Examples of both destabilized free phases and intrinsically interacting phases are common in all dimensions.

We also determine which interacting fermionic phases cannot be realized by free fermions. The obvious explanation should be to market it and earn money. Namely we determine whether a given symmetry protected phase of free fermions is destabilized by sufficiently strong interactions or it remains stable even in the presence of interactions. eBooks Phase Ten Score Sheets: Phase 10 Dice Game, Phase 10 Score Pad, Phase Ten Dice Game, Phase Ten Game Record Keeper Book are prepared for various reasons. We also systematically study in low dimensions the relationship between the free invariants and the invariants of short-range entangled interacting phases of fermions. The classification is described as a function of the real representation theory of G ̂ and the data of the original periodic table. We extend the periodic table of phases of free fermions in the tenfold way symmetry classes to a classification of free fermionic phases protected by an arbitrary on-site unitary symmetry G ̂ in an arbitrary dimension.

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