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,αβ}_{kln,χδ},n_i,d_i), which satisfy a set of nonlinear algebraic equations parameterized by phase factors Ξ^{ijm,αβ}_{kl}, Ξ^{ij}_{kln,χδ}, Ω^{kim,αβ}_{jl} and Ω^{ki}_{jln,χδ}. 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.

We analyze spectrum of Laplacian supported by a periodic honeycomb lattice with generally unequal edge lengths and a <i></i> type coupling in the vertices. Such a quantum graph has nonempty point spectrum with compactly supported eigenfunctions provided all the edge lengths are commensurate. We derive conditions determining the continuous spectral component and show that existence of gaps may depend on number-theoretic properties of edge lengths ratios. The case when two of the three lengths coincide is discussed in detail.

The classification and lattice model construction of symmetry-protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed-point wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group G_f = \mathbb{Z}^f_2 ×_{ω_2} G_b which is a central extension of bosonic symmetry group Gb (may contain time-reversal symmetry) by the fermion parity symmetry group \mathbb{Z}^f_2 = {1,P_f}. Our construction is based on the concept of an equivalence class of finite-depth fermionic symmetric local unitary transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We also discuss the systematical construction and classification of boundary anomalous SPT states which leads to a trivialization of the corresponding bulk FSPT states. Thus, we conjecture that the obstruction-free and trivialization-free constructions naturally lead to a classification of FSPT phases. Each fixed-point wave function admits an exactly solvable commuting-projector Hamiltonian. We believe that our classification scheme can be generalized to point and space group symmetry as well as continuum Lie group symmetry.

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We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the gap has a lower bound independent of the dimension. We also estimate the gap when the potential is not necessarily convex.