talk bristol uk-nl_2013_v01_for_web

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

UK-NL Condensed Matter Meeting, Bristol, UK, 2013(web version)

Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 1 / 69

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

PRELUDE - Symmetry

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

Unconventional superconductors

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PRELUDE - Topology


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 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) arXiv:1302.2161 Bristol 2013 6 / 69

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


Shallow nodes

Relax 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 (speculatively) 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)]. +


Shallow nodes



∆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 (speculatively) 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)]. +


Shallow nodes



∆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 (speculatively) 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)]. +


Shallow nodes



∆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



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 !!)


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 = 1Jorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 11 / 69

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



1 What are they?

2 How to get them

3 An example

4 Take-home message

A generic mechanismWe propose that shallow nodes will exist generically at topological phasetransitions in superocnductors with multi-component order parameters:

∆ 0

∆ 1Fermi Sea

∆ 0



∆ 1Fermi Sea

∆ 0



∆ 1Fermi Sea

∆ 0

Line

ar

node

s

Line

ar

node

sJorge Quintanilla (Kent and ISIS) arXiv:1302.2161 Bristol 2013 17 / 69


∆ 1Fermi Sea

∆ 0



∆ 1Fermi Sea

∆ 0

Sha

llow

no

de

Sha

llow

no

de



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:


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) arXiv:1302.2161 Bristol 2013 22 / 69





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






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


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)


Li2PdxPt3−xB: Phase diagramBogoliubov 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)

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

.

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)]}


Li2PdxPt3−xB: Phase diagramTreat A and B as in dependent 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.


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.


Li2PdxPt3−xB: Phase diagram


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:

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


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:

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



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


http://www.cond-mat.org


5 Additional details


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 n∫

dεε2+n−1e−ε︸︷︷︸a number



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

back


Shallow line nodes in pnictides

back



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; 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:

spin for

spin for

kk

kkk

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:

> 10-4

s ~

e

_

e

backward detector

forward detector

sample



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:

> 10-4

s ~

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), Li

2Pt

3B & Li

2Pd

3B

(2), ...

(1) Bauer et al. PRL’04 (2) Yuan et al. PRL’06



Character table




Character table


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




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


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



SO(3)xC2v

Gap function

(unitary)

Gap function

(non-unitary)

1A

1 (k)=1 -

1A

2 (k)=k

xk

Y -

1B

1 (k)=k

Xk

Z -

1B

2 (k)=k

Yk

Z -

3A

1 d(k)=(0,0,1)k

Z d(k)=(1,i,0)k

Z

3A

2 d(k)=(0,0,1)k

Xk

Yk

Z d(k)=(1,i,0)k

Xk

Yk

Z

3B

1 d(k)=(0,0,1)k

X d(k)=(1,i,0)k

X

3B

2 d(k)=(0,0,1)k

Y d(k)=(1,i,0)k

Y

Possible order parameters




SO(3)xC2v

Gap function

(unitary)

Gap function

(non-unitary)

1A

1 (k)=1 -

1A

2 (k)=k

xk

Y -

1B

1 (k)=k

Xk

Z -

1B

2 (k)=k

Yk

Z -

3A

1 d(k)=(0,0,1)k

Z d(k)=(1,i,0)k

Z

3A

2 d(k)=(0,0,1)k

Xk

Yk

Z d(k)=(1,i,0)k

Xk

Yk

Z

3B

1 d(k)=(0,0,1)k

X d(k)=(1,i,0)k

X

3B

2 d(k)=(0,0,1)k

Y d(k)=(1,i,0)k

Y





SO(3)xC2v

Gap function

(unitary)

Gap function

(non-unitary)

1A

1 (k)=1 -

1A

2 (k)=k

xk

Y -

1B

1 (k)=k

Xk

Z -

1B

2 (k)=k

Yk

Z -

3A

1 d(k)=(0,0,1)k

Z d(k)=(1,i,0)k

Z

3A

2 d(k)=(0,0,1)k

Xk

Yk

Z d(k)=(1,i,0)k

Xk

Yk

Z

3B

1 d(k)=(0,0,1)k

X d(k)=(1,i,0)k

X

3B

2 d(k)=(0,0,1)k

Y d(k)=(1,i,0)k

Y

Non-unitary d x d* ≠ 0





SO(3)xC2v

Gap function

(unitary)

Gap function

(non-unitary)

1A

1 (k)=1 -

1A

2 (k)=k

xk

Y -

1B

1 (k)=k

Xk

Z -

1B

2 (k)=k

Yk

Z -

3A

1 d(k)=(0,0,1)k

Z d(k)=(1,i,0)k

Z

3A

2 d(k)=(0,0,1)k

Xk

Yk

Z d(k)=(1,i,0)k

Xk

Yk

Z

3B

1 d(k)=(0,0,1)k

X d(k)=(1,i,0)k

X

3B

2 d(k)=(0,0,1)k

Y d(k)=(1,i,0)k

Y

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

*




Spin-up superfluid coexisting with spin-down Fermi liquid.

The A1 phase of liquid 3He.

Non-unitary pairing

0

00or

00

0ˆ

C.f.



C2v,Jno t

Gap function,

singlet component

Gap function,

triplet component

A1

(k) = A d(k) = (Bky,Ck

x,Dk

xk

yk

z)

A2

(k) = Akxk

Y d(k) = (Bk

x,Ck

y,Dk

z)

B1

(k) = AkXk

Z d(k) = (Bk

xk

yk

z,Ck

z,Dk

y)

B2

(k) = AkYk

Z d(k) = (Bk

z, Ck

xk

yk

z,Dk

x)

The role of spin-orbit coupling (SOC)

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,Ck

x,Dk

xk

yk

z)

A2

(k) = Akxk

Y d(k) = (Bk

x,Ck

y,Dk

z)

B1

(k) = AkXk

Z d(k) = (Bk

xk

yk

z,Ck

z,Dk

y)

B2

(k) = AkYk

Z d(k) = (Bk

z, Ck

xk

yk

z,Dk

x)

The role of spin-orbit coupling (SOC)

None of these break time-reversal symmetry!




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)



Li2PdxPt3−xB:order parameter

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