Research seminar given to the University of Birmingham Theory Group, England, 14 November 2013.

1. Thermodynamic signatures
of topological transitions
in nodal superconductors
arXiv:1302.2161
Bayan Mazidian1,2 , Jorge Quintanilla2,3
James F. Annett1 , Adrian D. Hillier2
1
University of Bristol
ISIS Facility, STFC Rutherford Appleton Laboratory
3
SEPnet and Hubbard Theory Consortium, University of Kent
2
Birmingham, UK, 14 November 2013
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
1 / 95

2. Anomalous thermodynamic power laws in nodal
superconductors
1
What are they?
2
How to get them
3
An example
4
Take-home message
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
2 / 95

3. Anomalous thermodynamic power laws in nodal
superconductors
1
What are they?
2
How to get them
3
An example
4
Take-home message

4. Power laws in nodal superconductors
Low-temperature specific heat of a superconductor gives information on the
spectrum of low-lying excitations:
Fully gapped
Point nodes
Cv ∼
T3
Line nodes
Cv ∼ T 2
∆
Cv ∼
e −∆/T
This simple idea has been around for a while.1
Widely used to fit experimental data on unconventional superconductors.2
1 Anderson
2 Sigrist,
& Morel (1961), Leggett (1975)
Ueda (’89), Annett (’90), MacKenzie & Maeno (’03)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
4 / 95

5. Linear nodes
It all comes from the density of states:
+
g (E ) ∼ E n−1 ⇒ Cv ∼ T n
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
5 / 95

6. Linear nodes
It all comes from the density of states:
+
g (E ) ∼ E n−1 ⇒ Cv ∼ T n
linear
point node
line node
y
x
∆2 = I1 k|| 2 + k||
k
g (E ) =
2
E2 √
2(2π )2 I1 I2
g (E ) =
n=3
Jorge Quintanilla (Kent and ISIS)
x
∆2 = I1 k|| 2
k
LE √
√
(2π )3 I1 I2
n=2
arXiv:1302.2161
B’ham 2013
5 / 95

7. Linear nodes
It all comes from the density of states:
+
g (E ) ∼ E n−1 ⇒ Cv ∼ T n
linear
point node
line node
y
x
∆2 = I1 k|| 2 + k||
k
g (E ) =
2
E2 √
2(2π )2 I1 I2
x
∆2 = I1 k|| 2
k
g (E ) =
n=3
LE √
√
(2π )3 I1 I2
n=2
Key assumption: linear increase of the gap away from the node
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
5 / 95

8. Shallow nodes
Relax the linear assumption and we also get different exponents:
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
6 / 95

9. Shallow nodes
Relax the linear assumption and we also get different exponents:
shallow
point node
line node
2
y
x
∆2 = I1 (k|| 2 + k|| )2
k
g (E ) =
E √
√
I1 I2
2(2π )2
g (E ) =
n=2
Jorge Quintanilla (Kent and ISIS)
x
∆2 = I1 k|| 4
k
√
L E
1√
(2π )3 I14
I2
n = 1.5
arXiv:1302.2161
B’ham 2013
6 / 95

10. Shallow nodes
Relax the linear assumption and we also get different exponents:
shallow
point node
line node
2
y
x
∆2 = I1 (k|| 2 + k|| )2
k
g (E ) =
E √
√
I1 I2
2(2π )2
x
∆2 = I1 k|| 4
k
g (E ) =
n=2
√
L E
1√
(2π )3 I14
I2
n = 1.5
Shallow point nodes first discussed (speculatively) by Leggett [1979].
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
6 / 95

11. Shallow nodes
Relax the linear assumption and we also get different exponents:
shallow
point node
line node
2
y
x
∆2 = I1 (k|| 2 + k|| )2
k
g (E ) =
E √
√
I1 I2
2(2π )2
x
∆2 = I1 k|| 4
k
g (E ) =
n=2
√
L E
1√
(2π )3 I14
I2
n = 1.5
Shallow point nodes first discussed (speculatively) by Leggett [1979].
A shallow point node may be required by symmetry e.g. the proposed E2u
pairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)].
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
6 / 95

12. Shallow nodes
Relax the linear assumption and we also get different exponents:
shallow
point node
line node
2
y
x
∆2 = I1 (k|| 2 + k|| )2
k
g (E ) =
E √
√
I1 I2
2(2π )2
x
∆2 = I1 k|| 4
k
g (E ) =
n=2
√
L E
1√
(2π )3 I14
I2
n = 1.5
Shallow point nodes first discussed (speculatively) by Leggett [1979].
A shallow point node may be required by symmetry e.g. the proposed E2u
pairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)].
A shallow line node may result at the boundary between gapless and line node
behaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
6 / 95

13. Shallow nodes
Relax the linear assumption and we also get different exponents:
shallow
point node
line node
2
y
x
∆2 = I1 (k|| 2 + k|| )2
k
g (E ) =
E √
√
I 1 I2
2(2π )2
x
∆2 = I1 k|| 4
k
g (E ) =
n=2
√
L E
1√
(2π )3 I14
I2
n = 1.5
Shallow point nodes first discussed (speculatively) by Leggett [1979].
A shallow point node may be required by symmetry e.g. the proposed E2u
pairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)].
A shallow line node may result at the boundary between gapless and line node
behaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
7 / 95

14. Line crossings
A different power law is expected at line crossings
(e.g. d-wave pairing on a spherical Fermi surface):
crossing
of linear line nodes
y
x
∆2 = I1 k|| 2 − k||
k
y
x
or I1 k|| 2 k||
2 2
2
1
√
4
L+ E /I1
1
√
E /I 4
√ 1
(2π )3 I1 I2
E (1+2ln|
g (E )
|)
=
∼ E 0 .8
n = 1.8 (< 2 !!)
+
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
8 / 95

15. Crossing of shallow line nodes
When shallow lines cross we get an even lower exponent:
crossing
of shallow line nodes
y
x
∆2 = I1 k|| 2 − k||
k
y
x
or I1 k|| 4 k||
√
g (E )
=
2 4
4
1 1
8
L+E 4 /I1
1 1
8
E 4 /I1
1√
3I 4 I
(2π ) 1
2
E (1+2ln|
|)
∼ E 0 .4
n = 1.4 *
* c.f. gapless excitations of a Fermi liquid: g (E ) = constant ⇒ n = 1
+
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
9 / 95

16. Numerics
n = d ln Cv /d ln T
4.5
linear point node
shallow point node
linear line node
crossing of linear line nodes
shallow line node
crossing of shallow line nodes
4
3.5
n
3
2.5
2
1.5
1
0
Jorge Quintanilla (Kent and ISIS)
0.05
0.1
0.15
0.2
T / Tc
arXiv:1302.2161
0.25
0.3
0.35
B’ham 2013
10 / 95

17. Anomalous thermodynamic power laws in nodal
superconductors
1
What are they?
2
How to get them
3
An example
4
Take-home message

18. A generic mechanism
∆0
∆1
We propose that shallow nodes will exist generically at topological phase
transitions in superocnductors with multi-component order parameters:
Fermi Sea
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
12 / 95

19. A generic mechanism
∆0
∆1
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
Fermi Sea
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
13 / 95

20. A generic mechanism
∆0
∆1
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
Fermi Sea
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
14 / 95

21. A generic mechanism
∆0
∆1
Linear
nodes
Linear
nodes
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
Fermi Sea
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
15 / 95

22. A generic mechanism
∆0
∆1
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
Fermi Sea
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
16 / 95

23. A generic mechanism
∆0
∆1
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
Fermi Sea
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
17 / 95

24. A generic mechanism
∆0
∆1
Shallow
node
Shallow
node
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
Fermi Sea
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
18 / 95

25. Note: no broken symmetry
Photo: Eddie Hui-Bon-Hoa, www.shiromi.com

Jorge Quintanilla (Kent and ISIS)

arXiv:1302.2161
B’ham 2013
19 / 95

26. Jorge Quintanilla (Kent and ISIS)
Photo: Kenneth G. Libbrecht, snowflakes.com
Photo: Eddie Hui-Bon-Hoa, www.shiromi.com
Note: no broken symmetry
arXiv:1302.2161




B’ham 2013
19 / 95

27. Photo: commons.wikimedia.org
Jorge Quintanilla (Kent and ISIS)
Photo: Kenneth G. Libbrecht, snowflakes.com
Photo: Eddie Hui-Bon-Hoa, www.shiromi.com
Note: no broken symmetry
arXiv:1302.2161






B’ham 2013
19 / 95

28. Photo: commons.wikimedia.org
Jorge Quintanilla (Kent and ISIS)
Photo: Kenneth G. Libbrecht, snowflakes.com
Photo: Eddie Hui-Bon-Hoa, www.shiromi.com
Note: no broken symmetry
arXiv:1302.2161






B’ham 2013
19 / 95

29. Photo: commons.wikimedia.org
Jorge Quintanilla (Kent and ISIS)
Photo: Kenneth G. Libbrecht, snowflakes.com
Photo: Eddie Hui-Bon-Hoa, www.shiromi.com
Note: no broken symmetry
arXiv:1302.2161






B’ham 2013
19 / 95

30. These are topological transitions
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
20 / 95

31. These are topological transitions
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
20 / 95

32. These are topological transitions
G. E. Volovik in “Quantum Analogues: From Phase Transitions to Black Holes and Cosmology“, William G.
Unruh and Ralf Schützhold (Eds.), Springer (2007).
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
20 / 95

33. These are topological transitions
G. E. Volovik in “Quantum Analogues: From Phase Transitions to Black Holes and Cosmology“, William G.
Unruh and Ralf Schützhold (Eds.), Springer (2007).
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
20 / 95

34. These are topological transitions
G. E. Volovik in “Quantum Analogues: From Phase Transitions to Black Holes and Cosmology“, William G.
Unruh and Ralf Schützhold (Eds.), Springer (2007).
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
20 / 95

35. These are topological transitions
G. E. Volovik in “Quantum Analogues: From Phase Transitions to Black Holes and Cosmology“, William G.
Unruh and Ralf Schützhold (Eds.), Springer (2007).
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
20 / 95

36. Anomalous thermodynamic power laws in nodal
superconductors
1
What are they?
2
How to get them
3
An example
4
Take-home message

37. Singlet-triplet mixing in noncentrosymmetric
superconductors
Non-centrosymmetric superconductors are the multi-component order
parameter supercondcutors par excellence:
 0  0  dx  id y
ˆ
k  
 
 0 0   dz
[ 0(k) even ]
singlet
฀
dz 

dx  id y 
triplet
[ d(k) odd ]
3 Batkova
et al. JPCM (2010)
et al. PRB (2007)
5 Adrian D. Hillier, JQ and R. Cywinski PRL (2009)
6 Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)
7 Bauer et al. PRL (2004)
4 Zuev
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
22 / 95

38. Singlet-triplet mixing in noncentrosymmetric
superconductors
Non-centrosymmetric superconductors are the multi-component order
parameter supercondcutors par excellence:
 0  0  dx  id y
ˆ
k  
 
 0 0   dz
[ 0(k) even ]
singlet
dz 

dx  id y 
triplet
[ d(k) odd ]
In practice, there is a varied phenomenology:
฀
3 Batkova
et al. JPCM (2010)
et al. PRB (2007)
5 Adrian D. Hillier, JQ and R. Cywinski PRL (2009)
6 Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)
7 Bauer et al. PRL (2004)
4 Zuev
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
22 / 95

39. Singlet-triplet mixing in noncentrosymmetric
superconductors
Non-centrosymmetric superconductors are the multi-component order
parameter supercondcutors par excellence:
 0  0  dx  id y
ˆ
k  
 
 0 0   dz
[ 0(k) even ]
singlet
dz 

dx  id y 
triplet
[ d(k) odd ]
In practice, there is a varied phenomenology:
฀
Some are conventional (singlet) superconductors:
BaPtSi33 , Re3W4 ,...
Others seem to be correlated, purely triplet superconductors:
LaNiC25 (c.f. centrosymmetric LaNiGa26 ) + , CePtr3Si (?)
+
7
3 Batkova
et al. JPCM (2010)
et al. PRB (2007)
5 Adrian D. Hillier, JQ and R. Cywinski PRL (2009)
6 Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)
7 Bauer et al. PRL (2004)
4 Zuev
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
22 / 95

40. Li2 Pdx Pt3−x B: tunable singlet-triplet mixing
The Li2 Pdx Pt3−x B family (0 ≤ x ≤ 3; cubic point group O) provides a tunable
realisation of this singlet-triplet mixing:
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
23 / 95

41. Li2 Pdx Pt3−x B: tunable singlet-triplet mixing
The Li2 Pdx Pt3−x B family (0 ≤ x ≤ 3; cubic point group O) provides a tunable
realisation of this singlet-triplet mixing:
Pd is a lighter element with weak spin-orbit coupling (Tc ∼ 7K)
Pt is a heavier element with strong spin orbit coupling (Tc ∼ 2.7K)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
23 / 95

42. Li2 Pdx Pt3−x B: tunable singlet-triplet mixing
The Li2 Pdx Pt3−x B family (0 ≤ x ≤ 3; cubic point group O) provides a tunable
realisation of this singlet-triplet mixing:
Pd is a lighter element with weak spin-orbit coupling (Tc ∼ 7K)
Pt is a heavier element with strong spin orbit coupling (Tc ∼ 2.7K)
The series goes from fully-gapped
(x = 3) to nodal (x = 0):
H.Q. Yuan et al.,
Phys. Rev. Lett. 97, 017006 (2006).
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
23 / 95

43. Li2 Pdx Pt3−x B: tunable singlet-triplet mixing
The Li2 Pdx Pt3−x B family (0 ≤ x ≤ 3; cubic point group O) provides a tunable
realisation of this singlet-triplet mixing:
Pd is a lighter element with weak spin-orbit coupling (Tc ∼ 7K)
Pt is a heavier element with strong spin orbit coupling (Tc ∼ 2.7K)
The series goes from fully-gapped
(x = 3) to nodal (x = 0):
NMR suggests nodal state a triplet:
H.Q. Yuan et al.,
Phys. Rev. Lett. 97, 017006 (2006).
Jorge Quintanilla (Kent and ISIS)
M.Nishiyama et al.,
Phys. Rev. Lett. 98, 047002 (2007)
arXiv:1302.2161
B’ham 2013
23 / 95

44. Li2 Pdx Pt3−x B: Phase diagram
Bogoliubov Hamiltonian with Rashba spin-orbit coupling:
H (k) =
ˆ
h (k)
ˆ † (k)
∆
ˆ
∆ (k)
ˆ T (−k)
−h
ˆ
h (k) = ε k I + γk · σ
ˆ
ˆ ˆ
∆ (k) = [∆0 (k) + d (k) · σ ] i σy (most general gap matrix)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
24 / 95

45. Li2 Pdx Pt3−x B: Phase diagram
Bogoliubov Hamiltonian with Rashba spin-orbit coupling:
H (k) =
ˆ
h (k)
ˆ † (k)
∆
ˆ
∆ (k)
ˆ T (−k)
−h
ˆ
h (k) = ε k I + γk · σ
ˆ
ˆ ˆ
∆ (k) = [∆0 (k) + d (k) · σ ] i σy (most general gap matrix)
Assuming |ε k | ≫ |γk | ≫ |d (k)| the quasi-particle spectrum is

 ± (ε − µ + |γ |)2 + (∆ (k) + |d (k)|)2 ; and
0
k
k
.
E =
 ± (ε − µ − |γ |)2 + (∆ (k) − |d (k)|)2
0
k
k
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
24 / 95

46. Li2 Pdx Pt3−x B: Phase diagram
Bogoliubov Hamiltonian with Rashba spin-orbit coupling:
H (k) =
ˆ
h (k)
ˆ † (k)
∆
ˆ
∆ (k)
ˆ T (−k)
−h
ˆ
h (k) = ε k I + γk · σ
ˆ
ˆ ˆ
∆ (k) = [∆0 (k) + d (k) · σ ] i σy (most general gap matrix)
Assuming |ε k | ≫ |γk | ≫ |d (k)| the quasi-particle spectrum is

 ± (ε − µ + |γ |)2 + (∆ (k) + |d (k)|)2 ; and
0
k
k
.
E =
 ± (ε − µ − |γ |)2 + (∆ (k) − |d (k)|)2
0
k
k
Take most symmetric (A1 ) irreducible representation:
+
∆0 (k) = ∆0
d(k) = ∆0 × {
2
2
2
2
2
2
A (x ) (kx , ky , kz ) − B (x ) kx ky + kz , ky kz + kx , kz kx + ky }
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
24 / 95

47. Li2 Pdx Pt3−x B: Phase diagram
Treat A and B as independent tuning parameters and study quasiparticle
spectrum. We find a very rich phase diagram with topollogically-distinct phases:8
8 C. Beri, PRB (2010); A. Schnyder, S. Ryu, PRB(R) (2011); A. Schnyder et al.,
PRB (2012); B. Mazidian, JQ, A.D. Hillier, J.F. Annett, arXiv:1302.2161.
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
25 / 95

48. Li2 Pdx Pt3−x B: Phase diagram
We find a very rich phase diagram with topollogically-distinct phases.9
9 C. Beri, PRB (2010); A. Schnyder, S. Ryu, PRB(R) (2011); A. Schnyder et al.,
PRB (2012); B. Mazidian, JQ, A.D. Hillier, J.F. Annett, arXiv:1302.2161.
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
26 / 95

49. Li2 Pdx Pt3−x B: Phase diagram
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
27 / 95

50. Li2 Pdx Pt3−x B: Phase diagram
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
28 / 95

51. Li2 Pdx Pt3−x B: Phase diagram
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
29 / 95

52. Li2 Pdx Pt3−x B: Phase diagram
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
30 / 95

53. Detecting the topological transitions
4
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
3
7
B’ham 2013
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54. Detecting the topological transitions
4
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
3
7
B’ham 2013
32 / 95

55. Li2 Pdx Pt3−x B: predicted specific heat power-laws
4
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
3
B’ham 2013
33 / 95

56. Li2 Pdx Pt3−x B: predicted specific heat power-laws
5
n=2
j
4
n=2
3
n = 1.8
11
n = 1.4
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
34 / 95

57. Li2 Pdx Pt3−x B: predicted specific heat power-laws
3
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
35 / 95

58. Li2 Pdx Pt3−x B: predicted specific heat power-laws
3
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
36 / 95

59. Li2 Pdx Pt3−x B: predicted specific heat power-laws
5
n=2
j
4
n=2
3
n = 1.8
11
n = 1.4
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
37 / 95

60. Anomalous power laws throughout the phase diagram
Does the observation of these effects require fine-tuning?
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
38 / 95

61. Anomalous power laws throughout the phase diagram
Does the observation of these effects require fine-tuning?
Let’s put these curves on a density plot:
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
38 / 95

62. Anomalous power laws throughout the phase diagram
Does the observation of these effects require fine-tuning?
Let’s put these curves on a density plot:
A=3
0.25
2.2
2.1
0.2
2
T/Tc
0.15
1.9
0.1
1.8
0.05
1.7
0
1.6
3.6
3.8
4
4.2
4.4
B
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
38 / 95

63. Anomalous power laws throughout the phase diagram
Does the observation of these effects require fine-tuning?
Let’s put these curves on a density plot:
A=3
0.25
2.2
2.1
0.2
2
T/Tc
0.15
1.9
0.1
1.8
0.05
1.7
0
1.6
3.6
3.8
4
4.2
4.4
B
The conventional exponent (n = 2 in this example) is only seen below a
temperature scale that converges to zero at the transition
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
38 / 95

64. Anomalous power laws throughout the phase diagram
Does the observation of these effects require fine-tuning?
Let’s put these curves on a density plot:
A=3
0.25
2.2
2.1
0.2
2
T/Tc
0.15
1.9
0.1
1.8
0.05
1.7
0
1.6
3.6
3.8
4
4.2
4.4
B
The conventional exponent (n = 2 in this example) is only seen below a
temperature scale that converges to zero at the transition
The anomalous exponent (here n = 1.8) is seen everywhere else
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
38 / 95

65. Anomalous power laws throughout the phase diagram
Does the observation of these effects require fine-tuning?
Let’s put these curves on a density plot:
A=3
0.25
2.2
2.1
0.2
2
T/Tc
0.15
1.9
0.1
1.8
0.05
1.7
0
1.6
3.6
3.8
4
4.2
4.4
B
The conventional exponent (n = 2 in this example) is only seen below a
temperature scale that converges to zero at the transition
The anomalous exponent (here n = 1.8) is seen everywhere else ⇒
the influence of the topo transition extends throughout the phase diagram
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
38 / 95

66. Anomalous power laws throughout the phase diagram
Does the observation of these effects require fine-tuning?
Let’s put these curves on a density plot:
A=3
0.25
2.2
2.1
0.2
2
T/Tc
0.15
1.9
0.1
1.8
0.05
1.7
0
1.6
3.6
3.8
4
4.2
4.4
B
The conventional exponent (n = 2 in this example) is only seen below a
temperature scale that converges to zero at the transition
The anomalous exponent (here n = 1.8) is seen everywhere else ⇒
the influence of the topo transition extends throughout the phase diagram
c.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
38 / 95

67. Anomalous thermodynamic power laws in nodal
superconductors
1
What are they?
2
How to get them
3
An example
4
Take-home message

68. Topological transitions in nodal superconductors
have clear signatures in bulk thermodynamic properties.
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
40 / 95

69. Topological transitions in nodal superconductors
have clear signatures in bulk thermodynamic properties.
THANKS!
www.cond-mat.org
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
40 / 95

70. Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
41 / 95

71. Power laws in nodal superconductors
Let’s remember where this came from:
dS
dT
Cv = T
∼ T −2
=
1
2kB T 2


∑  Ek − T

k

dEk 
 Ek sech2 Ek
dT 
2kB T
≈0
dEg (E ) E 2 e −E /kB T at low T
g (E ) ∼ E n−1 ⇒ Cv ∼ T n
≈4e −Ek /KB T
d ǫǫ2+n−1 e −ǫ
a number
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
41 / 95

72. Power laws in nodal superconductors
k_
|
Ek =
≈
∆(k||x,k||y)
2
ǫk + ∆ 2
k
y
2
x
I2 k⊥ + ∆ k|| , k||
k||y
2
on the Fermi surface
k||x
Compute density of states:
g (E ) =
δ(Ek − E )dkx dky dkz
back
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
42 / 95

73. Shallow line nodes in pnictides
back
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
43 / 95

74. Logarithm ⇒ power law (n − 1 = 0.8)
The power-law expression is asymptotically very good at E → 0:
back
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
44 / 95

75. Logarithm ⇒ power law (n − 1 = 0.4)
The power-law expression is asymptotically very good at E → 0:
back
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
45 / 95

76. Symmetry of pairing in NCS
Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
46 / 95

77. Symmetry of pairing in NCS
Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
47 / 95

78. Symmetry of pairing in NCS
Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Singlet and triplet representations of SO(3):
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
48 / 95

79. Symmetry of pairing in NCS
Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Singlet and triplet representations of SO(3):
Γns = - (Γns)T , Γnt = + (Γnt)T
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
49 / 95

80. Symmetry of pairing in NCS
Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Singlet and triplet representations of SO(3):
Γns = - (Γns)T , Γnt = + (Γnt)T
I pose Pauli s exclusio pri ciple:
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
50 / 95

81. Symmetry of pairing in NCS
Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Singlet and triplet representations of SO(3):
 , ' k   ', k
Γns = - (Γns)T , Γnt = + (Γnt)T
I pose Pauli s exclusio pri ciple:
฀
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
51 / 95

82. Symmetry of pairing in NCS
Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Singlet and triplet representations of SO(3):
 , ' k   ', k
Γns = - (Γns)T , Γnt = + (Γnt)T
I pose Pauli s exclusio pri ciple:

฀
ˆ
 k
Jorge Quintanilla (Kent and ISIS)
either singlet
฀
arXiv:1302.2161
B’ham 2013
52 / 95

83. Symmetry of pairing in NCS
Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Singlet and triplet representations of SO(3):
 , ' k   ', k
Γns = - (Γns)T , Γnt = + (Γnt)T
ˆ
 , ' k    0 k i y
I pose Pauli s exclusio pri ciple:

฀
ˆ
 k
Jorge Quintanilla (Kent and ISIS)
either singlet
฀
arXiv:1302.2161
B’ham 2013
53 / 95

84. Symmetry of pairing in NCS
Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Singlet and triplet representations of SO(3):
 , ' k   ', k
Γns = - (Γns)T , Γnt = + (Γnt)T
ˆ
 , ' k    0 k i y
I pose Pauli s exclusio pri ciple:

ˆ
 k
either singlet
฀
or triplet
฀
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
54 / 95

85. Symmetry of pairing in NCS
Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Singlet and triplet representations of SO(3):
 , ' k   ', k
Γns = - (Γns)T , Γnt = + (Γnt)T
ˆ
 , ' k    0 k i y
I pose Pauli s exclusio pri ciple:

ˆ
 k
either singlet
฀
or triplet
฀
Jorge Quintanilla (Kent and ISIS)
ˆ ˆ
 , ' k   dk .σi y
arXiv:1302.2161
B’ham 2013
55 / 95

86. Symmetry of pairing in NCS
The role of spin-orbit coupling (SOC)
 , ' k   spin ' orbit k 
 ,

Gap function may have both singlet and triplet components
 0
ˆ
k   
  0
 0   d x  id y

0   dz


d x  id y 

dz
• However, if we have a centre of inversion
basis functions either even or odd under inversion
 still have either singlet or triplet pairing (at Tc)
• No centre of inversion: may have singlet and triplet (even at Tc)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
back
56 / 95

87. LaNiC2 – a weakly-correlated, paramagnetic
superconductor?
W. H. Lee et al., Physica C 266, 138 (1996)
V. K. Pecharsky, L. L. Miller, and Zy, Physical Review B 58, 497 (1998)
specific heat
susceptibility
c 0 = 22.2 10-6 emu/mol
 0 = 6.5 mJ/mol K2
Tc=2.7 K
Jorge Quintanilla (Kent and ISIS)
ΔC/TC=1.26
(BCS: 1.43)
arXiv:1302.2161
B’ham 2013
57 / 95

88. ISIS
muSR
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
58 / 95

89. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Zero field muon spin relaxation
_

e
sample
e

forward
detector
backward
detector
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
59 / 95

90. Relaxation due to electronic moments
_

e
(longitudinal)

sample
e
forward
detector
backward
detector
Moment
size
~ 0.1G
(~ 0.01μB)
Timescale:
-4
>
~ 10 s
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
60 / 95

91. Relaxation due to electronic moments
_

e
(longitudinal)

sample
e
forward
detector
backward
detector
Moment
size
~ 0.1G
(~ 0.01μB)
Timescale:
-4
>
~ 10 s
Spontaneous, quasi-static fields appearing at Tc
⇒ superconducting state breaks time-reversal symmetry
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
[ c.f. Sr2RuO4 - Luke et al., Nature (1998) ]
arXiv:1302.2161
B’ham 2013
61 / 95

92. LaNiC2 is a non-ceontrsymmetric superconductor
Neutron diffraction
35000
30000
Intensity (arb units)
25000
Data from
D1B @ ILL
20000
15000
10000
5000
0
30
Orthorhombic Amm2 C2v
40
50
60
2  
70
80
o
Note no inversion centre.
C.f. CePt3Si (1), Li2Pt3B & Li2Pd3B
(2),
a=3.96 Å
b=4.58 Å
c=6.20 Å
...
(1) Bauer et al. PRL’04 (2) Yuan et al. PRL’06
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
62 / 95

93. Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
63 / 95

94. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Character table
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
64 / 95

95. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Character table
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
65 / 95

96. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Character table
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
66 / 95

97. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Character table
180o
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
67 / 95

98. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Character table
C2v
Symmetries and
their characters
Irreducible
E C2 v
representation
’v
Sample basis
functions
Even
Odd
A1
1 1
1
1
1
Z
A2
1 1
-1
-1
XY
XYZ
B1
1 -1
1
-1
XZ
X
B2
1 -1 -1
1
YZ
Y
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
68 / 95

99. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Character table
C2v
Symmetries and
their characters
Irreducible
E C2 v
representation
’v
Sample basis
functions
Even
Odd
A1
1 1
1
1
1
Z
A2
1 1
-1
-1
XY
XYZ
B1
1 -1
1
-1
XZ
X
B2
1 -1 -1
1
YZ
Y
These must be combined with the singlet and triplet
representations of SO(3).
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
69 / 95

100. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Possible order parameters
SO(3)xC2v
1A
1
1A
2
1B
1
1B
2
Gap function
(unitary)
Gap function
(non-unitary)
(k)=kxkY
-
(k)=1
(k)=kXkZ
-
(k)=kYkZ
-
3A
1
d(k)=(0,0,1)kZ
d(k)=(1,i,0)kZ
3A
3B
3B
2
d(k)=(0,0,1)kXkYkZ
d(k)=(1,i,0)kXkYkZ
1
d(k)=(0,0,1)kX
d(k)=(1,i,0)kX
2
d(k)=(0,0,1)kY
d(k)=(1,i,0)kY
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
70 / 95

101. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Possible order parameters
SO(3)xC2v
1A
1
1A
2
1B
1
1B
2
Gap function
(unitary)
Gap function
(non-unitary)
(k)=kxkY
-
(k)=1
(k)=kXkZ
-
(k)=kYkZ
-
3A
1
d(k)=(0,0,1)kZ
d(k)=(1,i,0)kZ
3A
3B
3B
2
d(k)=(0,0,1)kXkYkZ
d(k)=(1,i,0)kXkYkZ
1
d(k)=(0,0,1)kX
d(k)=(1,i,0)kX
2
d(k)=(0,0,1)kY
d(k)=(1,i,0)kY
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
71 / 95

102. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Possible order parameters
SO(3)xC2v
1A
1
1A
2
1B
1
1B
2
Gap function
(unitary)
Gap function
(non-unitary)
(k)=kxkY
-
(k)=1
(k)=kXkZ
-
(k)=kYkZ
-
3A
1
d(k)=(0,0,1)kZ
d(k)=(1,i,0)kZ
3A
3B
3B
2
d(k)=(0,0,1)kXkYkZ
d(k)=(1,i,0)kXkYkZ
1
d(k)=(0,0,1)kX
d(k)=(1,i,0)kX
2
d(k)=(0,0,1)kY
d(k)=(1,i,0)kY
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
72 / 95

103. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Possible order parameters
SO(3)xC2v
1A
1
1A
2
1B
1
1B
2
Gap function
(unitary)
Gap function
(non-unitary)
(k)=kxkY
-
(k)=1
(k)=kXkZ
-
(k)=kYkZ
-
3A
1
d(k)=(0,0,1)kZ
d(k)=(1,i,0)kZ
3A
3B
3B
2
d(k)=(0,0,1)kXkYkZ
d(k)=(1,i,0)kXkYkZ
1
d(k)=(0,0,1)kX
d(k)=(1,i,0)kX
2
d(k)=(0,0,1)kY
d(k)=(1,i,0)kY
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
Non-unitary
d x d* ≠ 0
arXiv:1302.2161
B’ham 2013
73 / 95

104. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Possible order parameters
SO(3)xC2v
1A
1
1A
2
1B
1
1B
2
Gap function
(unitary)
(k)=1
(k)=kxkY
(k)=kXkZ
(k)=kYkZ
Gap function
(non-unitary)
breaks only SO(3) x U(1) x T
-
3A
1
d(k)=(0,0,1)kZ
d(k)=(1,i,0)kZ
3A
3B
3B
2
d(k)=(0,0,1)kXkYkZ
d(k)=(1,i,0)kXkYkZ
1
d(k)=(0,0,1)kX
d(k)=(1,i,0)kX
2
d(k)=(0,0,1)kY
d(k)=(1,i,0)kY
* C.f. Li2Pd3B & Li2Pt3B,
H. Q. Yua et al. P‘L 0
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS)
*
Non-unitary
d x d* ≠ 0
arXiv:1302.2161
B’ham 2013
74 / 95

105. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Non-unitary pairing
Spin-up superfluid
coexisting with spindown Fermi liquid.
ˆ 
   
 0

0  0 0 

 or 
0   0   
 

C.f.
The A1 phase of
liquid 3He.
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
75 / 95

106. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = [SO(3)×Gc]×U(1)×T
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
76 / 95

107. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = [SO(3)×Gc]×U(1)×T
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
77 / 95

108. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = [SO(3)×Gc]×U(1)×T
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
78 / 95

109. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = [SO(3)×Gc]×U(1)×T
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
79 / 95

110. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = [SO(3)×Gc]×U(1)×T
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
80 / 95

111. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = Gc,J×U(1)×T
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
81 / 95

112. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = Gc,J×U(1)×T
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
82 / 95

113. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = Gc,J×U(1)×T
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
83 / 95

114. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
z
E.g. reflection through a vertical
plane perpendicular to the y axis:
 v, J  I C2y, J
y
x
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
84 / 95

115. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
z
E.g. reflection through a vertical
plane perpendicular to the y axis:
 v, J  I C2y, J
y
x
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
85 / 95

116. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
z
E.g. reflection through a vertical
plane perpendicular to the y axis:
 v, J  I C2y, J
y
x
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
86 / 95

117. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
z
E.g. reflection through a vertical
plane perpendicular to the y axis:
 v, J  I C2y, J
y
x
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
87 / 95

118. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
z
E.g. reflection through a vertical
plane perpendicular to the y axis:
 v, J  I C2y, J
This affects d(k) (a vector under
spin rotations).
y
x
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
88 / 95

119. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
z
E.g. reflection through a vertical
plane perpendicular to the y axis:
 v, J  I C2y, J
This affects d(k) (a vector under
spin rotations).
It does not affect 0(k) (a scalar).
y
x
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
89 / 95

120. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
C2v,Jno t
A1
A2
B1
B2
Gap function,
singlet component
(k) = A
Gap function,
triplet component
d(k) = (Bky,Ckx,Dkxkykz)
 (k) = AkxkY
d(k) = (Bkx,Cky,Dkz)
 (k) = AkYkZ
d(k) = (Bkz, Ckxkykz,Dkx)
 (k) = AkXkZ
d(k) = (Bkxkykz,Ckz,Dky)
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
90 / 95

121. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
C2v,Jno t
A1
A2
B1
B2
Gap function,
singlet component
(k) = A
Gap function,
triplet component
d(k) = (Bky,Ckx,Dkxkykz)
 (k) = AkxkY
d(k) = (Bkx,Cky,Dkz)
 (k) = AkYkZ
d(k) = (Bkz, Ckxkykz,Dkx)
 (k) = AkXkZ
d(k) = (Bkxkykz,Ckz,Dky)
None of these break time-reversal symmetry!
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
91 / 95

122. Virginia Tech, 18 March 2011
blogs.kent.ac.uk/strongcorrelations
Relativistic and non-relativistic
instabilities: a complex relationship
A1
singlet
Pairing
instabilities
A2
1A
1
non-unitary
triplet
pairing
instabilities
unitary
triplet
pairing
instabilities
1A
2
B1
B2
1B
1
3B (b)
1
3B (b)
2
3A (a)
1
1B
2
3A (b)
1
3A (b)
2
3A (a)
2
3B (a)
1
3B (a)
2
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
92 / 95

123. LaNiGa2 - a centrosymmetric cousin of LaNiC2
A similar muSR effect is seen in centrosymmetric LaNiGa2:
[A. D. Hillier, JQ, Mazidian, Annett & Cywinski, PRL 109, 097001 (2012)]
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
93 / 95

124. LaNiGa2 - a centrosymmetric cousin of LaNiC2
A similar muSR effect is seen in centrosymmetric LaNiGa2:
[A. D. Hillier, JQ, Mazidian, Annett & Cywinski, PRL 109, 097001 (2012)]
Simlarly to LaNiC2, we find only 1D irreps ⇒ non-unitary triplet pairing.
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
93 / 95

125. LaNiGa2 - a centrosymmetric cousin of LaNiC2
A similar muSR effect is seen in centrosymmetric LaNiGa2:
[A. D. Hillier, JQ, Mazidian, Annett & Cywinski, PRL 109, 097001 (2012)]
Simlarly to LaNiC2, we find only 1D irreps ⇒ non-unitary triplet pairing.
Lack of inversion symmetry seems to be a red herring in the case of LaNiC2.
back
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
93 / 95

126. Li2 Pdx Pt3−x B: Phase diagram
Bogoliubov Hamiltonian with Rashba spin-orbit coupling:
H (k) =
h (k)
∆ † (k)
∆ (k)
−hT (−k)
h (k) = ε k I + γk · σ
Assuming |ε k | ≫ |γk | ≫ |d (k)| the quasi-particle spectrum is

 ± (ε − µ + |γ |)2 + (∆ + |d (k )|)2 ; and
0
k
k
.
E =
 ± (ε − µ − |γ |)2 + (∆ − |d (k )|)2
0
k
k
Take the most symmetric (A1 ) irreducible representation
d(k)/∆0 = A (X , Y , Z ) − B X Y 2 + Z 2 , Y Z 2 + X 2 , Z X 2 + Y 2
back
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
94 / 95

127. Li2 Pdx Pt3−x B:
order parameter
back
Jorge Quintanilla (Kent and ISIS)
arXiv:1302.2161
B’ham 2013
95 / 95