Anomalous bulk behaviour in the free parafermion Z(N)Z(N) spin chain

We demonstrate using direct numerical diagonalization and extrapolation methods that boundary conditions have a profound effect on the bulk properties of a simple $Z(N)$ model for $N \ge 3$ for which the model hamiltonian is non-hermitian. For $N=2$ the model reduces to the well known quantum Ising model in a transverse field. For open boundary conditions the $Z(N)$ model is known to be solved exactly in terms of free parafermions. Once the ends of the open chain are connected by considering the model on a ring, the bulk properties, including the ground-state energy per site, are seen to differ dramatically with increasing $N$. Other properties, such as the leading finite-size corrections to the ground-state energy, the mass gap exponent and the specific heat exponent, are also seen to be dependent on the boundary conditions. We speculate that this anomalous bulk behaviour is a topological effect.Comment: 8 pages, 8 figures, minor change

The Bethe ansatz after 75 years

Hans Bethe introduced his now-famous ansatz to obtain the energy eigenstates of the one-dimensional version of Werner Heisenberg’s model of interacting, localized spins in a solid. Although it is among Bethe’s most cited works and has a wide range of applications, it is rarely included in the graduate physics curriculum except at the advanced level. The 75th anniversary of the Bethe ansatz is appropriately marked by reflecting on the impact of Bethe’s result on modern physics, ranging from its profound influence on the field of exactly solved models in statistical mechanics to insights into the subtle nature of quantum many-body effects observed in cold quantum gases

A coupled Temperley-Lieb algebra for the superintegrable chiral Potts chain

The hamiltonian of the $N$-state superintegrable chiral Potts (SICP) model is written in terms of a coupled algebra defined by $N-1$ types of Temperley-Lieb generators. This generalises a previous result for $N=3$ obtained by J. F. Fjelstad and T. M\r{a}nsson [J. Phys. A {\bf 45} (2012) 155208]. A pictorial representation of a related coupled algebra is given for the $N=3$ case which involves a generalisation of the pictorial presentation of the Temperley-Lieb algebra to include a pole around which loops can become entangled. For the two known representations of this algebra, the $N=3$ SICP chain and the staggered spin-1/2 XX chain, closed (contractible) loops have weight $\sqrt{3}$ and weight $2$, respectively. For both representations closed (non-contractible) loops around the pole have weight zero. The pictorial representation provides a graphical interpretation of the algebraic relations. A key ingredient in the resolution of diagrams is a crossing relation for loops encircling a pole which involves the parameter $\rho= e^{ 2\pi \mathrm{i}/3}$ for the SICP chain and $\rho=1$ for the staggered XX chain. These $\rho$ values are derived assuming the Kauffman bracket skein relation.Comment: 10 pages, 4 figures, further cubic relations adde