Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders

Inferring macroscopic properties of physical systems from their microscopic description is an ongoing work in many disciplines of physics, like condensed matter, ultra cold atoms or quantum chromo dynamics. The most drastic changes in the macroscopic properties of a physical system occur at phase transitions, which often involve a symmetry breaking process. The theory of such phase transitions was formulated by Landau as a phenomenological model [1] and later devised from microscopic principles using the renormalization group [2, 3]. One can identify phases by knowledge of an order parameter which is zero in the disordered phase and nonzero in the ordered phase. Whereas in many known models the order parameter can be determined by symmetry considerations of the underlying Hamiltonian, there are states of matter where such a parameter can only be defined in a complicated non-local way [4]. These systems include topological states like topological insulators, quantum spin hall states [5] or quantum spin liquids [6].