Citation: Sopenko, Nikita. "An index for invertible phases of two-dimensional quantum spin systems." arXiv preprint arXiv:2410.02059 (2024).
Web: https://arxiv.org/abs/2410.02059
Tags: Defects/boundaries, Conformal-field-theory, Fermionic-order, Operator-algebras, SPT/SETs
In this paper, Nikita Sopenko demonstrates how to compute an invariant of invertible phases which is (morally) equal to the Chiral central charge modulo 24.
The main idea of this paper, in my mind, is quite beautiful. How do you find invariants to distinguish invertible phases from the trivial phase? What you can do is consider a stack of N layers of the system. This has a canonical S_N symmetry from layer permutation. It can be that the phase is a non-trivial S_N SPT! SPTs are classified by cohomology, and thus there should be an invariant in H^3(S_N, U(1)) corresponding to the phase for all N. It just so happens that the cohomology groups H^3(S_N, U(1)) stabilize for large N. In particular, H^3(S_N,U(1)) is isomorphic to Z/12 + Z/2 + Z/2 for all N>=6. Thus, we get a canonical Z/12 + Z/2 + Z/2 -valued invariant for topological phases.
Much of this paper is devoted to a proper setup of invertible phases using the operator-algebra language. Using these operator algebras there is a nice way of computing a quantity which is more manifestly related to chiral central charge, and then relating this invariant to the H^3(S_N,U(1)) value. The conclusion is that the value in Z/12 + Z/2 + Z/2 exactly determines the chiral central charge modulo 24.
The idea that the chiral central charge is related to genons is not new. It is due to Andrey Gromov:
> Gromov, Andrey. "Geometric defects in quantum Hall states." Physical Review B 94.8 (2016): 085116.
Sopenko also discusses his invariant in the context of fermionic phases, and uses it to prove that free fermion systems with non-zero Chern number v mod 48 are in a non-trivial invertible phase.