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-1335215 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

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

Half-Heuslers: Unconventional Superconductivity Bay et al, PRB RC 2012 Kim et al, arXiv:1603.03375 (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

Half-Heuslers: LuPtBi and YPtBi Kim et al, arXiv:1603.03375 (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?

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)

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, 177001 2016

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:

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)

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

Pseudo-spin 1/2 Pseudo-spin 3/2

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)

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?

Superconductors without parity symmetry: Stability of different triplet-states

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

Example Rashba 2D

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

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, 097001 (2004)

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

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

Back to YPtBi

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 ?

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

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

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.