broken time-reversal symmetry and topological order in triplet superconductors

Broken Time-Reversal Symmetry and Topological Orderin Triplet Superconductors

Jorge Quintanilla1,2

1SEPnet and Hubbard Theory Consortium, University of Kent2ISIS Neutron and Muon Source, STFC Rutherford Appleton Laboratory

Dresden, 27 November 2014

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 1 / 119

People and Money

People: James F. Annett (Bristol) , Adrian D. Hillier (RAL)

Bayan Mazidian (RAL/Bristol) , Bob Cywinski (Huddersfield) .

Ravi P. Singh , Gheeta Balakrishnan , Don Paul ,

Martin Lees (Birmingham). Amitava Bhattacharyya ,

Devashibai Adroja (RAL). A. M. Strydom (Johannesburg) .

Naoki Kase, Jun Akimitsu (Aoyama Gakuin).

Money: STFC (UK) + HEFCE/SEPnet (UK) + UJ and NRF (South Africa) +Bristol + Kent.

The Hubbard Theory Consortium

Director: Piers Coleman (RHUL/Rutgers)SEPnet fellows: Matthias Eschrig (RHUL/RAL)

Claudio Castelnovo (RHUL/RAL)Jorge Quintanilla (Kent/RAL)

Associate: Jörg Schmalian (Karlsruhe)

+ several SEPnet PhD students.

Strong correlations theory in close collaboration with experiments at

• RAL (ISIS/Diamond)• London Centre for Nanotech.• RHUL

Director: Piers Coleman (RHUL/Rutgers)SEPnet fellows: Matthias Eschrig (RHUL/RAL)

Claudio Castelnovo (RHUL/RAL)Jorge Quintanilla (Kent/RAL)

Associate: Jörg Schmalian (Karlsruhe)

+ several SEPnet PhD students.

Strong correlations theory in close collaboration with experiments at

• RAL (ISIS/Diamond)• London Centre for Nanotech.• RHUL

Overview

Two Paradigms in Condensed Matter ...

Broken SymmetryTopological Transitions

... interlock via triplet pairing in superconductors:

Superconductors

Broken time-reversal

symmetry

Topological transitionsTriplet

pairing

SimpleTheories + Standard Measurements:Group Theory / Bogolibov QuasiparticlesNeutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth

Overview

Two Paradigms in Condensed Matter ...Broken Symmetry

Topological Transitions

Superconductors

symmetry

pairing

Overview

Two Paradigms in Condensed Matter ...Broken SymmetryTopological Transitions

Superconductors

symmetry

pairing

Overview

Superconductors

symmetry

pairing

Overview

Superconductors

symmetry

pairing

SimpleTheories + Standard Measurements:

Group Theory / Bogolibov QuasiparticlesNeutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth

Overview

Superconductors

symmetry

pairing

SimpleTheories + Standard Measurements:Group Theory / Bogolibov Quasiparticles

Neutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth

Overview

Superconductors

symmetry

pairing

Outline

1 Broken time-reversal symmetry in superconductors

2 Experimental evidence for broken TRS

3 Singlet, triplet, or both?

4 A symmetry zoo

5 Topological transitions in Superconductors

6 Topological transition state: Li2PdxPt3−xB

7 Take-home message

Quantum Materials Theory

4 A symmetry zoo

7 Take-home message

Broken symmetry

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

Unconventional superconductors

Broken symmetry

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

‘Unconventional’ superconductors:

Cuprates, Sr2RuO4, PrOs4Sb12, UPt3, (UTh)Be13 , ...

Time-reversal Symmetry

Classical time-reversal symmetry:t → −t equivalent to

r→ r and p→ −p

Also inverts angular momenta.True in the absence of friction/magnetic fields.

Quantum time-reversal symmetry:t → −t equivalent toψ→ ψ∗ and S→ −S.

True if H = H∗ and spin-invariant.

For quasi-particles in a superconductor:H = H0 + ∆c†

k c†−k + H.c. ⇒ TRS: ∆ = ∆∗

4 A symmetry zoo

7 Take-home message

Muon Spin Rotation

Adrian Hillier (Muons group leader, ISIS)

Muon Spin Rotation

Zero field muon spin relaxation

backward detector

forward detector

sample

Kubo-Toyabe x exponential

Asymmetry:

NF −NBNF + NB

= G (t)

σ : randomly-oriented fields (e.g. nuclear moments)Λ :smoothly-modulated fields (e.g. electronic moments)

The “classic” examples: UPt3 and Sr2RuO4

Luke et al. PRL (1993)

Sr2RuO4

Luke et al. Nature (1998)

Confirmed by Kerr effect

Schemm et al. Science (2014)

Sr2RuO4

Jing Xia et al. PRL (2006)Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 18 / 119

More recent finds: LaNiC2

Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

Relaxation due to electronic moments

Moment

~ 0.1G

(~ 0.01μB)

(longitudinal)

Timescale:

> 10-4

backward detector

forward detector

sample

More recent finds: LaNiGa2

A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett,Physical Review Letters 109, 097001 (2012).

More recent finds: Re6Zr

More recent finds: Lu5Rh6Sn18

4 A symmetry zoo

7 Take-home message

Singlet, triplet, or both?

It’s all in the gap function:

Symmetry of the gap function

See J.F. Annett Adv. Phys. 1990.

Pauli ⇒∆ (k) = −∆T (−k)Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations

ˆ k 0 0

dx idy dz

dz dx idy

singlet

[ 0(k) even ]

triplet

[ d(k) odd ]

ˆ k 0 0

dx idy dz

dz dx idy

singlet

[ 0(k) even ]

triplet

[ d(k) odd ]

Pauli ⇒∆ (k) = −∆T (−k)

ˆ k 0 0

dx idy dz

dz dx idy

singlet

[ 0(k) even ]

triplet

[ d(k) odd ]

ˆ k 0 0

dx idy dz

dz dx idy

singlet

[ 0(k) even ]

triplet

[ d(k) odd ]

Neglect (for now!) spin-orbit coupling:

Singlet and triplet representations of SO(3):

Γns = - (Γn

s)T , Γnt = + (Γn

Impose Pauli’s exclusion principle:

Γns = - (Γn

s)T , Γnt = + (Γn

, ' k ', k

Γns = - (Γn

s)T , Γnt = + (Γn

, ' k ', k

ˆ k either singlet

Γns = - (Γn

s)T , Γnt = + (Γn

, ' k ', k

ˆ k either singlet yiˆ

0', kk

Γns = - (Γn

s)T , Γnt = + (Γn

, ' k ', k

0', kk

or triplet

Γns = - (Γn

s)T , Γnt = + (Γn

, ' k ', k

0', kk

or triplet yiˆˆ.', σkdk

Γns = - (Γn

s)T , Γnt = + (Γn

The role of spin-orbit coupling (SOC)

Gap function may have both singlet and triplet components

kkorbitspin

• 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) Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 36 / 119

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:

JJv CI ,2,

This affects d(k) (a vector under spin rotations).

JJv CI ,2,

This affects d(k) (a vector under spin rotations).

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

When can we have singlet-triplet mixing?

We must now use basis functions of the double group:

∆ (k) =dΓ

∑n=1

ηnΓn (k)

Gap function may have both singlet and triplet components

kkorbitspin

• 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)

Crystal symmetryCentrosymmetric Non-centrosymmetric

Spin-orbit coupling Weak N NStrong N Y

When can we have broken time-reversal symmetry?

For ∆ (k) to be non-trivially complex, it must have more than onecomponent:

∆ (k) = η1Γ1 (k) + η2Γ2 (k) , arg η1 6= arg η2

The instability must therefore take place in an irrep with d > 1.Weak spin-orbit coupling: SO(3) × Gc

The singlet irrep of SO(3) has d = 1 ⇒ for singlet pairing, the point group Gcmust have a d > 1 irrep.The triplet irrep of SO(3) had d = 3 ⇒ for triplet pairing, broken TRS ispossible even for d = 1 irreps of Gc .

Strong spin-orbit coupling: Gc,J (double group)The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components

The instability must therefore take place in an irrep with d > 1.

Weak spin-orbit coupling: SO(3) × Gc

The singlet irrep of SO(3) has d = 1 ⇒ for singlet pairing, the point group Gcmust have a d > 1 irrep.

The triplet irrep of SO(3) had d = 3 ⇒ for triplet pairing, broken TRS ispossible even for d = 1 irreps of Gc .

Strong spin-orbit coupling: Gc,J (double group)

The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components

Strong spin-orbit coupling: Gc,J (double group)The dimensionality of the irreps is the same as for Gc therefore if all irreps ared = 1 then there can be no broken TRS.

Broken TRS involves always a d > 1 irrep and it requires both the singlet andtriplet components

4 A symmetry zoo

7 Take-home message

LaNiC2

Character table

LaNiC2

Character table

LaNiC2

Character table

LaNiC2

Character table

LaNiC2

Symmetries and

their characters

Sample basis

functions

Irreducible

representation

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

All irreps d = 1

⇒ weak SOC

LaNiC2

SO(3)xC2v

Gap function

(unitary)

Gap function

(non-unitary)

1 (k)=1 -

2 (k)=k

1 (k)=k

2 (k)=k

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

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

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

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

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

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

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

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

Possible order parameters

LaNiC2

SO(3)xC2v

Gap function

(unitary)

Gap function

(non-unitary)

1 (k)=1 -

2 (k)=k

1 (k)=k

2 (k)=k

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

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

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

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

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

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

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

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

LaNiC2

SO(3)xC2v

Gap function

(unitary)

Gap function

(non-unitary)

1 (k)=1 -

2 (k)=k

1 (k)=k

2 (k)=k

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

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

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

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

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

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

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

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

LaNiC2

SO(3)xC2v

Gap function

(unitary)

Gap function

(non-unitary)

1 (k)=1 -

2 (k)=k

1 (k)=k

2 (k)=k

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

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

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

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

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

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

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

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

Non-unitary d x d* ≠ 0

LaNiC2

SO(3)xC2v

Gap function

(unitary)

Gap function

(non-unitary)

1 (k)=1 -

2 (k)=k

1 (k)=k

2 (k)=k

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

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

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

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

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

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

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

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

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

LaNiC2Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations

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

The A1 phase of liquid 3He.

Non-unitary pairing

Also FM SC - but this is a paramagnet!

A. D. Hillier, J. Quintanilla and R. Cywinski, Physical Review Letters (2009).

J. Quintanilla, J. F. Annett, A. D. Hillier, R. Cywinski, Physical Review B (2010).

LaNiGa2

Centrosymmetric, but again all irreps d = 1 ⇒ again weak SOC and non-unitarytriplet

A.D. Hillier, J. Quintanilla, B. Mazidian, J.F. Annett, and R. Cywinski, PRL 109, 097001(2012).

⇒ A new family of superconductors?Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 65 / 119

Why non-unitary?Generic Landau theory for a triplet superconductor (1D irrep):

F = a |η|2 + b2∣∣η4∣∣

+ b′ |η× η∗|2 +m2

2χ+ b′′m · (iη× η∗) .

Magnetisation as a sub-dominant order parameter:

Temperature

magnetisation

Superconductivity

Theory (left): A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, PRL 109, 097001 (2012).

Experiment (right): Akihiko Sumiyama et al., JPSJ (2014).

F = a |η|2 + b2∣∣η4∣∣ + b′ |η× η∗|2

2χ+ b′′m · (iη× η∗) .

Temperature

magnetisation

Superconductivity