Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

Broken Time-Reversal Symmetry and Topological Order
in Triplet Superconductors
Jorge Quintanilla1,2
1SEPnet and Hubbard Theory Consortium, University of Kent
2ISIS Neutron and Muon Source, STFC Rutherford Appleton Laboratory
Dresden, 27 November 2014
Jorge Quintanilla (Kent and RAL) www.cond-mat.org Dresden 2014 1 / 119

People and Money
People: James F. Annett (Bristol) , Adrian D. Hillier (RAL)
Bayan Mazidian (RAL/Bristol) , Bob Cywinski (Huddersfield) .
Ravi P. Singh , Gheeta Balakrishnan , Don Paul ,
Martin Lees (Birmingham). Amitava Bhattacharyya ,
Devashibai Adroja (RAL). A. M. Strydom (Johannesburg) .
Naoki Kase, Jun Akimitsu (Aoyama Gakuin).
Money: STFC (UK) + HEFCE/SEPnet (UK) + UJ and NRF (South Africa) +
Bristol + Kent.

The Hubbard Theory Consortium
Director: Piers Coleman (RHUL/Rutgers)
SEPnet fellows: Matthias Eschrig (RHUL/RAL)
Claudio Castelnovo (RHUL/RAL)
Jorge Quintanilla (Kent/RAL)
Associate: Jörg Schmalian (Karlsruhe)
+ several SEPnet PhD students.
Strong correlations theory in close
collaboration with experiments at
• RAL (ISIS/Diamond)
• London Centre for Nanotech.
• RHUL
Coleman (RHUL/Rutgers)
Eschrig (RHUL/RAL)

Overview
Two Paradigms in Condensed Matter ...

Overview
Broken Symmetry

Overview
Broken Symmetry
Topological Transitions

Overview
Broken Symmetry
... interlock via triplet pairing in superconductors:
Superconductors
Broken time-reversal
symmetry
Topological
Triplet transitions
pairing

Overview
Broken Symmetry
Superconductors
symmetry
Topological
Triplet transitions
pairing
SimpleTheories + Standard Measurements:

Overview
Broken Symmetry
Superconductors
symmetry
Topological
Triplet transitions
pairing
Group Theory / Bogolibov Quasiparticles

Overview
Broken Symmetry
Superconductors
symmetry
Topological
Triplet transitions
pairing
Group Theory / Bogolibov Quasiparticles
Neutron diffraction / Muon Spin Rotation / Specific Heat / Penetration Depth

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: Li2Pdx Pt3xB
7 Take-home message

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: Li2Pdx Pt3xB
7 Take-home message

Broken symmetry


Photo: Eddie Hui-Bon-Hoa, www.shiromi.com

Broken symmetry


Photo: Kenneth G. Libbrecht, snowflakes.com



Broken symmetry


Photo: Kenneth G. Libbrecht, snowflakes.com


Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
Photo: commons.wikimedia.org
Unconventional superconductors



UVinrginciao Tench, v18e Mnarcth i2o011n al Superconductors blogs.kent.ac.uk/strongcorrelations



UVinrginciao Tench, v18e Mnarcth i2o011n al Superconductors blogs.kent.ac.uk/strongcorrelations
‘Unconventional’
superconductors:
Cuprates, Sr2RuO4,
PrOs4Sb12, UPt3,
(UTh)Be13 , ...

Time-reversal Symmetry

p
r
x
y
z

r -p
x
y
z

r -p
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.

r -p
x
y
z
t ! t equivalent to
r ! r and p ! p
Quantum time-reversal symmetry:
t ! t equivalent to
y ! y and S ! S.
True if Hˆ = Hˆ and spin-invariant.

r -p
x
y
z
t ! t equivalent to
r ! r and p ! p
Quantum time-reversal symmetry:
t ! t equivalent to
y ! y and S ! S.
True if Hˆ = Hˆ and spin-invariant.
For quasi-particles in a superconductor:
Hˆ = Hˆ 0 + Dcˆ †
k ˆc†
k + H.c. ) TRS: D = D

Muon Spin Rotation

Muon Spin Rotation
Adrian Hillier
(Muons group leader, ISIS)

Muon Spin Rotation
Zero field muon spin relaxation
_

e 
e
backward
detector
forward
detector
sample

Kubo-Toyabe x exponential
Asymmetry:
NF NB
NF + NB
= G (t)
s : randomly-oriented fields (e.g. nuclear moments)
L :smoothly-modulated fields (e.g. electronic moments)

The “classic” examples: UPt3 and Sr2RuO4
UPt3
Luke et al. PRL (1993)
Sr2RuO4
Luke et al. Nature (1998)

Confirmed by Kerr effect
UPt3
Schemm et al. Science (2014)
Sr2RuO4
Jing Xia et al. PRL (2006)

More recent finds: LaNiC2

More recent finds: LaNiC2
Relaxation due to electronic moments
Moment
size
~ 0.1G
(~ 0.01μB
)
Hillier, Quintanilla Cywinski,
PRL 102 117007 (2009)
(longitudinal)
_

 e
e
backward
detector
Timescale:
10-4s
~
forward
detector
sample
+

More recent finds: LaNiGa2
A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett,
Physical Review Letters 109, 097001 (2012).

More recent finds: Re6Zr

More recent finds: Lu5Rh6Sn18

Singlet, triplet, or both?
It’s all in the gap function:

Symmetry of the gap function
 

 k   k


   
 

 
 
 
k k
 
ˆ k
See J.F. Annett Adv. Phys. 1990.

 

 k   k


   
 

 
 
 
k k
 
ˆ k
See J.F. Annett Adv. Phys. 1990. Pauli )
ˆD
(k) = ˆD
T (k)

 

 k   k


   
 

 
 
 
k k
 
ˆ k
See J.F. Annett Adv. Phys. 1990. Pauli )
T (k) Virginia Tech, 18 March 2011 blogs.kent.ac.uk/strongcorrelations
ˆD
(k) = ˆD

ˆ k


0  0
 0 0



 
dx  idy dz
dz dx  idy






singlet
[ 0(k) even ]
triplet
[ d(k) odd ]

Neglect (for now!) spin-orbit coupling:

Singlet and triplet representations of SO(3):

Γn
s = - (Γn
s)T , Γn
t = + (Γn
t)T

Γn
s = - (Γn
s)T , Γn
Impose Pauli’s exclusion principle:
t = + (Γn
t)T


 , 'k  ', k
Γn
s = - (Γn
s)T , Γn
t = + (Γn
t)T

Γn
s = - (Γn

 , 'k  ', k


ˆ k either singlet
s)T , Γn
t = + (Γn
t)T

Γn
s = - (Γn
s)T , Γn

t = + (Γn
t)T
 , 'k  ', k


ˆ k either singlet     y i   ˆ , ' 0  k  k

Γn
s = - (Γn

 , 'k  ', k


or triplet
s)T , Γn
t = + (Γn
t)T

Γn
s = - (Γn
s)T , Γn

t = + (Γn
t)T
 , 'k  ', k


or triplet       y i   .ˆ ˆ , '  k  d k σ

The role of spin-orbit coupling (SOC)
Gap function may have both singlet and triplet components




 
d id d
x y z
d d id
  


 
 
0 k


 
z x y
0
0
ˆ
0
k k spin orbit
, ' , '       

• 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)
Jorge Quintanilla• (NKeont acnednRAtLr)e of inversion: mwawyw .hcoandv-mea ts.oirngglet and triplet (even at TDcre)s den 2014 36 / 119

G = [SO(3)×Gc]×U(1)×T

G = Gc,J×U(1)×T

E.g. reflection through a vertical
plane perpendicular to the y axis:
y
v J J I C , 2,  
Quintanilla, Hillier, Annett and Cywinski,
PRB 82, 174511 (2010)
z
y x

y
v J J I C , 2,  
PRB 82, 174511 (2010)
z
y x

y
v J J I C , 2,  
This affects d(k) (a vector under
spin rotations).
PRB 82, 174511 (2010)
z
y x

y
v J J I C , 2,  
This affects d(k) (a vector under
spin rotations).
It does not affect 0(k) (a scalar).
PRB 82, 174511 (2010)
z
y x

When can we have singlet-triplet mixing?
We must now use basis functions of the double group:
Gap function may have both singlet and triplet components
ˆD
(k) =
dGå
n=1
hnˆG
n (k)

 


 
d id d
x y z
d d id

  
 
 
0 k


 
z x y
0
0
ˆ
0
k k spin orbit
, ' , '       

• 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 symmetry
Centrosymmetric Non-centrosymmetric
Spin-orbit coupling Weak N N
Strong N Y

When can we have broken time-reversal symmetry?
For ˆD
(k) to be non-trivially complex, it must have more than one
component:
ˆD
(k) = h1ˆG
1 (k) + h2ˆG
2 (k) , arg h16= arg h2

For ˆD
component:
ˆD
(k) = h1ˆG
1 (k) + h2ˆG
2 (k) , arg h16= arg h2
The instability must therefore take place in an irrep with d 1.

For ˆD
component:
ˆD
(k) = h1ˆG
1 (k) + h2ˆG
2 (k) , arg h16= arg h2
Weak spin-orbit coupling: SO(3) Gc

For ˆD
component:
ˆD
(k) = h1ˆG
1 (k) + h2ˆG
2 (k) , arg h16= arg h2
The singlet irrep of SO(3) has d = 1 ) for singlet pairing, the point group Gc
must have a d 1 irrep.

For ˆD
component:
ˆD
(k) = h1ˆG
1 (k) + h2ˆG
2 (k) , arg h16= arg h2
The triplet irrep of SO(3) had d = 3 ) for triplet pairing, broken TRS is
possible even for d = 1 irreps of Gc .

For ˆD
component:
ˆD
(k) = h1ˆG
1 (k) + h2ˆG
2 (k) , arg h16= arg h2
Strong spin-orbit coupling: Gc,J (double group)

For ˆD
component:
ˆD
(k) = h1ˆG
1 (k) + h2ˆG
2 (k) , arg h16= arg h2
The dimensionality of the irreps is the same as for Gc therefore if all irreps are
d = 1 then there can be no broken TRS.

For ˆD
component:
ˆD
(k) = h1ˆG
1 (k) + h2ˆG
2 (k) , arg h16= arg h2
The dimensionality of the irreps is the same as for Gc therefore if all irreps are
d = 1 then there can be no broken TRS.
Broken TRS involves always a d 1 irrep and it requires both the singlet and
triplet components

LaNiC2
Character table
PRL 102 117007 (2009)

LaNiC2
Character table
PRL 102 117007 (2009)
180o

LaNiC2
C
2v
Character table
Symmetries and
their characters
Sample basis
functions
Irreducible
representation
E C
2
v
’
v
Even Odd
A
1
1 1 1 1 1 Z
A
2
1 1 -1 -1 XY XYZ
B
1
1 -1 1 -1 XZ X
B
2
1 -1 -1 1 YZ Y
PRL 102 117007 (2009)
All irreps d = 1
) weak SOC

LaNiC2
SO(3)xC
2v
Gap function
(unitary)
Gap function
(non-unitary)
1A
1 (k)=1 -
1A
2 (k)=k
k
x
Y
-
1B
1 (k)=k
k
X
Z
-
1B
2 (k)=k
k
Y
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
k
X
k
Y
Z
d(k)=(1,i,0)k
k
X
k
Y
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
PRL 102 117007 (2009)

LaNiC2
SO(3)xC
2v
Gap function
(unitary)
Gap function
(non-unitary)
1A
1 (k)=1 -
1A
2 (k)=k
k
x
Y
-
1B
1 (k)=k
k
X
Z
-
1B
2 (k)=k
k
Y
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
k
X
k
Y
Z
d(k)=(1,i,0)k
k
X
k
Y
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
PRL 102 117007 (2009)

LaNiC2
SO(3)xC
2v
Gap function
(unitary)
Gap function
(non-unitary)
1A
1 (k)=1 -
1A
2 (k)=k
k
x
Y
-
1B
1 (k)=k
k
X
Z
-
1B
2 (k)=k
k
Y
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
k
X
k
Y
Z
d(k)=(1,i,0)k
k
X
k
Y
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
PRL 102 117007 (2009)

LaNiC2
SO(3)xC
2v
Gap function
(unitary)
Gap function
(non-unitary)
1A
1 (k)=1 -
1A
2 (k)=k
k
x
Y
-
1B
1 (k)=k
k
X
Z
-
1B
2 (k)=k
k
Y
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
k
X
k
Y
Z
d(k)=(1,i,0)k
k
X
k
Y
Z
3B
1
d(k)=(0,0,1)k
X
d(k)=(1,i,0)k
X
3B
2
d(k)=(0,0,1)k
Y
d(k)=(1,i,0)k
Y
Non-unitary
d x d* ≠ 0
breaks only SO(3) x U(1) x T
* C.f. Li2Pd3B Li2Pt3B,
H. Q. Yuan et al. PRL’06
*
PRL 102 117007 (2009)

LaNiC2
Non-unitary pairing
Spin-up superfluid
coexisting with spin-down
Fermi liquid.

0 0
  

 
0
The A1 phase of
liquid 3He.

 

 


 

 


0
or
0 0
ˆ
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).

LaNiGa2
Centrosymmetric, but again all irreps d = 1 ) again weak SOC and non-unitary
triplet
A.D. Hillier, J. Quintanilla, B. Mazidian, J.F. Annett, and R. Cywinski, PRL 109, 097001
(2012).
A new family of superconductors?

Why non-unitary?
Generic Landau theory for a triplet superconductor (1D irrep):
F = a jhj2 +
b
2

+ b0 jh hj2

+ b0 jh hj2 +
m2
2c

+ b0 jh hj2 +
m2
2c
+ b00m (ih h) .

+ b0 jh hj2 +
m2
2c
+ b00m (ih h) .
Magnetisation as a sub-dominant order parameter:
Superconductivity
magnetisation
Temperature

+ b0 jh hj2 +
m2
2c
+ b00m (ih h) .
Magnetisation as a sub-dominant order parameter:
Superconductivity
magnetisation
Temperature
Theory (left): A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, PRL 109, 097001 (2012).
Experiment (right): Akihiko Sumiyama et al., JPSJ (2014).

Re6Zr
Td group:
noncentrosymmetric;
d = 1, 2, 3

Re6Zr
Td group:
noncentrosymmetric;
d = 1, 2, 3
) can have broken TRS with strong SOC

Re6Zr
Td group:
noncentrosymmetric;
d = 1, 2, 3
) broken TRS with singlet-triplet mixing

Re6Zr
Td group:
noncentrosymmetric;
d = 1, 2, 3
E irrep (d = 2) )

Re6Zr
Td group:
noncentrosymmetric;
d = 1, 2, 3
E irrep (d = 2) )
F1, F2 irreps (d = 3) ) several more mixed singlet-triplet states.

Lu5Rh6Sn18
Group D4h:
centrosymmetric ) no singlet-triplet mixing;1
d = 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). +

Lu5Rh6Sn18
Singlet state:
k k y x
kz
Triplet:
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 strong
spin-orbit coupling [see Veenstra et al. results + ].

A bowl is not a mug

A bowl is not a mug
?

A bowl is not a mug
?
Is there a thermodynamic signature?

Power laws in nodal superconductors
Low-temperature specific heat of a superconductor gives information on the
spectrum of low-lying excitations:
Fully gapped Point nodes Line nodes
Cv eD/T Cv T3 Cv T2
D
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)

Linear nodes
It all comes from the density of states: +
g (E) En1 ) Cv Tn

Linear nodes
g (E) En1 ) Cv Tn
linear
point node line node
D2k= I1

kx
jj
2 + ky
jj
2

D2k
= I1kx
jj
2
g(E) = E2
2(2p)2I1
pI2
g(E) = LE
(2p)3pI1
pI2
n = 3 n = 2

Linear nodes
g (E) En1 ) Cv Tn
linear
D2k= I1

kx
jj
2 + ky
jj
2

D2k
= I1kx
jj
2
g(E) = E2
2(2p)2I1
pI2
g(E) = LE
(2p)3pI1
pI2
n = 3 n = 2
Key assumption: linear increase of the gap away from the node

Shallow nodes
Relax the linear assumption and we also get different exponents:

Shallow nodes
shallow
D2k
= I1(kx
jj
2 + ky
jj
2
)2 D2k
= I1kx
jj
4
g(E) = E
2(2p)2pI1
pI2
g(E) = L
p
E
(2p)3I
1
4
1
pI2
n = 2 n = 1.5

Shallow nodes
shallow
D2k
= I1(kx
jj
2 + ky
jj
2
)2 D2k
= I1kx
jj
4
g(E) = E
2(2p)2pI1
pI2
g(E) = L
p
E
(2p)3I
1
4
1
pI2
n = 2 n = 1.5
Shallow point nodes first discussed (speculatively) by Leggett [1979].

Shallow nodes
shallow
D2k
= I1(kx
jj
2 + ky
jj
2
)2 D2k
= I1kx
jj
4
g(E) = E
2(2p)2pI1
pI2
g(E) = L
p
E
(2p)3I
1
4
1
pI2
n = 2 n = 1.5
A shallow point node may be required by symmetry e.g. the proposed E2u
pairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)] and our
own result for R5Rh6Sn18 [A. Bhattacharyya, D. T. Adroja, JQ et al.
(unpublished)].

Shallow nodes
shallow
D2k
= I1(kx
jj
2 + ky
jj
2
)2 D2k
= I1kx
jj
4
g(E) = E
2(2p)2pI1
pI2
g(E) = L
p
E
(2p)3I
1
4
1
pI2
n = 2 n = 1.5
A shallow point node may be required by symmetry e.g. the proposed E2u
pairing state in UPt3 [see J.A. Sauls, Adv. Phys. 43, 113-141 (1994)] and our
own 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 node
behaviour in pnictides [Fernandes and Schmalian, PRB 84, 012505 (’11)]. +

Line crossings
A different power law is expected at line crossings
(e.g. d-wave pairing on a spherical Fermi surface):
crossing
of linear line nodes
D2k
= I1

kx
jj
2 ky
jj
2
2
or I1kx
jj
2ky
jj
2
g(E) =
E(1+2lnj
L+
1
4
1
p
E/I
1
4
1
p
E/I
j)
(2p)3pI1I2
E0.8
n = 1.8 ( 2 !!)
+

Crossing of shallow line nodes
When shallow lines cross we get an even lower exponent:
crossing
of shallow line nodes
D2k= I1

kx
jj
2 ky
jj
2
4
or I1kx
jj
4ky
jj
4
g (E) =
p
E(1+2lnj
L+E
14
/I
18
1
E
14
/I
18
1
j)
(2p)3I
1
4
1
pI2
E0.4
n = 1.4 *
* c.f. gapless excitations of a Fermi liquid: g (E) = constant ) n = 1
+

Numerics
n = d ln Cv/d lnT
4.5
4
3.5
3
2.5
2
1.5
1
linear point node
shallow point node
linear line node
crossing of linear line nodes
shallow line node
crossing of shallow line nodes
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
n
T / T
c

A generic mechanism
We propose that shallow nodes will exist generically at topological phase
transitions in superocnductors with multi-component order parameters:
D0
D1
Fermi Sea

A generic mechanism
We propose that shallow nodes will exist generically at quantum phase
D1
Fermi Sea
D0

A generic mechanism
D1
Fermi Sea
D0

A generic mechanism
D1
Fermi Sea
D0
Linear
nodes
Linear
nodes

A generic mechanism
D1
Fermi Sea
D0

A generic mechanism
D1
Fermi Sea
D0
Shallow
node
Shallow
node

Singlet-triplet mixing in noncentrosymmetric
superconductors
Non-centrosymmetric superconductors are the multi-component order
parameter supercondcutors par excellence:

ˆ k


0  0
 0 0



 
dx  idy dz
dz dx  idy






singlet
[ 0(k) even ]
triplet
[ d(k) odd ]
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)

superconductors

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:

superconductors

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

Li2PdxPt3xB: tunable singlet-triplet mixing
The Li2Pdx Pt3xB family (0 x 3; cubic point group O) provides a tunable
realisation 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).

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)

Li2PdxPt3xB: Phase diagram
Bogoliubov Hamiltonian with Rashba spin-orbit coupling:
H(k) =
ˆh(k) ˆD
(k)
ˆD
†(k) ˆhT (k)

ˆh(k) = #kI + gk s
ˆD
(k) = [D0 (k) + d (k) ˆs
] iˆs
y (most general gap matrix)

H(k) =
ˆh(k) ˆD
(k)
ˆD
†(k) ˆhT (k)

ˆh(k) = #kI + gk s
ˆD
(k) = [D0 (k) + d (k) ˆs
] iˆs
Assuming j#kj jgkj jd (k)j the quasi-particle spectrum is
E =
8
:
q
2
(#k m + jg2 kj)+ (D0 (k) + jd (k)j); and

q
(#k m jgkj)2 + (D0 (k) jd (k)j)2 .

H(k) =
ˆh(k) ˆD
(k)
ˆD
†(k) ˆhT (k)

ˆh(k) = #kI + gk s
ˆD
(k) = [D0 (k) + d (k) ˆs
] iˆs
Assuming j#kj jgkj jd (k)j the quasi-particle spectrum is
E =
8
:
q
2
(#k m + jg2 kj)+ (D0 (k) + jd (k)j); and

q
(#k m jgkj)2 + (D0 (k) jd (k)j)2 .
Take most symmetric (A1) irreducible representation: +
D0 (k) = D0
d(k) = D0 f
A (x) (kx , ky , kz ) B (x)

kx

k2
y + k2
z

, ky

k2
z + k2
x

, kz

k2
x + k2
y

g

Treat A and B as independent tuning parameters and study quasiparticle
spectrum. 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.,
PRBJorg(e2Q0u1in2ta)n;illaB(.KeMntaanzdidRiAaLn) , JQ, A.D. Hillierw,wJw..cFo.ndA-mnatn.oergtt, arXiv:1302.2161. Dresden 2014 89 / 119

Detecting the topological transitions
4 33 7

Li2PdxPt3xB: predicted specific heat power-laws
4 3

5
n = 2 j
n = 2
n = 1.8
n = 1.4
4
3
11

3

Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors

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