# Topological Superconductivity in Doped Symmetry Protected Topological State

## TL;DR

This paper proposes a novel mechanism for topological superconductivity emerging from doping a 1D symmetry protected topological state, specifically a Haldane phase, leading to robust p-wave topological superconductivity confirmed by numerical simulations.

## Contribution

It introduces a new scenario where doping a bosonic SPT state in a fermionic system induces topological superconductivity, supported by DMRG simulations and theoretical analysis.

## Key findings

- Doping a Haldane phase leads to gapless charge and gapped spin channels.
- An instability toward p-wave topological superconductivity is observed.
- Stacking doped Haldane phases can realize a 2D exotic superconductor.

## Abstract

We propose an exotic scenario that topological superconductivity can emerge by doping strongly interacting fermionic systems whose spin degrees of freedom form bosonic symmetry protected topological (SPT) state. Specifically, we study a 1-dimensional (1D) example where the spin degrees of freedom form a spin-1 Haldane phase. {Before doping, the charge and spin degrees of freedom are both gapped.} Upon doping, {the charge channel becomes gapless and is described by a $c=1$ compactified bosonic conformal field theory (CFT), while the spin channel remains gapped and still form a bosonic SPT state. Interestingly,} an instability toward $p$-wave topological superconductivity is induced coexisting with the symmetry protected spin edge modes that are inherited from the Haldane phase. This scenario is confirmed by density-matrix renormalization group simulation of a concrete lattice model, where we find that topological superconductivity is robust against interactions. We further show that by stacking doped Haldane phases an exotic 2D anisotropic superconductor can be realized, {where the boundaries transverse to the {chain-}direction are either gapless or spontaneously symmetry-broken due to the Lieb-Schultz-Mattis (LSM) anomaly.}