Talk kent symposium_2013_v01_for_web

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

PRELUDE - Symmetry

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) 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

2 + ky||

∆2k = I1kx

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

Linear nodes

g (E ) ∼ En−1 ⇒ Cv ∼ T n

∆2k = I1

2 + ky||

∆2k = I1kx

g(E ) = E2

2(2π)2I1√

I2g(E ) = LE

(2π)3√I1√

I2n = 3 n = 2

Linear nodes

g (E ) ∼ En−1 ⇒ Cv ∼ T n

∆2k = I1

2 + ky||

∆2k = I1kx

g(E ) = E2

2(2π)2I1√

I2g(E ) = LE

(2π)3√I1√

I2n = 3 n = 2

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||

g(E ) = E2(2π)2√I1

g(E ) = L√

(2π)3I14

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)]. +

∆2k = I1(kx

||2 + ky

||2)2 ∆2

k = I1kx||

g(E ) = E2(2π)2√I1

g(E ) = L√

(2π)3I14

I2n = 2 n = 1.5

∆2k = I1(kx

||2 + ky

||2)2 ∆2

k = I1kx||

g(E ) = E2(2π)2√I1

g(E ) = L√

(2π)3I14

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

2 − ky||

or I1kx||

g(E ) =

E (1+2ln| L+√

E/I141

√E/I

(2π)3√I1I2∼ E0.8

n = 1.8 (< 2 !!)

Crossing of shallow line nodesWhen shallow lines cross we get an even lower exponent:

crossingof shallow line nodes

∆2k = I1

2 − ky||

or I1kx||

g (E ) =

√E (1+2ln| L+E

E14 /I

(2π)3I14

I2∼ E0.4

n = 1.4 *

* c.f. gapless excitations of a Fermi liquid: g (E ) = constant⇒ n = 1+

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:

∆ 1Fermi Sea

sJorge Quintanilla (Kent and ISIS) Anomalous supercond. power laws arXiv:1302.2161 Canterbury 2013 14 / 39

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

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

ˆ k 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

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 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

z + k2x), kz(k2

x + k2y)]}

Treat A and B as in dependent tuning parameters and study quasiparticlespectrum.

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.

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.

Detecting the topological transitions

Li2PdxPt3−xB: predicted specific heat power-laws

jn = 2

n = 1.8

n = 1.4

jn = 2

n = 1.8

n = 1.4

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)

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

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

THANKS!

www.cond-mat.org

ADDITIONAL INFORMATION

Let’s remember where this came from:

Cv = T(

12kBT 2 ∑

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

Ek =√

ε2k + ∆2

≈√

I2k2⊥ + ∆

(kx|| , k

on the Fermi surface k||

k|_ ∆(k

Compute density of states:

g(E ) =∫ ∫ ∫

δ(Ek − E )dkx dky dkz

Q.E.D.

Shallow line nodes in pnictides

Numerics

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

T / Tc

linear point nodeshallow point node

linear line nodecrossing of linear line nodes

shallow line nodecrossing of shallow line nodes

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))

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