The Bethe Ansatz

The axial Heisenberg antiferromagnet (XXZ with \(\Delta > 1\))

Let us specialize our equations to the case of the gapped antiferromagnet with anisotropy \(\Delta > 1\). We make use of parameter \(\eta\) such that \(\Delta = \cosh \eta\). We will adopt the convention that \(\eta > 0\).

We introduce rapidities \(\lambda\) defined in the physical strip

\begin{equation*} -\frac{\pi}{2} < \Re \lambda \leq \frac{\pi}{2}, \hspace{10mm} |\Im \lambda| < \infty \end{equation*}

and parametrize the bare momenta in terms of these as

\begin{equation} e^{ik(\lambda)} = \frac{\sinh (-i \lambda + \eta/2)}{\sinh (-i \lambda - \eta/2)} = \frac{\sin (\lambda + i\eta/2)}{\sin (\lambda - i\eta/2)}, \hspace{10mm} k (\lambda) = \pi - 2~\mbox{atan}~ \frac{\tan \lambda}{\tanh \frac{\eta}{2}}. \tag{a.l}\label{a.l} \end{equation}
Derivation \begin{equation*} \frac{\sin (\lambda + i\eta/2)}{\sin (\lambda - i\eta/2)} = \frac{\sin \lambda \cosh \eta/2 + i \cos \lambda \sinh \eta/2}{\sin \lambda \cosh \eta/2 - i \cos \lambda \sinh \eta/2} = -\frac{\tanh \eta/2 - i \tan \lambda}{\tanh \eta/2 + i \tan \lambda} \end{equation*}

and thus

\begin{equation*} k (\lambda) = \frac{1}{i} \ln \frac{\sin (\lambda + i\eta/2)}{\sin (\lambda - i\eta/2)} = \pi - 2 ~\mbox{atan}~ \frac{\tan \lambda}{\tanh \frac{\eta}{2}}. \end{equation*}

The Bethe equations in this regime take the form

\begin{equation} \left[ \frac{\sin (\lambda_j + i\eta/2)}{\sin(\lambda_j - i\eta/2)} \right]^N = \prod_{k \neq j}^M \frac{\sin(\lambda_j - \lambda_k + i\eta)}{\sin(\lambda_j - \lambda_k - i\eta)}, \hspace{1cm} j = 1, ..., M. \tag{}\label{} \end{equation} \begin{equation} \phi_1(\lambda_a) - \frac{1}{N} \sum_{b = 1}^M \phi_2(\lambda_a - \lambda_b) = 2\pi \frac{I_a}{N} \tag{a.bel}\label{a.bel} \end{equation}

where the kernel is now

\begin{equation} \phi_n (\lambda) \equiv 2~\mbox{atan} \left[ \frac{\tan(\lambda)}{\tanh(n\eta/2)} \right] + 2\pi \left \lfloor \frac{\lambda}{\pi} + \frac{1}{2} \right \rfloor. %\!\!-\! \frac{1}{N} \!\sum_{k = 1}^M %2\mbox{atan} \!\!\left[\frac{\tan(\lambda_j - \lambda_k)}{\tanh \eta}\right] \!\!=\! 2\pi \frac{I_j}{N}. \tag{a.phin}\label{a.phin} \end{equation}

The integer part of the kernel (second term) guarantees monotonicity for real \(\lambda\). This is important for the classification of states (you can find a partial discussion in 2008.Caux.JSTAT.P08006).

The energy of a state in an external magnetic field \(h\) is given as a function of the rapidities by

\begin{equation} E = J \sum_{j = 1}^M \frac{-\sinh^2 \eta}{\cosh \eta - \cos 2\lambda_j} - h(\frac{N}{2} - M), \tag{a.e}\label{a.e} \end{equation}

whereas the momentum, which is the sum over quasi-momenta associated to each rapidity, has a simple representation in terms of the quantum numbers:

\begin{equation} P = \sum_{j = 1}^M \frac{1}{i} \ln \left[\frac{\sin(\lambda_j + i\eta/2)}{\sin(\lambda_j - i\eta/2)}\right] = \pi M - \frac{2\pi}{N}\sum_{j = 1}^M I_j \hspace{0.5cm} \mbox{mod} \hspace{0.2cm}2\pi. \tag{a.p}\label{a.p} \end{equation}

In the presence of strings, the same construction applies as for \(XXX\). The Bethe equations again get reduced to a Bethe-Gaudin-Takahashi form, and the string-string scattering phase shift is still given by the form h.phijk.

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24