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Size-Driven Quantum Phase Transitions

Johannes Bausch, Toby S. Cubitt, Angelo Lucia, David Perez-Garcia, Michael M. Wolf

Proc Natl Acad Sci. 2018 Jan 2;115(1):19-23 , (2015) 1512.05687 Pfeil

Abstract: Can the properties of the thermodynamic limit of a many-body quantum system be extrapolated by analysing a sequence of finite-size cases? We present a model for which such an approach gives completely misleading results: a translationally invariant local Hamiltonian on a square lattice with constant spectral gap and classical product ground state for system sizes smaller than a threshold, and a ground state with topological degeneracy for larger system sizes. Starting from a minimal case with spins of dimension 5 and threshold lattice size 15×15, we show that the latter grows faster than any computable function with increasing local dimensions. The resulting effect may be viewed as a new type of quantum phase transition that is driven by the size of the system rather than by a local field or coupling. It also suggests that recently proven undecidability results in quantum many-body systems may affect spin lattices with relatively small local Hilbert space dimension.