# Why are Levin-Wen/Turaev-Viro models said to be non-chiral?

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I'd like to bring together the following two notions of "non-chiral":

On the abstract algebraic side, a modular fusion category describing the anyon content of some physical system is said to be non-chiral if it is the Drinfeld center of some fusion category, and thus has a vanishing chiral central charge. This is clearly fulfilled by the Levin-Wen string-net models aka TVBW state-sum.

On the other hand, in condensed matter physics, "non-chiral" refers to the presence of a "chiral symmetry", so a (anti-)unitary representation of the group $Z_2$ on the many-body Hilbert space that commutes with the Hamiltonian. Clearly, for any Hamiltonian one can find thousands of such representations, so in order to make "presence of a chiral symmetry" something meaningful, there have to be some conditions on the representation that make it a valid "chiral" symmetry.

Now my question: What kinds of representations of $Z_2$ are called a valid "chiral symmetry" and how is such a chiral symmetry implemented in each Levin-Wen model?

[Edit: I know that "chiral symmetry" is well-defined in the context of free-fermion systems. However in the Levin-Wen models there is not even a Fock space, we just have local (super-)Hilbert spaces distributed over a 2D lattice. I have no idea how to generalize the free fermion definition to this setting. This is the key point of the question I think.]

Edit: Meanwhile it came to my mind that "chiral" is a very generic word and the two notions of chiral might simply be different. My current guess of what a chiral symmetry could be on the many-body level is "complex conjugation in some local basis", and it seems like there are Turaev-Viro models that do not have such a symmetry.

This post imported from StackExchange MathOverflow at 2018-09-23 14:26 (UTC), posted by SE-user Andi Bauer
retagged Sep 23, 2018
I don't know anything about the Levin-Wen model nor anything about fusion categories but in condensed matter physics, when people discuss chiral symmetry (as well as particle-hole and time-reversal symmetries) of a Hamiltonian, the symmetry is assumed to be "local", so that if you write your Hamiltonian in a position representation, the matrix elements of the symmetry operator that relate components of the wavefunction at one point in space to those someplace else are required to vanish.

This post imported from StackExchange MathOverflow at 2018-09-23 14:27 (UTC), posted by SE-user j.c.
I'm having a hard time finding places where this is spelled out explicitly but see e.g. the start of section 3.3 of Freed's "Short-range entanglement and invertible field theories" arxiv.org/abs/1406.7278 .

This post imported from StackExchange MathOverflow at 2018-09-23 14:27 (UTC), posted by SE-user j.c.
I am a bit confused by the reference to a chiral symmetry as "commuting" with the Hamiltonian; that would allow one to block-diagonalize the Hamiltonian and the symmetry within each block would become trivial; instead, a chiral symmety is defined as a unitary operator that anticommutes with the Hamiltonian; then it cannot be removed by block-diagonalization; also note that the presence of both time-reversal and particle-hole symmetry implies a chiral symmetry, but not the other way around.

This post imported from StackExchange MathOverflow at 2018-09-23 14:27 (UTC), posted by SE-user Carlo Beenakker
Symmetries may anti-commute with a "Hamiltonian" of a free-fermion system on the single-particle level, but they always commute with the Hamiltonian on the many-body level. In the Levin-Wen models however, there is not even a notion of a single-particle Hilbert space.

This post imported from StackExchange MathOverflow at 2018-09-23 14:27 (UTC), posted by SE-user Andi Bauer
in my understanding the following statements are equivalent: 1) non-chiral topological phase; 2) chiral central charge = 0; 3) thermal quantum Hall conductance = 0; 4) presence of time-reversal symmetry ––– the string-net models satisfy all 4 criteria; a fifth statement "presence of chiral symmetry" means something else (a system may very well have chiral central charge $\neq 0$ in the presence of chiral symmetry, for example, graphene in a magnetic field); hence I do not subscribe to the statement "in condensed matter physics, non-chiral refers to the presence of a chiral symmetry".

This post imported from StackExchange MathOverflow at 2018-09-23 14:27 (UTC), posted by SE-user Carlo Beenakker
Ok that is good to know! Are you sure about all string-net models having a time-reversal symmetry? Do you have a reference where such a symmetry operation is explicitly spelled out for arbitrary Levin-Wen models? I always had the impression that only those that have F-symbols with tetrahedral symmetry (i.e. the ones in the original Levin-Wen paper) have a time-reversal symmetry, but not in general the more general ones (e.g. the twisted quantum doubles). E.g. title and abstract of arxiv.org/abs/1605.07194 sound to me like that is what they claim in this paper.

This post imported from StackExchange MathOverflow at 2018-09-23 14:27 (UTC), posted by SE-user Andi Bauer
On the other hand those string-net models without tetrahedral symmetry still have a Drinfeld center describing their quasiparticles and so have 0 central charge, and still have a gapped boundary. So they are non-chiral according to your points 1-3.

This post imported from StackExchange MathOverflow at 2018-09-23 14:27 (UTC), posted by SE-user Andi Bauer

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