# Séminaire du LPTMS: Patrick Azaria

## Topological States in Massless One-dimensional Fermionic Systems

### Patrick Azaria (LPTMC-Sorbonne Universités)

Topological states are characterized by the unique exponentially localized excitations they host at their ends in an open geometry. The topological protections of these systems relies on the bulk of these systems being gapped. This is the case in one-dimensional systems for superconductors proximity coupled to a bulk superconductor. However in a purely one-dimensional system, where superconductivity arises from intrinsic charge conserving attractive interactions, the bulk of the system remains gapless due to large quantum fluctuations of the phase. One may then naturally question the existence of such topological phases in the presence of massless bulk excitations. Indeed, systems of spinless fermions with either attractive or repulsive interactions are described by a Luttinger liquid and are topologically trivial: they display quasi-long range dominant superconducting or 2*k _{F}* charge density wave instabilities without exponentially localized modes at their ends. In this talk I shall show that in coupled chains, or in interacting multi-flavor fermionic systems, topological states with non trivial end modes may exist provided there is enough symmetry protection. This includes systems with tendency toward superconducting order as well as the more seemingly innocuous situations of coupled density waves with repulsive interactions. In the particular case with

*N*=3 flavors, I shall argue that a non trivial topological superconducting phase, corresponding to the gapless analogue of the Haldane phase in spin-1 chains, as well as non trivial chiral gapless spin density waves phases hosting Z

_{3}parafermionic edge modes may be stabilized.

**Date/Time : ** 26/02/2019 - *11:00 - 12:00*

**Location : ** LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay

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