From one-dimensional charge conserving superconductors to the gapless Haldane phase

Abstract

We develop a framework to analyze one-dimensional topological superconductors with charge conservation. In particular, we consider models with N flavors of fermions and (Z2) N symmetry, associated with the conservation of the fermionic parity of each flavor. For a single flavor, we recover the result that a distinct topological phase with exponentially localized zero modes does not exist due to absence of a gap to single particles in the bulk. For N > 1, however, we show that the ends of the system can host low-energy, exponentially-localized modes. The analysis can readily be generalized to systems in other symmetry classes. To illustrate these ideas, we focus on lattice models with SO (N ) symmetric interactions, and study the phase transition between the trivial and the topological gapless phases using bosonization and a weak-coupling renormalization group analysis. As a concrete example, we study in detail the case of N = 3. We show that in this case, the topologically non-trivial superconducting phase corresponds to a gapless analogue of the Haldane phase in spin-1 chains. In this phase, although the bulk is gapless to single particle excitations, the ends host spin-1/2 degrees of freedom which are exponentially localized and protected by the spin gap in the bulk. We obtain the full phase diagram of the model numerically, using density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B, 231(3), 445-480 (1984)], as a first-order transition line between the gapless Haldane phase and a trivial gapless phase. This allows us to identify the propagating spin-1/2 kinks in the Andrei-Destri model as the topological end-modes present at the domain walls between the two phases.

Topological transition on the conformal manifold

“…To characterize gapless phases without quasiparticles, inspired by the success in gapped phases, we start with the question of whether they can be topologically nontrivial. A far from exhaustive list of reference are [16][17][18][19][20][21][22][23][24][25][26][27]. To give an example, one construction is to impose symmetries and to decorate domain walls in gapless phases by symmetry charges, in analogy with the construction of symmetry protected topological phases [28,29].…”

Interplay between intrinsic and emergent topological protection on interacting helical modes

“…In this specialised case we uncover a number of non-trivial phases as function of interaction and crossovers as a function of temperature, which presumably one would see for any odd N , although to confirm or deny this conjecture remains work for the future. The work of Keselman et al [74] looks at a different model, concentrating on N = 3 channels in which the non-interacting model is non-topological, and like us finds a phase with TR symmetry breaking, and another phase with an emergent topology. While their TRSB phase is the same one that we find, they curiously find a different emergent topological phase, in the universality class of the Haldane spin-1 chain as opposed to our Z 3 parafermionic state.…”

“…Note added: When this manuscript was in preparation, we learned about preprints [28,74] with partly overlapping content. The work of Kagalovsky et al [28] discusses TRS breaking in the ground state leading to zero conductance at zero temperature, for any number of channels N ≥ 3.…”

Topology of critical chiral phases: multiband insulators and superconductors

“…phases which both support topological edge states. Earlier works [11][12][13][14][15][16][17][18][19][20][21][22][23][24] exploring specific models in 1D had anticipated that distinct topological phases − supporting robust edge states − may in fact form also at quantum critical points (and possibly also in higher dimensions in the presence of additional gapped degrees of freedom 25,26 ). However, why and how this happens was first unveiled in detail by Verresen et al 10 , providing important intuition.…”