Timeline for Why are Levin-Wen/Turaev-Viro models said to be non-chiral?
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| Aug 5, 2018 at 19:54 | comment | added | 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. | |
| Aug 5, 2018 at 19:50 | comment | added | Andi Bauer | 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. | |
| Jul 28, 2018 at 21:00 | comment | added | Carlo Beenakker | 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". | |
| Jul 28, 2018 at 18:51 | history | edited | Andi Bauer | CC BY-SA 4.0 |
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| May 21, 2018 at 19:02 | history | edited | Andi Bauer | CC BY-SA 4.0 |
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| May 21, 2018 at 18:57 | comment | added | Andi Bauer | 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. | |
| May 21, 2018 at 18:34 | comment | added | Carlo Beenakker | 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. | |
| May 21, 2018 at 18:28 | comment | added | 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 . | |
| May 21, 2018 at 18:27 | comment | added | j.c. | 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. | |
| May 21, 2018 at 16:43 | history | edited | Andi Bauer |
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| May 21, 2018 at 16:37 | history | asked | Andi Bauer | CC BY-SA 4.0 |