1. Pseudo-spin 3/2 pairing
Daniel F. Agterberg, University of Wisconsin - Milwaukee
Philip Brydon, Johnpierre Paglione, Liming Wang (Maryland), Paolo Frigeri, Manfred Sigrist (ETH-Zurich), Huiqiu Yuan, Myron Salamon (UIUC), Akihisha Koga (Osaka University), Mike Weinert (UWM), Carsten Timm (TU Dresden), Matthias Scheurer, Joerg Schmalian (KIT, Karlsruhe)
NSF DMREF
1- j=3/2 pairing in half-Heusler YPtBi - overview
2- pseudo-spin 1/2 and 3/2
3- physical consequences of pseudo-spin 3/2
4- examples: UPt3, KCr3As2, YPtBi

2. Half-Heuslers: Topological materials
Many Half-Heuslers show band s-p band inversion.
Tetrahedral Td symmetry:
H. Lin et al, Nature Materials (2010)
Feng et al, PRB (2010)
4-fold symmetry: j=3/2 state electrons

3. Half-Heuslers: Unconventional Superconductivity
Bay et al, PRB RC 2012
Kim et al, arXiv: (2016)
Low density metal, small Fermi energy (300K)
Meinert, PRL 2016: Theory e-p coupling maximum Tc=0.1 K<<1.5 K
Likely to be an unconventional superconductor with line-nodes

4. Half-Heuslers: LuPtBi and YPtBi
Kim et al, arXiv: (2016)
Consider pseudo-spin ½: consider local attractive interactions:
on-site, only s-wave pairing
Repulsive U: Kohn Luttinger can generate p-wave; however p-wave not allowed for j=1/2 (justified later). Perhaps d-wave?

5. j=3/2 pairing in tetrahedral materials
Four species to make Cooper pairs from: 3/2,1/2,-1/2,-3/2
on-site interactions: can make more than “s-wave” local Cooper pairs
Pauli exclusion: antisymmetry in m and m’ : 6 on-site Cooper pairs
Can also have p-wave states (justified later)

6. j=3/2 pairing in YPtBi Pairing occurs in pseudo-spin 3/2 states.
This is different than pseudo-spin 1/2
One proposal: one Fermi surface is fully gapped, the other has line nodes (from p-wave interactions)
Other proposal: on-site interactions create broken time-reversal “quintet” d-wave pairing state with line-nodes
Brydon, DFA, Wang, Weinert, PRL 116,

7. Pseudo-spin: Introduction
Assume time-reversal (T) and inversion (P) symmetries
Pseudo-spin: two degenerate states with same k:
Key assumption in superconductivity: pseudo-spin rotates the same as spin ½ :
Call this pseudo-spin ½
At each k can create single-particle Hamiltonian from:

8. Pseudo-spin -continued
Key assumption pseudo-spin rotates the same as spin ½ :
Consequences of Key assumption :
1- Pauli matrices rotate as spin ½ operators.
2- d-vector rotates as a spin-vector:
3- Coupling to Zeeman-field is:
Key assumption does not always hold when strong
spin-orbit coupling is present
Key assumption is that sigma_i rotates in the same way as the spin ½ Pauli matrices, sigma_i:
That is U(theta) = e^{-i\theta\cdot sigma}. When this is true, this is pseudospin 1/2. Claims here is that this is not always the case
Consequences of key assumptions:
1- Delta=(d\cdot sigma) T (T=isigma_2), so d rotates as a vector under spin rotations
2- coupling to Zeeman field is H\cdot \sigma (neglecting anisotropy)

9. Example: D3h
With strong spin-orbit need double group representations (Koster):
Three representations: construct Pauli matrices from these

10. Pseudo-spin 1/2
Pseudo-spin 3/2

11. Implications: Non-pseudo-spin ½ only matters in groups that
1- Pauli matrices do not rotate as a spin ½ operators.
2- d-vector does not rotate as a vector:
3- Coupling to Zeeman-field is:
Only paramagnetic limiting for field along three-fold symmetry axis
Non-pseudo-spin ½ only matters in groups that
have a 3-fold symmetry axis
Basis functions in these cases can always be chosen to include
jz=3/2,-3/2 (sometimes need all j=3/2 manifold)

12. K2Cr3As3 – D3h What happens when parity symmetry is also broken?
K2Cr3As3 Balakirev et al, PRB 2015 (interpreted as singlets made from spins that point along z?)
Similar critical field behavior in UPt3, (D6h) there it was interpreted
as proof of spin-triplet.
What happens when parity symmetry is also broken?

13. Superconductors without parity symmetry:
Stability of different triplet-states

14. Spin Orbit Coupling Broken parity exists through
Generalized Rashba Spin-Orbit Interaction:
Time reversal symmetry implies:
Parity symmetry would imply:
Single Particle excitations become:
With pseudo-spins polarized along

15. Example
Rashba 2D

16. Superconductivity Model
Broken Inversion exists solely through:
No additional assumptions on gk
Typically, a>>D and a<<m

17. Results on Stability spin-singlet spin-triplet
Tc for spin-triplet is not suppressed only for a single protected d vector (with g and d parallel).
PRL 92, (2004)

18. Single Protected d-vector
If d-vector is chosen along z, only opposite spins are paired.
If no inter-band pairing, then a state and the
time-reversed partner can pair, this implies only
opposite spins pair.
Hence g(k) and d (k) must be parallel
Other triplet states are suppressed

19. Singlet-triplet mixing
Parity symmetry is broken, so spin-triplet and spin-singlet mix in general:

20. Back to YPtBi

21. Half-Heuslers: LuPtBi and YPtBi
JP Paglione, M.A. Tanatar, R. Prozorov, unpublished
Claimed p-wave pairing not possible for pseudo-spin ½ why?
symmetry allows
This implies f-wave superconductivity
How about pseudo-spin 3/2 ?

22. j=3/2 pairing in YPtBi
Pairing occurs in pseudo-spin 3/2 states.
Spin-orbit is linear in k for j=3/2, not allowed for j=1/2

23. P-wave pairing in YPtBi
like
s-wave - p-wave mixing

24. Conclusions Pseudo-spin 3/2 is different than pseudo-spin 1/2.
This leads to new structures of dk and to different magnetic field response.
dk must be parallel to gk for spin-triplet superconductivity to exist in materials without parity symmetry.
Generalization of above to j=3/2 allows for p-wave pairing that was not allowed for j=1/2 in YPtBi.