Designing for Digital Learners (D4DL) UWE Bristol, UK, May 2014
Talk bristol uk-nl_2013_v01_for_web
-
Upload
jorge-quintanilla -
Category
Technology
-
view
123 -
download
0
description
Transcript of Talk bristol uk-nl_2013_v01_for_web
![Page 1: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/1.jpg)
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
UK-NL Condensed Matter Meeting, Bristol, UK, 2013(web version)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 1 / 69
![Page 2: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/2.jpg)
PRELUDE - 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 ISIS) arXiv:1302.2161 Bristol 2013 2 / 69
![Page 3: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/3.jpg)
PRELUDE - 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 ISIS) arXiv:1302.2161 Bristol 2013 2 / 69
![Page 4: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/4.jpg)
PRELUDE - 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 ISIS) arXiv:1302.2161 Bristol 2013 2 / 69
![Page 5: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/5.jpg)
PRELUDE - 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 ISIS) arXiv:1302.2161 Bristol 2013 2 / 69
![Page 6: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/6.jpg)
PRELUDE - 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 ISIS) arXiv:1302.2161 Bristol 2013 2 / 69
![Page 7: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/7.jpg)
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 3 / 69
![Page 8: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/8.jpg)
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 3 / 69
![Page 9: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/9.jpg)
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 3 / 69
![Page 10: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/10.jpg)
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 3 / 69
![Page 11: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/11.jpg)
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 Bristol 2013 4 / 69
![Page 12: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/12.jpg)
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
![Page 13: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/13.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 nodesCv ∼ 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 Bristol 2013 6 / 69
![Page 14: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/14.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 7 / 69
![Page 15: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/15.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 7 / 69
![Page 16: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/16.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 7 / 69
![Page 17: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/17.jpg)
Shallow nodes
Relax 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)].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 Bristol 2013 8 / 69
![Page 18: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/18.jpg)
Shallow nodes
Relax 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)].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 Bristol 2013 8 / 69
![Page 19: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/19.jpg)
Shallow nodes
Relax 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)].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 Bristol 2013 8 / 69
![Page 20: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/20.jpg)
Shallow nodes
Relax 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)].
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 Bristol 2013 8 / 69
![Page 21: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/21.jpg)
Shallow nodes
Relax 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)].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 Bristol 2013 8 / 69
![Page 22: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/22.jpg)
Shallow nodes
Relax 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)].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 Bristol 2013 9 / 69
![Page 23: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/23.jpg)
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/I141
√E/I
141
|)
(2π)3√I1I2∼ E0.8
n = 1.8 (< 2 !!)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 10 / 69
![Page 24: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/24.jpg)
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 = 1Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 11 / 69
![Page 25: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/25.jpg)
Numerics
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 Bristol 2013 12 / 69
![Page 26: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/26.jpg)
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
![Page 27: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/27.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 14 / 69
![Page 28: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/28.jpg)
A generic mechanismWe propose that shallow nodes will exist generically at topological phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 15 / 69
![Page 29: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/29.jpg)
A generic mechanismWe propose that shallow nodes will exist generically at topological phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 16 / 69
![Page 30: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/30.jpg)
A generic mechanismWe propose that shallow nodes will exist generically at topological phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Line
ar
node
s
Line
ar
node
sJorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 17 / 69
![Page 31: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/31.jpg)
A generic mechanismWe propose that shallow nodes will exist generically at topological phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 18 / 69
![Page 32: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/32.jpg)
A generic mechanismWe propose that shallow nodes will exist generically at topological phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 19 / 69
![Page 33: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/33.jpg)
A generic mechanismWe propose that shallow nodes will exist generically at topological phasetransitions in superocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Sha
llow
no
de
Sha
llow
no
de
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 20 / 69
![Page 34: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/34.jpg)
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
![Page 35: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/35.jpg)
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:BaPtSi33, Re3W4,...Others seem to be correlated 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 Bristol 2013 22 / 69
![Page 36: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/36.jpg)
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:BaPtSi33, Re3W4,...Others seem to be correlated 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 Bristol 2013 22 / 69
![Page 37: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/37.jpg)
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:BaPtSi33, Re3W4,...Others seem to be correlated 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 Bristol 2013 22 / 69
![Page 38: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/38.jpg)
Li2PdxPt3−xB:A superconductor with 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)
Experimentally, the series is found to gofrom fully-gapped (x = 3) to nodalbehaviour (x = 0):
H.Q. Yuan et al.,Phys. Rev. Lett. 97, 017006 (2006).
NMR suggests the nodal state is atriplet:
M.Nishiyama et al.,Phys. Rev. Lett. 98, 047002 (2007)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 23 / 69
![Page 39: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/39.jpg)
Li2PdxPt3−xB:A superconductor with 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)
Experimentally, the series is found to gofrom fully-gapped (x = 3) to nodalbehaviour (x = 0):
H.Q. Yuan et al.,Phys. Rev. Lett. 97, 017006 (2006).
NMR suggests the nodal state is atriplet:
M.Nishiyama et al.,Phys. Rev. Lett. 98, 047002 (2007)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 23 / 69
![Page 40: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/40.jpg)
Li2PdxPt3−xB:A superconductor with 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)
Experimentally, the series is found to gofrom fully-gapped (x = 3) to nodalbehaviour (x = 0):
H.Q. Yuan et al.,Phys. Rev. Lett. 97, 017006 (2006).
NMR suggests the nodal state is atriplet:
M.Nishiyama et al.,Phys. Rev. Lett. 98, 047002 (2007)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 23 / 69
![Page 41: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/41.jpg)
Li2PdxPt3−xB:A superconductor with 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)
Experimentally, the series is found to gofrom fully-gapped (x = 3) to nodalbehaviour (x = 0):
H.Q. Yuan et al.,Phys. Rev. Lett. 97, 017006 (2006).
NMR suggests the nodal state is atriplet:
M.Nishiyama et al.,Phys. Rev. Lett. 98, 047002 (2007)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 23 / 69
![Page 42: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/42.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 24 / 69
![Page 43: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/43.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 24 / 69
![Page 44: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/44.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 24 / 69
![Page 45: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/45.jpg)
Li2PdxPt3−xB: Phase diagramTreat A and B as in dependent 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.
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 25 / 69
![Page 46: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/46.jpg)
Li2PdxPt3−xB: Phase diagramWe 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 ISIS) arXiv:1302.2161 Bristol 2013 26 / 69
![Page 47: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/47.jpg)
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 27 / 69
![Page 48: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/48.jpg)
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 28 / 69
![Page 49: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/49.jpg)
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 29 / 69
![Page 50: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/50.jpg)
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 30 / 69
![Page 51: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/51.jpg)
Detecting the topological transitions
3 734
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 31 / 69
![Page 52: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/52.jpg)
Detecting the topological transitions
3 734
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 32 / 69
![Page 53: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/53.jpg)
Li2PdxPt3−xB: predicted specific heat power-laws
334
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 33 / 69
![Page 54: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/54.jpg)
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 Bristol 2013 34 / 69
![Page 55: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/55.jpg)
Li2PdxPt3−xB: predicted specific heat power-laws
3
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 35 / 69
![Page 56: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/56.jpg)
Li2PdxPt3−xB: predicted specific heat power-laws
3
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 36 / 69
![Page 57: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/57.jpg)
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 Bristol 2013 37 / 69
![Page 58: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/58.jpg)
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:
The influence of the topological transition extends throughout the phasediagram (c.f. quantum critical endpoints)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 38 / 69
![Page 59: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/59.jpg)
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:
The influence of the topological transition extends throughout the phasediagram (c.f. quantum critical endpoints)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 38 / 69
![Page 60: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/60.jpg)
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:
The influence of the topological transition extends throughout the phasediagram (c.f. quantum critical endpoints)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 38 / 69
![Page 61: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/61.jpg)
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:
The influence of the topological transition extends throughout the phasediagram (c.f. quantum critical endpoints)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 38 / 69
![Page 62: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/62.jpg)
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
![Page 63: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/63.jpg)
Topological transitions in nodal superconductorshave clear signatures in bulk thermodynamic properties.
THANKS!
www.cond-mat.org
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 40 / 69
![Page 64: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/64.jpg)
Topological transitions in nodal superconductorshave clear signatures in bulk thermodynamic properties.
THANKS!
www.cond-mat.org
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 40 / 69
![Page 65: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/65.jpg)
Anomalous thermodynamic power laws in nodalsuperconductors
5 Additional details
![Page 66: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/66.jpg)
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
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 42 / 69
![Page 67: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/67.jpg)
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
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 43 / 69
![Page 68: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/68.jpg)
Shallow line nodes in pnictides
back
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 44 / 69
![Page 69: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/69.jpg)
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
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 45 / 69
![Page 70: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/70.jpg)
Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC) • Simplest noncentrosymmetric system: a surface.
• Rashba term in the Hamiltonian:
• In general, form & strength of SOC depend on details of electronic structure.
• Split Fermi surface:
spin for
spin for
kk
kkk
Gor'kov & Rashba, PRL, 87, 037004 (2001)
• There’s a zoo of phenomenologies for noncentrosymmetric superconductors:
•Triplet: CePt3Si [1]
•Singlet (conventional): Li2Pd3B [2], BaPtSi3 [3], Re3W [4]
•Singlet-triplet admixture: Li2Pt3B [2]
[1] Bauer et al. PRL (2004); [2] Yuan et al PRL (2006); [3] Batkova et al. JPCM (2010); [4] Zuev et al. PRB (‘07)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 46 / 69
![Page 71: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/71.jpg)
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 Bristol 2013 47 / 69
![Page 72: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/72.jpg)
ISIS
muSR
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 48 / 69
![Page 73: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/73.jpg)
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 Bristol 2013 49 / 69
![Page 74: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/74.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 50 / 69
![Page 75: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/75.jpg)
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
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 51 / 69
![Page 76: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/76.jpg)
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
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 52 / 69
![Page 77: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/77.jpg)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 53 / 69
![Page 78: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/78.jpg)
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 Bristol 2013 54 / 69
![Page 79: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/79.jpg)
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 Bristol 2013 55 / 69
![Page 80: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/80.jpg)
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 Bristol 2013 56 / 69
![Page 81: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/81.jpg)
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 Bristol 2013 57 / 69
![Page 82: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/82.jpg)
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 Bristol 2013 58 / 69
![Page 83: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/83.jpg)
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 Bristol 2013 59 / 69
![Page 84: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/84.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 60 / 69
![Page 85: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/85.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 61 / 69
![Page 86: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/86.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 62 / 69
![Page 87: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/87.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 63 / 69
![Page 88: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/88.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 64 / 69
![Page 89: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/89.jpg)
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 Bristol 2013 65 / 69
![Page 90: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/90.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 66 / 69
![Page 91: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/91.jpg)
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 ISIS) arXiv:1302.2161 Bristol 2013 67 / 69
![Page 92: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/92.jpg)
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 Bristol 2013 68 / 69
![Page 93: Talk bristol uk-nl_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022051610/54921ddaac795949288b46a2/html5/thumbnails/93.jpg)
Li2PdxPt3−xB:order parameter
back
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 69 / 69