broken time-reversal symmetry and topological order in triplet superconductors

179
Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors Jorge Quintanilla 1,2 1 SEPnet and Hubbard Theory Consortium, University of Kent 2 ISIS 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

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Jorge Quintanilla, "Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors" - Research seminar, Max Planck Institute for the Physics of Complex Systems (Dresden), 27 November 2014 Abstract: The concept of broken symmetry is one of the cornerstones of modern physics, for which superconductors stand out as a major paradigm. In conventional superconductors electrons form isotropic singlet pairs that then condense into a coherent state, similar to that of photons in a laser. We understand this in terms of the breaking of global gauge symmetry, which is the invariance of a system under changes to the overall phase of its wave function. In unconventional superconductors, however, more complex forms of pairing are possible, leading to additional broken symmetries and even to topological forms of order that fall outside the broken-symmetry paradigm. In this talk I will discuss such phenomena, making emphasis on triplet pairing and the spontaneous breaking of time-reversal symmetry in some superconductors. I will pay particular attention to large-facility experiments using muons to detect tiny magnetic fields inside superconducting samples and group-theoretical arguments that enable us to constrain the type of pairing present in the light of such experiments. I will also address the possibility of mixed singlet-triplet pairing without broken time-reversal symmetry in superconductors whose crystal lattices lack a centre of inversion, and predict bulk experimental signatures of topological transitions expected to occur in such systems.

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Page 1: 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

Page 2: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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.

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 2 / 119

Page 3: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 3 / 119

Page 4: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119

Page 5: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Overview

Two Paradigms in Condensed Matter ...Broken Symmetry

Topological 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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119

Page 6: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119

Page 7: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119

Page 8: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119

Page 9: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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 Quasiparticles

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119

Page 10: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 4 / 119

Page 11: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 5 / 119

Page 12: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Quantum Materials Theory

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

Page 13: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Broken symmetry

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

Unconventional superconductors

Ph

oto

: Ed

die

Hu

i-B

on

-Ho

a, w

ww

.sh

iro

mi.c

om

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

Ph

oto

: Ken

net

h G

. Lib

bre

cht,

sn

ow

flak

es.c

om

Unconventional superconductors

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

Ph

oto

: co

mm

on

s.w

ikim

edia

.org

Unconventional superconductors

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 7 / 119

Page 14: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Broken symmetry

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

Unconventional superconductors

Ph

oto

: Ed

die

Hu

i-B

on

-Ho

a, w

ww

.sh

iro

mi.c

om

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

Ph

oto

: Ken

net

h G

. Lib

bre

cht,

sn

ow

flak

es.c

om

Unconventional superconductors

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

Ph

oto

: co

mm

on

s.w

ikim

edia

.org

Unconventional superconductors

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 7 / 119

Page 15: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Broken symmetry

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

Unconventional superconductors

Ph

oto

: Ed

die

Hu

i-B

on

-Ho

a, w

ww

.sh

iro

mi.c

om

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

Ph

oto

: Ken

net

h G

. Lib

bre

cht,

sn

ow

flak

es.c

om

Unconventional superconductors

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

Ph

oto

: co

mm

on

s.w

ikim

edia

.org

Unconventional superconductors

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 7 / 119

Page 16: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Broken symmetry

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

Unconventional superconductors

Ph

oto

: Ed

die

Hu

i-B

on

-Ho

a, w

ww

.sh

iro

mi.c

om

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

Ph

oto

: Ken

net

h G

. Lib

bre

cht,

sn

ow

flak

es.c

om

Unconventional superconductors

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

Ph

oto

: co

mm

on

s.w

ikim

edia

.org

Unconventional superconductors

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 7 / 119

Page 17: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Broken symmetry

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

Unconventional superconductors

Ph

oto

: Ed

die

Hu

i-B

on

-Ho

a, w

ww

.sh

iro

mi.c

om

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

Ph

oto

: Ken

net

h G

. Lib

bre

cht,

sn

ow

flak

es.c

om

Unconventional superconductors

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

Ph

oto

: co

mm

on

s.w

ikim

edia

.org

Unconventional superconductors

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 7 / 119

Page 18: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Ph

oto

: co

mm

on

s.w

ikim

edia

.org

Unconventional superconductors

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 8 / 119

Page 19: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Ph

oto

: co

mm

on

s.w

ikim

edia

.org

Unconventional superconductors

‘Unconventional’ superconductors:

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 9 / 119

Page 20: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Time-reversal Symmetry

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 10 / 119

Page 21: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Time-reversal Symmetry

p

r

x

y

z

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: ∆ = ∆∗

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 11 / 119

Page 22: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Time-reversal Symmetry

-pr

x

y

z

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: ∆ = ∆∗

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 11 / 119

Page 23: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Time-reversal Symmetry

-pr

x

y

z

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: ∆ = ∆∗

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 11 / 119

Page 24: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Time-reversal Symmetry

-pr

x

y

z

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: ∆ = ∆∗

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 11 / 119

Page 25: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Time-reversal Symmetry

-pr

x

y

z

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: ∆ = ∆∗

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 11 / 119

Page 26: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Quantum Materials Theory

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

Page 27: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Muon Spin Rotation

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 13 / 119

Page 28: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Muon Spin Rotation

Adrian Hillier (Muons group leader, ISIS)

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 14 / 119

Page 29: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Muon Spin Rotation

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

Zero field muon spin relaxation

e

_

e

backward detector

forward detector

sample

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 15 / 119

Page 30: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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)

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 16 / 119

Page 31: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

The “classic” examples: UPt3 and Sr2RuO4

UPt3

Luke et al. PRL (1993)

Sr2RuO4

Luke et al. Nature (1998)

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 17 / 119

Page 32: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Confirmed by Kerr effect

UPt3

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

Page 33: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

More recent finds: LaNiC2

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 19 / 119

Page 34: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

More recent finds: LaNiC2

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

+

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 20 / 119

Page 35: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

More recent finds: LaNiGa2

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 21 / 119

Page 36: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

More recent finds: Re6Zr

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 22 / 119

Page 37: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

More recent finds: Lu5Rh6Sn18

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 23 / 119

Page 38: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Quantum Materials Theory

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

Page 39: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Singlet, triplet, or both?

It’s all in the gap function:

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

Symmetry of the gap function

kk

kkkˆ

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

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

Singlet, triplet, or both?

ˆ k 0 0

0 0

dx idy dz

dz dx idy

singlet

[ 0(k) even ]

triplet

[ d(k) odd ]

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 25 / 119

Page 40: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Singlet, triplet, or both?

It’s all in the gap function:

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

Symmetry of the gap function

kk

kkkˆ

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

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

Singlet, triplet, or both?

ˆ k 0 0

0 0

dx idy dz

dz dx idy

singlet

[ 0(k) even ]

triplet

[ d(k) odd ]

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 25 / 119

Page 41: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Singlet, triplet, or both?

It’s all in the gap function:

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

Symmetry of the gap function

kk

kkkˆ

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

Pauli ⇒∆ (k) = −∆T (−k)

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

Singlet, triplet, or both?

ˆ k 0 0

0 0

dx idy dz

dz dx idy

singlet

[ 0(k) even ]

triplet

[ d(k) odd ]

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 25 / 119

Page 42: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Singlet, triplet, or both?

It’s all in the gap function:

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

Symmetry of the gap function

kk

kkkˆ

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

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

Singlet, triplet, or both?

ˆ k 0 0

0 0

dx idy dz

dz dx idy

singlet

[ 0(k) even ]

triplet

[ d(k) odd ]

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 25 / 119

Page 43: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Singlet, triplet, or both?

Neglect (for now!) spin-orbit coupling:

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 26 / 119

Page 44: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Singlet, triplet, or both?

Neglect (for now!) spin-orbit coupling:

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 27 / 119

Page 45: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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):

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 28 / 119

Page 46: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 29 / 119

Page 47: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Singlet, triplet, or both?

Impose Pauli’s exclusion principle:

Neglect (for now!) spin-orbit coupling:

Singlet and triplet representations of SO(3):

Γns = - (Γn

s)T , Γnt = + (Γn

t)T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 30 / 119

Page 48: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Singlet, triplet, or both?

Impose Pauli’s exclusion principle:

, ' k ', k

Neglect (for now!) spin-orbit coupling:

Singlet and triplet representations of SO(3):

Γns = - (Γn

s)T , Γnt = + (Γn

t)T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 31 / 119

Page 49: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Singlet, triplet, or both?

Impose Pauli’s exclusion principle:

, ' k ', k

Neglect (for now!) spin-orbit coupling:

ˆ k either singlet

Singlet and triplet representations of SO(3):

Γns = - (Γn

s)T , Γnt = + (Γn

t)T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 32 / 119

Page 50: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Singlet, triplet, or both?

Impose Pauli’s exclusion principle:

, ' k ', k

Neglect (for now!) spin-orbit coupling:

ˆ k either singlet yiˆ

0', kk

Singlet and triplet representations of SO(3):

Γns = - (Γn

s)T , Γnt = + (Γn

t)T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 33 / 119

Page 51: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Singlet, triplet, or both?

Impose Pauli’s exclusion principle:

, ' k ', k

Neglect (for now!) spin-orbit coupling:

ˆ k either singlet yiˆ

0', kk

or triplet

Singlet and triplet representations of SO(3):

Γns = - (Γn

s)T , Γnt = + (Γn

t)T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 34 / 119

Page 52: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Singlet, triplet, or both?

Impose Pauli’s exclusion principle:

, ' k ', k

Neglect (for now!) spin-orbit coupling:

ˆ k either singlet yiˆ

0', kk

or triplet yiˆˆ.', σkdk

Singlet and triplet representations of SO(3):

Γns = - (Γn

s)T , Γnt = + (Γn

t)T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 35 / 119

Page 53: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

yxz

zyx

iddd

didd

0

0

0k

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

Page 54: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

G = [SO(3)×Gc]×U(1)×T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 37 / 119

Page 55: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

G = [SO(3)×Gc]×U(1)×T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 38 / 119

Page 56: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

G = [SO(3)×Gc]×U(1)×T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 39 / 119

Page 57: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

G = [SO(3)×Gc]×U(1)×T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 40 / 119

Page 58: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

G = [SO(3)×Gc]×U(1)×T

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 41 / 119

Page 59: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

G = Gc,J×U(1)×T

The role of spin-orbit coupling (SOC)

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 42 / 119

Page 60: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

G = Gc,J×U(1)×T

The role of spin-orbit coupling (SOC)

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 43 / 119

Page 61: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

G = Gc,J×U(1)×T

The role of spin-orbit coupling (SOC)

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 44 / 119

Page 62: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

E.g. reflection through a vertical plane perpendicular to the y axis:

y

JJv CI ,2,

x y

z

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 45 / 119

Page 63: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

E.g. reflection through a vertical plane perpendicular to the y axis:

y

JJv CI ,2,

x y

z

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 46 / 119

Page 64: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

E.g. reflection through a vertical plane perpendicular to the y axis:

y

JJv CI ,2,

x y

z

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 47 / 119

Page 65: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

E.g. reflection through a vertical plane perpendicular to the y axis:

y

JJv CI ,2,

x y

z

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 48 / 119

Page 66: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

E.g. reflection through a vertical plane perpendicular to the y axis:

y

JJv CI ,2,

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

x y

z

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 49 / 119

Page 67: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

The role of spin-orbit coupling (SOC)

Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

E.g. reflection through a vertical plane perpendicular to the y axis:

y

JJv CI ,2,

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

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

z

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 50 / 119

Page 68: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

When can we have singlet-triplet mixing?

We must now use basis functions of the double group:

∆ (k) =dΓ

∑n=1

ηnΓn (k)

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

yxz

zyx

iddd

didd

0

0

0k

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)

Crystal symmetryCentrosymmetric Non-centrosymmetric

Spin-orbit coupling Weak N NStrong N Y

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 51 / 119

Page 69: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119

Page 70: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119

Page 71: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119

Page 72: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119

Page 73: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119

Page 74: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119

Page 75: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119

Page 76: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 52 / 119

Page 77: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Quantum Materials Theory

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

Page 78: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

LaNiC2

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

Character table

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 54 / 119

Page 79: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

LaNiC2

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

Character table

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 55 / 119

Page 80: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

LaNiC2

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

Character table

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 56 / 119

Page 81: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

LaNiC2

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

Character table

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

180o

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 57 / 119

Page 82: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

LaNiC2

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

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

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

All irreps d = 1

⇒ weak SOC

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 58 / 119

Page 83: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

LaNiC2

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

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

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 59 / 119

Page 84: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

LaNiC2

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

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

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 60 / 119

Page 85: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

LaNiC2

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

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

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 61 / 119

Page 86: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

LaNiC2

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

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

Possible order parameters

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 62 / 119

Page 87: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

LaNiC2

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

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

Possible order parameters

* C.f. Li2Pd3B & Li2Pt3B, H. Q. Yuan et al. PRL’06

*

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 63 / 119

Page 88: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

0

00or

00

C.f.

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).

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 64 / 119

Page 89: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Page 90: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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).

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119

Page 91: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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).

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119

Page 92: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

F = a |η|2 + b2∣∣η4∣∣ + b′ |η× η∗|2 +

m2

+ 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).

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119

Page 93: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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).

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119

Page 94: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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).

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119

Page 95: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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).

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119

Page 96: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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).Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 66 / 119

Page 97: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Re6Zr

Td group:noncentrosymmetric;d = 1, 2, 3

⇒ can have broken TRS with strong SOC⇒ broken TRS with singlet-triplet mixing

E irrep (d = 2) ⇒

F1,F2 irreps (d = 3) ⇒ several more mixed singlet-triplet states.

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 67 / 119

Page 98: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Re6Zr

Td group:noncentrosymmetric;d = 1, 2, 3⇒ can have broken TRS with strong SOC

⇒ broken TRS with singlet-triplet mixing

E irrep (d = 2) ⇒

F1,F2 irreps (d = 3) ⇒ several more mixed singlet-triplet states.

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 67 / 119

Page 99: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Re6Zr

Td group:noncentrosymmetric;d = 1, 2, 3⇒ can have broken TRS with strong SOC⇒ broken TRS with singlet-triplet mixing

E irrep (d = 2) ⇒

F1,F2 irreps (d = 3) ⇒ several more mixed singlet-triplet states.

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 67 / 119

Page 100: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Re6Zr

Td group:noncentrosymmetric;d = 1, 2, 3⇒ can have broken TRS with strong SOC⇒ broken TRS with singlet-triplet mixing

E irrep (d = 2) ⇒

F1,F2 irreps (d = 3) ⇒ several more mixed singlet-triplet states.

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 67 / 119

Page 101: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Re6Zr

Td group:noncentrosymmetric;d = 1, 2, 3⇒ can have broken TRS with strong SOC⇒ broken TRS with singlet-triplet mixing

E irrep (d = 2) ⇒

F1,F2 irreps (d = 3) ⇒ several more mixed singlet-triplet states.

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 67 / 119

Page 102: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Lu5Rh6Sn18

Group D4h:centrosymmetric ⇒ no singlet-triplet mixing;1d = 1, 2, 3⇒ can have broken TRS with strong SOC.

Only two states allowed: 1Eg (c) (singlet) and Eu(c) (triplet).

1c.f. recent ARPES-based claim for Sr2RuO4: C.N. Veenstra et al., PRL 112,127002 (2014). +

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 68 / 119

Page 103: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Lu5Rh6Sn18Singlet state:

kyk

x

kz

Triplet:

ky

kx

kz

kz

ky

kx

ky

kz

kx

ky

kx

kz

kz

ky

kx

ky

kz

kx

ky

kx

kz

kz

ky

kx

ky

kz

kx

N.B. “shallow” point nodes.These results should apply just as well to Sr2RuO4, in the regime of strongspin-orbit coupling [see Veenstra et al. results + ].

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 69 / 119

Page 104: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Quantum Materials Theory

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

Page 105: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

A bowl is not a mug

?

Is there a thermodynamic signature?

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119

Page 106: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

A bowl is not a mug

?

Is there a thermodynamic signature?

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119

Page 107: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

A bowl is not a mug

?

Is there a thermodynamic signature?

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119

Page 108: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

A bowl is not a mug

?

Is there a thermodynamic signature?

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119

Page 109: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

A bowl is not a mug

?

Is there a thermodynamic signature?

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119

Page 110: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

A bowl is not a mug

?

Is there a thermodynamic signature?

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119

Page 111: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

A bowl is not a mug

?

Is there a thermodynamic signature?

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 71 / 119

Page 112: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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.2

Widely used to fit experimental data on unconventional superconductors.3

2Anderson & Morel (1961), Leggett (1975)3Sigrist, Ueda (’89), Annett (’90), MacKenzie & Maeno (’03)Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 72 / 119

Page 113: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 73 / 119

Page 114: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 73 / 119

Page 115: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 73 / 119

Page 116: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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||

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)] and ourown result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.(unpublished)].A shallow line node may result at the boundary between gapless and line nodebehaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 74 / 119

Page 117: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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||

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)] and ourown result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.(unpublished)].A shallow line node may result at the boundary between gapless and line nodebehaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 74 / 119

Page 118: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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||

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)] and ourown result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.(unpublished)].A shallow line node may result at the boundary between gapless and line nodebehaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 74 / 119

Page 119: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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||

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)] and ourown result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.(unpublished)].

A shallow line node may result at the boundary between gapless and line nodebehaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 74 / 119

Page 120: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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||

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)] and ourown result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.(unpublished)].A shallow line node may result at the boundary between gapless and line nodebehaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 74 / 119

Page 121: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Line crossingsA 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 !!)

+

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 75 / 119

Page 122: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Crossing of shallow line nodesWhen 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+

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 76 / 119

Page 123: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Numerics

n = d lnCv /d lnT

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 77 / 119

Page 124: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

∆ 0

∆ 1Fermi Sea

∆ 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 78 / 119

Page 125: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

∆ 1Fermi Sea

∆ 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 79 / 119

Page 126: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

∆ 1Fermi Sea

∆ 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 80 / 119

Page 127: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

∆ 1Fermi Sea

∆ 0

Line

ar

node

s

Line

ar

node

sJorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 81 / 119

Page 128: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

∆ 1Fermi Sea

∆ 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 82 / 119

Page 129: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

∆ 1Fermi Sea

∆ 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 83 / 119

Page 130: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

∆ 1Fermi Sea

∆ 0

Sha

llow

no

de

Sha

llow

no

de

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 84 / 119

Page 131: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Quantum Materials Theory

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

Page 132: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Singlet-triplet mixing in noncentrosymmetricsuperconductors

Non-centrosymmetric superconductors are the multi-component orderparameter supercondcutors par excellence:

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

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:BaPtSi34, Re3W5,...Others seem to be correlated, purely triplet superconductors: +

LaNiC26 (c.f. centrosymmetric LaNiGa27) + , CePtr3Si (?) 8

4Batkova et al. JPCM (2010)5Zuev et al. PRB (2007)6Adrian D. Hillier, JQ and R. Cywinski PRL (2009)7Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)8Bauer et al. PRL (2004)Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 86 / 119

Page 133: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Singlet-triplet mixing in noncentrosymmetricsuperconductors

Non-centrosymmetric superconductors are the multi-component orderparameter supercondcutors par excellence:

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

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:BaPtSi34, Re3W5,...Others seem to be correlated, purely triplet superconductors: +

LaNiC26 (c.f. centrosymmetric LaNiGa27) + , CePtr3Si (?) 8

4Batkova et al. JPCM (2010)5Zuev et al. PRB (2007)6Adrian D. Hillier, JQ and R. Cywinski PRL (2009)7Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)8Bauer et al. PRL (2004)Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 86 / 119

Page 134: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Singlet-triplet mixing in noncentrosymmetricsuperconductors

Non-centrosymmetric superconductors are the multi-component orderparameter supercondcutors par excellence:

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

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:BaPtSi34, Re3W5,...Others seem to be correlated, purely triplet superconductors: +

LaNiC26 (c.f. centrosymmetric LaNiGa27) + , CePtr3Si (?) 84Batkova et al. JPCM (2010)5Zuev et al. PRB (2007)6Adrian D. Hillier, JQ and R. Cywinski PRL (2009)7Adrian D. Hillier, JQ, B. Mazidian, J. F. Annett, R. Cywinski PRL (2012)8Bauer et al. PRL (2004)Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 86 / 119

Page 135: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: 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)

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)

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 87 / 119

Page 136: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: 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)

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)

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 87 / 119

Page 137: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: 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)

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)

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 87 / 119

Page 138: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: 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)

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)

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 87 / 119

Page 139: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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)]}

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 88 / 119

Page 140: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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)]}

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 88 / 119

Page 141: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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)]}

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 88 / 119

Page 142: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: Phase diagramTreat A and B as independent tuning parameters and study quasiparticlespectrum. 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.Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 89 / 119

Page 143: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: Phase diagram

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 90 / 119

Page 144: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: Phase diagram

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 91 / 119

Page 145: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: Phase diagram

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 92 / 119

Page 146: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: Phase diagram

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Page 147: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Detecting the topological transitions

3 734

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 94 / 119

Page 148: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Detecting the topological transitions

3 734

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 95 / 119

Page 149: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: predicted specific heat power-laws

334

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 96 / 119

Page 150: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: predicted specific heat power-laws

jn = 2

n = 1.8

n = 1.4

n = 2

3

4

5

11

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 97 / 119

Page 151: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: predicted specific heat power-laws

3

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 98 / 119

Page 152: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: predicted specific heat power-laws

3

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 99 / 119

Page 153: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: predicted specific heat power-laws

jn = 2

n = 1.8

n = 1.4

n = 2

3

4

5

11

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 100 / 119

Page 154: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Anomalous power laws throughout the phase diagramDoes 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

0.25

T/T

c

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 transitionThe anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119

Page 155: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Anomalous power laws throughout the phase diagramDoes 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

0.25

T/T

c

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 transitionThe anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119

Page 156: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Anomalous power laws throughout the phase diagramDoes 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

The conventional exponent (n = 2 in this example) is only seen below atemperature scale that converges to zero at the transitionThe anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119

Page 157: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Anomalous power laws throughout the phase diagramDoes 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

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

The anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119

Page 158: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Anomalous power laws throughout the phase diagramDoes 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

The conventional exponent (n = 2 in this example) is only seen below atemperature scale that converges to zero at the transitionThe anomalous exponent (here n = 1.8) is seen everywhere else

⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119

Page 159: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Anomalous power laws throughout the phase diagramDoes 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

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

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119

Page 160: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Anomalous power laws throughout the phase diagramDoes 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

The conventional exponent (n = 2 in this example) is only seen below atemperature scale that converges to zero at the transitionThe anomalous exponent (here n = 1.8) is seen everywhere else ⇒the influence of the topo transition extends throughout the phase diagramc.f. quantum critical endpoints but here we did not have to fine-tune Tc → 0

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 101 / 119

Page 161: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Quantum Materials Theory

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

Page 162: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

What to take home

Superconductors

Broken time-reversal

symmetry

Topological transitionsTriplet

pairing

The relationship between triplet pairing and broken timre-reversal symmetryis complicatedNon-unitary triplet pairing breaks time-reversal symmetry and couples tomagnetism in a special waySinglet-triplet mixing may induce broken time-reversal symmetry ortopological transitionsThere are bulk signatures of topological transitionsThe thermodynamic transition is a distinct state

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 103 / 119

Page 163: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

THANKS!

www.cond-mat.org

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 104 / 119

Page 164: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 105 / 119

Page 165: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Spin-orbit coupling in Sr2RuO4

Recent spin-polarised ARPES find strong spin-orbit coupling in Sr2RuO4[C.N. Veenstra et al., PRL 112, 127002 (2014)]:

Veenstra et al.’s claim is that this will lead to singlet-triplet mixing.This seems at odds with our approach.

back

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Page 166: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Power laws in nodal superconductors

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

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Page 167: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Power laws in nodal superconductors

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

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Page 168: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Shallow line nodes in pnictides

back

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Page 169: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Logarithm ⇒ power law (n− 1 = 0.8)

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

back

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Page 170: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Logarithm ⇒ power law (n− 1 = 0.4)

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

back

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Page 171: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

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Page 172: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

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Page 173: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Hillier, Quintanilla & Cywinski, 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:

> 10-4

s ~

e

_

e

backward detector

forward detector

sample

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Page 174: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

back

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Page 175: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

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)

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Page 176: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

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!

Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

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Page 177: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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

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)

Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

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Page 178: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB: Phase diagram

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

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Page 179: Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Li2PdxPt3−xB:order parameter

back

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