Entanglement and the density matrix renormalisation group in the generalised Landau paradigm

Abstract

The fields of entanglement theory and tensor networks have recently emerged as central tools for characterising quantum phases of matter. In this article, we determine the entanglement structure of ground states of gapped symmetric quantum lattice models, and use this to obtain the most efficient tensor network representation of those ground states. We do this by showing that degeneracies in the entanglement spectrum arise through a duality transformation of the original model to the unique dual model where the entire dual (generalised) symmetry is spontaneously broken and subsequently no degeneracies are present. Physically, this duality transformation amounts to a (twisted) gauging of the unbroken symmetry in the original ground state. This result has strong implications for the complexity of simulating many-body systems using variational tensor network methods. For every phase in the phase diagram, the dual representation of the ground state that completely breaks the symmetry minimises both the entanglement entropy and the required number of variational parameters. We demonstrate the applicability of this idea by developing a generalised density matrix renormalisation group algorithm that works on (dual) constrained Hilbert spaces, and quantify the computational gains obtained over traditional tensor network methods in a perturbed Heisenberg model. Our work testifies to the usefulness of generalised non-invertible symmetries and their formal category theoretic description for the practical simulation of strongly correlated systems.