The Bethe Ansatz

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 $\mathrm{\infty }$), 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 $L\to \mathrm{\infty }$, $N\to \mathrm{\infty }$, 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 $\{x_j=\frac{I_j}{L}\}$, 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

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 $x\in \mathbb{R}$,

where we have defined the density distribution

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

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

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)

We thus rewrite the Bethe equations as

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

or more succinctly

where we have used the convolution notation

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