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Broken Time-Reversal Symmetry and Topological Orderin Triplet Superconductors
Jorge Quintanilla1,2
1SEPnet and Hubbard Theory Consortium, University of Kent2ISIS Neutron and Muon Source, STFC Rutherford Appleton Laboratory
Dresden, 27 November 2014
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 1 / 119
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People and Money
People: James F. Annett (Bristol) , Adrian D. Hillier (RAL)
Bayan Mazidian (RAL/Bristol) , Bob Cywinski (Huddersfield) .
Ravi P. Singh , Gheeta Balakrishnan , Don Paul ,
Martin Lees (Birmingham). Amitava Bhattacharyya ,
Devashibai Adroja (RAL). A. M. Strydom (Johannesburg) .
Naoki Kase, Jun Akimitsu (Aoyama Gakuin).
Money: STFC (UK) + HEFCE/SEPnet (UK) + UJ and NRF (South Africa) +Bristol + Kent.
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 2 / 119
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The Hubbard Theory Consortium
Director: Piers Coleman (RHUL/Rutgers)SEPnet fellows: Matthias Eschrig (RHUL/RAL)
Claudio Castelnovo (RHUL/RAL)Jorge Quintanilla (Kent/RAL)
Associate: Jörg Schmalian (Karlsruhe)
+ several SEPnet PhD students.
Strong correlations theory in close collaboration with experiments at
• RAL (ISIS/Diamond)• London Centre for Nanotech.• RHUL
Director: Piers Coleman (RHUL/Rutgers)SEPnet fellows: Matthias Eschrig (RHUL/RAL)
Claudio Castelnovo (RHUL/RAL)Jorge Quintanilla (Kent/RAL)
Associate: Jörg Schmalian (Karlsruhe)
+ several SEPnet PhD students.
Strong correlations theory in close collaboration with experiments at
• RAL (ISIS/Diamond)• London Centre for Nanotech.• RHUL
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 3 / 119
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Overview
Two Paradigms in Condensed Matter ...
Broken SymmetryTopological Transitions
... interlock via triplet pairing in superconductors:
Superconductors
Broken time-reversal
symmetry
Topological transitionsTriplet
pairing
SimpleTheories + Standard Measurements:Group Theory / Bogolibov QuasiparticlesNeutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119
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Overview
Two Paradigms in Condensed Matter ...Broken Symmetry
Topological Transitions
... interlock via triplet pairing in superconductors:
Superconductors
Broken time-reversal
symmetry
Topological transitionsTriplet
pairing
SimpleTheories + Standard Measurements:Group Theory / Bogolibov QuasiparticlesNeutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119
![Page 6: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/6.jpg)
Overview
Two Paradigms in Condensed Matter ...Broken SymmetryTopological Transitions
... interlock via triplet pairing in superconductors:
Superconductors
Broken time-reversal
symmetry
Topological transitionsTriplet
pairing
SimpleTheories + Standard Measurements:Group Theory / Bogolibov QuasiparticlesNeutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119
![Page 7: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/7.jpg)
Overview
Two Paradigms in Condensed Matter ...Broken SymmetryTopological Transitions
... interlock via triplet pairing in superconductors:
Superconductors
Broken time-reversal
symmetry
Topological transitionsTriplet
pairing
SimpleTheories + Standard Measurements:Group Theory / Bogolibov QuasiparticlesNeutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119
![Page 8: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/8.jpg)
Overview
Two Paradigms in Condensed Matter ...Broken SymmetryTopological Transitions
... interlock via triplet pairing in superconductors:
Superconductors
Broken time-reversal
symmetry
Topological transitionsTriplet
pairing
SimpleTheories + Standard Measurements:
Group Theory / Bogolibov QuasiparticlesNeutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119
![Page 9: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/9.jpg)
Overview
Two Paradigms in Condensed Matter ...Broken SymmetryTopological Transitions
... interlock via triplet pairing in superconductors:
Superconductors
Broken time-reversal
symmetry
Topological transitionsTriplet
pairing
SimpleTheories + Standard Measurements:Group Theory / Bogolibov Quasiparticles
Neutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119
![Page 10: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/10.jpg)
Overview
Two Paradigms in Condensed Matter ...Broken SymmetryTopological Transitions
... interlock via triplet pairing in superconductors:
Superconductors
Broken time-reversal
symmetry
Topological transitionsTriplet
pairing
SimpleTheories + Standard Measurements:Group Theory / Bogolibov QuasiparticlesNeutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119
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Outline
1 Broken time-reversal symmetry in superconductors
2 Experimental evidence for broken TRS
3 Singlet, triplet, or both?
4 A symmetry zoo
5 Topological transitions in Superconductors
6 Topological transition state: Li2PdxPt3−xB
7 Take-home message
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 5 / 119
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Quantum Materials Theory
1 Broken time-reversal symmetry in superconductors
2 Experimental evidence for broken TRS
3 Singlet, triplet, or both?
4 A symmetry zoo
5 Topological transitions in Superconductors
6 Topological transition state: Li2PdxPt3−xB
7 Take-home message
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Broken symmetry
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Unconventional superconductors
Ph
oto
: Ed
die
Hu
i-B
on
-Ho
a, w
ww
.sh
iro
mi.c
om
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: Ken
net
h G
. Lib
bre
cht,
sn
ow
flak
es.c
om
Unconventional superconductors
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: co
mm
on
s.w
ikim
edia
.org
Unconventional superconductors
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 7 / 119
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Broken symmetry
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Unconventional superconductors
Ph
oto
: Ed
die
Hu
i-B
on
-Ho
a, w
ww
.sh
iro
mi.c
om
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: Ken
net
h G
. Lib
bre
cht,
sn
ow
flak
es.c
om
Unconventional superconductors
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: co
mm
on
s.w
ikim
edia
.org
Unconventional superconductors
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 7 / 119
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Broken symmetry
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Unconventional superconductors
Ph
oto
: Ed
die
Hu
i-B
on
-Ho
a, w
ww
.sh
iro
mi.c
om
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: Ken
net
h G
. Lib
bre
cht,
sn
ow
flak
es.c
om
Unconventional superconductors
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: co
mm
on
s.w
ikim
edia
.org
Unconventional superconductors
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 7 / 119
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Broken symmetry
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Unconventional superconductors
Ph
oto
: Ed
die
Hu
i-B
on
-Ho
a, w
ww
.sh
iro
mi.c
om
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: Ken
net
h G
. Lib
bre
cht,
sn
ow
flak
es.c
om
Unconventional superconductors
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: co
mm
on
s.w
ikim
edia
.org
Unconventional superconductors
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 7 / 119
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Broken symmetry
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Unconventional superconductors
Ph
oto
: Ed
die
Hu
i-B
on
-Ho
a, w
ww
.sh
iro
mi.c
om
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: Ken
net
h G
. Lib
bre
cht,
sn
ow
flak
es.c
om
Unconventional superconductors
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: co
mm
on
s.w
ikim
edia
.org
Unconventional superconductors
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 7 / 119
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Unconventional SuperconductorsVirginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: co
mm
on
s.w
ikim
edia
.org
Unconventional superconductors
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 8 / 119
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Unconventional SuperconductorsVirginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Ph
oto
: co
mm
on
s.w
ikim
edia
.org
Unconventional superconductors
‘Unconventional’ superconductors:
Cuprates, Sr2RuO4, PrOs4Sb12, UPt3, (UTh)Be13 , ...
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 9 / 119
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Time-reversal Symmetry
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 10 / 119
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Time-reversal Symmetry
p
r
x
y
z
Classical time-reversal symmetry:t → −t equivalent to
r→ r and p→ −p
Also inverts angular momenta.True in the absence of friction/magnetic fields.
Quantum time-reversal symmetry:t → −t equivalent toψ→ ψ∗ and S→ −S.
True if H = H∗ and spin-invariant.
For quasi-particles in a superconductor:H = H0 + ∆c†
k c†−k + H.c. ⇒ TRS: ∆ = ∆∗
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 11 / 119
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Time-reversal Symmetry
-pr
x
y
z
Classical time-reversal symmetry:t → −t equivalent to
r→ r and p→ −p
Also inverts angular momenta.True in the absence of friction/magnetic fields.
Quantum time-reversal symmetry:t → −t equivalent toψ→ ψ∗ and S→ −S.
True if H = H∗ and spin-invariant.
For quasi-particles in a superconductor:H = H0 + ∆c†
k c†−k + H.c. ⇒ TRS: ∆ = ∆∗
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 11 / 119
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Time-reversal Symmetry
-pr
x
y
z
Classical time-reversal symmetry:t → −t equivalent to
r→ r and p→ −p
Also inverts angular momenta.True in the absence of friction/magnetic fields.
Quantum time-reversal symmetry:t → −t equivalent toψ→ ψ∗ and S→ −S.
True if H = H∗ and spin-invariant.
For quasi-particles in a superconductor:H = H0 + ∆c†
k c†−k + H.c. ⇒ TRS: ∆ = ∆∗
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 11 / 119
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Time-reversal Symmetry
-pr
x
y
z
Classical time-reversal symmetry:t → −t equivalent to
r→ r and p→ −p
Also inverts angular momenta.True in the absence of friction/magnetic fields.
Quantum time-reversal symmetry:t → −t equivalent toψ→ ψ∗ and S→ −S.
True if H = H∗ and spin-invariant.
For quasi-particles in a superconductor:H = H0 + ∆c†
k c†−k + H.c. ⇒ TRS: ∆ = ∆∗
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 11 / 119
![Page 25: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/25.jpg)
Time-reversal Symmetry
-pr
x
y
z
Classical time-reversal symmetry:t → −t equivalent to
r→ r and p→ −p
Also inverts angular momenta.True in the absence of friction/magnetic fields.
Quantum time-reversal symmetry:t → −t equivalent toψ→ ψ∗ and S→ −S.
True if H = H∗ and spin-invariant.
For quasi-particles in a superconductor:H = H0 + ∆c†
k c†−k + H.c. ⇒ TRS: ∆ = ∆∗
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 11 / 119
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Quantum Materials Theory
1 Broken time-reversal symmetry in superconductors
2 Experimental evidence for broken TRS
3 Singlet, triplet, or both?
4 A symmetry zoo
5 Topological transitions in Superconductors
6 Topological transition state: Li2PdxPt3−xB
7 Take-home message
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Muon Spin Rotation
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 13 / 119
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Muon Spin Rotation
Adrian Hillier (Muons group leader, ISIS)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 14 / 119
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Muon Spin Rotation
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 RAL) www.cond-mat.org Dresden 2014 15 / 119
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Kubo-Toyabe x exponential
Asymmetry:
NF −NBNF + NB
= G (t)
σ : randomly-oriented fields (e.g. nuclear moments)Λ :smoothly-modulated fields (e.g. electronic moments)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 16 / 119
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The “classic” examples: UPt3 and Sr2RuO4
UPt3
Luke et al. PRL (1993)
Sr2RuO4
Luke et al. Nature (1998)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 17 / 119
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Confirmed by Kerr effect
UPt3
Schemm et al. Science (2014)
Sr2RuO4
Jing Xia et al. PRL (2006)Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 18 / 119
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More recent finds: LaNiC2
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 19 / 119
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More recent finds: LaNiC2
Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
Relaxation due to electronic moments
Moment
size
~ 0.1G
(~ 0.01μB)
(longitudinal)
Timescale:
> 10-4
s ~
e
_
e
backward detector
forward detector
sample
+
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 20 / 119
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More recent finds: LaNiGa2
A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett,Physical Review Letters 109, 097001 (2012).
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 21 / 119
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More recent finds: Re6Zr
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 22 / 119
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More recent finds: Lu5Rh6Sn18
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 23 / 119
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Quantum Materials Theory
1 Broken time-reversal symmetry in superconductors
2 Experimental evidence for broken TRS
3 Singlet, triplet, or both?
4 A symmetry zoo
5 Topological transitions in Superconductors
6 Topological transition state: Li2PdxPt3−xB
7 Take-home message
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Singlet, triplet, or both?
It’s all in the gap function:
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Symmetry of the gap function
kk
kkkˆ
See J.F. Annett Adv. Phys. 1990.
Pauli ⇒∆ (k) = −∆T (−k)Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
ˆ k 0 0
0 0
dx idy dz
dz dx idy
singlet
[ 0(k) even ]
triplet
[ d(k) odd ]
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 25 / 119
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Singlet, triplet, or both?
It’s all in the gap function:
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Symmetry of the gap function
kk
kkkˆ
See J.F. Annett Adv. Phys. 1990.
Pauli ⇒∆ (k) = −∆T (−k)Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
ˆ k 0 0
0 0
dx idy dz
dz dx idy
singlet
[ 0(k) even ]
triplet
[ d(k) odd ]
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 25 / 119
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Singlet, triplet, or both?
It’s all in the gap function:
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Symmetry of the gap function
kk
kkkˆ
See J.F. Annett Adv. Phys. 1990.
Pauli ⇒∆ (k) = −∆T (−k)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
ˆ k 0 0
0 0
dx idy dz
dz dx idy
singlet
[ 0(k) even ]
triplet
[ d(k) odd ]
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 25 / 119
![Page 42: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/42.jpg)
Singlet, triplet, or both?
It’s all in the gap function:
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Symmetry of the gap function
kk
kkkˆ
See J.F. Annett Adv. Phys. 1990.
Pauli ⇒∆ (k) = −∆T (−k)Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
ˆ k 0 0
0 0
dx idy dz
dz dx idy
singlet
[ 0(k) even ]
triplet
[ d(k) odd ]
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 25 / 119
![Page 43: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/43.jpg)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 26 / 119
![Page 44: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/44.jpg)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 27 / 119
![Page 45: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/45.jpg)
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 RAL) www.cond-mat.org Dresden 2014 28 / 119
![Page 46: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/46.jpg)
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 RAL) www.cond-mat.org Dresden 2014 29 / 119
![Page 47: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/47.jpg)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Impose Pauli’s exclusion principle:
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 RAL) www.cond-mat.org Dresden 2014 30 / 119
![Page 48: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/48.jpg)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Impose Pauli’s exclusion principle:
, ' 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 RAL) www.cond-mat.org Dresden 2014 31 / 119
![Page 49: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/49.jpg)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Impose Pauli’s exclusion principle:
, ' 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 RAL) www.cond-mat.org Dresden 2014 32 / 119
![Page 50: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/50.jpg)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Impose Pauli’s exclusion principle:
, ' k ', k
Neglect (for now!) spin-orbit coupling:
ˆ k either singlet yiˆ
0', kk
Singlet and triplet representations of SO(3):
Γns = - (Γn
s)T , Γnt = + (Γn
t)T
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 33 / 119
![Page 51: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/51.jpg)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Impose Pauli’s exclusion principle:
, ' k ', k
Neglect (for now!) spin-orbit coupling:
ˆ k either singlet yiˆ
0', kk
or triplet
Singlet and triplet representations of SO(3):
Γns = - (Γn
s)T , Γnt = + (Γn
t)T
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 34 / 119
![Page 52: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/52.jpg)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
Impose Pauli’s exclusion principle:
, ' k ', k
Neglect (for now!) spin-orbit coupling:
ˆ k either singlet yiˆ
0', kk
or triplet yiˆˆ.', σkdk
Singlet and triplet representations of SO(3):
Γns = - (Γn
s)T , Γnt = + (Γn
t)T
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 35 / 119
![Page 53: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/53.jpg)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
yxz
zyx
iddd
didd
0
0ˆ
0
0k
The role of spin-orbit coupling (SOC)
Gap function may have both singlet and triplet components
kkorbitspin
',',
• 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 RAL) www.cond-mat.org Dresden 2014 36 / 119
![Page 54: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/54.jpg)
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 RAL) www.cond-mat.org Dresden 2014 37 / 119
![Page 55: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/55.jpg)
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 RAL) www.cond-mat.org Dresden 2014 38 / 119
![Page 56: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/56.jpg)
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 RAL) www.cond-mat.org Dresden 2014 39 / 119
![Page 57: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/57.jpg)
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 RAL) www.cond-mat.org Dresden 2014 40 / 119
![Page 58: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/58.jpg)
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 RAL) www.cond-mat.org Dresden 2014 41 / 119
![Page 59: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/59.jpg)
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 RAL) www.cond-mat.org Dresden 2014 42 / 119
![Page 60: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/60.jpg)
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 RAL) www.cond-mat.org Dresden 2014 43 / 119
![Page 61: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/61.jpg)
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 RAL) www.cond-mat.org Dresden 2014 44 / 119
![Page 62: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/62.jpg)
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:
y
JJv CI ,2,
x y
z
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 45 / 119
![Page 63: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/63.jpg)
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:
y
JJv CI ,2,
x y
z
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 46 / 119
![Page 64: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/64.jpg)
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:
y
JJv CI ,2,
x y
z
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 47 / 119
![Page 65: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/65.jpg)
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:
y
JJv CI ,2,
x y
z
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 48 / 119
![Page 66: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/66.jpg)
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:
y
JJv CI ,2,
This affects d(k) (a vector under spin rotations).
x y
z
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 49 / 119
![Page 67: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/67.jpg)
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:
y
JJv 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 RAL) www.cond-mat.org Dresden 2014 50 / 119
![Page 68: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/68.jpg)
When can we have singlet-triplet mixing?
We must now use basis functions of the double group:
∆ (k) =dΓ
∑n=1
ηnΓn (k)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
yxz
zyx
iddd
didd
0
0ˆ
0
0k
The role of spin-orbit coupling (SOC)
Gap function may have both singlet and triplet components
kkorbitspin
',',
• 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)
Crystal symmetryCentrosymmetric Non-centrosymmetric
Spin-orbit coupling Weak N NStrong N Y
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 51 / 119
![Page 69: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/69.jpg)
When can we have broken time-reversal symmetry?
For ∆ (k) to be non-trivially complex, it must have more than onecomponent:
∆ (k) = η1Γ1 (k) + η2Γ2 (k) , arg η1 6= arg η2
The instability must therefore take place in an irrep with d > 1.Weak spin-orbit coupling: SO(3) × Gc
The singlet irrep of SO(3) has d = 1 ⇒ for singlet pairing, the point group Gcmust have a d > 1 irrep.The triplet irrep of SO(3) had d = 3 ⇒ for triplet pairing, broken TRS ispossible even for d = 1 irreps of Gc .
Strong spin-orbit coupling: Gc,J (double group)The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119
![Page 70: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/70.jpg)
When can we have broken time-reversal symmetry?
For ∆ (k) to be non-trivially complex, it must have more than onecomponent:
∆ (k) = η1Γ1 (k) + η2Γ2 (k) , arg η1 6= arg η2
The instability must therefore take place in an irrep with d > 1.
Weak spin-orbit coupling: SO(3) × Gc
The singlet irrep of SO(3) has d = 1 ⇒ for singlet pairing, the point group Gcmust have a d > 1 irrep.The triplet irrep of SO(3) had d = 3 ⇒ for triplet pairing, broken TRS ispossible even for d = 1 irreps of Gc .
Strong spin-orbit coupling: Gc,J (double group)The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119
![Page 71: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/71.jpg)
When can we have broken time-reversal symmetry?
For ∆ (k) to be non-trivially complex, it must have more than onecomponent:
∆ (k) = η1Γ1 (k) + η2Γ2 (k) , arg η1 6= arg η2
The instability must therefore take place in an irrep with d > 1.Weak spin-orbit coupling: SO(3) × Gc
The singlet irrep of SO(3) has d = 1 ⇒ for singlet pairing, the point group Gcmust have a d > 1 irrep.The triplet irrep of SO(3) had d = 3 ⇒ for triplet pairing, broken TRS ispossible even for d = 1 irreps of Gc .
Strong spin-orbit coupling: Gc,J (double group)The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119
![Page 72: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/72.jpg)
When can we have broken time-reversal symmetry?
For ∆ (k) to be non-trivially complex, it must have more than onecomponent:
∆ (k) = η1Γ1 (k) + η2Γ2 (k) , arg η1 6= arg η2
The instability must therefore take place in an irrep with d > 1.Weak spin-orbit coupling: SO(3) × Gc
The singlet irrep of SO(3) has d = 1 ⇒ for singlet pairing, the point group Gcmust have a d > 1 irrep.
The triplet irrep of SO(3) had d = 3 ⇒ for triplet pairing, broken TRS ispossible even for d = 1 irreps of Gc .
Strong spin-orbit coupling: Gc,J (double group)The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119
![Page 73: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/73.jpg)
When can we have broken time-reversal symmetry?
For ∆ (k) to be non-trivially complex, it must have more than onecomponent:
∆ (k) = η1Γ1 (k) + η2Γ2 (k) , arg η1 6= arg η2
The instability must therefore take place in an irrep with d > 1.Weak spin-orbit coupling: SO(3) × Gc
The singlet irrep of SO(3) has d = 1 ⇒ for singlet pairing, the point group Gcmust have a d > 1 irrep.The triplet irrep of SO(3) had d = 3 ⇒ for triplet pairing, broken TRS ispossible even for d = 1 irreps of Gc .
Strong spin-orbit coupling: Gc,J (double group)The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119
![Page 74: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/74.jpg)
When can we have broken time-reversal symmetry?
For ∆ (k) to be non-trivially complex, it must have more than onecomponent:
∆ (k) = η1Γ1 (k) + η2Γ2 (k) , arg η1 6= arg η2
The instability must therefore take place in an irrep with d > 1.Weak spin-orbit coupling: SO(3) × Gc
The singlet irrep of SO(3) has d = 1 ⇒ for singlet pairing, the point group Gcmust have a d > 1 irrep.The triplet irrep of SO(3) had d = 3 ⇒ for triplet pairing, broken TRS ispossible even for d = 1 irreps of Gc .
Strong spin-orbit coupling: Gc,J (double group)
The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119
![Page 75: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/75.jpg)
When can we have broken time-reversal symmetry?
For ∆ (k) to be non-trivially complex, it must have more than onecomponent:
∆ (k) = η1Γ1 (k) + η2Γ2 (k) , arg η1 6= arg η2
The instability must therefore take place in an irrep with d > 1.Weak spin-orbit coupling: SO(3) × Gc
The singlet irrep of SO(3) has d = 1 ⇒ for singlet pairing, the point group Gcmust have a d > 1 irrep.The triplet irrep of SO(3) had d = 3 ⇒ for triplet pairing, broken TRS ispossible even for d = 1 irreps of Gc .
Strong spin-orbit coupling: Gc,J (double group)The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.
Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119
![Page 76: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/76.jpg)
When can we have broken time-reversal symmetry?
For ∆ (k) to be non-trivially complex, it must have more than onecomponent:
∆ (k) = η1Γ1 (k) + η2Γ2 (k) , arg η1 6= arg η2
The instability must therefore take place in an irrep with d > 1.Weak spin-orbit coupling: SO(3) × Gc
The singlet irrep of SO(3) has d = 1 ⇒ for singlet pairing, the point group Gcmust have a d > 1 irrep.The triplet irrep of SO(3) had d = 3 ⇒ for triplet pairing, broken TRS ispossible even for d = 1 irreps of Gc .
Strong spin-orbit coupling: Gc,J (double group)The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119
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Quantum Materials Theory
1 Broken time-reversal symmetry in superconductors
2 Experimental evidence for broken TRS
3 Singlet, triplet, or both?
4 A symmetry zoo
5 Topological transitions in Superconductors
6 Topological transition state: Li2PdxPt3−xB
7 Take-home message
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LaNiC2
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Character table
Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 54 / 119
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LaNiC2
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Character table
Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 55 / 119
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LaNiC2
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Character table
Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 56 / 119
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LaNiC2
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Character table
Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
180o
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 57 / 119
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LaNiC2
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)
All irreps d = 1
⇒ weak SOC
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 58 / 119
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LaNiC2
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
SO(3)xC2v
Gap function
(unitary)
Gap function
(non-unitary)
1A
1 (k)=1 -
1A
2 (k)=k
xk
Y -
1B
1 (k)=k
Xk
Z -
1B
2 (k)=k
Yk
Z -
3A
1 d(k)=(0,0,1)k
Z d(k)=(1,i,0)k
Z
3A
2 d(k)=(0,0,1)k
Xk
Yk
Z d(k)=(1,i,0)k
Xk
Yk
Z
3B
1 d(k)=(0,0,1)k
X d(k)=(1,i,0)k
X
3B
2 d(k)=(0,0,1)k
Y d(k)=(1,i,0)k
Y
Possible order parameters
Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 59 / 119
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LaNiC2
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
SO(3)xC2v
Gap function
(unitary)
Gap function
(non-unitary)
1A
1 (k)=1 -
1A
2 (k)=k
xk
Y -
1B
1 (k)=k
Xk
Z -
1B
2 (k)=k
Yk
Z -
3A
1 d(k)=(0,0,1)k
Z d(k)=(1,i,0)k
Z
3A
2 d(k)=(0,0,1)k
Xk
Yk
Z d(k)=(1,i,0)k
Xk
Yk
Z
3B
1 d(k)=(0,0,1)k
X d(k)=(1,i,0)k
X
3B
2 d(k)=(0,0,1)k
Y d(k)=(1,i,0)k
Y
Possible order parameters
Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 60 / 119
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LaNiC2
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
SO(3)xC2v
Gap function
(unitary)
Gap function
(non-unitary)
1A
1 (k)=1 -
1A
2 (k)=k
xk
Y -
1B
1 (k)=k
Xk
Z -
1B
2 (k)=k
Yk
Z -
3A
1 d(k)=(0,0,1)k
Z d(k)=(1,i,0)k
Z
3A
2 d(k)=(0,0,1)k
Xk
Yk
Z d(k)=(1,i,0)k
Xk
Yk
Z
3B
1 d(k)=(0,0,1)k
X d(k)=(1,i,0)k
X
3B
2 d(k)=(0,0,1)k
Y d(k)=(1,i,0)k
Y
Possible order parameters
Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 61 / 119
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LaNiC2
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
SO(3)xC2v
Gap function
(unitary)
Gap function
(non-unitary)
1A
1 (k)=1 -
1A
2 (k)=k
xk
Y -
1B
1 (k)=k
Xk
Z -
1B
2 (k)=k
Yk
Z -
3A
1 d(k)=(0,0,1)k
Z d(k)=(1,i,0)k
Z
3A
2 d(k)=(0,0,1)k
Xk
Yk
Z d(k)=(1,i,0)k
Xk
Yk
Z
3B
1 d(k)=(0,0,1)k
X d(k)=(1,i,0)k
X
3B
2 d(k)=(0,0,1)k
Y d(k)=(1,i,0)k
Y
Non-unitary d x d* ≠ 0
Possible order parameters
Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 62 / 119
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LaNiC2
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
SO(3)xC2v
Gap function
(unitary)
Gap function
(non-unitary)
1A
1 (k)=1 -
1A
2 (k)=k
xk
Y -
1B
1 (k)=k
Xk
Z -
1B
2 (k)=k
Yk
Z -
3A
1 d(k)=(0,0,1)k
Z d(k)=(1,i,0)k
Z
3A
2 d(k)=(0,0,1)k
Xk
Yk
Z d(k)=(1,i,0)k
Xk
Yk
Z
3B
1 d(k)=(0,0,1)k
X d(k)=(1,i,0)k
X
3B
2 d(k)=(0,0,1)k
Y d(k)=(1,i,0)k
Y
Non-unitary d x d* ≠ 0
breaks only SO(3) x U(1) x T
Possible order parameters
* C.f. Li2Pd3B & Li2Pt3B, H. Q. Yuan et al. PRL’06
*
Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 63 / 119
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LaNiC2Virginia 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.
Also FM SC - but this is a paramagnet!
A. D. Hillier, J. Quintanilla and R. Cywinski, Physical Review Letters (2009).
J. Quintanilla, J. F. Annett, A. D. Hillier, R. Cywinski, Physical Review B (2010).
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 64 / 119
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LaNiGa2
Centrosymmetric, but again all irreps d = 1 ⇒ again weak SOC and non-unitarytriplet
A.D. Hillier, J. Quintanilla, B. Mazidian, J.F. Annett, and R. Cywinski, PRL 109, 097001(2012).
⇒ A new family of superconductors?Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 65 / 119
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Why non-unitary?Generic Landau theory for a triplet superconductor (1D irrep):
F = a |η|2 + b2∣∣η4∣∣
+ b′ |η× η∗|2 +m2
2χ+ b′′m · (iη× η∗) .
Magnetisation as a sub-dominant order parameter:
Temperature
magnetisation
Superconductivity
Theory (left): A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, PRL 109, 097001 (2012).
Experiment (right): Akihiko Sumiyama et al., JPSJ (2014).
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119
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Why non-unitary?Generic Landau theory for a triplet superconductor (1D irrep):
F = a |η|2 + b2∣∣η4∣∣ + b′ |η× η∗|2
+m2
2χ+ b′′m · (iη× η∗) .
Magnetisation as a sub-dominant order parameter:
Temperature
magnetisation
Superconductivity
Theory (left): A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, PRL 109, 097001 (2012).
Experiment (right): Akihiko Sumiyama et al., JPSJ (2014).
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119
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Why non-unitary?Generic Landau theory for a triplet superconductor (1D irrep):
F = a |η|2 + b2∣∣η4∣∣ + b′ |η× η∗|2 +
m2
2χ
+ b′′m · (iη× η∗) .
Magnetisation as a sub-dominant order parameter:
Temperature
magnetisation
Superconductivity
Theory (left): A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, PRL 109, 097001 (2012).
Experiment (right): Akihiko Sumiyama et al., JPSJ (2014).
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119
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Why non-unitary?Generic Landau theory for a triplet superconductor (1D irrep):
F = a |η|2 + b2∣∣η4∣∣ + b′ |η× η∗|2 +
m2
2χ+ b′′m · (iη× η∗) .
Magnetisation as a sub-dominant order parameter:
Temperature
magnetisation
Superconductivity
Theory (left): A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, PRL 109, 097001 (2012).
Experiment (right): Akihiko Sumiyama et al., JPSJ (2014).
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119
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Why non-unitary?Generic Landau theory for a triplet superconductor (1D irrep):
F = a |η|2 + b2∣∣η4∣∣ + b′ |η× η∗|2 +
m2
2χ+ b′′m · (iη× η∗) .
Magnetisation as a sub-dominant order parameter:
Temperature
magnetisation
Superconductivity
Theory (left): A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, PRL 109, 097001 (2012).
Experiment (right): Akihiko Sumiyama et al., JPSJ (2014).
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119
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Why non-unitary?Generic Landau theory for a triplet superconductor (1D irrep):
F = a |η|2 + b2∣∣η4∣∣ + b′ |η× η∗|2 +
m2
2χ+ b′′m · (iη× η∗) .
Magnetisation as a sub-dominant order parameter:
Temperature
magnetisation
Superconductivity
Theory (left): A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, PRL 109, 097001 (2012).
Experiment (right): Akihiko Sumiyama et al., JPSJ (2014).
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119
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Why non-unitary?Generic Landau theory for a triplet superconductor (1D irrep):
F = a |η|2 + b2∣∣η4∣∣ + b′ |η× η∗|2 +
m2
2χ+ b′′m · (iη× η∗) .
Magnetisation as a sub-dominant order parameter:
Temperature
magnetisation
Superconductivity
Theory (left): A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, PRL 109, 097001 (2012).
Experiment (right): Akihiko Sumiyama et al., JPSJ (2014).Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119
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Re6Zr
Td group:noncentrosymmetric;d = 1, 2, 3
⇒ can have broken TRS with strong SOC⇒ broken TRS with singlet-triplet mixing
E irrep (d = 2) ⇒
F1,F2 irreps (d = 3) ⇒ several more mixed singlet-triplet states.
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 67 / 119
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Re6Zr
Td group:noncentrosymmetric;d = 1, 2, 3⇒ can have broken TRS with strong SOC
⇒ broken TRS with singlet-triplet mixing
E irrep (d = 2) ⇒
F1,F2 irreps (d = 3) ⇒ several more mixed singlet-triplet states.
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 67 / 119
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Re6Zr
Td group:noncentrosymmetric;d = 1, 2, 3⇒ can have broken TRS with strong SOC⇒ broken TRS with singlet-triplet mixing
E irrep (d = 2) ⇒
F1,F2 irreps (d = 3) ⇒ several more mixed singlet-triplet states.
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 67 / 119
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Re6Zr
Td group:noncentrosymmetric;d = 1, 2, 3⇒ can have broken TRS with strong SOC⇒ broken TRS with singlet-triplet mixing
E irrep (d = 2) ⇒
F1,F2 irreps (d = 3) ⇒ several more mixed singlet-triplet states.
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 67 / 119
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Re6Zr
Td group:noncentrosymmetric;d = 1, 2, 3⇒ can have broken TRS with strong SOC⇒ broken TRS with singlet-triplet mixing
E irrep (d = 2) ⇒
F1,F2 irreps (d = 3) ⇒ several more mixed singlet-triplet states.
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 67 / 119
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Lu5Rh6Sn18
Group D4h:centrosymmetric ⇒ no singlet-triplet mixing;1d = 1, 2, 3⇒ can have broken TRS with strong SOC.
Only two states allowed: 1Eg (c) (singlet) and Eu(c) (triplet).
1c.f. recent ARPES-based claim for Sr2RuO4: C.N. Veenstra et al., PRL 112,127002 (2014). +
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 68 / 119
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Lu5Rh6Sn18Singlet state:
kyk
x
kz
Triplet:
ky
kx
kz
kz
ky
kx
ky
kz
kx
ky
kx
kz
kz
ky
kx
ky
kz
kx
ky
kx
kz
kz
ky
kx
ky
kz
kx
N.B. “shallow” point nodes.These results should apply just as well to Sr2RuO4, in the regime of strongspin-orbit coupling [see Veenstra et al. results + ].
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 69 / 119
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Quantum Materials Theory
1 Broken time-reversal symmetry in superconductors
2 Experimental evidence for broken TRS
3 Singlet, triplet, or both?
4 A symmetry zoo
5 Topological transitions in Superconductors
6 Topological transition state: Li2PdxPt3−xB
7 Take-home message
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A bowl is not a mug
?
Is there a thermodynamic signature?
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119
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A bowl is not a mug
?
Is there a thermodynamic signature?
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119
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A bowl is not a mug
?
Is there a thermodynamic signature?
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119
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A bowl is not a mug
?
Is there a thermodynamic signature?
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119
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A bowl is not a mug
?
Is there a thermodynamic signature?
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119
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A bowl is not a mug
?
Is there a thermodynamic signature?
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119
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A bowl is not a mug
?
Is there a thermodynamic signature?
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119
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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 nodesCv ∼ e−∆/T Cv ∼ T 3 Cv ∼ T 2
∆
This simple idea has been around for a while.2
Widely used to fit experimental data on unconventional superconductors.3
2Anderson & Morel (1961), Leggett (1975)3Sigrist, Ueda (’89), Annett (’90), MacKenzie & Maeno (’03)Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 72 / 119
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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√
I2n = 3 n = 2
Key assumption: linear increase of the gap away from the node
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 73 / 119
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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√
I2n = 3 n = 2
Key assumption: linear increase of the gap away from the node
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 73 / 119
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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√
I2n = 3 n = 2
Key assumption: linear increase of the gap away from the node
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 73 / 119
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Shallow nodesRelax the linear assumption and we also get different exponents:
shallowpoint node line node
∆2k = I1(kx
||2 + ky
||2)2 ∆2
k = I1kx||
4
g(E ) = E2(2π)2√I1
√I2
g(E ) = L√
E
(2π)3I14
1√
I2n = 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 E2upairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)] and ourown result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.(unpublished)].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 RAL) www.cond-mat.org Dresden 2014 74 / 119
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Shallow nodesRelax the linear assumption and we also get different exponents:
shallowpoint node line node
∆2k = I1(kx
||2 + ky
||2)2 ∆2
k = I1kx||
4
g(E ) = E2(2π)2√I1
√I2
g(E ) = L√
E
(2π)3I14
1√
I2n = 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 E2upairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)] and ourown result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.(unpublished)].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 RAL) www.cond-mat.org Dresden 2014 74 / 119
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Shallow nodesRelax the linear assumption and we also get different exponents:
shallowpoint node line node
∆2k = I1(kx
||2 + ky
||2)2 ∆2
k = I1kx||
4
g(E ) = E2(2π)2√I1
√I2
g(E ) = L√
E
(2π)3I14
1√
I2n = 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 E2upairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)] and ourown result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.(unpublished)].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 RAL) www.cond-mat.org Dresden 2014 74 / 119
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Shallow nodesRelax the linear assumption and we also get different exponents:
shallowpoint node line node
∆2k = I1(kx
||2 + ky
||2)2 ∆2
k = I1kx||
4
g(E ) = E2(2π)2√I1
√I2
g(E ) = L√
E
(2π)3I14
1√
I2n = 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 E2upairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)] and ourown result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.(unpublished)].
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 RAL) www.cond-mat.org Dresden 2014 74 / 119
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Shallow nodesRelax the linear assumption and we also get different exponents:
shallowpoint node line node
∆2k = I1(kx
||2 + ky
||2)2 ∆2
k = I1kx||
4
g(E ) = E2(2π)2√I1
√I2
g(E ) = L√
E
(2π)3I14
1√
I2n = 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 E2upairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)] and ourown result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.(unpublished)].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 RAL) www.cond-mat.org Dresden 2014 74 / 119
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Line crossingsA 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/I141
√E/I
141
|)
(2π)3√I1I2∼ E0.8
n = 1.8 (< 2 !!)
+
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 75 / 119
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Crossing of shallow line nodesWhen 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+
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Numerics
n = d lnCv /d lnT
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
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A generic mechanismWe propose that shallow nodes will exist generically at topological phasetransitions in superocnductors with multi-component order parameters:
∆ 0
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 78 / 119
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A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 79 / 119
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A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 80 / 119
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A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Line
ar
node
s
Line
ar
node
sJorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 81 / 119
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A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 82 / 119
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A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 83 / 119
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A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Sha
llow
no
de
Sha
llow
no
de
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 84 / 119
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Quantum Materials Theory
1 Broken time-reversal symmetry in superconductors
2 Experimental evidence for broken TRS
3 Singlet, triplet, or both?
4 A symmetry zoo
5 Topological transitions in Superconductors
6 Topological transition state: Li2PdxPt3−xB
7 Take-home message
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Singlet-triplet mixing in noncentrosymmetricsuperconductors
Non-centrosymmetric superconductors are the multi-component orderparameter supercondcutors par excellence:
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
ˆ k 0 0
0 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:BaPtSi34, Re3W5,...Others seem to be correlated, purely triplet superconductors: +
LaNiC26 (c.f. centrosymmetric LaNiGa27) + , CePtr3Si (?) 8
4Batkova et al. JPCM (2010)5Zuev et al. PRB (2007)6Adrian D. Hillier, JQ and R. Cywinski PRL (2009)7Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)8Bauer et al. PRL (2004)Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 86 / 119
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Singlet-triplet mixing in noncentrosymmetricsuperconductors
Non-centrosymmetric superconductors are the multi-component orderparameter supercondcutors par excellence:
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
ˆ k 0 0
0 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:BaPtSi34, Re3W5,...Others seem to be correlated, purely triplet superconductors: +
LaNiC26 (c.f. centrosymmetric LaNiGa27) + , CePtr3Si (?) 8
4Batkova et al. JPCM (2010)5Zuev et al. PRB (2007)6Adrian D. Hillier, JQ and R. Cywinski PRL (2009)7Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)8Bauer et al. PRL (2004)Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 86 / 119
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Singlet-triplet mixing in noncentrosymmetricsuperconductors
Non-centrosymmetric superconductors are the multi-component orderparameter supercondcutors par excellence:
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
ˆ k 0 0
0 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:BaPtSi34, Re3W5,...Others seem to be correlated, purely triplet superconductors: +
LaNiC26 (c.f. centrosymmetric LaNiGa27) + , CePtr3Si (?) 84Batkova et al. JPCM (2010)5Zuev et al. PRB (2007)6Adrian D. Hillier, JQ and R. Cywinski PRL (2009)7Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)8Bauer et al. PRL (2004)Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 86 / 119
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Li2PdxPt3−xB: tunable singlet-triplet mixingThe Li2PdxPt3−xB 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)
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Li2PdxPt3−xB: tunable singlet-triplet mixingThe Li2PdxPt3−xB 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 RAL) www.cond-mat.org Dresden 2014 87 / 119
![Page 137: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/137.jpg)
Li2PdxPt3−xB: tunable singlet-triplet mixingThe Li2PdxPt3−xB 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 RAL) www.cond-mat.org Dresden 2014 87 / 119
![Page 138: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/138.jpg)
Li2PdxPt3−xB: tunable singlet-triplet mixingThe Li2PdxPt3−xB 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 RAL) www.cond-mat.org Dresden 2014 87 / 119
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Li2PdxPt3−xB: Phase diagramBogoliubov 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)]}
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Li2PdxPt3−xB: Phase diagramBogoliubov 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 RAL) www.cond-mat.org Dresden 2014 88 / 119
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Li2PdxPt3−xB: Phase diagramBogoliubov 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)]}
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Li2PdxPt3−xB: Phase diagramTreat A and B as independent tuning parameters and study quasiparticlespectrum. 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.Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 89 / 119
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Li2PdxPt3−xB: Phase diagram
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Li2PdxPt3−xB: Phase diagram
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Li2PdxPt3−xB: Phase diagram
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Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 93 / 119
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Detecting the topological transitions
3 734
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Detecting the topological transitions
3 734
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 95 / 119
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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 RAL) www.cond-mat.org Dresden 2014 97 / 119
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Li2PdxPt3−xB: predicted specific heat power-laws
3
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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 RAL) www.cond-mat.org Dresden 2014 100 / 119
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Anomalous power laws throughout the phase diagramDoes 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
0.25
T/T
c
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 transitionThe anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119
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Anomalous power laws throughout the phase diagramDoes 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
0.25
T/T
c
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 transitionThe anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119
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Anomalous power laws throughout the phase diagramDoes 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 transitionThe anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119
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Anomalous power laws throughout the phase diagramDoes 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 diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119
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Anomalous power laws throughout the phase diagramDoes 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 transitionThe anomalous exponent (here n = 1.8) is seen everywhere else
⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119
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Anomalous power laws throughout the phase diagramDoes 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 transitionThe 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 RAL) www.cond-mat.org Dresden 2014 101 / 119
![Page 160: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors](https://reader031.vdocuments.site/reader031/viewer/2022020105/559c194f1a28ab22598b465f/html5/thumbnails/160.jpg)
Anomalous power laws throughout the phase diagramDoes 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 transitionThe anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119
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Quantum Materials Theory
1 Broken time-reversal symmetry in superconductors
2 Experimental evidence for broken TRS
3 Singlet, triplet, or both?
4 A symmetry zoo
5 Topological transitions in Superconductors
6 Topological transition state: Li2PdxPt3−xB
7 Take-home message
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What to take home
Superconductors
Broken time-reversal
symmetry
Topological transitionsTriplet
pairing
The relationship between triplet pairing and broken timre-reversal symmetryis complicatedNon-unitary triplet pairing breaks time-reversal symmetry and couples tomagnetism in a special waySinglet-triplet mixing may induce broken time-reversal symmetry ortopological transitionsThere are bulk signatures of topological transitionsThe thermodynamic transition is a distinct state
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THANKS!
www.cond-mat.org
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Spin-orbit coupling in Sr2RuO4
Recent spin-polarised ARPES find strong spin-orbit coupling in Sr2RuO4[C.N. Veenstra et al., PRL 112, 127002 (2014)]:
Veenstra et al.’s claim is that this will lead to singlet-triplet mixing.This seems at odds with our approach.
back
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Power laws in nodal superconductors
Let’s remember where this came from:
Cv = T(
dSdT
)=
12kBT 2 ∑
k
Ek − T dEkdT︸︷︷︸≈0
Ek sech2 Ek2kBT︸ ︷︷ ︸
≈4e−Ek /KBT
∼ T−2∫
dEg (E )E2e−E/kBT at low T
g (E ) ∼ En−1 ⇒ Cv ∼ T n∫
dεε2+n−1e−ε︸ ︷︷ ︸a number
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Power laws in nodal superconductors
Ek =√
ε2k + ∆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
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Shallow line nodes in pnictides
back
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Logarithm ⇒ power law (n− 1 = 0.8)
The power-law expression is asymptotically very good at E → 0:
back
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Logarithm ⇒ power law (n− 1 = 0.4)
The power-law expression is asymptotically very good at E → 0:
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 RAL) www.cond-mat.org Dresden 2014 111 / 119
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Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)
Relaxation due to electronic moments
Moment
size
~ 0.1G
(~ 0.01μB)
(longitudinal)
Timescale:
> 10-4
s ~
e
_
e
backward detector
forward detector
sample
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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:
> 10-4
s ~
e
_
e
backward detector
forward detector
sample
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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), Li
2Pt
3B & Li
2Pd
3B
(2), ...
(1) Bauer et al. PRL’04 (2) Yuan et al. PRL’06
back
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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,Ck
x,Dk
xk
yk
z)
A2
(k) = Akxk
Y d(k) = (Bk
x,Ck
y,Dk
z)
B1
(k) = AkXk
Z d(k) = (Bk
xk
yk
z,Ck
z,Dk
y)
B2
(k) = AkYk
Z d(k) = (Bk
z, Ck
xk
yk
z,Dk
x)
The role of spin-orbit coupling (SOC)
Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 115 / 119
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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,Ck
x,Dk
xk
yk
z)
A2
(k) = Akxk
Y d(k) = (Bk
x,Ck
y,Dk
z)
B1
(k) = AkXk
Z d(k) = (Bk
xk
yk
z,Ck
z,Dk
y)
B2
(k) = AkYk
Z d(k) = (Bk
z, Ck
xk
yk
z,Dk
x)
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 RAL) www.cond-mat.org Dresden 2014 116 / 119
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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)
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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))
back
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Li2PdxPt3−xB:order parameter
back
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