thermodynamic signatures of topological transitions in nodal superconductors
DESCRIPTION
Research seminar given to the University of Birmingham Theory Group, England, 14 November 2013.TRANSCRIPT
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Thermodynamic signaturesof topological transitionsin nodal superconductors
arXiv:1302.2161
Bayan Mazidian1,2, Jorge Quintanilla2,3
James F. Annett1, Adrian D. Hillier2
1University of Bristol2ISIS Facility, STFC Rutherford Appleton Laboratory
3SEPnet and Hubbard Theory Consortium, University of Kent
Birmingham, UK, 14 November 2013
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 1 / 95
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Anomalous thermodynamic power laws in nodalsuperconductors
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
![Page 3: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/3.jpg)
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
![Page 4: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/4.jpg)
Power laws in nodal superconductors
Low-temperature specific heat of a superconductor gives information on thespectrum of low-lying excitations:
Fully gapped Point nodes Line nodes
Cv ∼ e−∆/T Cv ∼ T 3 Cv ∼ T 2
∆
This simple idea has been around for a while.1
Widely used to fit experimental data on unconventional superconductors.2
1Anderson & Morel (1961), Leggett (1975)2Sigrist, Ueda (’89), Annett (’90), MacKenzie & Maeno (’03)Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 4 / 95
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Linear nodes
It all comes from the density of states: +
g (E ) ∼ En−1 ⇒ Cv ∼ T n
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 5 / 95
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Linear nodes
It all comes from the density of states: +
g (E ) ∼ En−1 ⇒ Cv ∼ T n
linearpoint node line node
∆2k = I1
(
kx||
2 + ky
||2)
∆2k = I1kx
||2
g(E ) = E2
2(2π)2I1√
I2g(E ) = LE
(2π)3√
I1√
I2
n = 3 n = 2
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 5 / 95
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Linear nodes
It all comes from the density of states: +
g (E ) ∼ En−1 ⇒ Cv ∼ T n
linearpoint node line node
∆2k = I1
(
kx||
2 + ky
||2)
∆2k = I1kx
||2
g(E ) = E2
2(2π)2I1√
I2g(E ) = LE
(2π)3√
I1√
I2
n = 3 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
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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
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Shallow nodes
Relax the linear assumption and we also get different exponents:
shallowpoint node line node
∆2k = I1(kx
||2 + k
y
||2)2 ∆2
k = I1kx||
4
g(E ) = E
2(2π)2√
I1√
I2g(E ) = L
√E
(2π)3I14
1
√I2
n = 2 n = 1.5
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 6 / 95
![Page 10: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/10.jpg)
Shallow nodes
Relax the linear assumption and we also get different exponents:
shallowpoint node line node
∆2k = I1(kx
||2 + k
y
||2)2 ∆2
k = I1kx||
4
g(E ) = E
2(2π)2√
I1√
I2g(E ) = L
√E
(2π)3I14
1
√I2
n = 2 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
![Page 11: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/11.jpg)
Shallow nodes
Relax the linear assumption and we also get different exponents:
shallowpoint node line node
∆2k = I1(kx
||2 + k
y
||2)2 ∆2
k = I1kx||
4
g(E ) = E
2(2π)2√
I1√
I2g(E ) = L
√E
(2π)3I14
1
√I2
n = 2 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
![Page 12: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/12.jpg)
Shallow nodes
Relax the linear assumption and we also get different exponents:
shallowpoint node line node
∆2k = I1(kx
||2 + k
y
||2)2 ∆2
k = I1kx||
4
g(E ) = E
2(2π)2√
I1√
I2g(E ) = L
√E
(2π)3I14
1
√I2
n = 2 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 nodebehaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 6 / 95
![Page 13: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/13.jpg)
Shallow nodes
Relax the linear assumption and we also get different exponents:
shallowpoint node line node
∆2k = I1(kx
||2 + k
y
||2)2 ∆2
k = I1kx||
4
g(E ) = E
2(2π)2√
I1√
I2g(E ) = L
√E
(2π)3I14
1
√I2
n = 2 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 nodebehaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 7 / 95
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Line crossings
A different power law is expected at line crossings(e.g. d-wave pairing on a spherical Fermi surface):
crossingof linear line nodes
∆2k = I1
(
kx||
2 − ky
||2)2
or I1kx||
2ky
||2
g(E ) =
E (1+2ln| L+√
E/I
141
√E/I
141
|)
(2π)3√
I1I2
∼ E0.8
n = 1.8 (< 2 !!)
+
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 8 / 95
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Crossing of shallow line nodes
When shallow lines cross we get an even lower exponent:
crossingof shallow line nodes
∆2k = I1
(
kx||
2 − ky
||2)4
or I1kx||
4ky
||4
g (E ) =
√E (1+2ln| L+E
14 /I
181
E14 /I
181
|)
(2π)3I14
1
√I2
∼ E0.4
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
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Numerics
n = d ln Cv /d ln T
1
1.5
2
2.5
3
3.5
4
4.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
n
T / Tc
linear point nodeshallow point node
linear line nodecrossing of linear line nodes
shallow line nodecrossing of shallow line nodes
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 10 / 95
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Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
![Page 18: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/18.jpg)
A generic mechanism
We propose that shallow nodes will exist generically at topological phase
transitions in superocnductors with multi-component order parameters:
∆ 0
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 12 / 95
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A generic mechanism
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 13 / 95
![Page 20: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/20.jpg)
A generic mechanism
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 14 / 95
![Page 21: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/21.jpg)
A generic mechanism
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Lin
ear
nodes
Lin
ear
nodes
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 15 / 95
![Page 22: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/22.jpg)
A generic mechanism
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 16 / 95
![Page 23: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/23.jpg)
A generic mechanism
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 17 / 95
![Page 24: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/24.jpg)
A generic mechanism
We propose that shallow nodes will exist generically at quantum phase
transitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Shallow
node
Shallow
node
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 18 / 95
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Note: no broken symmetry
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Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 19 / 95
![Page 26: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/26.jpg)
Note: no broken symmetry
Ph
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Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 19 / 95
![Page 27: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/27.jpg)
Note: no broken symmetry
Ph
oto
: E
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Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 19 / 95
![Page 28: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/28.jpg)
Note: no broken symmetry
Ph
oto
: E
dd
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ui-
Bo
n-H
oa
, w
ww
.sh
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mi.
com
Ph
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en
ne
th G
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t, s
no
wfl
ak
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Ph
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: co
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s.w
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Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 19 / 95
![Page 29: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/29.jpg)
Note: no broken symmetry
Ph
oto
: E
dd
ie H
ui-
Bo
n-H
oa
, w
ww
.sh
iro
mi.
com
Ph
oto
: K
en
ne
th G
. Li
bb
rech
t, s
no
wfl
ak
es.
com
Ph
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: co
mm
on
s.w
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Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 19 / 95
![Page 30: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/30.jpg)
These are topological transitions
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 20 / 95
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These are topological transitions
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 20 / 95
![Page 32: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/32.jpg)
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
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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
![Page 34: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/34.jpg)
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
![Page 35: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/35.jpg)
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
![Page 36: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/36.jpg)
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
![Page 37: Thermodynamic signatures of topological transitions in nodal superconductors](https://reader033.vdocuments.net/reader033/viewer/2022042521/55836d16d8b42a360c8b514b/html5/thumbnails/37.jpg)
Singlet-triplet mixing in noncentrosymmetricsuperconductors
Non-centrosymmetric superconductors are the multi-component orderparameter supercondcutors par excellence:
ˆ k 0 00 0
dx idy dz
dz dx idy
singlet
[ 0(k) even ]
triplet
[ d(k) odd ]
3Batkova et al. JPCM (2010)4Zuev et al. PRB (2007)5Adrian D. Hillier, JQ and R. Cywinski PRL (2009)6Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)7Bauer et al. PRL (2004)Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 22 / 95
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Singlet-triplet mixing in noncentrosymmetricsuperconductors
Non-centrosymmetric superconductors are the multi-component orderparameter supercondcutors par excellence:
ˆ k 0 00 0
dx idy dz
dz dx idy
singlet
[ 0(k) even ]
triplet
[ d(k) odd ]
In practice, there is a varied phenomenology:
3Batkova et al. JPCM (2010)4Zuev et al. PRB (2007)5Adrian D. Hillier, JQ and R. Cywinski PRL (2009)6Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)7Bauer et al. PRL (2004)Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 22 / 95
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Singlet-triplet mixing in noncentrosymmetricsuperconductors
Non-centrosymmetric superconductors are the multi-component orderparameter supercondcutors par excellence:
ˆ k 0 00 0
dx idy dz
dz dx idy
singlet
[ 0(k) even ]
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
3Batkova et al. JPCM (2010)4Zuev et al. PRB (2007)5Adrian D. Hillier, JQ and R. Cywinski PRL (2009)6Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)7Bauer et al. PRL (2004)Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 22 / 95
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Li2PdxPt3−xB: tunable singlet-triplet mixing
The Li2Pdx Pt3−x B family (0 ≤ x ≤ 3; cubic point group O) provides a tunablerealisation of this singlet-triplet mixing:
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 23 / 95
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Li2PdxPt3−xB: tunable singlet-triplet mixing
The Li2Pdx Pt3−x B family (0 ≤ x ≤ 3; cubic point group O) provides a tunablerealisation 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
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Li2PdxPt3−xB: tunable singlet-triplet mixing
The Li2Pdx Pt3−x B family (0 ≤ x ≤ 3; cubic point group O) provides a tunablerealisation 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).
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Li2PdxPt3−xB: tunable singlet-triplet mixing
The Li2Pdx Pt3−x B family (0 ≤ x ≤ 3; cubic point group O) provides a tunablerealisation 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).
NMR suggests nodal state a triplet:
M.Nishiyama et al.,
Phys. Rev. Lett. 98, 047002 (2007)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 23 / 95
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Li2PdxPt3−xB: Phase diagram
Bogoliubov Hamiltonian with Rashba spin-orbit coupling:
H(k) =
(h(k) ∆(k)
∆†(k) −hT (−k)
)
h(k) = εkI + γk · σ
∆ (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
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Li2PdxPt3−xB: Phase diagram
Bogoliubov Hamiltonian with Rashba spin-orbit coupling:
H(k) =
(h(k) ∆(k)
∆†(k) −hT (−k)
)
h(k) = εkI + γk · σ
∆ (k) = [∆0 (k) + d (k) · σ] i σy (most general gap matrix)
Assuming |εk| ≫ |γk| ≫ |d (k)| the quasi-particle spectrum is
E =
±√
(εk − µ + |γk|)2 + (∆0 (k) + |d (k)|)2; and
±√
(εk − µ − |γk|)2 + (∆0 (k)− |d (k)|)2.
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 24 / 95
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Li2PdxPt3−xB: Phase diagram
Bogoliubov Hamiltonian with Rashba spin-orbit coupling:
H(k) =
(h(k) ∆(k)
∆†(k) −hT (−k)
)
h(k) = εkI + γk · σ
∆ (k) = [∆0 (k) + d (k) · σ] i σy (most general gap matrix)
Assuming |εk| ≫ |γk| ≫ |d (k)| the quasi-particle spectrum is
E =
±√
(εk − µ + |γk|)2 + (∆0 (k) + |d (k)|)2; and
±√
(εk − µ − |γk|)2 + (∆0 (k)− |d (k)|)2.
Take most symmetric (A1) irreducible representation: +
∆0 (k) = ∆0
d(k) = ∆0 × {A (x) (kx , ky , kz )− B (x)
[kx
(k2
y + k2z
), ky
(k2
z + k2x
), kz
(k2
x + k2y
)]}
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 24 / 95
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Li2PdxPt3−xB: Phase diagram
Treat A and B as independent tuning parameters and study quasiparticlespectrum. We find a very rich phase diagram with topollogically-distinct phases:8
8C. 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.
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Li2PdxPt3−xB: Phase diagram
We find a very rich phase diagram with topollogically-distinct phases.9
9C. 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.
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Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 27 / 95
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Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 28 / 95
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Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 29 / 95
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Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 30 / 95
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Detecting the topological transitions
3 734
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 31 / 95
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Detecting the topological transitions
3 734
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 32 / 95
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Li2PdxPt3−xB: predicted specific heat power-laws
334
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Li2PdxPt3−xB: predicted specific heat power-laws
jn = 2
n = 1.8
n = 1.4
n = 2
3
4
5
11
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 34 / 95
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Li2PdxPt3−xB: predicted specific heat power-laws
3
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 35 / 95
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Li2PdxPt3−xB: predicted specific heat power-laws
3
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Li2PdxPt3−xB: predicted specific heat power-laws
jn = 2
n = 1.8
n = 1.4
n = 2
3
4
5
11
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 37 / 95
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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
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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
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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
3.6 3.8 4 4.2 4.4
B
0
0.05
0.1
0.15
0.2
0.25T
/Tc
1.6
1.7
1.8
1.9
2
2.1
2.2
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 38 / 95
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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
3.6 3.8 4 4.2 4.4
B
0
0.05
0.1
0.15
0.2
0.25T
/Tc
1.6
1.7
1.8
1.9
2
2.1
2.2
The conventional exponent (n = 2 in this example) is only seen below atemperature scale that converges to zero at the transition
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 38 / 95
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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
3.6 3.8 4 4.2 4.4
B
0
0.05
0.1
0.15
0.2
0.25T
/Tc
1.6
1.7
1.8
1.9
2
2.1
2.2
The conventional exponent (n = 2 in this example) is only seen below atemperature 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
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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
3.6 3.8 4 4.2 4.4
B
0
0.05
0.1
0.15
0.2
0.25T
/Tc
1.6
1.7
1.8
1.9
2
2.1
2.2
The conventional exponent (n = 2 in this example) is only seen below atemperature 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
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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
3.6 3.8 4 4.2 4.4
B
0
0.05
0.1
0.15
0.2
0.25T
/Tc
1.6
1.7
1.8
1.9
2
2.1
2.2
The conventional exponent (n = 2 in this example) is only seen below atemperature 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
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Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
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Topological transitions in nodal superconductorshave clear signatures in bulk thermodynamic properties.
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 40 / 95
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Topological transitions in nodal superconductorshave clear signatures in bulk thermodynamic properties.
THANKS!
www.cond-mat.org
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 40 / 95
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Power laws in nodal superconductors
Let’s remember where this came from:
Cv = T
(dS
dT
)
=1
2kBT 2 ∑k
Ek − TdEk
dT︸︷︷︸
≈0
Ek sech2 Ek
2kBT︸ ︷︷ ︸
≈4e−Ek /KBT
∼ T−2∫
dEg (E )E2e−E/kBT at low T
g (E ) ∼ En−1 ⇒ Cv ∼ T n∫
dǫǫ2+n−1e−ǫ
︸ ︷︷ ︸
a number
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 41 / 95
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Power laws in nodal superconductors
Ek =√
ǫ2k+ ∆2
k
≈√
I2k2⊥ + ∆
(
kx|| , k
y
||
)2
on the Fermi surface k||
x
k||
y
k|_ ∆(k
||
x,k||
y)
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
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Shallow line nodes in pnictides
back
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 43 / 95
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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
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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
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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
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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
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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
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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 = - (Γn
s)T , Γnt = + (Γn
t)T
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 49 / 95
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Symmetry of pairing in NCS
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Iマpose Pauliげs exclusioミ priミciple:
Neglect (for now!) spin-orbit coupling:
Singlet and triplet representations of SO(3):
Γns = - (Γn
s)T , Γnt = + (Γn
t)T
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 50 / 95
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Symmetry of pairing in NCS
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Iマpose Pauliげs exclusioミ priミciple:
, ' k ', k
Neglect (for now!) spin-orbit coupling:
Singlet and triplet representations of SO(3):
Γns = - (Γn
s)T , Γnt = + (Γn
t)T
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 51 / 95
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Symmetry of pairing in NCS
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Iマpose Pauliげs exclusioミ priミciple:
, ' k ', k
Neglect (for now!) spin-orbit coupling:
ˆ k either singlet
Singlet and triplet representations of SO(3):
Γns = - (Γn
s)T , Γnt = + (Γn
t)T
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 52 / 95
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Symmetry of pairing in NCS
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Iマpose Pauliげs exclusioミ priミciple:
, ' k ', k
Neglect (for now!) spin-orbit coupling:
ˆ k either singlet yi ˆ0', kk
Singlet and triplet representations of SO(3):
Γns = - (Γn
s)T , Γnt = + (Γn
t)T
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 53 / 95
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Symmetry of pairing in NCS
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Iマpose Pauliげs exclusioミ priミciple:
, ' k ', k
Neglect (for now!) spin-orbit coupling:
ˆ k either singlet yi ˆ0', kk or triplet
Singlet and triplet representations of SO(3):
Γns = - (Γn
s)T , Γnt = + (Γn
t)T
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 54 / 95
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Symmetry of pairing in NCS
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Iマpose Pauliげs exclusioミ priミciple:
, ' k ', k
Neglect (for now!) spin-orbit coupling:
ˆ k either singlet yi ˆ0', kk or triplet yi ˆˆ.', σkdk
Singlet and triplet representations of SO(3):
Γns = - (Γn
s)T , Γnt = + (Γn
t)T
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Symmetry of pairing in NCS
yxz
zyx
iddd
didd
0
0ˆ0
0k
The role of spin-orbit coupling (SOC)
Gap function may have both singlet and triplet components
kk orbitspin',',
• 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) back
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LaNiC2 – a weakly-correlated, paramagnetic
superconductor?
Tc=2.7 K
W. H. Lee et al., Physica C 266, 138 (1996)
V. K. Pecharsky, L. L. Miller, and Zy, Physical Review B 58, 497 (1998)
ΔC/TC=1.26
(BCS: 1.43)
specific heat susceptibility
0 = 6.5 mJ/mol K2
c 0 = 22.2 10-6 emu/mol
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 57 / 95
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ISIS
muSR
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 58 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Zero field muon spin relaxation
e
_
e
backward
detector
forward
detector
sample
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 59 / 95
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Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Relaxation due to electronic moments
Moment size ~ 0.1G (~ 0.01μB)
(longitudinal)
Timescale: > 10-4s ~
e
_
e
backward
detector
forward
detector
sample
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 60 / 95
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Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Relaxation due to electronic moments
Moment size ~ 0.1G (~ 0.01μB)
Spontaneous, quasi-static fields appearing at Tc ⇒ superconducting state breaks time-reversal symmetry [ c.f. Sr2RuO4 - Luke et al., Nature (1998) ]
(longitudinal)
Timescale: > 10-4s ~
e
_
e
backward
detector
forward
detector
sample
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 61 / 95
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LaNiC2 is a non-ceontrsymmetric superconductor
Neutron diffraction
30 40 50 60 70 800
5000
10000
15000
20000
25000
30000
35000
Inte
nsity (
arb
un
its)
2 o
Orthorhombic Amm2 C2v
a=3.96 Å b=4.58 Å c=6.20 Å
Data from D1B @ ILL
Note no inversion centre.
C.f. CePt3Si (1), Li2Pt3B & Li2Pd3B (2), ... (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
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Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 63 / 95
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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
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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
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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
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Character table
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
180o
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 67 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
C2v Symmetries and their characters
Sample basis functions
Irreducible representation
E C2 v ’v 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
Character table
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 68 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
C2v Symmetries and their characters
Sample basis functions
Irreducible representation
E C2 v ’v 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
Character table
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
These must be combined with the singlet and triplet
representations of SO(3).
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 69 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
SO(3)xC2v Gap function (unitary)
Gap function (non-unitary)
1A1 (k)=1 -
1A2 (k)=kxkY -
1B1 (k)=kXkZ -
1B2 (k)=kYkZ -
3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3A2 d(k)=(0,0,1)kXkYkZ d(k)=(1,i,0)kXkYkZ
3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
Possible order parameters
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 70 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
SO(3)xC2v Gap function (unitary)
Gap function (non-unitary)
1A1 (k)=1 -
1A2 (k)=kxkY -
1B1 (k)=kXkZ -
1B2 (k)=kYkZ -
3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3A2 d(k)=(0,0,1)kXkYkZ d(k)=(1,i,0)kXkYkZ
3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
Possible order parameters
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 71 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
SO(3)xC2v Gap function (unitary)
Gap function (non-unitary)
1A1 (k)=1 -
1A2 (k)=kxkY -
1B1 (k)=kXkZ -
1B2 (k)=kYkZ -
3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3A2 d(k)=(0,0,1)kXkYkZ d(k)=(1,i,0)kXkYkZ
3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
Possible order parameters
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 72 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
SO(3)xC2v Gap function (unitary)
Gap function (non-unitary)
1A1 (k)=1 -
1A2 (k)=kxkY -
1B1 (k)=kXkZ -
1B2 (k)=kYkZ -
3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3A2 d(k)=(0,0,1)kXkYkZ d(k)=(1,i,0)kXkYkZ
3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
Non-unitary
d x d* ≠ 0
Possible order parameters
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 73 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
SO(3)xC2v Gap function (unitary)
Gap function (non-unitary)
1A1 (k)=1 -
1A2 (k)=kxkY -
1B1 (k)=kXkZ -
1B2 (k)=kYkZ -
3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3A2 d(k)=(0,0,1)kXkYkZ d(k)=(1,i,0)kXkYkZ
3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
Non-unitary
d x d* ≠ 0
breaks only SO(3) x U(1) x T
Possible order parameters
* C.f. Li2Pd3B & Li2Pt3B,
H. Q. Yuaミ et al. P‘Lげ0ヶ
*
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 74 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Spin-up superfluid
coexisting with spin-
down Fermi liquid.
The A1 phase of
liquid 3He.
Non-unitary pairing
0
00or
00
0ˆ
C.f.
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 75 / 95
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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
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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
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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
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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
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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
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
G = Gc,J×U(1)×T
The role of spin-orbit coupling (SOC)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 81 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
G = Gc,J×U(1)×T
The role of spin-orbit coupling (SOC)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 82 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
G = Gc,J×U(1)×T
The role of spin-orbit coupling (SOC)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 83 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
E.g. reflection through a vertical
plane perpendicular to the y axis:
yJJv CI ,2,
x y
z
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 84 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
E.g. reflection through a vertical
plane perpendicular to the y axis:
yJJv CI ,2,
x y
z
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 85 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
E.g. reflection through a vertical
plane perpendicular to the y axis:
yJJv CI ,2,
x y
z
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 86 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
E.g. reflection through a vertical
plane perpendicular to the y axis:
yJJv CI ,2,
x y
z
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 87 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
E.g. reflection through a vertical
plane perpendicular to the y axis:
yJJv CI ,2,
This affects d(k) (a vector under
spin rotations).
x y
z
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 88 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
E.g. reflection through a vertical
plane perpendicular to the y axis:
yJJv CI ,2,
This affects d(k) (a vector under
spin rotations).
It does not affect 0(k) (a scalar). x y
z
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 89 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
C2v,Jno t Gap function, singlet component
Gap function, triplet component
A1 (k) = A d(k) = (Bky,Ckx,Dkxkykz)
A2 (k) = AkxkY d(k) = (Bkx,Cky,Dkz)
B1 (k) = AkXkZ d(k) = (Bkxkykz,Ckz,Dky)
B2 (k) = AkYkZ d(k) = (Bkz, Ckxkykz,Dkx)
The role of spin-orbit coupling (SOC)
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 90 / 95
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
C2v,Jno t Gap function, singlet component
Gap function, triplet component
A1 (k) = A d(k) = (Bky,Ckx,Dkxkykz)
A2 (k) = AkxkY d(k) = (Bkx,Cky,Dkz)
B1 (k) = AkXkZ d(k) = (Bkxkykz,Ckz,Dky)
B2 (k) = AkYkZ d(k) = (Bkz, Ckxkykz,Dkx)
The role of spin-orbit coupling (SOC)
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
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Relativistic and non-relativistic
instabilities: a complex relationship
singlet
Pairing
instabilities
non-unitary
triplet
pairing
instabilities
unitary
triplet
pairing
instabilities
A1 B1
3B1(b) 3B2(b)
1A1 1A2
3A1(a) 3A2(a)
A2 B2
1B1 1B2
3B1(a) 3B2(a)
3A1(b) 3A2(b)
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 92 / 95
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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
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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
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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
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Li2PdxPt3−xB: Phase diagram
Bogoliubov Hamiltonian with Rashba spin-orbit coupling:
H(k) =
(h(k) ∆(k)
∆†(k) −hT (−k)
)
h(k) = εk I + γk · σ
Assuming |εk| ≫ |γk| ≫ |d (k)| the quasi-particle spectrum is
E =
±√
(εk − µ + |γk |)2 + (∆0 + |d(k)|)2; and
±√
(εk − µ − |γk |)2 + (∆0 − |d(k)|)2.
Take the most symmetric (A1) irreducible representation
d(k)/∆0 = A (X ,Y ,Z )− B(X(Y 2 + Z2
),Y
(Z2 + X2
),Z
(X2 + Y 2
))
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Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 94 / 95
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Li2PdxPt3−xB:order parameter
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Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 95 / 95