Spontaneous symmetry breaking in projected entangled pair state models

In this thesis, we consider projected entangled pair state (PEPS) models as a framework for two-dimensional strongly-correlated quantum many body systems, where the global properties of the system are concisely encoded in one local tensor. While it was well known how to construct PEPS models with desired symmetries by manifestly encoding the symmetry in the local tensor, our goal in this work is to investigate whether this manifestly encoded symmetry can be spontaneously broken. By defining long-range order as a good criterion for symmetry breaking in a finite volume PEPS, we answer this question in the affirmative. A particularly attractive feature of PEPS models is that they have an exact holographic mapping, which is to say that the large system behavior is described by the ''PEPS boundary'', given by the fixed point(s) of the so called transfer operator. We investigate the implications of long-range order and prove that it leads to a particular symmetry breaking pattern in the fixed point space. We find this pattern by first proving that long-range order implies an asymptotic degeneracy in the transfer operator and then identifying the symmetry breaking mechanism as the one which, from all possible boundary (-matrix) configurations, selects the configurations that correspond to positive-semi-definite matrices, i.e. one-dimensional virtual quantum states. We prove that these correspond to boundary configurations that do not mix under arbitrary perturbations and are thus stable. We study the entanglement properties of these virtual quantum states and show that they can be described by local entanglement Hamiltonians, establishing that also in phases with symmetry breaking, gapped quantum phases have an associated local entanglement Hamiltonian