-\frac{L}{2} -\delta n\biggr ), \end{aligned}$$

(D.17)

thus in this case the wave function vanishes without any restrictions.

The third and fourth root require another repeated application of the quasi-periodicity rule: for the third root (which is parity-dual to the first one) we find after another L applications that

$$\begin{aligned} \sigma \biggl ( \frac{(I_\text {tot}+L) \,\omega }{2} -\delta n\biggr ) = (-1)^{I_\text {tot}+L} \, \mathrm {e}^{-\delta n (I_\text {tot}+L) \eta _2} \, \sigma \biggl ( -\frac{(I_\text {tot}+L)\, \omega }{2} -\delta n\biggr ) \end{aligned}$$

(D.18)

and when evaluated at the third root in (D.14) the first term in the brackets of the $M=2$ wave function becomes

$$\begin{aligned} (-1)^{I_{\mathrm{tot}}+L} \exp \biggl (\frac{2\pi {\mathrm {i}} \, I_{\mathrm{tot}} \, n_2}{L} +2\pi {\mathrm {i}} \, \delta n \biggr ) \, \sigma \biggl ( -\frac{(I_{\mathrm{tot}}+L) \, \omega }{2} -\delta n\biggr ), \end{aligned}$$

(D.19)

which when evaluated for integer $\delta n$ vanishes against the second factor in the brackets of (D.15) as long as $L+I_\text {tot}$ is odd, which is exactly what is necessary to avoid the pole coming from the overall factor.

Finally, for the fourth root $\gamma = [(I_\text {tot}+L) \, \omega +L]/2$ we can follow the approach of its parity-dual, the second root, and rewrite the first term to cancel the second term without any restrictions on L and $I_\text {tot}$.

Thus we see that the wave function vanishes precisely when these roots also occur as solutions to the constraint (4.75) and can thus safely be removed from this equation to ultimately yield the constraint (4.78). It would be interesting to see this analysis for higher M as it could reveal what would be suitable coordinates for the spectral problem in those sectors.