Bethe ansatz

Definition

Let $N \ge 2$ and $J \in \R$. Consider the spin-$1/2$ Heisenberg XXX chain on $N$ sites with periodic boundary conditions. It is the operator on $(\C^2)^{\otimes N}$ given by $$H = J \sum_{j=1}^{N} \left( \mathbf{S}_j \cdot \mathbf{S}_{j+1} - \tfrac{1}{4} \right), \qquad \mathbf{S}_{N+1} = \mathbf{S}_1,$$ where $\mathbf{S}_j$ is one half of the vector of Pauli matrices acting on site $j$. For $0 \le M \le N/2$, let $| x_1, \dots, x_M \rangle$ with $1 \le x_1 < \dots < x_M \le N$ be the basis vector with spins down at the sites $x_i$ and up elsewhere. Let $S_M$ be the symmetric group, let $k_1, \dots, k_M$ be complex numbers and let $A_P$, for $P \in S_M$, be complex amplitudes. The coordinate Bethe ansatz is the trial vector $\sum \psi(x_1, \dots, x_M) | x_1, \dots, x_M \rangle$ with $$\psi(x_1, \dots, x_M) = \sum_{P \in S_M} A_P \exp\left( i \sum_{j=1}^{M} k_{P(j)} x_j \right).$$ The ratios of the amplitudes are fixed by the two-body scattering of the momenta. Write $e^{i k_j} = (\lambda_j + i/2)/(\lambda_j - i/2)$. Suppose that the $\lambda_j$ are distinct and satisfy the Bethe equations $$\left( \frac{\lambda_j + i/2}{\lambda_j - i/2} \right)^{N} = \prod_{l \ne j} \frac{\lambda_j - \lambda_l + i}{\lambda_j - \lambda_l - i}, \qquad j = 1, \dots, M.$$ Then the vector, when it is nonzero, is an eigenvector of $H$. Its eigenvalue is then $E = -\frac{J}{2} \sum_{j=1}^{M} \frac{1}{\lambda_j^2 + 1/4}$.