Kieran Ryan

The quantum Heisenberg XXZ model and the Brauer algebra

Several quantum spin systems have probabilistic representations as
interchange processes. Some of these processes can be thought of as
continuous time random walks on the symmetric group, which in turn has
led to study using representation theory. We will in particular
discuss the spin-1/2; quantum Heisenberg XXZ model (and analogous
models in higher spins, which are not the XXZ). This model, in a
similar way to study using the symmetric group for other models, can
be studied using the Brauer algebra. We will introduce this algebra,
and its representation theory. Using this we obtain the free energy of
the system when the underlying graph is the complete graph, which
further lets us determine phase diagrams.