I will discuss the interplay between gapped phases and dualities of symmetric quantum lattice models and demonstrate that every phase is efficiently characterised by the maximal breaking of the dual (genereralised) symmetry: for every symmetric Hamiltonian, there is a unique equivalent dual Hamiltonian whose ground states break all dual symmetries. Besides the fundamental importance of this result for setting up a generalised Landau paradigm, it has strong implications for the construction of state of the art tensor network algorithms for simulating quantum lattice Hamiltonians. (work with L. Lootens and C. Delcamp)
