The Bethe Ansatz

Equipped with conjugacy relations for monodromy matrix entries as well as with Gaudin's norm formula, we are now in position to make conjugacy relations for physical eigenstates explicit.

In view of slight regime-specific details worth underlining, we adapt the formulas to each regime, using the conventions a.xxzreg.

Axial antiferromagnet \(\Delta > 1\)

Factoring out \(-\sinh \eta\) in gmx, we adopt the convention (the dot meaning "adapted to this regime")

\begin{equation} \dot{{\cal G}}_{ab} = \left\{ \begin{array}{ll} \frac{N}{\sin^2 \lambda_a + \sinh^2 \frac{\eta}{2}} - \sum_{c \neq a} \frac{2\cosh \eta}{\sin^2 \lambda_{ac} + \sinh^2 \eta}, \hspace{5mm} & a=b,\\ \frac{2 \cosh \eta}{\sin^2 \lambda_{ab} + \sinh^2 \eta} & a \neq b \end{array}\right. \tag{xxz.aa.gm}\label{xxz.aa.gm} \end{equation}

We therefore have

\[ (-1)^M \langle 0 | C(\bar{\lambda}) B(\bar{\lambda}) | 0 \rangle = (\sinh \eta)^{2M} \,\prod_{1 \leq a < b \leq M} \left(1 + \frac{\sinh^2 \eta}{\sin^2 \lambda_{ab}} \right) ~\text{det} ~\dot{\cal G} (\{ \lambda \}) \]

whose right-hand side is a positive quantity for any eigenstate.

In this regime, eigenstates are characterized by self-conjugate sets of rapidities \( \{ \lambda^* \} = \{ \lambda \}\). From a.xxzcl we thus have (using the fact that \( d(\bar{\lambda}) = 1 \) for an eigenstate)

\[ \left[ B(\bar{\lambda}) \right]^\dagger = B^\dagger (\bar{\lambda^*}) = \frac{(-1)^M}{d(\bar{\lambda^*})} C (\bar{\lambda^*}) = (-1)^M C(\bar{\lambda}). \]

Our orthonormalized physical eigenstates and their conjugates are thus defined as

\begin{align*} ~| \Psi (\{ \lambda \}) \rangle &\equiv \frac{1}{\sqrt{\cal N}} \,| \{ \lambda \} \rangle = \frac{1}{\sqrt{\cal N}} \,B(\bar{\lambda}) | 0 \rangle \\ \langle \Psi (\{ \lambda \}) | &\equiv \frac{(-1)^M}{\sqrt{\cal N}} \,\langle \{ \lambda \} | = \frac{(-1)^M}{\sqrt{\cal N}} \,\langle 0 | C(\bar{\lambda}) \\ {\cal N} (\{ \lambda \}) &\equiv (\sinh \eta)^{2M} \prod_{1 \leq a < b \leq M} \left(1 + \frac{\sinh^2 \eta}{\sin^2 \lambda_{ab}} \right) ~\text{det} ~\dot{\cal G} (\{ \lambda \}) \end{align*}

Planar antiferromagnet \(0 < \Delta < 1\)

This regime is characterized by the presence of strings of positive and negative parity. The set \( \{ \lambda \}_M \) of cardinality \( M \) thus splits into fixed-parity sets,

\[ \{ \lambda \}_M = \{ \lambda^{\scriptscriptstyle (+)} \}_{M^{\scriptscriptstyle (+)}} + \{ \lambda^{\scriptscriptstyle (-)} \}_{M^{\scriptscriptstyle (-)}} \]

For negative-parity 1-strings, \( \sinh^2 \lambda_a \) takes negative values (higher negative-parity strings, composed of rapidities which are conjugated pairwise about the \(i\pi/2\) line, can be onboarded in the following reasoning). In order to obtain a manifestly positive determinant, we thus include a factor of \( \nu_a \) (this being the parity of rapidity \( \lambda_a \) into the definition of the Gaudin matrix:

\begin{equation} \dot{{\cal G}}_{ab} = \nu_a \left\{ \begin{array}{ll} \frac{N}{\sinh^2 \lambda_a + \sin^2 \frac{\zeta}{2}} - \sum_{c \neq a} \frac{2\cos \zeta}{\sinh^2 \lambda_{ac} + \sin^2 \zeta}, \hspace{5mm} & a=b,\\ \frac{2 \cos \zeta}{\sinh^2 \lambda_{ab} + \sin^2 \zeta} & a \neq b \end{array}\right. \tag{xxz.pa.gm}\label{xxz.pa.gm} \end{equation}

so \[ \text{det}\,{\cal G} = (-1)^{M^{\scriptscriptstyle (-)}} (\sin \zeta)^M \,\text{det}\,{\dot{\cal G}}, \hspace{5mm} \text{det}\,{\dot{\cal G}} > 0. \]

We therefore have

\[ (-1)^{M^{\scriptscriptstyle (+)}} \langle 0 | C(\bar{\lambda}) B(\bar{\lambda}) | 0 \rangle = (\sin \zeta)^{2M} \,\prod_{1 \leq a < b \leq M} \left(1 + \frac{\sin^2 \zeta}{\sinh^2 \lambda_{ab}} \right) ~\text{det} ~\dot{\cal G} (\{ \lambda \}) \]

whose right-hand side is a positive quantity for any eigenstate.

Let us now look at the conjugacy of monodromy matrix entries. Under conjugacy, and remembering that the rapidities are defined modulo \( i\pi \), we have that

\begin{align*} &\{ (\lambda^{\scriptscriptstyle (+)})^* \} = \{ \lambda^{\scriptscriptstyle (+)} \}, \\ &\{ (\lambda^{\scriptscriptstyle (-)})^* \} = \{ \lambda^{\scriptscriptstyle (-)} - i\pi \} \end{align*}

Using the conjugacy and periodicity of the monodromy matrix entries, we thus get

\begin{align*} &\left[ B(\lambda^{\scriptscriptstyle (+)}) \right]^\dagger = B^\dagger ((\lambda^{\scriptscriptstyle (+)})^*) = \frac{-C ((\lambda^{\scriptscriptstyle (+)})^*)}{d((\lambda^{\scriptscriptstyle (+)})^*)} = \frac{-C (\lambda^{\scriptscriptstyle (+)})}{d(\lambda^{\scriptscriptstyle (+)})}, \\ &\left[ B(\lambda^{\scriptscriptstyle (-)}) \right]^\dagger = B^\dagger ((\lambda^{\scriptscriptstyle (-)})^*) = \frac{-C ((\lambda^{\scriptscriptstyle (-)})^*)}{d((\lambda^{\scriptscriptstyle (-)})^*)} = \frac{-C (\lambda^{\scriptscriptstyle (-)} -i\pi)}{d((\lambda^{\scriptscriptstyle (-)})^*)} = \frac{+C (\lambda^{\scriptscriptstyle (-)})}{d((\lambda^{\scriptscriptstyle (-)})^*)}. \end{align*}

For a physical eigenstate, we thus get

\[ \left[ B(\bar{\lambda}) \right]^\dagger = \left[ B(\bar{\lambda^{\scriptscriptstyle (+)}}) B(\bar{\lambda^{\scriptscriptstyle (-)}}) \right]^\dagger = \frac{(-1)^{M^{\scriptscriptstyle (+)}} C(\bar{\lambda})}{d(\bar{\lambda^*})} = (-1)^{M^{\scriptscriptstyle (+)}} C(\bar{\lambda}) \]

in which we have used the fact that \( d(\bar{\lambda}) = 1 \) for an eigenstate.

Putting things together, the orthonormalized physical eigenstates and their conjugates are thus defined as

\begin{align*} ~| \Psi (\{ \lambda \}) \rangle &\equiv \frac{1}{\sqrt{\cal N}} \,| \{ \lambda \} \rangle = \frac{1}{\sqrt{\cal N}} \,B(\bar{\lambda}) | 0 \rangle \\ \langle \Psi (\{ \lambda \}) | &\equiv \frac{(-1)^{M^{\scriptscriptstyle (+)}}}{\sqrt{\cal N}} \,\langle \{ \lambda \} | = \frac{(-1)^{M^{\scriptscriptstyle (+)}}}{\sqrt{\cal N}} \,\langle 0 | C(\bar{\lambda}) \\ {\cal N} (\{ \lambda \}) &\equiv (\sin \zeta)^{2M} \prod_{1 \leq a < b \leq M} \left(1 + \frac{\sin^2 \zeta}{\sinh^2 \lambda_{ab}} \right) ~\text{det} ~\dot{\cal G} (\{ \lambda \}) \end{align*}

Planar ferromagnet \(-1 < \Delta < 0\)

While we could adapt equations to this regime as well, in practice, it is easier to make use the of duality relation to obtain the eigenstates from those of the the planar antiferromagnetic regime.

Axial ferromagnet \(\Delta < -1\)

In this regime as well, we use the duality relations to obtain the states from those of the axial antiferromagnetic regime.