New Broken Time-reversal Symmetry Superconductors: Theoretical Constraints on Pairing States and Mechanisms

The UK’s European university
New Broken Time-reversal Symmetry
Superconductors / Theoretical Constraints
on Pairing States and Mechanisms
Jorge Quintanilla

Jorge Quintanilla (www.cond-mat.org) SCES 2016 (Hangzhou, 13 May 2016)Page 2
Superconductivity
Triplet
pairing
Broken time-
reversal
symmetry
Topological
transitions
LaNiC2
,
LaNiGa2
UPt3
,Sr2
RuO4
Re6
Zr Topological
insulators
Ferro-
magnets
Antiferro-
magnets
R5
Rh6
Sn18
Cu-based
HTSC
Fe-based
HTSC
URhGe,UGe2
Conventional
superconductors
Li2
Pt3-x
Pdx
B
Noncentrosymmetric
superconductors

People
Theory: James F. Annett (Bristol)
Bayan Mazidian (RAL/Bristol) / LaNiGa2, Re6Zr
Experiment: Adrian Hillier (RAL)
Bob Cywinski (Huddersfield) / LaNiC2, LaNiGa2
Michael Smidman, Huiqiu Yuan,
C. F. Weng, J. L. Zhang, T. Shang, G. M. Pang,
L. Jiao, W. B. Jiang & Y. Chen (Zhejiang),
M. Nicklas & F. Steglich (MPI-CPS, Dresden) / LaNiGa2
Ravi P. Singh, Martin Lees,
Gheeta Balakrishnan & Don Paul (Warwick) / Re6Zr
Amitava Bhattacharyya & D. T. Adroja (RAL),
Naoki Kase & J. Akimitsu (Aoyama-Gakuin),
A. M. Strydom (Dresden) / Lu5Rh6Sn18
Additional discussions:
V. Mineev (CEA Grenoble), Susumu Katano (Saitama), Akihiko Sumiyama (Hyogo),
Takashi Yanagisawa (Tokyo), David Singh (ORNL),
Silvia Ramos, Paul Strange, Phil Whittlesea & Mark A. Green (Kent)
Funding:
STFC
SEPnet (HEFCE)
EPSRC
AWM (EU-RDF)
SA-NRF
& FRC of UJ

Unconventional
superconductors

Unconventional
superconductorsPhoto:EddieHui-Bon-Hoa,www.shiromi.com

Photo:KennethG.Libbrecht,snowflakes.com
Unconventional
superconductors

Photo:commons.wikimedia.org
Unconventional
superconductors

Unconventional
superconductors
‘Unconventional’
superconductors:

Unconventional
superconductors
‘Unconventional’
superconductors:
Cuprates,
Sr2RuO4,
PrOs4Sb12, UPt3,
(UTh)Be13 , ...

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
 0 = 22.2 10-6
emu/mol

Time-reversal symmetry breaking

Detecting broken time-reversal symmetry:
Zero-field muSR
+

Zero-field muSR
Field on:
measure 
+

Zero-field muSR
Field on:
measure 
Field off:
measure m
+

Zero-field muSR
Field on:
measure 
Field off:
measure m
Gives yes/no answer to the
question:
“Does this system respect
time-reversal symmetry?”
+

The “classics”

Confirmed by the Kerr effect

Symmetry of the gap function
See J.F. Annett Adv. Phys. 1990.

Neutron diffraction
30 40 50 60 70 80
0
5000
10000
15000
20000
25000
30000
35000
Intensity(arbunits)
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)
, Li2Pt3B & Li2Pd3B (2)
, ...
(1) Bauer et al. PRL’04 (2) Yuan et al. PRL’06

SEPnet Conference, 13 Sep 2012 blogs.kent.ac.uk/strongcorrelations
Character table
Hillier, Quintanilla & Cywinski,
PRL 102 117007 (2009)

Character table
PRL 102 117007 (2009)
180
o

C2v
Symmetries
and their
characters
Sample basis
functions
Irreducible
representatio
n
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
PRL 102 117007 (2009)

C2v
Symmetries
and their
characters
Sample basis
functions
Irreducible
representatio
n
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
PRL 102 117007 (2009)
These must be combined with the singlet and
triplet representations of SO(3).

SO(3)xC2v Gap function
(unitary)
Gap function
(non-unitary)
1
A1
(k)=1 -
1
A2
(k)=kxkY -
1
B1
(k)=kXkZ -
1
B2
(k)=kYkZ -
3
A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3
A2 d(k)=(0,0,1)kXkYkZ d(k)=(1,i,0)kXkYkZ
3
B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3
B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
Possible order parameters
PRL 102 117007 (2009)

(unitary)
Gap function
(non-unitary)
1
A1
(k)=1 -
1
A2
(k)=kxkY -
1
B1
(k)=kXkZ -
1
B2
(k)=kYkZ -
3
A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3
3
B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3
B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
Non-unitary
d x d* ≠ 0
PRL 102 117007 (2009)

(unitary)
Gap function
(non-unitary)
1
A1
(k)=1 -
1
A2
(k)=kxkY -
1
B1
(k)=kXkZ -
1
B2
(k)=kYkZ -
3
A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3
3
B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX
3
B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY
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)

Spin-up superfluid
coexisting with
spin-down Fermi
liquid.
Non-unitary pairing

Spin-up superfluid
coexisting with
spin-down Fermi
liquid.
Non-unitary pairing
C.f.

Spin-up superfluid
coexisting with
spin-down Fermi
liquid.
The A1 phase of
liquid 3
He.
Non-unitary pairing
C.f.

Spin-up superfluid
coexisting with
spin-down Fermi
liquid.
The A1 phase of
liquid 3
He.
Non-unitary pairing
C.f.
Ferromagnetic
superconductors.
F. Hardy et al., Physica B
359-61, 1111-13 (2005)
[ See A. de Visser in Encyclopedia of
Materials: Science and Technology
(Eds. K. H. J. Buschow et al.),
Elsevier, 2010 ]
[ See A. de Visser in Encyclopedia of
Materials: Science and Technology
(Eds. K. H. J. Buschow et al.),
Elsevier, 2010 ]

Ferromagnetic
superconductors
A. de Visser in Encyclopedia of Materials: Science and Technology
(Eds. K. H. J. Buschow et al.), Elsevier, 2010

But LaNiC2 is a paramagnet !

What about spin-orbit
coupling?

Gap function may have both singlet and triplet components

• 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)
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:

82, 174511 (2010)
axis:
This affects d(k) (a vector
under spin rotations).

82, 174511 (2010)
axis:
This affects d(k) (a vector
under spin rotations).
It does not affect 0(k)
(a scalar).

C2v,Jno t Gap function,
singlet component
Gap function,
triplet component
A1
(k) = A d(k) = (Bky,Ckx,Dkxkykz)
A2
(k) = AkxkY d(k) = (Bkx,Cky,Dkz)
B1
(k) = AkXkZ d(k) = (Bkxkykz,Ckz,Dky)
B2
(k) = AkYkZ d(k) = (Bkz, Ckxkykz,Dkx)
82, 174511 (2010)

C2v,Jno t Gap function,
singlet component
Gap function,
triplet component
A1
(k) = A d(k) = (Bky,Ckx,Dkxkykz)
A2
(k) = AkxkY d(k) = (Bkx,Cky,Dkz)
B1
(k) = AkXkZ d(k) = (Bkxkykz,Ckz,Dky)
B2
(k) = AkYkZ d(k) = (Bkz, Ckxkykz,Dkx)
None of these break time-reversal
symmetry!
82, 174511 (2010)

Zeng et al,
Phys. Rev B 66 092503 (2002)
Cmmm (centrosymmetric)
a=4.27 Å
b=17.70 Å
c=4.23 Å
ΔC/TC=1.3
(BCS: 1.43)
LaNiGa2: physical properties

LaNiGa2: muSR
0 2 4 6 8 10 12 14
0.0
0.1
0.2
0.3
Asymmetry
Time (s)
Hillier, Quintanilla, Mazidian, Annett & Cywinski,
PRL 109, 097001 (2012)

SO(3)xD2h Gap function
(unitary)
Gap function
(non-unitary)
1
A1
(k)=1 -
1
B1
(k)=kxkY -
1
B2
(k)=kXkZ -
1
B3
(k)=kYkZ -
3
3
B1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3
B2 d(k)=(0,0,1)ky d(k)=(1,i,0)ky
3
B3 d(k)=(0,0,1)kx d(k)=(1,i,0)kx
PRL 109, 097001 (2012)

SO(3)xD2h Gap function
(unitary)
Gap function
(non-unitary)
1
A1
(k)=1 -
1
B1
(k)=kxkY -
1
B2
(k)=kXkZ -
1
B3
(k)=kYkZ -
3
3
B1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
3
B2 d(k)=(0,0,1)ky d(k)=(1,i,0)ky
3
B3 d(k)=(0,0,1)kx d(k)=(1,i,0)kx
Non-unitary
d x d* ≠ 0
PRL 109, 097001 (2012)

Why non-unitary?
(Let's make a prediction!)

Superconducting magnetism
Free energy of a triplet superconductor:
T
T=Tc
(a=0)
 > 0 unitary
< 0 nonunitary
AD Hillier, JQ, B Mazidian
and JF Annett, PRL (2012)

T
T=Tc
(a=0)
Add magnetism:
> 0 unitary
< 0 nonunitary


T
T=Tc
(a=0)
Add magnetism:
> 0 unitary
< 0 nonunitary
m


Confirmed (weh-hey!) by bulk SQUID measurements on LaNiC2: [1,2]
Note: Scanning [3] and bulk [2] SQUID measurements on Sr2RuO4
were negative.
In LaNiC2 it is easier because of Rashba spin-orbit coupling
(Sumiyama, private communication).
[1] Sumiyama, A. et al.
JPSP 84, 13702
(2015).
[2] Sumiyama et al.,
JPS Conf. Proc.,
015017 (2014)
[3] Hicks et al.,
PRB (2010).

Two different gaps consistent with (one for spin-up, one for spin-down).
But our non-unitary triplet states are all nodal.
J Chen, L Jiao, J L Zhang, Y Chen, L Yang, M Nicklas, F Steglich, and H Q Yuan
”Evidence for two-gap superconductivity in the non-centrosymmetric compound LaNiC2”,
New J. Phys. 15, 53005 (2013).
More data:

LaNiC2 – a weakly-correlated, paramagnetic
superconductor?
Tc=2.7 K
ΔC/TC=1.26
(BCS: 1.43)
specific heat susceptibility
 0 = 6.5 mJ/mol K2
 0 = 22.2 10-6
emu/mol
Wilson ratio
RW = (1+F0
a
)-1
 0.3
Wilson ratio
RW = (1+F0
a
)-1
 0.3

At the Hartree-Fock level F0
a > 0 implies a net-attractive interaction
e.g. in a 3D continuum, [Quintanilla & Schofield PRB (2006)]
Wilson ratio
RW = (1+F0
a
)-1
 0.3
Wilson ratio
RW = (1+F0
a
)-1
 0.3

Wilson ratio
RW = (1+F0
a
)-1
 0.3
Wilson ratio
RW = (1+F0
a
)-1
 0.3
Suggests a negative-U, equal-spin interaction (driven, for example,
by Hund on Ni):

A
B
Must involve
two different
orbitals A,B
Wilson ratio
RW = (1+F0
a
)-1
 0.3
Wilson ratio
RW = (1+F0
a
)-1
 0.3
Suggests a negative-U, equal-spin interaction (driven, for example,
by Hund on Ni):

Construct variational mean field theory:
Hamiltonian:
Two bands:
Interaction:
Mean fields:

Hamiltonian:
Two bands:
Interaction:
Mean fields:
Non-unitary triplet pairing

Fully-gapped + equal-spin + orbital-
antisymmetric –similar to [1,2] but

Broken time-reversal symmetry
(nonunitary)
[1] X Dai, Z
Fang, Y Zhou,
& F-C Zhang,
PRL (2008).
[2] T Tzen
Ong, P
Coleman, & J
Schmalian
(2014).

Hamiltonian:
Two bands:
Interaction:
Mean fields:
Non-unitary triplet pairing

Fully-gapped + equal-spin + orbital-
antisymmetric –similar to [1,2] but

Broken time-reversal symmetry
(nonunitary)
Magnetisation

Discusssed
before for
Sr2RuO4 [3]
[1] X Dai, Z
Fang, Y Zhou,
& F-C Zhang,
PRL (2008).
[2] T Tzen
Ong, P
Coleman, & J
Schmalian
(2014).

Bogoliubov-de Gennes Hamiltonian:
Note spins completely decoupled.
ZF Weng, JL Zhang, M Smidman, T Shang, J Quintanilla, JF Annett, M Nicklas,
GM Pang, L Jiao, WB Jiang, Y Chen, F Steglich, and HQ Yuan, Phys. Rev. Lett. (submitted)

A simple example:
EAk
= k2
-, EBk
= k2
-+
A
A
B
B

Note: very different from two-band superconductivity!

2-band
pairing:

Non-
unitary
triplet
pairing:

Note: very different from two-band superconductivity!

2-band
pairing:

Non-
unitary
triplet
pairing:
The band splitting emerges spontaneously -similar to Stoner.

Two-gap behaviour results from non-unitary triplet
 generic feature of this type of superconductor

Two-gap behaviour results from non-unitary triplet
 generic feature of this type of superconductor
Indeed it is observed for LaNiGa2
as well:

LaNiC2 and LaNiGa2 are two examples of a new class
of La-Ni superconductors with nonunitary triplet
pairing
This type of pairing induces a magnetisation as a
subdominant order parameter
This is achieved via spontaneous spin-splitting of
the bands similar to Stoner –but driven by the
superconductivity
Where did the correlations come from?

Epilogue
S Katano et al., PRB(R) (2014)

Epilogue
S Katano et al., PRB(R) (2014)
Is this the first
“backward
discovery” of a
quantum-critical
superconductor?
blogs.kent.ac.uk/strongcorrelations

THE UK’S
EUROPEAN
UNIVERSITY
www.kent.ac.uk
Places

THE UK’S
EUROPEAN
UNIVERSITY
www.kent.ac.uk
Places
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