7.3.1
Lagrangian subgroup formalism and fermion condensation
In this section, we consider the topologically distinct gapped edges of a bosonic abelian topological phase interfaced with a fermionic vacuum. Luckily most of the analysis follows from a small modification of the arguments given in [89], however we will spell out some of the details and give a physical picture behind the processes leading to a fermionic gapped edge.
We can model this set-up as follows: let KT O be the K-matrix corresponding to a bosonic abelian topological state living in the left half-plane, and let Kt describe K-matrix of the topologically trivial fermionic gapped system system living in the right half-plane. This condition requires KT O to have only even numbers along the diagonal and vanishing sign. The fermionic vacuum can be easily characterized by the following K-matrix; Kt=
1 0
0 −1
, since |Kt| = 0, the sign vanishes and the diagonal contains odd numbers (thus there are local fermionic degrees of freedom).
Although this combined system of fermionic and bosonic locality seems strange, we can directly map into a problem we know how to deal with. We can attempt to use the folding trick, which instructs us to view the bilayer system KT O⊕ Kt as having fermionic topological order since the combined K-matrix is odd. We therefore have to use the fermionic version of finding Lagrangian subgroups in Sec. 7.2.2, which leads to possibly more Lagrangian subgroups and hence more gapped edges. However, the folding trick does not give much insight into the physics involved – clearly, the actual topological order (in the non-folded system) is still bosonic in nature, so what does it really mean to treat the
combined system as having fermionic topological order? We would like a physical reason as to the appearance of more gapped edges without reference to the folding trick. What precisely about the locality nature of the vacuum allows for more quasiparticles in the topological phase to condense?
The key point is that in the original, unfolded system, while the self and mutual statis- tics of the quasiparticles in the bosonic topological phase indeed remain unchanged, the commutation relations of the exponentially localized operators Ua and Ub in Eqn. 7.6 that annihilates the quasiparticles on the boundary at points a and b respectively can change if we pair each operator up with a localized fermionic operator from the vacuum near the boundary, such as χ, a Majorana fermion operator. Precisely, we can have
˜
Uξ:= Uξ⊗ χξ, ξ = a, b, (7.7)
which is still an exponentially localized operator at the boundary that removes the quasi- particle and creates a real fermion in the vacuum at the same time. Also, more importantly, if [Ua, Ub] = 0, then { ˜Ua, ˜Ub} = 0. Using ˜U , this change in the commutation relation does not affect condition 1 and condition 2 to find M as described in Sec.7.2.2. However, it does affect condition 3. First let us understand condition 3 better. Let the self-statistics of the m particle be θm. We want the order in which we condense two pairs of m quasiparticles on the boundary to be irrelevant, so consider the following process:
Ub Ud × Ua Uc = (UbUdWm,bd)(UaUcWm,ac) (7.8)
which should bring the system back to the vacuum state. Here Wm,ij is a Wilson loop from point i to j on the boundary for the particle of type m. Then, Eq. (7.8) is equivalent to
(UbUd)(UaUc) a b c d = eiθm(U aUb)(UcUd) a b c d = eiθm Ua Ub × Uc Ud (7.9)
However, there is a phase factor eiθm due to the self-statistics of m, and so Eqn. (7.6) and (7.9) are inconsistent if θm 6= 0 – that is, if they are not self-bosons. Now consider if we use ˜U instead of U to annihilate the quasiparticles on the boundary. Then we have
( ˜UbU˜d)( ˜UaU˜c) a b c d = −eiθm( ˜U aU˜b)( ˜UcU˜d) a b c d ˜ Ub ˜ Ud × ˜ Ua ˜ Uc = −eiθm ˜ Ua ˜ Ub × ˜ Uc ˜ Ud (7.10)
where once again ( ˜UaU˜cWm,ac) is understood to happen before ( ˜UbU˜dWm,bd). Crucially now, the RHS of Eqn. (7.10) now has a phase factor (−1) × eiθm, which can made equal to unity if θm = π – that is, if the quasiparticles are self-fermions, they can self-consistently condense. We see that condition 3 should then be relaxed to allow both self-bosons and self-fermions in the Lagrangian subgroup.
In principle, this argument just shows that if the boundary between a bosonic topolog- ical order and a fermionic vacuum is gapped, it must satisfy the criterion of Lagrangian subgroups without the self-boson condition. One still needs to show that having a bosonic topological order with a fermionic Lagrangian subgroup, one will be able to gap out the edge modes. This part of the analysis is however identical to the purely fermionic case and shown in ref. [89].
It is not hard to see that the modification of condition 3 to allow both self-bosons and self-fermions to be in the Lagrangian subgroup is actually equivalent to the situation if we compute self-statistics in the bosonic TO using the fermionic version of Eqn.7.3 – that is, if we consider the bosonic topological phase as having fermionic topological order instead. Therefore, the folding trick gives an equivalent result to our analysis, except that in our approach we have explicitly elucidated the process of fermion condensation as the physical mechanism for fermionic gapped edges.
7.3.2
Modular invariance formalism
In this section1 we give an alternative argument for the presence of fermionic gapped edges,
related to the discussion given in Ref. [216]. This description has the advantage that it works for non-abelian theories too and can be computed from properties that in principle are available from the ground states[1, 2]. More details regarding the bosonic case can be found in Ref. [226].
Imagine a domain wall between two gapped systems, each described by an effective Chern-Simons theory. It is well-known that the boundary theory will be a conformal field theory [61, 62], with conformal families labeled by each topological particle in the bulk. If there are iL = 1, . . . , nL topological excitations of the left and iR = 1, . . . , nR on the right, we have the corresponding characters χR
iL and χ L
iR. The first quasiparticles, iL = 1 and iR = 1, will always be taken to be the identity quasiparticles on the L or R part respectively.
In order to be able to gap the conformal field theory (and regularize it on a lattice), a necessary condition is that there are no global gravitational anomalies, i.e. there are no anomalies with respect to large diffeomorphims. In other words, the partition function on the torus Z(τ ) = X iLiR MiLiRχ L iL(τ )χ R iR(τ ), (7.11)
must be modular invariant[89], where MiLiR ∈ N0 and τ is the complex-valued modular parameter characterizing the torus. Under modular transformations S : τ → −1/τ and T : τ → τ + 1, the characters transform as
χL(−1/τ ) = SLχL(τ ), χL(τ + 1) = TLχL(τ ), χR(−1/τ ) = S−1
R χR(τ ), χ
L(τ + 1) = T−1
R χR(τ ),
where (SL, TL) and (SR, TR) are the modular matrices[56, 89] for the anyon theory on the left and right, respectively. The condition for modular invariance Z(τ ) = Z(−1/τ ) = Z(τ + 1) gives us the constraints
SLM = MSR, TLM = MTR, M11= 1. (7.12)
The last condition comes from the fact that the theory must always contain the conformal family corresponding to the identity operator and that it must be unique [59]. These
conditions are similar to the ones stated in Ref. [216]. Furthermore it was shown that these conditions are equivalent to the notion of Lagrangian subgroups for bosonic abelian topological order, and thus are also sufficient in this case. The discussion however changes for a theory containing fermions: the condition Z(τ ) = Z(τ + 1) must be replaced by the weaker condition Z(τ ) = Z(τ + 2)[89, 227]. With these weaker conditions we arrive at the following constraints on M:
SLM = MSR, TL2M = MTR2, M11= 1. (7.13)
Any solution M will correspond to a possible gapped boundary, solutions of Eqn. (7.12) correspond to bosonic gapped edges while the extra solutions of Eqn. (7.13) correspond to fermionic gapped edges.
Using these weaker conditions in the proof given in the appendix of Ref. [216] we conclude that for bosonic abelian topological order, these conditions for fermionic gapped edges are equivalent to the ones given in section7.3.1.