An interesting topic in the theory of quantum phase transitions is that of the transition between a classically ordered state, such as a Néel state, and a quantum ordered state, such as a valence bond crystal. On the classically ordered side of the transition, we have a well developed understanding based on the spontaneous breaking of a continuous symmetry and the occurrence of associated Goldstone modes (spin waves). On the quantum ordered side, things are rather less clear. Even the description of the quantum ordered state itself is not straightforward, since it is not a saddle-point of the usual spin-coherent-state path integral.
In this talk, I shall present an approach designed to address this problem by constructing the path integral over matrix product states rather than spin coherent states. This allows both classically ordered states (which are matrix product states of bond dimension 1) and quantum ordered states (which are matrix product states of higher bond dimension) to be captured on an equal footing. I shall use this approach to show that, for at least one example of such a transition, a Landau-Ginzburg-Wilson-type description can be given, where the ‘order parameter’ is a field representing the nearest-neighbour entanglement of the spins.
The talk is designed to be entirely self-contained: in particular, no prior knowledge of matrix product states or quantum ordered phases is assumed. The work about which I shall speak was undertaken in collaboration with Andrew Green (University College London), Jonathan Keeling (St Andrews), and Steve Simon (Oxford), under the auspices of the TOPNES programme. Most of the details can be found at https://arxiv.org/abs/1607.01778.