Thermodynamic signatures of topological transitions in nodal superconductors

Thermodynamic signatures
of topological transitions
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
UK-NL Condensed Matter Meeting, Bristol, UK, 2013
(web version)
Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 1 / 69

PRELUDE - Symmetry

Photo:EddieHui-Bon-Hoa,www.shiromi.com

PRELUDE - Symmetry

Photo:KennethG.Libbrecht,snowflakes.com



PRELUDE - Symmetry

Photo:KennethG.Libbrecht,snowflakes.com


Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Photo:commons.wikimedia.org
Unconventional superconductors



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

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 ∼ T2
∆
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)

Linear nodes
It all comes from the density of states: +
g (E) ∼ En−1
⇒ Cv ∼ Tn

Linear nodes
g (E) ∼ En−1
⇒ Cv ∼ Tn
linear
point node line node
∆2
k = I1 kx
||
2
+ ky
||
2
∆2
k = I1kx
||
2
g(E) = E2
2(2π)2I1
√
I2
g(E) = LE
(2π)3
√
I1
√
I2
n = 3 n = 2

Linear nodes
g (E) ∼ En−1
⇒ Cv ∼ Tn
linear
∆2
k = I1 kx
||
2
+ ky
||
2
∆2
k = I1kx
||
2
g(E) = E2
2(2π)2I1
√
I2
g(E) = LE
(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
∆2
k = I1(kx
||
2
+ ky
||
2
)2 ∆2
k = I1kx
||
4
g(E) = E
2(2π)2
√
I1
√
I2
g(E) = L
√
E
(2π)3I
1
4
1
√
I2
n = 2 n = 1.5

Shallow nodes
shallow
∆2
k = I1(kx
||
2
+ ky
||
2
)2 ∆2
k = I1kx
||
4
g(E) = E
2(2π)2
√
I1
√
I2
g(E) = L
√
E
(2π)3I
1
4
1
√
I2
n = 2 n = 1.5
Shallow point nodes ﬁrst discussed (speculatively) by Leggett [1979].

Shallow nodes
shallow
∆2
k = I1(kx
||
2
+ ky
||
2
)2 ∆2
k = I1kx
||
4
g(E) = E
2(2π)2
√
I1
√
I2
g(E) = L
√
E
(2π)3I
1
4
1
√
I2
n = 2 n = 1.5
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)].

Shallow nodes
shallow
∆2
k = I1(kx
||
2
+ ky
||
2
)2 ∆2
k = I1kx
||
4
g(E) = E
2(2π)2
√
I1
√
I2
g(E) = L
√
E
(2π)3I
1
4
1
√
I2
n = 2 n = 1.5
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
∆2
k = I1 kx
||
2
− ky
||
2 2
or I1kx
||
2
ky
||
2
g(E) =
E(1+2ln|
L+
√
E/I
1
4
1
√
E/I
1
4
1
|)
(2π)3
√
I1I2
∼ E0.8
n = 1.8 (< 2 !!)

Crossing of shallow line nodes
When shallow lines cross we get an even lower exponent:
crossing
of shallow line nodes
∆2
k = I1 kx
||
2
− ky
||
2 4
or I1kx
||
4
ky
||
4
g (E) =
√
E(1+2ln|
L+E
1
4 /I
1
8
1
E
1
4 /I
1
8
1
|)
(2π)3I
1
4
1
√
I2
∼ E0.4
n = 1.4 *
* c.f. gapless excitations of a Fermi liquid: g (E) = constant ⇒ n = 1

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 node
shallow point node
linear line node
crossing of linear line nodes
shallow line node
crossing of shallow line nodes

A generic mechanism
We propose that shallow nodes will exist generically at topological phase
transitions in superocnductors with multi-component order parameters:
∆0
∆1Fermi Sea
∆0

A generic mechanism
∆1Fermi Sea
∆0

A generic mechanism
∆1Fermi Sea
∆0
Linear
nodes
Linear
nodesJorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 17 / 69

A generic mechanism
∆1Fermi Sea
∆0

A generic mechanism
∆1Fermi Sea
∆0
Shallow
node
Shallow
nodeJorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 20 / 69

Singlet-triplet mixing in noncentrosymmetric
superconductors
Non-centrosymmetric superconductors are the multi-component order
parameter supercondcutors par excellence:
Singlet, triplet, or both?

ˆ k 
0 0
0 0






dx  idy dz
dz dx  idy






singlet
[ 0(k) even ]
triplet
[ d(k) odd ]
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)

superconductors

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:

superconductors

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

Li2Pdx Pt3−x B:
A superconductor with tunable singlet-triplet mixing
The Li2Pdx Pt3−x B family (0 ≤ x ≤ 3; cubic point group O) provides a tunable
realisation of this singlet-triplet mixing:

Li2Pdx 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)

Li2Pdx Pt3−x B:
Experimentally, the series is found to go
from fully-gapped (x = 3) to nodal
behaviour (x = 0):

Li2Pdx Pt3−x B:
Experimentally, the series is found to go
from fully-gapped (x = 3) to nodal
behaviour (x = 0):
NMR suggests the nodal state is a
triplet:

Li2Pdx Pt3−x B: Phase diagram
Bogoliubov Hamiltonian with Rashba spin-orbit coupling:
H(k) =
ˆh(k) ˆ∆(k)
ˆ∆†(k) −ˆhT (−k)
ˆh(k) = εkI + γk · σ
ˆ∆ (k) = [∆0 (k) + d (k) · ˆσ] i ˆσy (most general gap matrix)

H(k) =
ˆh(k) ˆ∆(k)
ˆ∆†(k) −ˆhT (−k)
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
.

H(k) =
ˆh(k) ˆ∆(k)
ˆ∆†(k) −ˆhT (−k)
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 + k2
z , ky k2
z + k2
x , kz k2
x + k2
y }

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
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.

We ﬁnd a very rich phase diagram with topollogically-distinct phases.9
9C. Beri, PRB (2010); A. Schnyder, S. Ryu, PRB(R) (2011); A. Schnyder et al.,
PRB (2012); B. Mazidian, JQ, A.D. Hillier, J.F. Annett, arXiv:1302.2161.

Detecting the topological transitions
3 734

Li2Pdx Pt3−x B: predicted speciﬁc heat power-laws
334

jn = 2
n = 1.8
n = 1.4
n = 2
3
4
5
11

3

jn = 2
n = 1.8
n = 1.4
n = 2
3
4
5
11

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:

Let’s put 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

superconductors
5 Additional details

Let’s remember where this came from:
Cv = T
dS
dT
=
1
2kBT2 ∑
k



Ek − T
dEk
dT
≈0



 Ek sech2 Ek
2kBT
≈4e−Ek /KBT
∼ T−2
dEg (E) E2
e−E/kBT
at low T
g (E) ∼ En−1
⇒ Cv ∼ Tn
d 2+n−1
e−
a number

Ek = 2
k + ∆2
k
≈ I2k2
⊥ + ∆ kx
||
, ky
||
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

Shallow line nodes in pnictides
back

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

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:
 
 
 





spinfor
spinfor
kk
kk
k



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)

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

ISIS
muSR

Zero field muon spin relaxation
e
_

e
backward
detector
forward
detector
sample

Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)
Relaxation due to electronic moments
Moment
size
~ 0.1G
(~ 0.01μB)
(longitudinal)
Timescale:
> 10-4s~
e
_

e
backward
detector
forward
detector
sample

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:
> 10-4s~
e
_

e
backward
detector
forward
detector
sample

LaNiC2 is a non-ceontrsymmetric superconductor
Neutron diffraction
30 40 50 60 70 80
0
5000
10000
15000
20000
25000
30000
35000
Intensity(arbunits)
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), Li2Pt3B & Li2Pd3B (2), ...
(1) Bauer et al. PRL’04 (2) Yuan et al. PRL’06

Character table
PRL 102 117007 (2009)

Character table
PRL 102 117007 (2009)
180o

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
PRL 102 117007 (2009)

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
PRL 102 117007 (2009)
These must be combined with the singlet and triplet
representations of SO(3).

SO(3)xC2v Gap function
(unitary)
Gap function
(non-unitary)
1A1 (k)=1 -
1A2 (k)=kxkY -
1B1 (k)=kXkZ -
1B2 (k)=kYkZ -
3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3A2 d(k)=(0,0,1)kXkYkZ d(k)=(1,i,0)kXkYkZ
3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
Possible order parameters
PRL 102 117007 (2009)

(unitary)
Gap function
(non-unitary)
1A1 (k)=1 -
1A2 (k)=kxkY -
1B1 (k)=kXkZ -
1B2 (k)=kYkZ -
3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
PRL 102 117007 (2009)

(unitary)
Gap function
(non-unitary)
1A1 (k)=1 -
1A2 (k)=kxkY -
1B1 (k)=kXkZ -
1B2 (k)=kYkZ -
3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
Non-unitary
d x d* ≠ 0
PRL 102 117007 (2009)

(unitary)
Gap function
(non-unitary)
1A1 (k)=1 -
1A2 (k)=kxkY -
1B1 (k)=kXkZ -
1B2 (k)=kYkZ -
3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
Non-unitary
d x d* ≠ 0
breaks only SO(3) x U(1) x T
* C.f. Li2Pd3B & Li2Pt3B,
H. Q. Yuan et al. PRL’06
*
PRL 102 117007 (2009)

Spin-up superfluid
coexisting with spin-
down Fermi liquid.
The A1 phase of
liquid 3He.
Non-unitary pairing















0
00
or
00
0ˆ
C.f.

C2v,Jno t Gap function,
singlet component
Gap function,
triplet component
A1 (k) = A d(k) = (Bky,Ckx,Dkxkykz)
A2  (k) = AkxkY d(k) = (Bkx,Cky,Dkz)
B1  (k) = AkXkZ d(k) = (Bkxkykz,Ckz,Dky)
B2  (k) = AkYkZ d(k) = (Bkz, Ckxkykz,Dkx)
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)

C2v,Jno t Gap function,
singlet component
Gap function,
triplet component
A1 (k) = A d(k) = (Bky,Ckx,Dkxkykz)
A2  (k) = AkxkY d(k) = (Bkx,Cky,Dkz)
B1  (k) = AkXkZ d(k) = (Bkxkykz,Ckz,Dky)
B2  (k) = AkYkZ d(k) = (Bkz, Ckxkykz,Dkx)
None of these break time-reversal symmetry!
PRB 82, 174511 (2010)

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)
PRB 82, 174511 (2010)

Li2Pdx Pt3−x B:
order parameter
back

Thermodynamic signatures of topological transitions in nodal superconductors

Recommended

Recommended

More Related Content

Similar to Thermodynamic signatures of topological transitions in nodal superconductors

Similar to Thermodynamic signatures of topological transitions in nodal superconductors (20)

More from Jorge Quintanilla

More from Jorge Quintanilla (12)

Recently uploaded

Recently uploaded (20)

Thermodynamic signatures of topological transitions in nodal superconductors