The Bethe Ansatz

Bethe equations for strings

The Bethe equation in terms of the string rapidities are

\begin{align*} e^{i \lambda^{j,a}_{\alpha} L} &= \prod_{(k, \beta, b) \neq (j, \alpha, a)} \frac{\lambda^{j, a}_{\alpha} - \lambda^{k, b}_{\beta} - i\bar{c}}{\lambda^{j, a}_{\alpha} - \lambda^{k, b}_{\beta} + i\bar{c}} \nonumber \\ &= \prod_{(k, \beta) \neq (j, \alpha)} \prod_{b=1}^k \frac{\lambda^j_{\alpha} - \lambda^k_{\beta} + i\bar{c} (\frac{j-k}{2} - a + b - 1)} {\lambda^j_{\alpha} - \lambda^k_{\beta} + i\bar{c} (\frac{j-k}{2} - a + b + 1)} \prod_{b \neq a} \frac{\bar{c} (-a + b - 1) + \delta^{j, (a,b)}_{\alpha}}{\bar{c} (-a + b + 1) + \delta^{j, (a,b)}_{\alpha}} \hspace{10mm} \end{align*}

where we have separated inter- and intra-string parts, dropped all string deviations for inter-string factors, and denoted \(\delta^{j, a}_{\alpha} - \delta^{j, b}_{\alpha} = \delta^{j, (a,b)}_{\alpha}\) in the intra-string part. Simplified Bethe equations are obtained by taking the product of these equations within the string considered. The left-hand side becomes

\begin{equation*} \prod_{a=1}^j e^{i \lambda^{j,a}_{\alpha} L} = e^{i j\lambda^j_{\alpha} L}. \end{equation*}

On the right-hand side, we have

\begin{equation*} \prod_{a=1}^j \prod_{b \neq a} \frac{\bar{c} (-a + b - 1) + \delta^{j, (a,b)}_{\alpha}}{\bar{c} (-a + b + 1) + \delta^{j, (a,b)}_{\alpha}} = (-1)^{j(j+1)} = 1 \end{equation*}

for the intra-string part, and (writing \(\lambda = \lambda^j_{\alpha} - \lambda^k_{\beta}\))

\begin{equation*} \prod_{a=1}^j \prod_{b=1}^k \frac{\lambda + i\bar{c} (\frac{j-k}{2} - a + b - 1)} {\lambda + i\bar{c} (\frac{j-k}{2} - a + b + 1)} = e_{|j - k|} (\lambda) e_{|j-k| + 2}^2 (\lambda) e_{|j - k| + 4}^2 (\lambda) ... e_{j+k - 2}^2 (\lambda) e_{j+k} (\lambda) \equiv E_{jk} (\lambda) \end{equation*}


\begin{equation*} e_j (\lambda) = \frac{\lambda - i\bar{c}j/2}{\lambda + i\bar{c}j/2}. \end{equation*}

The exponential form of the Bethe equations has thus been reduced to the set of \(N_s\) coupled equations for the string centers \(\lambda^j_{\alpha}\),

\begin{equation*} e^{i j \lambda^j_{\alpha}L} = \prod_{(k, \beta) \neq (j, \alpha)} E_{jk} (\lambda^j_{\alpha} - \lambda^k_{\beta}). \end{equation*}

Taking the logarithm and defining

\begin{equation*} \phi_j (\lambda) = 2 ~\mbox{atan}~ \frac{2\lambda}{\bar{c} j} \end{equation*}

such that \(-i \log (-e_j (\lambda)) = \phi_j(\lambda)\), we find the reduced Bethe equations

\begin{equation} j \lambda_{\alpha}^j L - \sum_{(k, \beta)} \Phi_{jk} (\lambda_{\alpha}^j - \lambda_{\beta}^k) = 2\pi I_{\alpha}^j \tag{al.bel}\label{al.bel} \end{equation}

with \(I_{\alpha}^j\) half-odd integer (integer) if \(N_j\) is even (odd), and where the scattering phase shifts are

\begin{equation} \Phi_{jk} (\lambda) = (1 - \delta_{jk}) \phi_{|j-k|} (\lambda) + 2 \phi_{|j-k| + 2}(\lambda) + ... + 2\phi_{j+k - 2} (\lambda) + \phi_{j+k} (\lambda). \tag{al.phijk}\label{al.phijk} \end{equation}

These strings are stable particles under scattering with one another, and are therefore soliton-like objects. One point worth emphasizing is that the scattering phase shifts al.phijk are simply those of breathers in the classical limit \(\beta \rightarrow 0\) of the sine-Gordon model after a trivial reparametrization of the rapidity. The sine-Gordon soliton's mass is in this limit much higher than that of the breathers, which have an evenly-spaced rest mass.

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24