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

▶
complex tensor order and quantum criticality in half
SMART_READER_LITE
LIVE PREVIEW

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

Dec 01, 2023 313 likes •816 views

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,

slide-1
SLIDE 1

Complex tensor order and quantum criticality in half-Heusler superconductors

Igor Boettcher Simon Fraser U Vancouver

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 Joint work with Igor Herbut

slide-2
SLIDE 2

Outline

Superconducting quantum criticality Complex tensor order half-Heusler superconductors

slide-3
SLIDE 3

Novel phases from electronic band crossings

Dirac semimetals Weyl semimetals

Pics: MPIKS Dresden

Luttinger semimetals

slide-4
SLIDE 4

Novel phases from electronic band crossings

Dirac semimetals Weyl semimetals

Pics: MPIKS Dresden

Luttinger semimetals

half-Heusler superconductors:

  • 3D quadratic band touching
  • superconducting below 1K
slide-5
SLIDE 5

Half-Heusler superconductors

Yan, de Visser, MRS Bulletin 39, 859 (2014)

Heavy elements Y and Z form zincblende structure: structural and electronic properties similar to CdTe, HgTe, ...

slide-6
SLIDE 6

Half-Heusler superconductors

Chadov, Qi, Kübler, Fecher, Felser, Zhang, Nat. Mater. 9, 541 (2010)

CdTe HgTe

Γ6, j=1/2 Γ6, j=1/2 Γ8, j=3/2 Γ8, j=3/2

Spin-orbit coupling: Hg heavier than Cd

  • causes band inversion

Cubic and TR symmetry

  • 4 bands touch at the Γ point
  • quadratic band touching
  • Fermi level at touching point
  • Topological insulator state

induced by strain or quantum confinement

slide-7
SLIDE 7

Half-Heusler superconductors

Chadov, Qi, Kübler, Fecher, Felser, Zhang, Nat. Mater. 9, 541 (2010)

CdTe HgTe

Γ6, j=1/2 Γ6, j=1/2 Γ8, j=3/2 Γ8, j=3/2

ScPtSb ScPtBi

slide-8
SLIDE 8

Half-Heusler superconductors

Chadov, Qi, Kübler, Fecher, Felser, Zhang, Nat. Mater. 9, 541 (2010)

normal band structure inverted band structure

slide-9
SLIDE 9

Half-Heusler superconductors

Chadov, Qi, Kübler, Fecher, Felser, Zhang, Nat. Mater. 9, 541 (2010)

normal band structure inverted band structure

Yan, de Visser, MRS Bulletin 39, 859 (2014)

slide-10
SLIDE 10

Half-Heusler superconductors

Hints of unconventional superconductivity

upper critical field exceeds s-wave models

Bay et al,PRB 86, 064515 (2012) Pan et al, EPL 104, 27001 (2013)

coexistence with magnetism in ErPdBi

Kim et al, arXiv:1603.03375

line nodes of the gap in YPtBi

slide-11
SLIDE 11

Quadratic band touching

4 x 4 Luttinger Hamiltonian

4x4 spin-3/2 matrices

GaAs Sn

slide-12
SLIDE 12

Quadratic band touching

4 x 4 Luttinger Hamiltonian

4x4 spin-3/2 matrices

GaAs Sn

rotation invariant SO(3) cubic invariant Oh ≈ permutations of x,y,z

slide-13
SLIDE 13

Quadratic band touching

4 x 4 Luttinger Hamiltonian

4x4 spin-3/2 matrices

GaAs Sn

particle-hole asymmetry cubic anisotropy

slide-14
SLIDE 14

Quadratic band touching

4 x 4 Luttinger Hamiltonian

five 4x4 gamma matrices five L=2 spherical harmonics

slide-15
SLIDE 15

Quadratic band touching

Kim et al, arXiv:1603.03375

YPtBi: x = 0.17 δ = -0.19

4 x 4 Luttinger Hamiltonian

slide-16
SLIDE 16

Part I Superconducting quantum criticality

IB, Herbut, PRB 93, 205138 (2016) IB, Herbut, PRB 95, 075149 (2017)

slide-17
SLIDE 17

Superconductivity

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

slide-18
SLIDE 18

Superconductivity

Short-range interactions This can be exactly rewritten as

slide-19
SLIDE 19

Superconductivity

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

slide-20
SLIDE 20

Superconducting quantum criticality

QCP

quantum critical properties?

Yukawa-type theory for fermions and Cooper pairs

slide-21
SLIDE 21

Superconducting quantum criticality

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

slide-22
SLIDE 22

Superconducting quantum criticality

s-wave particle-particle pairing 3D ultracold atoms at a Feshbach resonance

Diehl, Wetterich; Sachdev, Nikolic

= 0

slide-23
SLIDE 23

Superconducting quantum criticality

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

IB, Herbut, PRB 93, 205138 (2016)

slide-24
SLIDE 24

Superconducting quantum criticality

IB, Herbut, PRB 93, 205138 (2016)

slide-25
SLIDE 25

Superconducting quantum criticality

IB, Herbut, PRB 93, 205138 (2016)

exceptionally slow!

slide-26
SLIDE 26

Anisotropy, NFL, and tensor order

Flow of the anisotropy Anisotropy constant for all practical purposes

slide-27
SLIDE 27

Anisotropy, NFL, and tensor order

RG fixed points - possible 2nd order quantum phase transitions

IB, Herbut, PRB 95, 075149 (2017)

slide-28
SLIDE 28

Tensor orders

Magnetic order

  • rank 1 under SO(3)
  • breaks TRS

Nematic order

  • rank 2 under SO(3)
  • preserves TRS

think of coarse-grained microscopic orders

slide-29
SLIDE 29

Tensor orders

Magnetic order

  • rank 1 under SO(3)
  • breaks TRS

Nematic order

  • rank 2 under SO(3)
  • preserves TRS

Nemagnetic order

  • rank 3 under SO(3)
  • breaks TRS

All-In-All-Out Spin Ice

*electrons on the pyrochlore lattice think of coarse-grained microscopic orders

Spin Pics: Goswami, Roy, Das Sarma, PRB 95, 085120 (2017)

slide-30
SLIDE 30

Part II Complex tensor order

IB, Herbut, arXiv:1707.03444 IB, Herbut, arXiv:1712.03981

slide-31
SLIDE 31

Superconductivity

Short-range interactions This can be exactly rewritten as

slide-32
SLIDE 32

Superconductivity

Short-range interactions This can be exactly rewritten as

slide-33
SLIDE 33

Superconductivity

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

a=1,2,3,4,5

slide-34
SLIDE 34

Complex tensor order

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

slide-35
SLIDE 35

Complex tensor order

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

slide-36
SLIDE 36

Complex tensor order

The bigger context: higher-spin Cooper pairing

j=3/2 fermions Cooper pairs

Brydon, Wang, Meinert, Agterberg, PRL 116, 177001 (2016) IB, Herbut, PRB 93, 205138 (2016) Kim et al, arXiv:1603.03375

slide-37
SLIDE 37

Complex tensor order

The bigger context: higher-spin Cooper pairing

j=3/2 fermions Cooper pairs s-wave superconductor Cooper pairs of spin 0

IB, Herbut, PRB 93, 205138 (2016)

complex tensor order Cooper pairs of spin 2

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

slide-38
SLIDE 38

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

slide-39
SLIDE 39

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:

slide-40
SLIDE 40

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:

slide-41
SLIDE 41

Complex tensor order

Accidental SO(5) x U(1) symmetry

slide-42
SLIDE 42

Complex tensor order

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

slide-43
SLIDE 43

Complex tensor order

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

slide-44
SLIDE 44

Complex tensor order

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

slide-45
SLIDE 45

Complex tensor order

IB, Herbut, arXiv:1707.03444, PRL in press

Strong-coupling transition for μ=0 real order parameter line nodes preserves TRS

slide-46
SLIDE 46

Complex tensor order

Weak-coupling transition for μ>0 complex order parameter point nodes breaks TRS real order parameter line nodes preserves TRS

IB, Herbut, arXiv:1707.03444, PRL in press

slide-47
SLIDE 47

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

slide-48
SLIDE 48

Outlook

Spin-1/2 electrons Scalar BEC Spin-3/2 electrons Spinor BEC

Cooper pairing Cooper pairing Spin-Orbit- Coupling

  • Luttinger semimetals
  • relativistic dispersion

(Rarita-Schwinger-Weyl,

  • topol. crystalline insulators),
  • synthetic spin-orbit coupling
  • BCS-BEC crossover
  • Ultracold Bose gases
  • Frustrated magnetism
  • Topology & defects
slide-49
SLIDE 49

Outlook

Spin-1/2 electrons Scalar BEC Spin-3/2 electrons Spinor BEC

Cooper pairing Cooper pairing Spin-Orbit- Coupling

  • Luttinger semimetals
  • relativistic dispersion

(Rarita-Schwinger-Weyl,

  • topol. crystalline insulators),
  • synthetic spin-orbit coupling
  • BCS-BEC crossover
  • Ultracold Bose gases
  • Frustrated magnetism
  • Topology & defects

Thank you