Talk kent symposium_2013_v01_for_web
-
Upload
jorge-quintanilla -
Category
Documents
-
view
611 -
download
0
Transcript of Talk kent symposium_2013_v01_for_web
![Page 1: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/1.jpg)
Anomalous thermodynamic power lawsin 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
Functional Materials Symposium, University of Kent, Canterbury 2013
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 1 / 39
![Page 2: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 2 / 39
![Page 3: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 2 / 39
![Page 4: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 2 / 39
![Page 5: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 2 / 39
![Page 6: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 2 / 39
![Page 7: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/7.jpg)
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 3 / 39
![Page 8: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/8.jpg)
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 3 / 39
![Page 9: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/9.jpg)
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 3 / 39
![Page 10: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/10.jpg)
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 3 / 39
![Page 11: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 4 / 39
![Page 12: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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 kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 6 / 39
![Page 14: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 7 / 39
![Page 15: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 7 / 39
![Page 16: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 7 / 39
![Page 17: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/17.jpg)
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 (speculativebeamer reveal one at atimely) 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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 8 / 39
![Page 18: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/18.jpg)
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 (speculativebeamer reveal one at atimely) 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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 8 / 39
![Page 19: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/19.jpg)
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 (speculativebeamer reveal one at atimely) 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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 8 / 39
![Page 20: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/20.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 9 / 39
![Page 21: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/21.jpg)
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+
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 10 / 39
![Page 22: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/22.jpg)
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
![Page 23: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/23.jpg)
A generic mechanismMore generically, we expect this to happen at topological phase transitions insuperocnductors with multi-component order parameters:
∆ 0
∆ 1Fermi Sea
∆ 0
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 12 / 39
![Page 24: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/24.jpg)
A generic mechanismMore generically, we expect this to happen at topological phase transitions insuperocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Sha
llow
no
de
Sha
llow
no
de
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 13 / 39
![Page 25: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/25.jpg)
A generic mechanismMore generically, we expect this to happen at topological phase transitions insuperocnductors with multi-component order parameters:
∆ 1Fermi Sea
∆ 0
Line
ar
node
s
Line
ar
node
sJorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 14 / 39
![Page 26: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/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 kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/27.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 16 / 39
![Page 28: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/28.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 16 / 39
![Page 29: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/29.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 16 / 39
![Page 30: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/30.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 17 / 39
![Page 31: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/31.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 17 / 39
![Page 32: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/32.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 17 / 39
![Page 33: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/33.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 17 / 39
![Page 34: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/34.jpg)
Li2PdxPt3−xB: Phase diagram
Assume the order parameter corresponds to the 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)]}
Treat A and B as in dependent tuning parameters and study quasiparticlespectrum.
+
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 18 / 39
![Page 35: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/35.jpg)
Li2PdxPt3−xB: Phase diagramWe 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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 19 / 39
![Page 36: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/36.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 20 / 39
![Page 37: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/37.jpg)
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 21 / 39
![Page 38: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/38.jpg)
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 22 / 39
![Page 39: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/39.jpg)
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 23 / 39
![Page 40: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/40.jpg)
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 24 / 39
![Page 41: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/41.jpg)
Detecting the topological transitions
3 734
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 25 / 39
![Page 42: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/42.jpg)
Detecting the topological transitions
3 734
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 26 / 39
![Page 43: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/43.jpg)
Li2PdxPt3−xB: predicted specific heat power-laws
334
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 27 / 39
![Page 44: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/44.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 28 / 39
![Page 45: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/45.jpg)
Li2PdxPt3−xB: predicted specific heat power-laws
3
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 29 / 39
![Page 46: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/46.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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 30 / 39
![Page 47: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/47.jpg)
Anomalous power laws throughout the phase diagrampPut 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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 31 / 39
![Page 48: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/48.jpg)
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
![Page 49: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/49.jpg)
Topological transitions in nodal superconductorshave clear signatures in bulk thermodynamic properties.
THANKS!
www.cond-mat.org
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 33 / 39
![Page 50: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/50.jpg)
Topological transitions in nodal superconductorshave clear signatures in bulk thermodynamic properties.
THANKS!
www.cond-mat.org
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 33 / 39
![Page 51: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/51.jpg)
ADDITIONAL INFORMATION
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 34 / 39
![Page 52: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/52.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−2T 1+2+n−1∫
dεε2+n−1e−ε︸ ︷︷ ︸a number
∼ T n
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 35 / 39
![Page 53: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/53.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
Q.E.D.
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 36 / 39
![Page 54: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/54.jpg)
Shallow line nodes in pnictides
back
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 37 / 39
![Page 55: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/55.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
back
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 38 / 39
![Page 56: Talk kent symposium_2013_v01_for_web](https://reader034.fdocuments.us/reader034/viewer/2022042715/5599ffc91a28ab02098b4795/html5/thumbnails/56.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.
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) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 39 / 39