Pseudo-spin 3/2 pairing
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
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
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
Meinert, PRL 2016: Theory e-p coupling maximum Tc=0.1 K<<1.5 K
Bay et al, PRB RC 2012 Kim et al, arXiv:1603.03375 (2016)
Likely to be an unconventional superconductor with line-nodes
3
10
18 cm n
Low density metal, small Fermi energy (300K)
Half-Heuslers: LuPtBi and YPtBi
• Consider pseudo-spin ½: consider local attractive interactions:
Kim et al, arXiv:1603.03375 (2016)
i i i
t i t
i
c c c
c
U ( )( )
on-site, only s-wavepairing
• 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
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)
Four species to make Cooper pairs from: 3/2,1/2,-1/2,-3/2
0 1
2
2
3
2 3
3
j=3/2 pairing in YPtBi
Brydon, DFA, Wang, Weinert, PRL 116, 177001 2016
One proposal: one Fermi surface is fully gapped, the other has line nodes (from p-wave interactions)
Pairing occurs in pseudo- spin 3/2 states.
This is different than pseudo-spin 1/2
Other proposal: on-site interactions create broken time-reversal “quintet” d- wave pairing state with line- nodes
Pseudo-spin: Introduction
, | , | ,
,
| k PT k k
Pseudo-spin: two degenerate states with same k:
Assume time-reversal (T) and inversion (P) symmetries
At each k can create single-particle Hamiltonian from:
s e k s
k 0, i | 0,
| /2
Key assumption in superconductivity: pseudo-spin rotates the same as spin ½ : Call this pseudo-spin ½
) ( ) (
,
k k
H i
i k
i
Pseudo-spin -continued
s e k s
k 0, i | 0,
| /2
Key assumption pseudo-spin rotates the same as spin ½ :
Consequences of Key assumption :
i y
d
)
(
2- d-vector rotates as a spin-vector:
3- Coupling to Zeeman-field is:
k B
z B k
H
( )Key assumption does not always hold when strong spin-orbit coupling is present
1- Pauli matrices rotate as spin ½ operators.
Example: D
3hWith strong spin-orbit need double group representations (Koster):
Three representations: construct Pauli matrices from these
'' '
'
21 8
8 7
7
A A E
'' ''
'
'
2 1 21 9
9
A A A A
0
z
x,
y
0
z? ?
Pseudo-spin 1/2
Pseudo-spin 3/2
Implications:
'' ''
'
'
2 1 21 9
9
A A A A
2- d-vector does not rotate as a vector:
3- Coupling to Zeeman-field is:
k
z z B
z B k
H
( )Non-pseudo-spin ½ only matters in groups that have a 3-fold symmetry axis
Only paramagnetic limiting for field along three-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)
1- Pauli matrices do not rotate as a spin ½ operators.
i y
d
)
(
0
z
x
yK
2Cr
3As
3– D
3hK2Cr3As3 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:
k s
s k
k
p
g
H
' , ,
k
k
g
g
Single Particle excitations become:
|
|
kk
k
g
With pseudo-spins polarized along
g
kTime reversal symmetry implies:
Parity symmetry would imply:
g
k g
kx y
k
x k y k
g ˆ ˆ
Rashba
Example
2D
Superconductivity Model
s k s k t
ks t
ks s
s k k
k k ks
s k
t ks
k
c c V c c c c
H
' ' ' '' , ,' ,
' ,
,
2
1
Broken Inversion exists solely through:
' '
, '
, ,
ks t
ks s
s s
s k
k
p
g c c
H g
k g
k No additional assumptions on gkTypically, >> and <<
Results on Stability
22ln
F cs
c
O
T T
spin-singlet
2 | |
2| ˆ |
2( )
22ln
k F k
k k
k ct
c
d g d f O
T T
Tc for spin-triplet is not suppressed only for a single protected d vector (with g and d parallel).
z y
x k
y x
k
y x
k
k z k
y k
x d
k x k
y d
k x k
y d
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
PRL 92, 097001 (2004)
spin-triplet
Single Protected d-vector
( k ) ( k ) ( k )
) k ( )
k ( )
k ) (
k (
y x
z
z y
x
id d
d
d id
d
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:
) (
| ) ˆ (
| )
2
(
,
1
k d g k k
| |
2
| 1 2 |
| 1
| |
2
| 1 2 |
| 1
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?
T
symmetry allowsd gk xˆkx(ky2 kz2) yˆky (kz2 kx2) zˆkz(kx2 ky2)How about pseudo-spin 3/2 ?
This implies f-wave superconductivity
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
) ˆ (
) ˆ (
)
ˆ x( y2 z2 y z2 x2 z x2 y2
k xk k k yk k k zk k k
g
i
i i i
i i i
i
J J J J J J T
k
k ) ( )
(
1 1 2 2s-wave - p-wave mixing
P-wave pairing in YPtBi
T g ) (
like (T i
