The Bethe Ansatz

The Bethe equations h.bel represent a mapping between sets of quantum numbers \(\{ I_a \}\) and sets of rapidities \(\{ \lambda_a \}\). The nonlinearity of this mapping, together with its fully-coupled nature (any single rapidity is an implicit function of all the quantum numbers) mean that there is no universally robust algorithm to generate the solutions of the Bethe equations. Nevertheless, some analytical results can be obtained by going to the thermodynamic limit.

We proceed as follows. Consider a set of rapididites \(\{ \lambda_a \}\), solution to the Bethe equations for a set \(\{ I_a \}\) of quantum numbers. For simplicity, we consider \(\lambda_a \in {\mathbb R}\) for the moment. Let us first introduce a function \(\lambda(x)\) of a real parameter \(x\), having the property that it takes the value of the rapidity \(\lambda_a\) when evaluated at argument \(I_a/N\), \(\lambda(I_a/N) = \lambda_a\) for \(a = 1, ..., M\). Second, let us introduce a density function

\begin{equation*} \rho(x) = \frac{1}{N} \sum_{a=1}^M \delta (x - I_a/N) \end{equation*}

which is again fully specified by the given eigenstate. The Bethe equations h.bel can now be written

\begin{equation*} \left(\phi_1 (\lambda(x)) - \int_{-x_-}^{x_+} dy~\phi_2(\lambda(x) - \lambda(y)) \rho(y) - 2\pi x\right) |_{x = I_a/N} = 0, \end{equation*}

where \(x_+\) and \(x_-\) are real numbers.% such that \(\rho(x > x_+) = \rho(x < -x_-) = 0\).

The following step is to generalize this to the whole real axis of the \(x\) variable. That is, we pose that the value of \(\lambda(x)\) for any \(x \in \mathbb{R}\) is given by the solution to the integral equation

\begin{equation} \phi_1 (\lambda(x)) - \int_{-\lambda_-}^{\lambda_+} dy~\phi_2(\lambda(x) - \lambda(y)) \rho(y) = 2\pi x. \tag{h.bee}\label{h.bee} \end{equation}

In particular, for a given \(\rho(x)\), this defines \(\lambda(x)\) on the set of points \(x = \bar{I}/N\) defined by the complementary quantum numbers (in other words, the unoccupied quantum numbers) \(\{ \bar{I} \}]\) defined by \(\{ I \} + \{\bar{I}\} = \{ T \}\) with \(\{ T \} = \mathbb{Z} + 1/2\) if \(N - M\) is even and \(\{ T \} = \mathbb{Z}\) if \(N - M\) is odd. We call particle quantum number an occupied quantum number, and hole quantum number an unoccupied one. Densities of particles and holes can now be defined in \(x\)-space:

\begin{equation*} \rho(x) = \frac{1}{N} \sum_{n \in \{I\}} \delta (x - \frac{n}{N}), \hspace{0.5cm} \rho_h(x) = \frac{1}{N} \sum_{m \in \{\bar{I}\}} \delta (x - \frac{m}{N}), \end{equation*}

as well as the total (sum of particle and hole) density

\begin{equation*} \rho_t (x) = \rho (x) + \rho_h (x). \end{equation*}

In the thermodynamic limit \(N \rightarrow \infty\), the densities become smooth functions. In particular, the total density becomes

\begin{equation*} \rho_t (x) \rightarrow_{T.L.} \int_{-\infty}^{\infty} dy ~\delta (x - y) = 1. \end{equation*}

The continuum extension of the Bethe equations, l.bee defines \(\lambda(x)\) as a mapping from \(x\)-space to rapidity space. We suppose that this mapping is invertible, in other words that the function \(x(\lambda)\) exists and is differentiable. We can rewrite the three densities in rapidity space by using the transformation rules for delta functions (we abuse notations and use the same symbol),

\begin{align*} \rho (\lambda) &= \rho (x(\lambda)) \frac{d x(\lambda)}{d\lambda}, \\ \rho_h (\lambda) &= \rho_h (x(\lambda)) \frac{d x(\lambda)}{d\lambda}, \\ \rho_t (\lambda) &= \rho_t (x(\lambda)) \frac{d x(\lambda)}{d\lambda} = \frac{ d x(\lambda)}{d\lambda}, \end{align*}

with \(\rho (\lambda) + \rho_h (\lambda) = \rho_t (\lambda)\). This allows us to rewrite l.bee as

\begin{equation*} \phi_1 (\lambda) - \int_{-\lambda_-}^{\lambda_+} d\lambda' ~\phi_2(\lambda - \lambda') \rho(\lambda') = 2\pi x(\lambda) \end{equation*}

where \(\lambda_{\pm} = \lambda(x_{\pm})\). Differentiating this equation with respect to \(\lambda\), and defining the differential kernels \(a_n\) as

\begin{equation} a_n (\lambda) = \frac{1}{2\pi} \frac{d}{d\lambda}\phi_n(\lambda) = \frac{1}{2\pi} \frac{n}{\lambda^2 + n^2/4}. \tag{h.an}\label{h.an} \end{equation}

yields the integral equation

\begin{equation} a_1 (\lambda) - \int_{-\lambda_-}^{\lambda_+} d\lambda' ~a_2(\lambda - \lambda') \rho(\lambda') = \rho(\lambda) + \rho_h(\lambda) \tag{h.bec}\label{h.bec} \end{equation}

linking the particle and hole densities of Bethe eigenstates. In other words: this equation should be interpreted as the continuum limit of the Bethe equations for a particular eigenstate, whose distribution of rapidities is given by the density function \(\rho(\lambda)\).

The energy and momentum of a state with rapidity distribution \(\rho(\lambda)\) is readily obtained in view of the definition of the density:

\begin{align} &E = -NJ \pi \int_{-\lambda_-}^{\lambda_+} d\lambda a_1(\lambda) \rho(\lambda), \nonumber \\ &P = N \int_{-\lambda_-}^{\lambda_+} d\lambda (\pi - \phi_1(\lambda)) \rho(\lambda). \tag{h.ep}\label{h.ep} \end{align}