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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Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors
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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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
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
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
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
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
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
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
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
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
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 + ].
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)]. +
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)]. +
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)]. +
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)]. +
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)]. +
A generic mechanismWe propose that shallow nodes will exist generically at topological phasetransitions in superocnductors with multi-component order parameters:
A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
A generic mechanismWe propose that shallow nodes will exist generically at quantum phasetransitions in superocnductors with multi-component order parameters:
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
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: +
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
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
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
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)
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)
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)
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)
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
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
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
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
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
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
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
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
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
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