View
45
Download
0
Category
Tags:
Preview:
Citation preview
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
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
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
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
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
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
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 3 / 39
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 3 / 39
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 3 / 39
PRELUDE - Topology
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 3 / 39
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
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
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
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
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
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
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
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
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
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
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
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
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
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
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
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
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
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
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
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
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
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
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
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
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
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
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 21 / 39
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 22 / 39
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 23 / 39
Li2PdxPt3−xB: Phase diagram
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 24 / 39
Detecting the topological transitions
3 734
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 25 / 39
Detecting the topological transitions
3 734
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 26 / 39
Li2PdxPt3−xB: predicted specific heat power-laws
334
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 27 / 39
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
Li2PdxPt3−xB: predicted specific heat power-laws
3
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 29 / 39
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
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
Anomalous thermodynamic power laws in nodalsuperconductors
1 What are they?
2 How to get them
3 An example
4 Take-home message
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
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
ADDITIONAL INFORMATION
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 34 / 39
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
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
Shallow line nodes in pnictides
back
Jorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 37 / 39
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
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
Recommended