The Bethe Ansatz

Let us take a first plunge into hunting for and counting solutions. Using the notation \(z_j = e^{ik_j}\), the Bethe equations are

\begin{align*} &(z_1 z_2)^N = e^{-2i\alpha}, \\ &e^{i\alpha} z_1^N (1 + z_1 z_2 -2\Delta z_2) + 1 + z_1 z_2 - 2\Delta z_1 = 0. \end{align*}

The first equation above means that we can set

\[ z_2 = \frac{w_\alpha^a}{z_1}, \hspace{5mm} a = 0, 1, \ldots N-1, \hspace{5mm}w_\alpha \equiv e^{i \frac{2(\pi-\alpha)}{N}} \tag{xxz.z2}\label{xxz.z2} \]

separating the set of solutions into sectors distinguished by parameter \(a\) (which, as we will detail later, is related to total momentum).

For a given sector \(a\), \(z_2\) can then be substituted back into the second equation, giving the condition which our roots \(z_1\) (here written as \(z\)) must obey in that sector:

\begin{align*} p^{\scriptscriptstyle (N,2)}_a (z; \alpha) &= 0, \tag{xxz.nm2bpe}\label{xxz.nm2pbe} \end{align*}

where

\begin{align*} p^{\scriptscriptstyle (N,2)}_a (z; \alpha) &\equiv (1 + w_\alpha^a) (e^{i\alpha} z^N + 1) -2\Delta \,(w_\alpha^a e^{i\alpha} z^{N-2} + 1)\,z \tag{xxz.nm2bp}\label{xxz.nm2pb} \end{align*}

which we shall call the XXZ \(N,M=2\) Bethe polynomial.

For each of the \(N\) possible values for parameter \(a\) in xxz.z2, equation xxz.nm2bpe is an order \(N\) (with exceptions) polynomial equation, each root of which provides, with xxz.z2, a pair \(z_1, z_2 \rightarrow k_1, k_2\) to be substituted in xxz.ba2.

Let us seek the solutions of xxz.nm2bpe. For later convenience, let us write the first two derivatives of the Bethe polynomial:

\begin{align*} \frac{d}{dz} p^{\scriptscriptstyle (N,2)}_a(z;\alpha) &= (1 + w_\alpha^a) e^{i\alpha}N z^{N-1} -2\Delta (w_\alpha^a e^{i\alpha}(N-1) z^{N-2} + 1), \\ \frac{d^2}{dz^2} p^{\scriptscriptstyle (N,2)}_a(z;\alpha) &= (1 + w_\alpha^a) e^{i\alpha}N (N-1) z^{N-2} - 2\Delta w_\alpha^a e^{i\alpha} (N-1)(N-2) z^{N-3} \\ &= e^{i\alpha} (N-1) z^{N-3} \left( (1+w_\alpha^a) N z - 2\Delta w_\alpha^a (N-2)\right). \tag{xxz.nm2bp}\label{xxz.nm2bp} \end{align*}

We need to consider a number of cases, which we differentiate by making various starting assumptions. To be clear, on the right of each paragraph heading, we put in parentheses the assumptions which are applicable.

Remark: here under, we have assumed \(\alpha = 0\).

Assumption 1: \(\Delta \neq 0\)

For the time being, we will assume a nonzero anisotropy. We will later consider the limit \(\Delta \rightarrow 0\) of the resulting constructions.

Assumption 2: \(w^a \neq -1\) (A1)

For \(w^a \neq -1\), \(p^{\scriptscriptstyle (N,2)}_a(z)\) is a polynomial of degree \(N\) with depressed-like character (powers \(2, 3, \ldots, N-2\) have zero coefficient). Despite this simplified character, the Abel-Ruffini theorem teaches us that no solution to \(p^{\scriptscriptstyle (N,2)}_a(z) = 0\) in the form or radicals exists for \(N \geq 5\) (we shall treat the \(N=3,4\) cases explicitly later on).

Invariance; factorization (A1,A2)

To start our discussion, note that \(z=0\) is not a root of \(p^{\scriptscriptstyle (N,2)}_a(z)\). More interestingly, if \(z\neq 0\) is a root of \(p^{\scriptscriptstyle (N,2)}_a(z)\), then so is \(w^a/z\). The cardinality-\(N\) set of roots of \(p^{\scriptscriptstyle (N,2)}_a(z)\) is thus invariant under the mapping \(z\rightarrow w^a/z\),

\[ \left\{ z_1, \ldots \right\}_N = \left\{ w^a/z_1, \ldots \right\}_N. \]

Nonzero roots thus come either in pairs \((z_i, w^a/z_i \neq z_i)\) or as self-invariant roots \(z_i = w^a/z_i\) whose two possible values are

\[ z_\pm = \pm w^{a/2}. \tag{xxz.nm2sir}\label{xxz.nm2sir} \]

For a non-self-invariant pair \((z_i, w^a/z_i \neq z_i)\), we fix the labelling ambiguity by requiring \(\text{Arg} (z_i) \in [-\pi + \text{Arg}(w^a), \text{Arg}(w^a)]\), which we call the base arg region.

Allowing for multiplicity \(n_\pm\) of the self-invariant roots \(z_\pm\), and letting \(n_p\) be the number of non-self-invariant pairs (with \(\{ z_\alpha \}_{n_p}\) being the cardinality-\(n_p\) set of non-self-invariant roots in the base arg region), we can thus formally factorize the Bethe polynomial in this sector as

\begin{align*} p^{\scriptscriptstyle (N,2)}_a(z) &= (1 + w^a) (z - z_+)^{n_+} (z - z_-)^{n_-} \prod_{\alpha=1}^{n_p} (z - z_\alpha) (z - w^\alpha/z_\alpha), \\ &n_+ + n_- + 2n_p = N \end{align*}

(the prefactor is fixed by considering the \(N\)-th power of \(z\) and matching with xxz.nm2bp).

Self-invariant roots

Substituting xxz.nm2sir into xxz.nm2bp shows that a self-invariant root \(z_\sigma\) occurs when

\[ (\sigma^N w^{aN/2} + 1) (w^a - 2\sigma \Delta w^{a/2} + 1) = 0. \tag{xxz.nm2sirc}\label{xxz.nm2sirc} \]

Case 1

Let us first look at the first possibility: \[ \sigma^N w^{aN/2} + 1 = 0 ~\longrightarrow ~\sigma w^{a/2} = e^{i \frac{\pi}{N} (2b+1)}, ~b = 0, 1, \ldots, N-1, \]

which leads to

\[ a = 2b+1 + \frac{N}{2} (\sigma - 1) ~~\text{mod}(2N) \]

Thus, we reach the conclusion that

• \(z_+\) is a root for \(a\) odd
• \(z_-\) is a root for \(a+N\) odd.

To check the multiplicity of \(z_\pm\), we evaluate \(\frac{d}{dz} p^{\scriptscriptstyle (N,2)}_a(z_\pm)\), giving

\[ {\scriptstyle \frac{d}{dz}} p^{\scriptscriptstyle (N,2)}_a(z_\sigma) = 2(-1)^a \sigma^N (N \sigma \cos\frac{\pi a}{N} -(N-1)\Delta) -2\Delta. \]

When this is nonvanishing (that's the generic case), the multiplicity of \(z_\sigma\) is one. We get multiplicity (at least) two for \(z_+\) (resp. \(z_-\)) in the special case that

\begin{align*} &\left. {\scriptstyle \frac{d}{dz}} p^{\scriptscriptstyle (N,2)}_a(z_+) \right|_{a \,\text{odd}} = 0 ~~~~~~\longrightarrow~~ \Delta = \Delta_{N,a} \equiv \frac{N}{N-2} \cos \frac{\pi a}{N}, \\ &\left. {\scriptstyle \frac{d}{dz}} p^{\scriptscriptstyle (N,2)}_a(z_-) \right|_{a+N \,\text{odd}} = 0 ~~\longrightarrow~~ \Delta = -\Delta_{N,a}. \end{align*}

Going further, one can show that

\[ \left. {\scriptstyle \frac{d^2}{dz^2}} p^{\scriptscriptstyle (N,2)}_a(z_+) \right|_{a \,\text{odd}, \Delta = \Delta_{N,a}} = 0, \hspace{10mm} \left. {\scriptstyle \frac{d^2}{dz^2}} p^{\scriptscriptstyle (N,2)}_a(z_-) \right|_{a+N \,\text{odd}, \Delta = -\Delta_{N,a}} = 0 \]

which (together with the easily shown fact that the third derivative is nonvanishing) shows that the multiplicity of the self-invariant roots, when they occur, is either one (generic case) or three (special case).

Case 2

The second possibility for a self-invariant root \(z_\sigma\) is if \[ w^a - 2\Delta \sigma w^{a/2} + 1 = 0 \rightarrow w^{a/2} = \sigma (\Delta + \sigma' \sqrt{\Delta^2 - 1}), \hspace{3mm} \sigma, \sigma' = \pm 1. \]

For \(|\Delta| > 1\), this equation cannot be satisfied since \(|w|=1\) while the right-hand side is not unimodular. When \(|\Delta| \leq 1\) however, we get that xxz.nm2sir is satisfied if (using the notation \(\Delta = \cos \zeta\))

\[ w^{a/2} = \sigma (\Delta + i \sigma' \sqrt{1 - \Delta^2}) = \sigma e^{i\sigma' \zeta} \]

or \[ \frac{\pi}{N} a = \pi \sigma + \sigma' \zeta ~\text{mod}(2\pi) \rightarrow a = N\sigma + N\sigma' \frac{\zeta}{\pi} ~\text{mod}(2N) \]

or more simply \[ \Delta = -\cos \frac{\pi a}{N} \]