Fracton models exhibit a variety of exotic properties and lie beyond the
conventional framework of gapped topological order. In a previous work, we
generalized the notion of gapped phase to one of foliated fracton phase by
allowing the addition of layers of gapped two-dimensional resources in the
adiabatic evolution between gapped three-dimensional models. Moreover, we
showed that the X-cube model is a fixed point of one such phase. In this paper,
according to this definition, we look for universal properties of such phases
which remain invariant throughout the entire phase. We propose multi-partite
entanglement quantities, generalizing the proposal of topological entanglement
entropy designed for conventional topological phases. We present arguments for
the universality of these quantities and show that they attain non-zero
constant value in non-trivial foliated fracton phases.
We have made the following changes in accordance with Referee #2's suggestions.
1) to improve the accessibility of the paper to a broader audience, the authors could perhaps spend a few words to explain what they mean by the double arrow "adiabatic evolution" in Eq.(1)
We have added the following sentence to explain what is meant by adiabatic evolution: "Here, adiabatic evolution refers to a smooth deformation of the Hamiltonian that preserves the energy gap, i.e. one that does not pass through an intervening gapless phase."
2) proofreading for typos: "the the" and "fracon". Missing period at the end of Eq.(5).
These typos have been fixed.
3) I think that the von Neumann entropy in Eq.(6) has the wrong sign
This error has been corrected.
4) at the end of Sec.3.1, the authors cite Ref.32 on two occasions. I wonder if earlier references may be more appropriate here (at least in addition to Ref.32). For example, in relation to non-universal contributions due to the Euler characteristic, PRL 97, 050404 (2006); and for entanglement signatures of gapped 3D topological phases, Ref.33
We have added citation of the suggested references on both occasions.