thermodynamic signatures of topological transitions in nodal superconductors

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

Birmingham, UK, 14 November 2013

Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 1 / 95

http://xxx.arxiv.org/abs/1302.2161

Anomalous thermodynamic power laws in nodalsuperconductors

1 What are they?

2 How to get them

3 An example

4 Take-home message



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 nodes

Cv ∼ 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 B’ham 2013 4 / 95

Linear nodes

It all comes from the density of states: +

g (E ) ∼ En−1 ⇒ Cv ∼ T n


Linear nodes


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√

I2

n = 3 n = 2


Linear nodes


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√

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


shallowpoint node line node

∆2k = I1(kx

||2 + k

y

||2)2 ∆2

k = I1kx||

4

g(E ) = E

2(2π)2√

I1√

I2g(E ) = L

√E

(2π)3I14

1

√I2

n = 2 n = 1.5


Shallow nodes



∆2k = I1(kx

||2 + k

y

||2)2 ∆2

k = I1kx||

4

g(E ) = E

2(2π)2√

I1√

I2g(E ) = L

√E

(2π)3I14

1

√I2

n = 2 n = 1.5

Shallow point nodes first discussed (speculatively) by Leggett [1979].


Shallow nodes



∆2k = I1(kx

||2 + k

y

||2)2 ∆2

k = I1kx||

4

g(E ) = E

2(2π)2√

I1√

I2g(E ) = L

√E

(2π)3I14

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



∆2k = I1(kx

||2 + k

y

||2)2 ∆2

k = I1kx||

4

g(E ) = E

2(2π)2√

I1√

I2g(E ) = L

√E

(2π)3I14

1

√I2

n = 2 n = 1.5




A shallow line node may result at the boundary between gapless and line nodebehaviour 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):

crossingof linear line nodes

∆2k = I1

(

kx||

2 − ky

||2)2

or I1kx||

2ky

||2

g(E ) =

E (1+2ln| L+√

E/I

141

√E/I

141

|)

(2π)3√

I1I2

∼ E0.8

n = 1.8 (< 2 !!)

+


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 = 1

+


Numerics

n = d ln Cv /d ln T

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



1 What are they?

2 How to get them

3 An example

4 Take-home message

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

We propose that shallow nodes will exist generically at quantum phase


∆ 1Fermi Sea

∆ 0


A generic mechanism



∆ 1Fermi Sea

∆ 0


A generic mechanism



∆ 1Fermi Sea

∆ 0

Lin

ear

nodes

Lin

ear

nodes


A generic mechanism



∆ 1Fermi Sea

∆ 0


A generic mechanism



∆ 1Fermi Sea

∆ 0

Shallow

node

Shallow

node


Note: no broken symmetry

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These are topological transitions


These are topological transitions

G. E. Volovik in “Quantum Analogues: From Phase Transitions to Black Holes and Cosmology“, William G.

Unruh and Ralf Schützhold (Eds.), Springer (2007).



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:

ˆ k 0 00 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)Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 B’ham 2013 22 / 95



ˆ k 0 00 0

dx idy dz

dz dx idy

singlet

[ 0(k) even ]

triplet

[ d(k) odd ]

In practice, there is a varied phenomenology:




ˆ k 0 00 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, purely triplet superconductors: +

LaNiC25 (c.f. centrosymmetric LaNiGa26) + , CePtr3Si (?) 7


Li2PdxPt3−xB: tunable singlet-triplet mixing

The Li2Pdx Pt3−x B 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)






The series goes from fully-gapped(x = 3) to nodal (x = 0):

H.Q. Yuan et al.,

Phys. Rev. Lett. 97, 017006 (2006).






The series goes from fully-gapped(x = 3) to nodal (x = 0):

H.Q. Yuan et al.,

Phys. Rev. Lett. 97, 017006 (2006).

NMR suggests nodal state a triplet:

M.Nishiyama et al.,

Phys. Rev. Lett. 98, 047002 (2007)


Li2PdxPt3−xB: Phase diagram

Bogoliubov 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)




H(k) =

(h(k) ∆(k)

∆†(k) −hT (−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) =

(h(k) ∆(k)

∆†(k) −hT (−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 + k2z

), ky

(k2

z + k2x

), kz

(k2

x + k2y

)]}



Treat A and B as independent 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.



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


Li2PdxPt3−xB: 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:





A = 3

3.6 3.8 4 4.2 4.4

B

0

0.05

0.1

0.15

0.2

0.25T

/Tc

1.6

1.7

1.8

1.9

2

2.1

2.2





A = 3

3.6 3.8 4 4.2 4.4

B

0

0.05

0.1

0.15

0.2

0.25T

/Tc

1.6

1.7

1.8

1.9

2

2.1

2.2

The conventional exponent (n = 2 in this example) is only seen below atemperature scale that converges to zero at the transition





A = 3

3.6 3.8 4 4.2 4.4

B

0

0.05

0.1

0.15

0.2

0.25T

/Tc

1.6

1.7

1.8

1.9

2

2.1

2.2


The anomalous exponent (here n = 1.8) is seen everywhere else





A = 3

3.6 3.8 4 4.2 4.4

B

0

0.05

0.1

0.15

0.2

0.25T

/Tc

1.6

1.7

1.8

1.9

2

2.1

2.2


The anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagram





A = 3

3.6 3.8 4 4.2 4.4

B

0

0.05

0.1

0.15

0.2

0.25T

/Tc

1.6

1.7

1.8

1.9

2

2.1

2.2


The anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagram

c.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0



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.


http://www.cond-mat.org

Topological transitions in nodal superconductorshave clear signatures in bulk thermodynamic properties.

THANKS!

www.cond-mat.org


http://www.cond-mat.org


Let’s remember where this came from:

Cv = T

(dS

dT

)

=1

2kBT 2 ∑k

Ek − TdEk

dT︸︷︷︸

≈0

Ek sech2 Ek

2kBT︸︷︷︸

≈4e−Ek /KBT

∼ T−2∫

dEg (E )E2e−E/kBT at low T

g (E ) ∼ En−1 ⇒ Cv ∼ T n∫

dǫǫ2+n−1e−ǫ

︸︷︷︸

a number



Ek =√

ǫ2k+ ∆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


Shallow line nodes in pnictides

back


Logarithm ⇒ power law (n − 1 = 0.8)

The power-law expression is asymptotically very good at E → 0:

back


Logarithm ⇒ power law (n − 1 = 0.4)

The power-law expression is asymptotically very good at E → 0:

back


Symmetry of pairing in NCS

Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations

Singlet, triplet, or both?

Neglect (for now!) spin-orbit coupling:






Singlet and triplet representations of SO(3):







Γns = - (Γn

s)T , Γnt = + (Γn

t)T





Iﾏpose Pauliげs exclusioﾐ priﾐciple:



Γns = - (Γn

s)T , Γnt = + (Γn

t)T






, ' k ', k



Γns = - (Γn

s)T , Γnt = + (Γn

t)T






, ' k ', k


ˆ k either singlet


Γns = - (Γn

s)T , Γnt = + (Γn

t)T






, ' k ', k


ˆ k either singlet yi ˆ0', kk


Γns = - (Γn

s)T , Γnt = + (Γn

t)T






, ' k ', k


ˆ k either singlet yi ˆ0', kk or triplet


Γns = - (Γn

s)T , Γnt = + (Γn

t)T






, ' k ', k


ˆ k either singlet yi ˆ0', kk or triplet yi ˆˆ.', σkdk


Γns = - (Γn

s)T , Γnt = + (Γn

t)T



yxz

zyx

iddd

didd

0

0ˆ0

0k

The role of spin-orbit coupling (SOC)

Gap function may have both singlet and triplet components

kk orbitspin',',

• However, if we have a centre of inversion

basis functions either even or odd under inversion

still have either singlet or triplet pairing (at Tc)

• No centre of inversion: may have singlet and triplet (even at Tc) back


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 70 800

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), 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)





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)





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)





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. Yuaﾐ et al. P‘Lげ0ヶ

*


PRL 102 117007 (2009)



Spin-up superfluid

coexisting with spin-

down Fermi liquid.

The A1 phase of

liquid 3He.

Non-unitary pairing

0

00or

00

0ˆ

C.f.




G = [SO(3)×Gc]×U(1)×T



G = Gc,J×U(1)×T





Quintanilla, Hillier, Annett and Cywinski,

PRB 82, 174511 (2010)

E.g. reflection through a vertical

plane perpendicular to the y axis:

yJJv CI ,2,

x y

z





PRB 82, 174511 (2010)



yJJv CI ,2,

x y

z





PRB 82, 174511 (2010)



yJJv CI ,2,

This affects d(k) (a vector under

spin rotations).

x y

z





PRB 82, 174511 (2010)



yJJv CI ,2,

This affects d(k) (a vector under

spin rotations).

It does not affect 0(k) (a scalar). x y

z



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)



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)


LaNiGa2 - a centrosymmetric cousin of LaNiC2

A similar muSR effect is seen in centrosymmetric LaNiGa2:

[A. D. Hillier, JQ, Mazidian, Annett & Cywinski, PRL 109, 097001 (2012)]





Simlarly to LaNiC2, we find only 1D irreps ⇒ non-unitary triplet pairing.





Simlarly to LaNiC2, we find only 1D irreps ⇒ non-unitary triplet pairing.

Lack of inversion symmetry seems to be a red herring in the case of LaNiC2.

back




H(k) =

(h(k) ∆(k)

∆†(k) −hT (−k)

)

h(k) = εk I + 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


Li2PdxPt3−xB:order parameter

back


thermodynamic signatures of topological transitions in nodal superconductors

Technology

e i1 i222

i1 i2g e

shallow nodes

shallow point nodes

linear nodes

i1 i2n

i1 i2x

i1 i2e