Fig. 3: Edge transition and the renormlization group flow.
From: Quantum restoration of symmetry protected topological phases
a Eigenvalues of the edge Hamiltonian, Eq. (7), as a function of Δ for t = 0, Ng = 2. Note the edge transition from gapless spin edge modes to gapless charge edge modes at Δ = EC. N1 represents the number of bosons on the first bosonic site. Ntotal = 2N1 + n1 represents the total charge on the left edge. b Edge states correspond to kink-like field configurations within the effective field theory near the free fermion critical point, Eq. (8). The blue and yellow regions are used to distinguish the trivial and topological insulating regimes. Φρ/s stand for the bosonic fields corresponding to charge (red curve) and spin (blue curve) degrees of freedom, respectively. The charge (red curve) and spin (blue curve) densities, Eq. (11) and Eq. (12), exhibit the localization of respective densities close to the boundary of the trivial and non-trivial insulating states, which is the character of edge states. c Renormalization Group (RG) flow obtained using the flow equations given in Eq. (9). The black arrows represent flow towards an insulating state, whereas the red arrows represent flow towards a state without a charge gap.
