Size-driven quantum phase transitions

Abstract

Significance In this work we construct simple examples of 2D quantum spin-lattice models with small local state spaces which exhibit very unusual finite-size effects that we term “size-driven phase transitions”: For all system sizes smaller than some threshold N, the low-energy physics are classical; for all sizes larger than N, the system exhibits topological quantum order. Most quantum many-body models are too complex to be solved analytically; our theoretical understanding comes from extrapolating numerical simulations of finite systems to the thermodynamic limit. However, this approach cannot work for the size-driven effects revealed here. We give explicit examples where the threshold size is beyond the capability of any foreseeable numerics.

Abstract Can the properties of the thermodynamic limit of a many-body quantum system be extrapolated by analyzing a sequence of finite-size cases? We present models for which such an approach gives completely misleading results: translationally invariant, local Hamiltonians on a square lattice with open boundary conditions and constant spectral gap, which have a classical product ground state for all system sizes smaller than a particular threshold size, but a ground state with topological degeneracy for all system sizes larger than this threshold. Starting from a minimal case with spins of dimension 6 and threshold lattice size 15*15, we show that the latter grows faster than any computable function with increasing local spin dimension. The resulting effect may be viewed as a unique type of quantum phase transition that is driven by the size of the system rather than by an external field or coupling strength. We prove that the construction is thermally robust, showing that these effects are in principle accessible to experimental observation.