The Bethe Ansatz

The Hilbert space; operators; the reference state

We consider a system of \(N\) lattice sites (labeled by an index \(j\)). On each site \(j\), we install a spin-\(1/2\) degree of freedom, associated to an on-site two-dimensional Hilbert space \({\cal H}_j\). The full Hilbert space \({\cal H}^{N}\) is obtained by tensoring all the on-site spaces, \({\cal H}^{N} = \otimes_{j=1}^N {\cal H}_j\).

All operators in the full Hilbert space can be expressed in terms of fundamental spin operators \(S^{\alpha}_j\) with indices \(\alpha = x, y, z\) acting nontrivially on site \(j\) only and obeying canonical on-site \(su(2)\) commutation relations

\begin{equation*} \left[ S^{\alpha}_j, S^{\beta}_k \right] = i \hbar \delta_{jk} \epsilon^{\alpha \beta \gamma} S^{\gamma}_j \end{equation*}

where \(\epsilon^{\alpha \beta \gamma}\) is the completely antisymmetric tensor, and the Kronecker symbol \(\delta_{jk}\) ensures commutation of operators on different sites. Convenient for calculations are the spin raising and lowering operators

\begin{equation*} S^{\pm}_j = S^x_j \pm i S^y_j \end{equation*}

with commutation relations

\begin{equation*} \left[S^z_j, S^{\pm}_k \right] %= \left[ S^z_j, S^x_k \pm i S^y_k \right] = \pm \hbar \delta_{jk} S^{\pm}_j, \hspace{1cm} \left[ S^+_j, S^-_k \right] = 2\hbar \delta_{jk} S^z_j. \end{equation*}

These spin-\(1/2\) operators can be represented in matrix form using Pauli spin matrices,

\begin{equation*} S^{\alpha}_j = \frac{\hbar}{2} \sigma^{\alpha}_j, \end{equation*}

with the standard definitions used for each site

\begin{equation*} \sigma^x = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right), \hspace{1cm} \sigma^y = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right), \hspace{1cm} \sigma^z = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right). \end{equation*}

The Pauli ladder operators are defined as

\begin{equation*} \sigma^+ = \frac{\sigma^x + i \sigma^y}{2} = \left(\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right) \equiv S^+, \hspace{1cm} \sigma^- = \frac{\sigma^x - i \sigma^y}{2} = \left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right) \equiv S^-. \end{equation*}

The Hilbert space \({\cal H}_j\) on each lattice site \(j\) is then spanned by the two states \(| \pm \rangle_j\), chosen as eigenstates of the \(S^z_j\) operator (N.B.: from now on, we take \(\hbar = 1\)):

\begin{equation*} S^z_j | \pm \rangle_j = \pm \frac{1}{2} | \pm \rangle_j, \hspace{1cm} S^{\pm}_j | \mp \rangle_j = | \pm \rangle_j, \hspace{1cm} S^{\pm}_j | \pm \rangle_j = 0. \end{equation*}

The full Hilbert space \({\cal H}^{N}\) is spanned by the set of \(2^N\) basis states \(\{ | \epsilon_1, ..., \epsilon_N \rangle \}\) with \(\epsilon_j = \{ +, -\} ~\forall j\). One particular member of this set will be of importance later on: the state with all spins pointing up along \(\hat{z}\),

\begin{equation} | 0 \rangle = \otimes_{j = 1}^N | + \rangle_j. \tag{xxz.r}\label{xxz.r} \end{equation}

We will refer to this state as the reference state.

Hamiltonian

Let us define a coupling between two spin-\(1/2\) degrees of freedom as

\begin{align*} h_{j,l} &\equiv J \left[ S^x_j S^x_l + S^y_j S^y_l + \Delta \left(S^z_j S^z_l - 1/4\right) \right] \\ &= J \left[ \frac{1}{2} \left(S^+_j S^-_l + S^-_j S^+_l \right) + \Delta \left(S^z_j S^z_l - 1/4\right) \right]. \tag{xxz.hjl}\label{xxz.hjl} \end{align*}

Here, \(J\) is a real parameter representing the exchange coupling, with \(J > 0\) (resp. \(J < 0\)) being the antiferromagnetic (resp. ferromagnetic) case. The parameter \(\Delta \in {\mathbb R}\) is called the anisotropy.

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

\[ H = H_{\scriptscriptstyle bulk} + H_{\scriptscriptstyle bdry}, \tag{xxz.h}\label{xxz.h} \]

in which we have separately bulk and boundary components. The bulk term is given by coupling each neighbouring pairs of spins along the chain,

\[ H_{\scriptscriptstyle bulk} = \sum_{j=1}^{N-1} h_{j, j+1}. \tag{xxz.hbulk}\label{xxz.hbulk} \]

The boundary term depends on which boundary conditions we wish to impose on our system. The simplest case (and the one we will consider in detail first) is that of periodic boundary conditions, where we couple both ends of the chain together in precisely the same way as bulk neighbouring sites are coupled:

\[ H_{\scriptscriptstyle bdry} \rightarrow H_{\scriptscriptstyle pbc} \equiv h_{N, 1}. \tag{xxz.hpbc}\label{xxz.hpbc} \]

Magnetization sectors

The \(XXZ\) Hamiltonian xxz.h with periodic boundary conditions xxz.hpbc commutes with the \(\hat{z}\)-projection of the total spin operator, \(S^z_{\rm tot} = \sum_{j=1}^N S^z_j\), \[ \left[ H, S^z_{\rm tot} \right] = 0, \] so that the Hilbert space separates into subspaces of fixed magnetization along the \(\hat{z}\) axis.

Letting the integer \(M \in \{0, 1, ..., N \}\) represent the number of down spins, we denote each of these subspaces as \({\cal H}^N_M\). The magnetization in such a subspace is \(S^z_{\rm tot} = \frac{N}{2} - M\). The full Hilbert space can be written as the direct sum \({\cal H}^N = \oplus_{M=0}^N {\cal H}^N_M\). The dimensionality of each subspace is given by the binomial coefficient \(\mbox{dim} ({\cal H}^N_M) = \left( \begin{array}{c} N \\ M \end{array} \right)\), fulfilling the requirement \(\sum_{M = 0}^N \mbox{dim} ({\cal H}^N_M) = 2^N\).