[Paper Review] Topological d+s wave superconductors in a multi-orbital quadratic band touching system

GiBaik Sim, Archana Mishra+4 more|arXiv (Cornell University)|Nov 9, 2018

Topological Materials and Phenomena4 citations

TL;DR

This paper proposes a novel route to realize topological superconductors with non-trivial Fermi surfaces in multi-orbital quadratic band-touching systems with finite chemical potential. Using Landau theory with SO(3) symmetry, it demonstrates that broken particle-hole symmetry induces parasitic s-wave pairing, stabilizing two distinct topological superconducting phases: a uniaxial nematic $d_{(3z^2-r^2)}+s$ phase and a time-reversal-symmetry-broken $d_{(3z^2-r^2,xy)}+id_{x^2-y^2}+s$ phase, both hosting gapless Bogoliubov quasiparticles with winding numbers $\pm 2$ and exotic surface states.

ABSTRACT

Realization of topological superconductors is one of the most important goals in studies of topological phases in quantum materials. In this work, we theoretically propose a novel way to attain topological superconductors with non-trivial Fermi surfaces of Bogoliubov quasiparticles. Considering the interacting Luttinger model with $j\!=\!3/2$ electrons, we investigate the dominant superconducting channels for a multi-orbital quadratic band-touching system with finite chemical potential, which breaks the particle-hole symmetry in the normal state. Notably, while the system generally favors d-wave pairing, the absence of the particle-hole symmetry necessarily induces parasitic s-wave pairing. Based on the Landau theory with $SO(3)$ symmetry, we demonstrate that two kinds of topological superconductors are energetically favored; uniaxial nematic phase with parasitic $s$ wave pairing ($d_{(3z^2-r^2)}\!+\!s$) and time-reversal-symmetry broken phase with parasitic $s$ wave pairing ($d_{(3z^2-r^2,xy)}\!+\!id_{x^2-y^2}\!+\!s$). These superconductors contain either nodal lines or Fermi pockets of gapless Bogoliubov quasiparticles and moreover exhibit topological winding numbers $\pm2$, leading to non-trivial surface states such as drumhead-like surface states or Fermi arcs. We discuss applications of our theory to relevant families of materials, especially half-heusler compound YPtBi, and suggest possible future experiments.

Motivation & Objective

• To identify generic material platforms for topological superconductors with non-trivial Fermi surfaces of Bogoliubov quasiparticles.
• To understand the role of broken particle-hole symmetry in inducing parasitic s-wave pairing in multi-orbital systems.
• To classify stable topological superconducting phases in a $j=3/2$ Luttinger model with finite chemical potential.
• To predict exotic surface states such as drumhead modes and Fermi arcs in these phases.
• To suggest experimental control parameters like doping, pressure, and temperature for tuning superconducting states.

Proposed method

• Formulates a multi-orbital Hamiltonian based on the Luttinger model with $j=3/2$ electrons and quadratic band-touching at the Brillouin zone center.
• Applies Landau theory of complex tensor order parameters to describe coexisting s-wave and d-wave pairing channels.
• Constructs Landau free energy functionals invariant under SO(3) symmetry to analyze phase competition.
• Calculates topological invariants such as Chern numbers and winding numbers for nodal lines and Fermi pockets.
• Performs numerical diagonalization of the Bogoliubov-de Gennes Hamiltonian to map gap structures and Fermi surface topology.
• Uses phase diagrams in $\Delta_s$-$\Delta_{TSB}$ parameter space to track topological phase transitions and gap evolution.

Experimental results

Research questions

• RQ1 How does broken particle-hole symmetry in a multi-orbital quadratic band-touching system influence the pairing symmetry of superconducting order parameters?
• RQ2 What topological superconducting phases emerge when both s-wave and d-wave pairing are present, and what symmetries are preserved?
• RQ3 What is the role of the chemical potential in stabilizing gapless Bogoliubov quasiparticles with non-trivial winding numbers?
• RQ4 How do the surface states of these topological superconductors differ from those of conventional d-wave superconductors?
• RQ5 Can the nature of the superconducting state be tuned via external parameters such as chemical doping or hydrostatic pressure?

Key findings

• Two distinct topological superconducting phases are stabilized: a uniaxial nematic $d_{(3z^2-r^2)}+s$ phase and a time-reversal-symmetry-broken $d_{(3z^2-r^2,xy)}+id_{x^2-y^2}+s$ phase.
• The $d_{(3z^2-r^2)}+s$ phase hosts nodal lines with winding number $\pm 2$, supporting drumhead-like surface states.
• The $d_{(3z^2-r^2,xy)}+id_{x^2-y^2}+s$ phase exhibits Fermi pockets with Chern number $\pm 2$, hosting Fermi arc surface states.
• Topological phase transitions occur via merging and splitting of Fermi pockets as $\Delta_s$ is tuned, with Chern number changing from $\pm 2$ to $0$.
• Nodal rings persist when $-\Delta_2 < \Delta_s < \Delta_2\sqrt{(\mu^2 + \Delta_2^2)/(4\mu^2 + \Delta_2^2)}$ and $\Delta_{TSB}=0$.
• The system transitions from fully gapped to gapless superconducting states at critical values of $\Delta_s$ and $\Delta_{TSB}$, with phase boundaries determined by the minimum gap magnitude.