Towards a complete classification of non-chiral topological phases in 2D
fermion systems
- URL: http://arxiv.org/abs/2112.06124v2
- Date: Thu, 28 Dec 2023 04:24:55 GMT
- Title: Towards a complete classification of non-chiral topological phases in 2D
fermion systems
- Authors: Jing-Ren Zhou and Qing-Rui Wang and Zheng-Cheng Gu
- Abstract summary: We argue that all non-chiral fermionic topological phases in 2+1D are characterized by a set of tensors $(Nij_k,Fij_k,Fijm,alphabeta_kln,chidelta,n_i,d_i)$.
Several examples with q-type anyon excitations are discussed, including the Fermionic topological phase from Tambara-gami category for $mathbbZ_2N$.
- Score: 33.49184078479579
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: In recent years, fermionic topological phases of quantum matter has attracted
a lot of attention. In a pioneer work by Gu, Wang and Wen, the concept of
equivalence classes of fermionic local unitary(FLU) transformations was
proposed to systematically understand non-chiral topological phases in 2D
fermion systems and an incomplete classification was obtained. On the other
hand, the physical picture of fermion condensation and its corresponding super
pivotal categories give rise to a generic mathematical framework to describe
fermionic topological phases of quantum matter. In particular, it has been
pointed out that in certain fermionic topological phases, there exists the
so-called q-type anyon excitations, which have no analogues in bosonic
theories. In this paper, we generalize the Gu, Wang and Wen construction to
include those fermionic topological phases with q-type anyon excitations. We
argue that all non-chiral fermionic topological phases in 2+1D are
characterized by a set of tensors
$(N^{ij}_{k},F^{ij}_{k},F^{ijm,\alpha\beta}_{kln,\chi\delta},n_{i},d_{i})$,
which satisfy a set of nonlinear algebraic equations parameterized by phase
factors $\Xi^{ijm,\alpha\beta}_{kl}$, $\Xi^{ij}_{kln,\chi\delta}$,
$\Omega^{kim,\alpha\beta}_{jl}$ and $\Omega^{ki}_{jln,\chi\delta}$. Moreover,
consistency conditions among algebraic equations give rise to additional
constraints on these phase factors which allow us to construct a topological
invariant partition for an arbitrary triangulation of 3D spin manifold.
Finally, several examples with q-type anyon excitations are discussed,
including the Fermionic topological phase from Tambara-Yamagami category for
$\mathbb{Z}_{2N}$, which can be regarded as the $\mathbb{Z}_{2N}$ parafermion
generalization of Ising fermionic topological phase.
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