The Bethe Ansatz

Excitations above the ground state are then obtained by either simply giving momentum to the ground state \(N\) string, or more elaborately by partitioning it into smaller strings to which individual momenta can be given. We consider here only the \(N\) atom sector relevant for the dynamical structure factor. The case of states with \(N-1\) (or less) atoms trivially follows.

We will label the string content of eigenstates by column-separated entries specifying the length and number of each different string type. For example \(N-M:M\) will be a state with a \(N-M\) string and a \(M\) string, and \(N-M_1-2M_2:M_1:(M_2)_2\) a state with an \(N - M_1 - 2M_2\) string, an \(M_1\) string and two \(M_2\) strings.