Complex tensor order and quantum criticality in half-Heusler - PowerPoint PPT Presentation

Complex tensor order and quantum criticality in half-Heusler superconductors Igor Boettcher Simon Fraser U Vancouver Joint work with Igor Herbut IB, Herbut, PRB 93, 205138 (2016) IB, Herbut, PRB 95, 075149 (2017) IB, Herbut, arXiv:1707.03444, PRL in press IB, Herbut, arXiv:1712.03981

Outline half-Heusler superconductors Superconducting quantum criticality Complex tensor order

Novel phases from electronic band crossings Pics: MPIKS Dresden Dirac semimetals Weyl semimetals Luttinger semimetals

Novel phases from electronic band crossings Pics: MPIKS Dresden Dirac semimetals Weyl semimetals half-Heusler superconductors: ● 3D quadratic band touching ● superconducting below 1K Luttinger semimetals

Half-Heusler superconductors Heavy elements Y and Z form zincblende structure: structural and electronic properties similar to CdTe, HgTe, ... Yan, de Visser, MRS Bulletin 39, 859 (2014)

Half-Heusler superconductors Γ6, j=1/2 Spin-orbit coupling: Hg heavier than Cd CdTe ● causes band inversion Γ8, j=3/2 Cubic and TR symmetry ● 4 bands touch at the Γ point ● quadratic band touching ● Fermi level at touching point Γ8, j=3/2 HgTe ● Topological insulator state Γ6, j=1/2 induced by strain or quantum confinement Chadov, Qi, Kübler, Fecher, Felser, Zhang, Nat. Mater. 9, 541 (2010)

Half-Heusler superconductors Γ6, j=1/2 ScPtSb CdTe Γ8, j=3/2 Γ8, j=3/2 ScPtBi HgTe Γ6, j=1/2 Chadov, Qi, Kübler, Fecher, Felser, Zhang, Nat. Mater. 9, 541 (2010)

Half-Heusler superconductors normal band structure inverted band structure Chadov, Qi, Kübler, Fecher, Felser, Zhang, Nat. Mater. 9, 541 (2010)

Half-Heusler superconductors normal band structure inverted band Yan, de Visser, MRS structure Bulletin 39, 859 (2014) Chadov, Qi, Kübler, Fecher, Felser, Zhang, Nat. Mater. 9, 541 (2010)

Half-Heusler superconductors Hints of unconventional superconductivity upper critical field exceeds s-wave models line nodes of the Bay et al,PRB 86, gap in YPtBi 064515 (2012) Pan et al, EPL 104, 27001 (2013) coexistence with magnetism in ErPdBi Kim et al, arXiv:1603.03375

Quadratic band touching 4 x 4 Luttinger Hamiltonian GaAs Sn 4x4 spin-3/2 matrices

Quadratic band touching 4 x 4 Luttinger Hamiltonian cubic invariant Oh GaAs rotation invariant SO(3) ≈ permutations of x,y,z Sn 4x4 spin-3/2 matrices

Quadratic band touching 4 x 4 Luttinger Hamiltonian particle-hole asymmetry GaAs cubic anisotropy Sn 4x4 spin-3/2 matrices

Quadratic band touching 4 x 4 Luttinger Hamiltonian five L=2 spherical harmonics five 4x4 gamma matrices

Quadratic band touching 4 x 4 Luttinger Hamiltonian YPtBi: x = 0.17 δ = -0.19 Kim et al, arXiv:1603.03375

Part I Superconducting quantum criticality IB, Herbut, PRB 93, 205138 (2016) IB, Herbut, PRB 95, 075149 (2017)

Superconductivity Short-range interactions Fierz-complete: further terms contain derivatives / momenta

Superconductivity Short-range interactions This can be exactly rewritten as

Superconductivity Short-range interactions This can be exactly rewritten as s-wave superconducting gap

Superconducting quantum criticality quantum critical properties? QCP Yukawa-type theory for fermions and Cooper pairs

Superconducting quantum criticality s-wave particle-particle pairing 3D Luttinger semimetals 3D ultracold atoms at at a superconducting QCP a Feshbach resonance

Superconducting quantum criticality s-wave particle-particle pairing = 0 3D ultracold atoms at a Feshbach resonance Diehl, Wetterich; Sachdev, Nikolic

Superconducting quantum criticality s-wave particle-particle pairing 3D Luttinger semimetals 3D ultracold atoms at at a superconducting QCP a Feshbach resonance IB, Herbut, PRB 93, 205138 (2016)

Superconducting quantum criticality IB, Herbut, PRB 93, 205138 (2016)

Superconducting quantum criticality exceptionally slow! IB, Herbut, PRB 93, 205138 (2016)

Anisotropy, NFL, and tensor order Flow of the anisotropy Anisotropy constant for all practical purposes

Anisotropy, NFL, and tensor order RG fixed points - possible 2 nd order quantum phase transitions IB, Herbut, PRB 95, 075149 (2017)

Tensor orders think of coarse-grained microscopic orders Nematic order Magnetic order ● rank 2 under SO(3) ● rank 1 under SO(3) ● preserves TRS ● breaks TRS

Tensor orders think of coarse-grained microscopic orders Nematic order Magnetic order ● rank 2 under SO(3) ● rank 1 under SO(3) ● preserves TRS ● breaks TRS Nemagnetic order ● rank 3 under SO(3) ● breaks TRS *electrons on the Spin Ice All-In-All-Out pyrochlore lattice Spin Pics: Goswami, Roy, Das Sarma, PRB 95, 085120 (2017)

Part II Complex tensor order IB, Herbut, arXiv:1707.03444 IB, Herbut, arXiv:1712.03981

Superconductivity Short-range interactions This can be exactly rewritten as

Superconductivity Short-range interactions This can be exactly rewritten as

Superconductivity Short-range interactions This can be exactly rewritten as a=1,2,3,4,5 d-wave superconducting gap

Complex tensor order transforms under five-dimensional (S=2) representation of SO(3)

Complex tensor order transforms under five-dimensional (S=2) representation of SO(3) irreducible 2nd-rank complex tensor Φ is symmetric & traceless

Complex tensor order The bigger context: higher-spin Cooper pairing j=3/2 Cooper pairs fermions Brydon, Wang, Meinert, Agterberg, PRL 116, 177001 (2016) IB, Herbut, PRB 93, 205138 (2016) Kim et al, arXiv:1603.03375

Complex tensor order The bigger context: higher-spin Cooper pairing j=3/2 Cooper pairs fermions s-wave superconductor complex tensor order Cooper pairs of spin 0 Cooper pairs of spin 2 IB, Herbut, PRB 93, 205138 (2016) IB, Herbut,arXiv:1707.03444 Flurry of activity this summer: Timm, Schnyder, Agterberg, Brydon, PRB 96, 094526 (2017) Savary, Ruhman, Venderbos, Fu, Lee, arXiv:1707.03831 Yang, Xiang, Wu, PRB 96, 144514 (2017) Roy, Ghorashi, Foster, Nevidomskyy, arXiv:1708.07825 Venderbos, Savary, Ruhman, Lee, Fu, arXiv:1709.04487

Complex tensor order Free energy F(Φ) with SO(3) x U(1) symmetry Invariant theory: The most general F(Φ) is polynomial in the eight invariants

Complex tensor order Free energy F(Φ) with SO(3) x U(1) symmetry Invariant theory: The most general F(Φ) is polynomial in the eight invariants to quartic order:

Complex tensor order Free energy F(Φ) with SO(3) x U(1) symmetry Invariant theory: The most general F(Φ) is polynomial in the eight invariants to quartic order:

Complex tensor order Accidental SO(5) x U(1) symmetry

Complex tensor order Accidental SO(5) x U(1) symmetry prefers real due to biaxial nematic uniaxial nematic

Complex tensor order Accidental SO(5) x U(1) symmetry prefers genuinely complex : break TRS

Complex tensor order The conditions or leave a huge degeneracy of potential ground states degeneracy lifted by terms of sextic order in Φ: with

Complex tensor order Strong-coupling transition for μ=0 real order parameter line nodes preserves TRS IB, Herbut, arXiv:1707.03444, PRL in press

Complex tensor order Weak-coupling transition for μ>0 real order parameter line nodes preserves TRS complex order parameter point nodes breaks TRS IB, Herbut, arXiv:1707.03444, PRL in press

Complex tensor order Critical properties of complex tensor order transition ● fluctuation-induced first-order transition ● cubic half-Heusler YPtBi: two fluctuating complex components, maps to Frustrated magnetism with O(2)xO(2) symmetry IB, Herbut, arXiv:1712.03981

Outlook Cooper pairing Spin-1/2 electrons Scalar BEC Spin-Orbit- Coupling Cooper pairing Spin-3/2 electrons Spinor BEC ● Luttinger semimetals ● Ultracold Bose gases ● relativistic dispersion ● Frustrated magnetism (Rarita-Schwinger-Weyl, ● Topology & defects topol. crystalline insulators), ● synthetic spin-orbit coupling ● BCS-BEC crossover

Outlook Cooper pairing Spin-1/2 electrons Scalar BEC Spin-Orbit- Coupling Cooper pairing Thank you Spin-3/2 electrons Spinor BEC ● Luttinger semimetals ● Ultracold Bose gases ● relativistic dispersion ● Frustrated magnetism (Rarita-Schwinger-Weyl, ● Topology & defects topol. crystalline insulators), ● synthetic spin-orbit coupling ● BCS-BEC crossover