The Bethe Ansatz

The \(M = 1\) subsector is simple to treat. This \(\left(\begin{array}{c} N \\ 1 \end{array} \right) = N\)-dimensional subspace is spanned by the following set of states:

\[ | j \rangle \equiv S^-_j | 0 \rangle, \hspace{10mm} 1 \leq j \leq N. \]

Let us use the Ansatz \[ ~|\Psi_1 \rangle = \sum_{j=1}^N \Psi_j | j \rangle \]

in which each wavefunction is characterized by \(N\) complex amplitudes \(\Psi_j\) with \(1 \leq j \leq N\).

Projecting the Schrödinger equation \(H |\Psi_1 \rangle = E_1 |\Psi_1 \rangle\) onto bras \(\langle j |\) gives us constraints on the amplitudes.

In the bulk

For \(1 < j < N\), only xxz.hbulk acts nontrivially, yielding the \(N-2\) conditions

\[ \frac{J}{2} \left(\Psi_{j - 1} + \Psi_{j + 1}\right) = (E_1 + J\Delta) \Psi_j, \hspace{10mm} {\scriptstyle 1 < j < N}. \]

The free wave (not yet Bethe) Ansatz

The bulk equations are simultaneously solved by the free wave Ansatz

\[ \Psi_j \rightarrow \Psi_j (k) \equiv {\cal N}(k)\,e^{i k j} \tag{xxz.psi1}\label{xxz.psi1} \]

for \(1 \leq j \leq N\) (in which \({\cal N}(k)\) is some normalization constant), provided the energy is fixed to

\begin{equation*} E_1 = e(k) \equiv J (\cos k -\Delta). \tag{xxz.e1}\label{xxz.e1} \end{equation*}

At the boundary (periodic)

Projecting onto the bras \( \langle j | \) for \(j=1\) and \(j=N\) (bearing in mind our adopted periodic coupling xxz.hpbc) gives the remaining two conditions

\begin{align*} &\frac{J}{2} \left(\Psi_N + \Psi_2\right) = (E_1 + J\Delta) \Psi_1, \\ &\frac{J}{2} \left(\Psi_{N-1} + \Psi_1\right) = (E_1 + J\Delta) \Psi_N \\ \end{align*}

for a total of \(N\) equations as required to fully determine our \(N\) amplitudes.

These boundary equations are most conveniently solved by introducing "ghost" sites at \(j = 0, N+1\), extending xxz.psi1 to those and requiring

\[\Psi_N = \Psi_0, \hspace{5mm} \Psi_1 = \Psi_{N+1}.\]

Quantization (not yet Bethe) equations

These last two requirements are in turn simultaneously fulfilled provided the parameter \(k\) is quantized according to \(e^{ikN} = 1\). The allowed values are therefore given by

\begin{equation*} k = k_{\tilde{I}} \equiv 2\pi \tilde{I}/N, \hspace{1cm} \tilde{I} = 0, 1, ..., N - 1 \tag{xxz.be1}\label{xxz.be1} \end{equation*}

(or any equivalent coverage of the Brillouin zone).

State counting; subspace eigenbasis

For lattice size \(N\), there are thus \(N\) linearly independent solutions for \(k\), a number corresponding to the Hilbert space dimensionality \(\left( \begin{array}{c} N \\ 1 \end{array} \right) = N\).

To each individual solution for \(k\), identified by the single quantum number \(\tilde{I}\), we thus associated wavefunction

\[ ~| k \rangle \equiv \sum_{j=1}^N \Psi_j (k) \,| j \rangle = {\cal N}(k)\, \sum_{j=1}^N e^{i k j} \, | j\rangle \]

whose normalization constant can be fixed to \(1/\sqrt{N}\) (we will return to the question of normalizing Bethe states later on).

The simple Ansatz xxz.psi1 used for the \(N\) separate solutions of xxz.be1 therefore generates all the wavefunctions, and thus a complete state basis, in this subspace.

Remark: as far as information condensation is concerned, note the important reduction that xxz.psi1 achieves: the single parameter \(k\) is all that is needed to set all the required \(N\) amplitudes \(\Psi_j (k)\) for the corresponding eigenstate.