TY - JOUR

T1 - Topological crystalline superconductivity and second-order topological superconductivity in nodal-loop materials

AU - Shapourian, Hassan

AU - Wang, Yuxuan

AU - Ryu, Shinsei

N1 - Funding Information:
We would like to thank Taylor Hughes, Rahul Nandkishore, and Luka Trifunovic for insightful discussions. Computational resources were provided by the Taub campus cluster at the University of Illinois at Urbana-Champaign. This work was supported by the NSF under Grant No. DMR-1455296 (H.S. and S.R.) and by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant No. GBMF4305 at the University of Illinois (Y.W.).
Funding Information:
We would like to thank Taylor Hughes, Rahul Nandkishore, and Luka Trifunovic for insightful discussions. Computational resources were provided by the Taub campus cluster at the University of Illinois at Urbana-Champaign. This work was supported by the NSF under Grant No. DMR-1455296 (H.S. and S.R.) and by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant No. GBMF4305 at the University of Illinois (Y.W.).

PY - 2018/3/14

Y1 - 2018/3/14

N2 - We study the intrinsic fully gapped odd-parity superconducting order in doped nodal-loop materials with a torus-shaped Fermi surface. We show that the mirror symmetry, which protects the nodal loop in the normal state, also protects the superconducting state as a topological crystalline superconductor. As a result, the surfaces preserving the mirror symmetry host gapless Majorana cones. Moreover, for a Weyl-loop system (twofold degenerate at the nodal loop), the surfaces that break the mirror symmetry (those parallel to the bulk nodal loop) contribute a Chern (winding) number to the quasi-two-dimensional system in a slab geometry, which leads to a quantized thermal Hall effect and a single Majorana zero mode bound at a vortex line penetrating the system. This Chern number can be viewed as a higher-order topological invariant, which supports hinge modes in a cubic sample when mirror symmetry is broken. For a Dirac-loop system (fourfold degenerate at the nodal loop), the fully gapped odd-parity state can be either time-reversal symmetry-breaking or symmetric, similar to the A and B phases of He3. In a slab geometry, the A phase has a Chern number two, while the B phase carries a nontrivial Z2 invariant. We discuss the experimental relevance of our results to nodal-loop materials such as CaAgAs.

AB - We study the intrinsic fully gapped odd-parity superconducting order in doped nodal-loop materials with a torus-shaped Fermi surface. We show that the mirror symmetry, which protects the nodal loop in the normal state, also protects the superconducting state as a topological crystalline superconductor. As a result, the surfaces preserving the mirror symmetry host gapless Majorana cones. Moreover, for a Weyl-loop system (twofold degenerate at the nodal loop), the surfaces that break the mirror symmetry (those parallel to the bulk nodal loop) contribute a Chern (winding) number to the quasi-two-dimensional system in a slab geometry, which leads to a quantized thermal Hall effect and a single Majorana zero mode bound at a vortex line penetrating the system. This Chern number can be viewed as a higher-order topological invariant, which supports hinge modes in a cubic sample when mirror symmetry is broken. For a Dirac-loop system (fourfold degenerate at the nodal loop), the fully gapped odd-parity state can be either time-reversal symmetry-breaking or symmetric, similar to the A and B phases of He3. In a slab geometry, the A phase has a Chern number two, while the B phase carries a nontrivial Z2 invariant. We discuss the experimental relevance of our results to nodal-loop materials such as CaAgAs.

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U2 - 10.1103/PhysRevB.97.094508

DO - 10.1103/PhysRevB.97.094508

M3 - Article

AN - SCOPUS:85043975566

VL - 97

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 9

M1 - 094508

ER -