# The Bethe Ansatz

#### The thermodynamic limitg.l.tl

The Bethe equations represent a mapping between a proper set of quantum numbers $$\{I_j\}$$ and a set of momenta (rapidities) $$\{\lambda_j\}$$. This map is nonlinear and fully-coupled: changing one quantum number changes all the rapidities. For nontrivial values of the interaction (namely: $$c$$ neither $$0$$ nor $$\infty$$), the Bethe equations form a transcendental system of equations which cannot be solved in closed form. As is often the case in statistical mechanics, one can however hope that going to a thermodynamically large system allows to simplify matters somewhat. This chapter provides details of how to implement such a limit.

For large $$L$$ and finite quantum number difference, the rapidities $$\lambda_j$$ are always separated by intervals of order $$1/L$$, as can be seen from inequality l.bd. In the thermodynamic limit, defined as the limit $$L \rightarrow \infty$$, $$N \rightarrow \infty$$, with $$N/L$$ fixed and finite, they become dense on (subsets of) the real line.

Let us begin by considering a given proper set of quantum numbers $$I_j$$. Let us then introduce a function $$\lambda(x)$$ for an argument $$x$$ on the real line $$\mathbb{R}$$, which we interpret as a continuum version of the space of quantum numbers. On the discrete set of points $$\left\{ x_j = \frac{I_j}{L} \right\}$$, we fix the values of $$\lambda(x)$$ to $$\lambda(x_j) = \lambda_j$$, the $$\{ \lambda_j \}$$ being the solution to the Bethe equations for a given specific proper set $$\{ I_j\}$$. The Bethe equations can then be rewritten as

\begin{equation*} \lambda(x_j) + \frac{1}{L} \sum_{l=1}^N \phi(\lambda(x_j) - \lambda(x_l)) = 2\pi x_j. \end{equation*}

The value of the function $$\lambda(x)$$ remains arbitrary outside of the discrete set of points $$\{ x_j\}$$. A natural extension to the whole of $$\mathbb{R}$$ can be obtained by requiring that $$\lambda(x)$$ fulfill the following equation for any $$x \in \mathbb{R}$$,

$$\lambda(x) + \int_{-\infty}^{\infty} dy ~\phi (\lambda(x) - \lambda(y)) \rho(y) = 2\pi x \tag{l.bee}\label{l.bee}$$

where we have defined the density distribution

\begin{equation*} \rho (x) = \frac{1}{L} \sum_{j=1}^N \delta (x - \frac{I_j}{L}) \end{equation*}

which is exactly determined by the chosen $$\{ I_j \}$$. The function $$\lambda(x)$$ defined by l.bee is a single-valued, monotonically increasing function of $$x$$ associated to a given $$\rho(x)$$, and defines a one-to-one mapping $$\mathbb{R} \to \mathbb{R}$$.

In particular, l.bee defines the function $$\lambda(x)$$ for points $$x = \tilde{I}/L$$ with $$\tilde{I} \notin \{I_j \}$$, and $$\tilde{I}$$ half-odd integer if $$N$$ is even and integers if $$N$$ is odd (in other words on the unoccupied quantum numbers of the allowed set). The sets $$\{I \}$$ and $$\{ \tilde{I} \}$$ are complementary, in the sense that $$\{ I \} + \{\tilde{I}\} = \{ T \}$$ with $$\{ T \} = {\mathbb Z} + 1/2$$ if $$N$$ is even and $$\{ T \} = {\mathbb Z}$$ if $$N$$ is odd. We call particle rapidity a rapidity $$\lambda(\frac{n}{L})$$ with $$n \in \{I \}$$, and a hole rapidity one with $$n \in \{ \tilde{I} \}$$. Particle, hole and total densities are then defined as

\begin{align} &\rho(x) = \frac{1}{L} \sum_{n \in \{I\}} \delta (x - n/L), \hspace{0.5cm} &\rho_h(x) = \frac{1}{L} \sum_{m \in \{\tilde{I}\}} \delta (x - m/L), \nonumber \\ &\rho_t(x) = \frac{1}{L} \sum_{t \in \{T\}} \delta (x - t/L), \hspace{5mm} &\rho (x) + \rho_h (x) = \rho_t (x). \tag{l.rx}\label{l.rx} \end{align}

In the thermodynamic limit, these densities become smooth functions of $$x$$. In particular,

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

The continuum version of the Bethe equations l.bex define a continuous differentiable mapping between the $$x$$- and $$\lambda$$-spaces. This allows us to rewrite the densities as functions in $$\lambda$$-space by simply using the transformation rule for $$\delta$$ functions (we abuse the notation and use the same symbol, the argument determining which density we are using)

\begin{align} &\rho (\lambda) = \rho (x(\lambda)) \frac{d x(\lambda)}{d\lambda}, \hspace{0.5cm} &\rho_h (\lambda) = \rho_h (x(\lambda)) \frac{d x(\lambda)}{d\lambda}, \nonumber \\ &\rho_t (\lambda) = \rho_t (x(\lambda)) \frac{d x(\lambda)}{d\lambda} = \frac{ d x(\lambda)}{d\lambda}, \hspace{5mm} &\rho (\lambda) + \rho_h (\lambda) = \rho_t (\lambda). \tag{l.r}\label{l.r} \end{align}

We thus rewrite the Bethe equations as

$$\lambda + \int_{-\infty}^{\infty} d\lambda' \phi (\lambda - \lambda') \rho (\lambda') = 2\pi x(\lambda). \tag{l.bex}\label{l.bex}$$

Differentiating with respect to $$\lambda$$ gives (using the definition of the fundamental Cauchy kernel l.ck)

\begin{equation*} 1 + 2\pi \int_{-\infty}^{\infty} d\lambda' ~{\cal C} (\lambda - \lambda') \rho (\lambda') = 2\pi (\rho (\lambda) + \rho_h (\lambda)), \end{equation*}

or more succinctly

$$\rho(\lambda) + \rho_h(\lambda) = \frac{1}{2\pi} + {\cal C} * \rho (\lambda) \tag{l.bec}\label{l.bec}$$

where we have used the convolution notation

$$f * g (\lambda) \equiv \int_{-\infty}^\infty d\lambda' f(\lambda-\lambda') g(\lambda'). \tag{convo}\label{convo}$$

Equation l.bec is a functional relation between two functions. It can thus be interpreted as follows: given any function $$\rho_h (\lambda)$$, it determines a function $$\rho(\lambda)$$ and thus an eigenstate. Alternately, given a $$\rho(\lambda)$$, a $$\rho_h(\lambda)$$ can be calculated. Given the distribution $$\rho (\lambda)$$, the particle, momentum and energy (linear) densities of the state are

$$n = \frac{N}{L} = \int_{-\infty}^\infty d\lambda \rho(\lambda), \hspace{5mm} p = \frac{P}{L} = \int_{-\infty}^\infty d\lambda \lambda \rho (\lambda), \hspace{5mm} e = \frac{E}{L} = \int_{-\infty}^\infty d\lambda \lambda^2 \rho(\lambda). \tag{l.npe}\label{l.npe}$$