Introduction to the Bethe ansatz I

Abstract

The Bethe ansatz for the one-dimensional s=1/2 Heisenberg ferromagnet is introduced at an elementary level. The presentation follows Bethe's original work very closely. A detailed description and a complete classification of all two-magnon scattering states and two-magnon bound states are given for finite and infinite chains. The paper is designed as a tutorial for beginning graduate students. It includes 10 problems for further study.

Figures (4)

• Figure 1: The allowed pairs of Bethe quantum numbers $(\lambda_1,\lambda_2)$ that characterize the $N(N-1)/2$ eigenstates in the $r=2$ subspace for $N=32$. The states of class $C_1,C_2$, and $C_3$ are colored red, white, and blue, respectively.
• Figure 2: Excitation energy $(E-E_0)/J$ versus wave number $k$ of all $N(N-1)/2$ eigenstates in the invariant subspace with $r=2$ overturned spins for a system with $N=32$. States of class $C_1$ are denoted by red circles, states of class $C_2$ by open black circles, and states of class $C_3$ by blue squares if $\lambda_2=\lambda_1$, or blue diamonds if $\lambda_2=\lambda_1+1$.
• Figure 3: Excitation energy $(E-E_0)/J$ versus wave number $k$ of all two-magnon scattering states (classes $C_1$ and $C_2$ from Fig. 2) for a system with $N=32$ in comparison with the noninteracting magnon pairs (+).
• Figure 4: Weight distribution $|a(n_1,n_2)|$ versus distance $n_2-n_1$ of the two down spins of class $C_3$ states at $k=(2\pi/N)n,\;n=4,8,\ldots,N/2$ for $N=128$.