The Bethe Ansatz

Following Takahashi and Suzuki 1972.Takahashi.PTP.48, we take \(\zeta/\pi\) to be a real number between \(0\) and \(1\), which is expressed as a continued fraction of real positive integers as

\begin{equation*} \frac{\zeta}{\pi} = \frac{1|}{|\nu_1} + \frac{1|}{|\nu_2} + ... \frac{1|}{|\nu_l}, \hspace{1cm} \nu_1, ..., \nu_{l-1} \geq 1, \nu_l \geq 2. \end{equation*}

For large \(N\), rapidities congregate to form strings centered either on the real line (for positive parity strings) or on the axis \(i\pi/2\) (for negative parity strings),

\begin{equation} \lambda^{n_j, a}_{\alpha} = \lambda^{n_j}_{\alpha} + i\frac{\zeta}{2} (n_j + 1 - 2a) + i\frac{\pi}{4} (1 - v_j) + i \delta^{n_j, a}_{\alpha}, \hspace{1cm} a = 1, ..., n_j \tag{p.sh}\label{p.sh} \end{equation}

where the allowable lengths \(n_j\) and parities \(v_j = \pm 1\) are to be determined. In a string configuration, the parameters \(\delta^{n_j, a}_{\alpha}\) are exponentially suppressed with system size.

The classification of allowable string types in the thermodynamic limit proceeds according to the following algorithm. First, the positive integer series \(y_{-1}, y_0, y_1, \ldots, y_l\) and \(m_0, m_1, \ldots, m_l\) are defined as

\begin{align*} &y_{-1} = 0, ~~y_0 = 1, ~~y_1 = \nu_1, ~~\mbox{and}~~ y_i = y_{i-2} + \nu_i y_{i-1}, ~~i = 2, \ldots, l, \nonumber \\ &m_0 = 0, ~~m_i = \sum_{k = 1}^i \nu_k. \end{align*}

Lengths and parities are then given by (our conventions here have the advantage of giving a proper ordering of string lengths, \(n_j > n_k\), \(j > k\))

\begin{equation*} n_j = y_{i-1} + (j - m_i)y_i, ~~v_j = (-1)^{\lfloor (n_j - 1)\zeta/\pi \rfloor}, \hspace{1cm} m_i \leq j < m_{i+1}. \end{equation*}

The total number of possible strings is \(N_s = m_l + 1\), and the index \(j\) runs over the set \(1, ..., N_s\). The real parameters \(\lambda_{\alpha}^{n_j}\) represent the centers of strings with length \(n_j\) and parity \(v_j\), and are hereafter noted as \(\lambda^j_{\alpha}\), \(\alpha = 1, ..., M_j\), where \(M_j\) is the number of strings of length \(n_j\) in the eigenstate under consideration. We therefore have the constraint \(\sum_{j = 1}^{N_s} n_j M_j = M\).

In a string configuration, many factors appearing in the Bethe equations become of the indeterminate form \(\delta/\delta\). Remultiplying p.be for each member of a particular string gets rid of these factors, and allows one to rewrite the whole set of Bethe equations in terms of a reduced set involving only the string centers \(\lambda_{\alpha}^j\). Doing this, one finds (for \(N\) even) the reduced set of Bethe-Takahashi equations 1972.Takahashi.PTP.48

\begin{equation} \bar{e}_j^N (\lambda_{\alpha}^j) = (-1)^{M_j-1}\prod_{(k,\beta) \neq (j,\alpha)} \bar{E}_{jk} (\lambda_{\alpha}^j - \lambda_{\beta}^k), \tag{p.bgt}\label{p.bgt} \end{equation}

where

\begin{equation} \bar{e}_j(\lambda) = -v_j\frac{\sinh(\lambda + i\frac{\pi}{4}(1 - v_j) + i n_j\zeta/2 )}{\sinh(\lambda + i\frac{\pi}{4}(1 - v_j) - i n_j\zeta/2)}, \tag{p.ej}\label{p.ej} \end{equation}

and

\begin{equation} \bar{E}_{jk}(\lambda) = \bar{e}_{|n_j-n_k|}^{1-\delta_{n_j n_k}} (\lambda) \bar{e}_{|n_j-n_k|+2}^2(\lambda) ... \bar{e}_{n_j+n_k-2}^2 (\lambda) \bar{e}_{n_j+n_k}(\lambda). \tag{p.ejk}\label{p.ejk} \end{equation}

For the state classification and computation, it is preferable to work with the logarithmic form

\begin{equation} N\phi_j(\lambda^j_{\alpha}) - \sum_{k=1}^{N_s} \sum_{\beta=1}^{M_k} \Phi_{jk}(\lambda^j_{\alpha} - \lambda^k_{\beta}) = 2\pi I^j_{\alpha} \tag{p.bgtl}\label{p.bgtl} \end{equation}

where \(I^j_{\alpha}\) is integer if \(M_j\) is odd, and half-integer if \(M_j\) is even. The dispersion kernels and scattering phases appearing here are

\begin{align*} \phi_j (\lambda) &= 2 v_j ~\mbox{atan} \left[(\tan n_j\zeta/2)^{-v_j}\tanh \lambda \right], \nonumber \\ \Phi_{jk}(\lambda) &= (1-\delta_{n_j n_k}) \phi_{|n_j-n_k|}(\lambda) + 2\phi_{|n_j-n_k|+2}(\lambda) + \ldots + 2\phi_{n_j+n_k-2} (\lambda) + \phi_{n_j+n_k} (\lambda). \end{align*}

The energy and momentum of a string are given by

\begin{align} &E^{j}_{\alpha} = - J \frac{\sin \zeta \sin n_j \zeta}{v_j \cosh 2\lambda^j_{\alpha} - \cos n_j \zeta}, \nonumber \\ &p_0 (\lambda^j_\alpha) = i \ln \left[\frac{\sinh(\lambda^j_{\alpha} + i \frac{\pi}{4}(1 - v_j) + in_j \zeta/2)} {\sinh(\lambda^j_{\alpha} + i \frac{\pi}{4}(1 - v_j) - in_j \zeta/2)} \right] \tag{p.eps}\label{p.eps} \end{align}

so the total momentum is again expressible in terms of the string quantum numbers as

\begin{equation} P = \pi T_{v = +} - \frac{2\pi}{N} \sum_{j=1}^{N_s}\sum_{\alpha = 1}^{M_j} I_{\alpha}^j \hspace{0.5cm}\mbox{mod} ~~2\pi \tag{p.p}\label{p.p} \end{equation}

in which \(T_{v=+}\) is the total number of excitations with positive parity.