The Bethe Ansatz

The thermodynamic limit

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 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 \(x \in \mathbb{R}\),

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

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

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

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

\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

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

where we have used the convolution notation

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

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

\begin{equation} 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} \end{equation}

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24