*Citation:* Bonderson, Parsa, and Chetan Nayak. "Quasi-topological phases of matter and topological protection." Physical Review B 87.19 (2013): 195451.

*Web:* https://arxiv.org/abs/1212.6395

*Tags:* Physical, TQFT, Modular-tensor-categories, Defects/boundaries, Majorana-fermions

This paper attempts to give a reality-check to the field of topological phases of matter. When people are pressed to define topological order, they generality say this: a topologically ordered system in which one has a degenerate ground state, determined only by the topology of the system. Moreover, there should be an energy gap between the ground state and 1st excited state which is independent of system size.

When pressed on this definition, most people would admit that this is an idealization. In a real system all of this should be true up to exponentially small error. However, even this is an idealization. Every real system will have non-topological gapless excitations. For instance, any solid will always have vibrational excitations (phonons). There will also always be photons emitted from various processes, and stray electrons coming from gates and wires. In many cases, the gapless vibrational excitations (phonons) can even be key to the mechanism underlying the topological material so the topological degrees of freedom cannot be extracted as an independent subsystem.

The goal of this paper is thus as follows, in the words of the authors: "Do actual physical systems ever satisfy the definition of a topological phase? Real solids are never fully gapped. At the very least, they will have gapless phonons and, usually, gapless photons. These gapless excitations can modify some of the predictions of TQFTs while leaving others intact. Therefore, it is important to exercise caution in applying the abstract concept of a topological phase and its mathematical formulation as a TQFT to real systems".

Of these takeaways, here is one: "Quasi-topological phases that are not 'strong' have a pseudo-ground state subspace that is not described by a TQFT. This pseudo-ground state subspace may still exhibit exponentially-suppressed splitting of degeneracies and, hence, topological-protection of qubits and topological gate operations acting within this subspace. The algebraic structure of these protected subspaces and operations within them will depend on the details of the system.".

In the end, the authors leave us with a very uplifting message: "This does not mean that one should discard TQFTs altogether. In fact some of their predictions hold (possibly with modifications) in a wide variety of systems. Therefore, it is important to distinguish between the different possible types of discrepancies between a physical system and the predictions of a TQFT".

One relevant advance that has been made since the publication of this paper is Zhenghan Wang's introduction of mixed state TQFTs, which seek to describe topological order within a larger non-topologically ordered system:

> Zini, Modjtaba Shokrian, and Zhenghan Wang. "Mixed-state TQFTs." arXiv preprint arXiv:2110.13946 (2021).