The existence of gapped boundaries of bosonic topological orders can be tested in terms of the vanishing of higher central charges, which can be easily computed in terms of the modular data. For fermionic topological orders, even the chiral central charge admits no simple expression in terms of the modular data. Using the congruence property of representations formed by the modular data, we develop a method that tests whether the higher central charges of a fermionic topological order, including the chiral central charge, vanish. The test can be carried out entirely in terms of the modular data of the super-modular tensor category describing the fermionic topological order, and does not require explicit computation of modular extensions. This leads to a stringent set of easily computable necessary conditions for a fermionic topological order to admit a gapped boundary. We apply our test to known examples of fermionic topological orders to determine which of them potentially admit a gapped boundary.
The existence of gapped boundaries of bosonic topological orders can be tested in terms of the vanishing of higher central charges, which can be easily computed in terms of the modular data. For fermionic topological orders, even the chiral central charge admits no simple expression in terms of the modular data. Using the congruence property of representations formed by the modular data, we develop a method that tests whether the higher central charges of a fermionic topological order, including the chiral central charge, vanish. The test can be carried out entirely in terms of the modular data of the super-modular tensor category describing the fermionic topological order, and does not require explicit computation of modular extensions. This leads to a stringent set of easily computable necessary conditions for a fermionic topological order to admit a gapped boundary. We apply our test to known examples of fermionic topological orders to determine which of them potentially admit a gapped boundary.
Two-dimensional topologically ordered states such as fractional quantum Hall fluids host anyonic excitations, which are relevant for realizing fault-tolerant topological quantum computers. Classification and characterization of topological orders have been intensely pursued in both the condensed matter and mathematics literature. These topological orders can be bosonic or fermionic depending on whether the system hosts fundamental fermionic excitations or not. In particular, emergent topological orders in usual solid state systems are fermionic topological orders because the electron is a fermion. Recently, bosonic topological orders have been extensively completely classified up to rank 6 using representation theory. Inspired by their method, we provide in this paper a systematic method to classify the fermionic topological orders by explicitly building their modular data, which encodes the self and mutual statistics between anyons. Our construction of the modular data relies on the fact that the modular data of a fermionic topological order forms a projective representation of the ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\theta}}$ subgroup of the modular group SL2(Z). We carry out the classification up to rank 10 and obtain both unitary and nonunitary modular data. This includes all previously known unitary modular data, and also two new classes of modular data of rank 10. We also determine the chiral central charges (mod 21) via a novel method, which does not require the explicit computation of modular extensions.
2+1D bosonic topological orders can be characterized by the S,T matrices that encode the statistics of topological excitations. In particular, the S,T matrices can be used to systematically obtain the gapped boundaries of bosonic topological orders. Such an approach, however, does not naively apply to fermionic topological orders (FTOs). In this work, we propose a systematic approach to obtain the gapped boundaries of 2+1D abelian FTOs. The main trick is to construct a bosonic extension in which the fermionic excitation is "condensed" to form the associated FTOs. Here we choose the parent bosonic topological order to be the Z2 topological order, which indeed has a fermionic excitation. Such a construction allows us to find an explicit correspondence between abelian FTOs (described by odd K-matrix KF) and the "fermion-" condensed Z2 topological orders (described by even K-matrix KB). This provides a systematic algorithm to obtain the modular covariant boundary partition functions as well as the boundary topological excitations of abelian FTOs. For example, the ν=1−m1 Laughlin's states have exactly one type of gapped boundary when m is a square, whose boundary excitations form a Z2×Zm fusion ring. Our approach can be easily generalized to obtain gapped and gapless boundaries of non-abelian fermionic topological orders.
A (2+1)D topologically ordered phase may or may not have a gappable edge, even if its chiral central charge c_{-} is vanishing. Recently, it was discovered that a quantity regarded as a "higher" version of chiral central charge gives a further obstruction beyond c_{-} to gapping out the edge. In this Letter, we show that the higher central charges can be characterized by the expectation value of the partial rotation operator acting on the wave function of the topologically ordered state. This allows us to extract the higher central charge from a single wave function, which can be evaluated on a quantum computer. Our characterization of the higher central charge is analytically derived from the modular properties of edge conformal field theory, as well as the numerical results with the ν=1/2 bosonic Laughlin state and the non-Abelian gapped phase of the Kitaev honeycomb model, which corresponds to U(1)_{2} and Ising topological order, respectively. The Letter establishes a numerical method to obtain a set of obstructions to the gappable edge of (2+1)D bosonic topological order beyond c_{-}, which enables us to completely determine if a (2+1)D bosonic Abelian topological order has a gappable edge or not. We also point out that the expectation values of the partial rotation on a single wave function put a constraint on the low-energy spectrum of the bulk-boundary system of (2+1)D bosonic topological order, reminiscent of the Lieb-Schultz-Mattis-type theorems.
We present an efficient method to compute the modular extension of both fermionic topological orders and Z2-symmetric bosonic topological orders in two spatial dimensions, from congruence representations of SL2(Z) and its subgroups. To demonstrate the validity of our approach, we provide explicit calculations for topological orders with ranks up to 10 for the fermionic cases and up to 6 for the bosonic cases. Along the way, we clarify the relation between fermionic rational conformal field theories, which live on the boundary of the corresponding fermionic topological orders, and modular extensions. We show that the SL2(Z) representation of the Ramond-Ramond sector can be determined from the Nuveu-Schwarz-Nuveu-Schwarz sector using the modular extensions.
In this short paper, we argue that the chiral central charge c -of a (2 + 1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of antiunitary global symmetry.For example, if a (2 + 1)d fermionic topological quantum field theory (TQFT) has a time-reversal anomaly with T 2 = (-1) F labeled as ν ∈ Z 16 , the TQFT must have c -= 1/4 mod 1/2 for odd ν, while c -= 0 mod 1/2 for even ν.This generalizes the fact that the bosonic TQFT with T anomaly in a particular class must carry c -= 4 mod 8 to fermionic cases.We also study such a constraint for fermionic TQFT with U (1) × CT symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.
In Phys. Rev. Lett. 128, 176402 (2022), we argued that the chiral central charge---a topologically protected quantity characterizing the edge theory of a gapped (2+1)-dimensional system---can be extracted from the bulk by using an order parameter called the modular commutator. In this paper, we reveal general properties of the modular commutator and strengthen its relationship with the chiral central charge. First, we identify connections between the modular commutator and conditional mutual information, time reversal, and modular flow. Second, we prove, within the framework of the entanglement bootstrap program, that two topologically ordered media connected by a gapped domain wall must have the same modular commutator in their respective bulk. Third, we numerically calculate the value of the modular commutator for a bosonic lattice Laughlin state for finite sizes and extrapolate to the infinite-volume limit. The result of this extrapolation is consistent with the proposed formula up to an error of about 0.7.