The Bethe Ansatz

The Hamiltonian of the Heisenberg magnet which we will use throughout is

$J$ is the exchange coupling, with $J>0$ (resp. $J<0$) being the antiferromagnetic (resp. ferromagnetic) case. The parameter $\mathrm{\Delta }\in \mathbb{R}$ is called the anisotropy of the model. The spin-$1/2$ operators $S_j^{\alpha }$ are equipped with indices $\alpha =x,y,z$ and $j$ which labels the lattice site. We consider a closed periodic chain, so that

These operators obey canonical $su(2)$ commutation relations

where ${ϵ}^{\alpha \beta \gamma }$ is the completely antisymmetric tensor, and the Kronecker symbol ${\delta }_{jk}$ ensures commutation of operators on different sites. More convenient operators for calculations are the spin raising and lowering operators

with commutation relations

These provide an equivalent form of xxz.h,

Most of our review will concern the spin-$1/2$ chain, for which the spin operators can be represented using Pauli spin matrices,

with the standard definitions used for each site

The Pauli ladder operators are defined as

The Hilbert space ${\mathcal{H}}_j$ on each lattice site $j$ is then spanned by the two states $|\pm {⟩}_j$, chosen as eigenstates of the $S_j^z$ operator (N.B.: from now on, we take $\hslash =1$):

The full Hilbert space is obtained by tensoring all the on-site spaces, $\mathcal{H}={\oplus }_{j=1}^N{\mathcal{H}}_j$. It is spanned by the set of $2^N$ basis states $\{|{ϵ}_1,...,{ϵ}_N⟩\}$ with ${ϵ}_j=\{+,-\}\mathrm{\forall }j$. One particular member of this set will be of importance later on: the state with all spins pointing up along $\hat{z}$,

We will refer to this state as the reference state.

The $XXZ$ Hamiltonian xxz.h commutes with the $\hat{z}$-projection of the total spin operator, $S_{\mathrm{t}\mathrm{o}\mathrm{t}}^z=\sum _{j=1}^N{S}_j^z$, $$[H,S_{\mathrm{t}\mathrm{o}\mathrm{t}}^z]=0,$$ so that the Hilbert space separates into subspaces of fixed magnetization along the $\hat{z}$ axis. We label each of these subspaces ${\mathcal{H}}_M$ by the integer $M\in \{0,1,...,N\}$ representing the number of down spins, i.e. $S_{\mathrm{t}\mathrm{o}\mathrm{t}}^z=\frac{N}{2}-M$. The dimensionality of each subspace is then given by the binomial coefficient $\text{dim}({\mathcal{H}}_M)=\left(\begin{array}{c}N\\ M\end{array}\right)$, fulfilling the requirement $\sum _{M=0}^N\text{dim}({\mathcal{H}}_M)=2^N$.