Talk kent symposium_2013_v01_for_web

Anomalous thermodynamic power laws
in nodal superconductors
arXiv:1302.2161

Bayan Mazidian1,2 , Jorge Quintanilla2,3
James F. Annett1 , Adrian D. Hillier2

1
University of Bristol
2
ISIS Facility, STFC Rutherford Appleton Laboratory
3
SEPnet 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

Photo: Eddie Hui-Bon-Hoa, www.shiromi.com
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PRELUDE - Symmetry

Photo: Kenneth G. Libbrecht, snowflakes.com
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PRELUDE - Symmetry

Photo: Kenneth G. Libbrecht, snowflakes.com
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Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations

Unconventional superconductors

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Photo: commons.wikimedia.org

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PRELUDE - Topology


Anomalous thermodynamic power laws in nodal
superconductors

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 the
spectrum of low-lying excitations:
Fully gapped Point nodes Line nodes
Cv ∼ e −∆/T Cv ∼ T3 Cv ∼ T 2
∆

This simple idea has been around for a while.1
Widely used to ﬁt experimental data on unconventional superconductors.2

1 Anderson & Morel (1961), Leggett (1975)
2 Sigrist, Ueda (’89), Annett (’90), MacKenzie & Maeno (’03)

Linear nodes

It all comes from the density of states: +

g (E ) ∼ E n−1 ⇒ Cv ∼ T n


Linear nodes


g (E ) ∼ E n−1 ⇒ Cv ∼ T n

linear
point node line node

y 2
∆2 = I1 k|| 2 + k||
k
x ∆2 = I1 k|| 2
k
x

E2 √ LE √
g (E ) = 2(2π )2 I1 I2
g (E ) = √
(2π )3 I1 I2
n=3 n=2


Linear nodes


g (E ) ∼ E n−1 ⇒ Cv ∼ T n

linear

y 2
∆2 = I1 k|| 2 + k||
k
x ∆2 = I1 k|| 2
k
x

E2 √ LE √
g (E ) = 2(2π )2 I1 I2
g (E ) = √
(2π )3 I1 I2
n=3 n=2

Key assumption: linear increase of the gap away from the node


Shallow nodes
Relax the linear assumption and we also get diﬀerent exponents:


Shallow nodes
shallow

y 2
∆2 = I1 (k|| 2 + k|| )2
k
x ∆2 = I1 k|| 4
k
x
√
E √ L E
g (E ) = 2(2π )2
√
I1 I2
g (E ) = 1√
(2π )3 I14 I2
n=2 n = 1.5


Shallow nodes
shallow

y 2
∆2 = I1 (k|| 2 + k|| )2
k
x ∆2 = I1 k|| 4
k
x
√
E √ L E
g (E ) = 2(2π )2
√
I1 I2
g (E ) = 1√
(2π )3 I14 I2
n=2 n = 1.5

Shallow point nodes ﬁrst discussed (speculativebeamer reveal one at a
timely) by Leggett [1979].
A shallow point node may be required by symmetry e.g. the proposed E2u
pairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)].
A shallow line node may result at the boundary between gapless and line node
behaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +

Line crossings

A diﬀerent power law is expected at line crossings
(e.g. d-wave pairing on a spherical Fermi surface):

crossing
of linear line nodes

y 2 2
∆2 = I1 k|| 2 − k||
k
x

y 2
or I1 k|| 2 k||
x

√ 1
L+ E /I1 4
E (1+2ln| 1 |)
√
E /I 4
g (E ) = √ 1
(2π )3 I1 I2
∼ E 0 .8
n = 1.8 (< 2 !!)


Crossing of shallow line nodes
When shallow lines cross we get an even lower exponent:
crossing
of shallow line nodes

y 2 4
∆2 = I1 k|| 2 − k||
k
x

y 4
or I1 k|| 4 k||
x

1 1
√ L+E 4 /I1 8
E (1+2ln| |)
1 1
E 4 /I18
g (E ) = 1√
(2π )3 I14 I2
∼ E 0 .4
n = 1.4 *
* c.f. gapless excitations of a Fermi liquid: g (E ) = constant ⇒ n = 1
+


A generic mechanism
More generically, we expect this to happen at topological phase transitions in
superocnductors with multi-component order parameters:

∆1
∆0

Fermi Sea


A generic mechanism

Shallow
Shallow
node

node
∆1
∆0

Fermi Sea


A generic mechanism

Linear

Linear
nodes

nodes
∆1
∆0

Fermi Sea


Singlet-triplet mixing in noncentrosymmetric

superconductors Singlet, triplet, or both?
Non-centrosymmetric superconductors are the multi-component order
parameter supercondcutors par excellence:

 0  0  dx  id y dz 
k  
ˆ   
 0 0   dz dx  id y 
singlet triplet
[ 0(k) even ] [ d(k) odd ]



3 Batkova et al. JPCM (2010)
4 Zuev et al. PRB (2007)
5 Adrian D. Hillier, JQ and R. Cywinski PRL (2009)
6 Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)
7 Bauer et al. PRL (2004)



 0  0  dx  id y dz 
k  
ˆ   
 0 0   dz dx  id y 
singlet triplet
[ 0(k) even ] [ d(k) odd ]


In practice, there is a varied phenomenology:




 0  0  dx  id y dz 
k  
ˆ   
 0 0   dz dx  id y 
singlet triplet
[ 0(k) even ] [ 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

Li2 Pdx Pt3−x B:
A superconductor with tunable singlet-triplet mixing
The Li2 Pdx Pt3−x B family (0 ≤ x ≤ 3; cubic point group O) provides a tunable
realisation of this singlet-triplet mixing:


Li2 Pdx Pt3−x B:
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)


Li2 Pdx Pt3−x B:
Experimentally, the series is found to go
from fully-gapped (x = 3) to nodal
behaviour (x = 0):


Li2 Pdx Pt3−x B:
Experimentally, the series is found to go
from fully-gapped (x = 3) to nodal NMR suggests the nodal state is a
behaviour (x = 0): triplet:


Li2 Pdx Pt3−x B: Phase diagram

Assume the order parameter corresponds to the most symmetric (A1 )
irreducible representation:

∆0 (k) = ∆0
d(k) = ∆0 × {
2 2 2 2 2 2
A (x ) (kx , ky , kz ) − B (x ) kx ky + kz , ky kz + kx , kz kx + ky }

Treat A and B as in dependent tuning parameters and study quasiparticle
spectrum.
+


We ﬁnd a very rich phase diagram with topollogically-distinct phases.8

8 C. 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.

We ﬁnd a very rich phase diagram with topollogically-distinct phases.9

9 C. 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.

Detecting the topological transitions

4 3 7


Li2 Pdx Pt3−x B: predicted speciﬁc heat power-laws

4 3



5

n=2 j

4

n=2

3

n = 1.8

11

n = 1.4



3



5

n=2 j

4

n=2

3

n = 1.8

11

n = 1.4


Anomalous power laws throughout the phase diagram
pPut these curves on a density plot:

The inﬂuence of the topological transition extends throughout the phase
diagram (c.f. quantum critical endpoints)

Topological transitions in nodal superconductors
have clear signatures in bulk thermodynamic properties.


Topological transitions in nodal superconductors
have clear signatures in bulk thermodynamic properties.

THANKS!

www.cond-mat.org


ADDITIONAL INFORMATION



Let’s remember where this came from:

 
dS 1 dEk 
 Ek sech2 Ek
∑  Ek − T

Cv = T =
dT 2kB T 2 k
 dT  2kB T
≈0
≈4e −Ek /KB T

∼ T −2 dEg (E ) E 2 e −E /kB T at low T

g (E ) ∼ E n−1 ⇒ Cv ∼ T −2 T 1+2+n−1 d 2+n −1 −
e ∼ Tn

a number



k_
| ∆(k||x,k||y)
Ek = 2
k + ∆2
k
k||y
y 2
≈ I2 k⊥ + ∆ k|| , k||
2 x

on the Fermi surface
k||x

Compute density of states:

g (E ) = δ(Ek − E )dkx dky dkz

Q.E.D.


Shallow line nodes in pnictides

back


Numerics

4.5
linear point node
shallow point node
linear line node
4 crossing of linear line nodes
shallow line node
crossing of shallow line nodes
3.5

3
n

2.5

2

1.5

1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
T / Tc

back



Bogoliubov Hamiltonian with Rashba spin-orbit coupling:

h (k) ∆ (k)
H (k) =
∆ † (k) −hT (−k)
h (k) = ε k I + γk · σ

Assuming |ε k | | γk | |d (k)| the quasi-particle spectrum is

E =± (ε k − 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 + Z 2 , Y Z 2 + X 2 , Z X 2 + Y 2

back


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