Unconventional Pairings and Nodal Topology in Inversion Symmetry

Institute of
Theoretical &
Applied
Physics
Unconventional Pairings and Nodal Topology
in
Inversion Symmetry Broken Superconductors
Tuğrul Hakioğlu 1,2,3 and Mehmet Günay1,2
1
Physics Department, Bilkent University
2
Institute of Theoretical and Applied Physics (ITAP)
and
3
Center for Quantum Technologies in Energy (QTECH),
İstanbul Technical University
Quantum Metamaterials Conference, Spetses, Greece (1-5 June, 2015)
Center for Quantum Technologies in Energy
- İstanbul
ITAP and İTU collaboration:
1) Research:
quantum science & technology
2) Research Training:
International Diploma and Graduate Programs on QTECH
3) Dissemination
Abstract: Since the discovery of non-BCS superconductivity beyond the
conventional s-wave singlet pairing symmetry, a large number of unconventional
superconductors (USCs) with other pairing symmetries have been found. Due to
the Pauli principle, superconductivity is allowed to occur not only in even orbital
angular momenta with spin singlets (e.g. s and d-wave pairing), but also in odd
orbital angular momenta with spin triplets (e.g. p-wave pairing). This talk focuses
on the factors affecting the symmetry of the order parameter(s) in the physics of
USC.
Surprising enough, the exciton condensates (EC) in semiconductor DQWs are
valuable sources for understanding USC. The large spin-orbit coupling and strong
electronic correlations classify the EC as a laboratory for unconventional pairing
symmetries. On the other hand, under special conditions, EC can also be classified
as a Topological Superconductor (TSC). In short, EC, provides a unifying bridge
that is crucially needed for understanding of USC and TSC in the same broad picture
This talk will concentrate on the even/odd superconducting pairing symmetries and
the possible mechanisms leading to these symmetries using a unifying picture
between USCs, ECs and TSCs. We will quickly discuss the case of pairing in even
orbital angular momenta and the tetragonal Fermi surface nesting leading to
strongly repulsive electronic short range correlations in High Tc superconductors.
An odd angular momentum pairing is less understood in this context. We will
propose other crystal structures that can lead to preferred nesting directions which
can then lead to a pure triplet p-wave pairing.
A comprehensive understanding of the mechanism driving the pairing symmetry
of the order parameter in unconventional superconductivity still remains to be a
challenge. That is largely due to the mysterious nodal structure in the a) order
parameters, b) gap and c) excitation energies as well as the complexity of the
crystal and the orbital structure. We will discuss the types of nodes and the
mechanisms driving them separately for these types in a, b and c. We will
demonstrate the conditions of observing linear dispersion in the energy bands
(Dirac cones) and discuss the topology of the bands (spin Hall insulator versus
Z2 insulator). In this context, a thorough investigation of the nodal structure may
yield evidence on the attractive part of the pairing interaction being in the short
wavelength or the long wavelength sector which may give further evidence on
the pairing mechanism. In addition to that, in the s-p mixed USC, we will provide
a condition for the emergence of the Majorana bound states. We will finally
examine how the time reversal symmetry can be spontaneously broken in
some USCs.
Özet: İlk BCS-dışı üstüniletkenlerin keşfinden bu yana, tekil s-dalga çiftlenimi dışında
simetrilerde düzen parametresine sahip çok sayıda sıradışı üstüniletken (SDÜ) bulunmuştur.
Pauli ilkesi nedeniyle üstüniletkenlerde, s ve d-dalgası olarak bilinen + pariteli yörüngesel
açısal momentum çiftlenimine ek olarak, p-dalgası olarak bilinen – pariteli üçül çiftlenim
simetrileri oluşabilmektedir. Bu sunumda, SDÜ fiziğinde özellikle simetri merkezi olmayan
üstüniletkenlerde s-p karışımı düzen parametrelerinin simetrisini etkileyen faktörlere değineceğiz.
Diğer yandan, yarı-iletken çift kuantum-kuyularda egziton yoğuşkanı (EY) sıradışı üstüniletkenlik
için bulunmaz bir laboratuvar oluşturmaktadır. Bu sistemlerde görülen spin-yörünge etkileşimi
ve güçlü elektronik korelasyonların, sıradışı çiftlenme simetrilerini oluşturan mekanizmaların
incelenmesi açısından önemli bir kaynak teşkil ettiği çok bilinmemektedir. Diğer taraftan, özel
koşullar altında, EY bir topolojik üstüniletken (TÜ) olarak sınıflandırılabilir. EY'nin SDÜ ve TÜ'i
tek bir çerçeve içinde anlamak için gerekli olan bütünleştirici bir köprü görevi görebilmesinin
sonuçları tartışılacaktır.
Bu konuşmada bir yandan bu bütünleştirici dil kullanılırken, ana tema kapsamında üstüniletken
çiftlenim simetrileri (özellikle s-p dalgası) ve bu simetrilere yol açabilecek olan mekanizmalar
üzerinde durulacaktır. Yüksek sıcaklık üstüniletkenlerindeki tetragonal simetride itici kısa erimli
korelasyonların neden olduğu Fermi yüzeyi yuvalanması ve bunun sonucu olarak ortaya çıkan
+ pariteli d-dalga açısal momentum çiftlenmesi örnek olarak verilerek, - pariteli tek açısal
momentum çiftlenim simetrisinin (p-dalga) başka tür örgülerde nasıl oluşturulabileceği
gösterilecektir.
Özellikle s-p sıradışı çiftlenme simetrilerini oluşturan mekanizmalar gizemini büyük ölçüde
korumaktadır. Bunda temel neden düzen parametrelerinde üç farklı geometride ortaya çıkan
(noktasal, açısal ve yeni gözlemlediğimiz dairesel) düğümlerin temelde üç farklı fiziksel
büyüklükte (düzen parametresi, enerji aralığı ve uyarılma enerjisi) farklı şekillerde
görünebilmesidir. Bu düğümlerin etrafındaki band topolojisi tartışılacaktır. Bu bağlamda,
özellikle noktasal ve dairesel düğümlerin, çekici potansiyelin kısa veya uzun dalga sektörüne
bağlılığını gösteren yeni sonuçlar sunulacaktır. Bunlara ek olarak, s-p karışık SDÜ'lerde sıfır
enerjili ve Majorana bağlı durumlarının oluşturulabilmesi için bir koşul önerilecek ve en son olarak,
SDÜ'lerde zaman tersinme simetrisinin kırık olduğu çözümlere örnekler sunulacaktır.
Outline
- Pairing Symmetries and Mechanisms in inversion preserved symmetries
- Conventional BCS
- What decides in the pairing symmetry?, pairing under repulsive potentials
- Unconventional d-wave (repulsive V), sign changing s-wave (sign changing V)
- Unconventional p-wave
- Fund. Symmetries: Time reversal, inversion, fermion exchange, particle-hole
- Parity non-conserving (mixed)
- u.c pairing in inversion broken systems (Non-centrosymmetric supercond.'s)
- Excitonic Condensates as NCS
- Knotting the energy gap: Topological Superconductors
- Edge states
- Protected versus unprotected edge states
- zero energy states in the gap (Majorana BS)
- Which is a broader class? Topological superconductors or NCS
- Exciton Condensates between TS and NCS
- Nodal structure in unconv. SC
- Point, angular and line nodes in OP's, Δk and Ek
- Breaking the prejudice: non-phononic versus phononic mechanisms
Electronic States of Condensed Matter (A)
Moore, Physics World, Feb. 2011, 32
Insulators
A perfect insulator at T=0
Trivial topology
Quantum Hall system
Quantum Spin Hall system
(2D Topological Insulator)
3D Topological Insulator
✓
✓
✓
X
Time reversal invariance
Spin degeneracy
No B-field required
Topologically unprotected
X Time reversal symm. (Broken)
X Needs strong B-field
X very low temperatures
 Odd number of Dirac cones
✓ Topologically protected
✓
✓
✓



Time reversal symm.
Needs no B-field
Robust to high temperatures
Strong spin-orbit coupling
Odd # of Dirac cones
Topologically protected
Electronic States of Condensed Matter (B)
Superconductors
BCS ✓ Time reversal invariance
✓ Spin degeneracy
 low En Phonon, attractive
✓ Inversion symmetry
X Topology (trivial)
→ s-wave singlet
High-Tc
✓
✓
✓
✓
X
✓
Time reversal symmetry
Inversion symmetry
High Energy Repulsive
Spin-non degenerate
Topology (trivial)
dwave singlet
NCS, mixed parity, ECs
✓
X
✓
✓
✓
Time reversal (yes and no)
Broken Inversion symmetry
Spin-non degenerate
s-p mixing, pure s pure p
nontrivial Topology in nodes
Topological SC
✓ Time reversal (yes and no)
X Broken Inversion symmetry
✓ Spin-non degenerate
✓ s-p mixing, pure s, pure p
✓ nodes in TRI points
IS manifested
TRS manifested
Superconducting Pairing
s-wave singlet
2
BCS type
d-wave singlet
A, B, C, D
A
B
Point node at K=0 and
Angular line nodes in
the gap
--
OR
C
-
+
+
+
+
-
--
D
=
+
+
+ ...
Considering Spinless V0(q)=U
U is spin dependent !!!!
Splitting of the singlet & triplet
Hubbard U can lead to a triplet channel.
Can we form a p-wave triplet the same way?
A
Less angular line
nodes than
the d-wave
Energetically
preferable
+
-
+
Stable provided this
symmetry exists !!!
B
Fundamental symmetries of pairing
in
Unconventional superconductors
orbital
spin
Most common superconducting symmetries
under exchange of fermion quantum numbers
If spin and orbital D.o.F are uncoupled the SC usually
chooses one of the pairing types.
IS broken
TRS manifested
!!!
A superposition of all possibly odd pairings
A spin-orbit coupling can do this:
coupling orbit and spin by breaking
of inversion symmetry
Solution depends on
- the crystal point group symmetry
- the symmetry of the pairing potential (s)
Non-centrosymmetric superconductivity
Nodal Topology in the
Order Parameters
TH, M. Günay, arXiv:1411.4273
Pure singlet, mixed
(almost together)
pure ESP triplet
All possible time-reversal
invariant solutions
Energy bands of the
mixed solution
Almost gap closing
Time reversal invariant
Pure ESP triplet
Pure OSP triplet
Time reversal invariance
broken spontaneously
Both solutions have
half-spin quantum vortex topology
quantum spin Hall insulator (QSHI)
Nodal superconductivity (Topological)
Nodal structure in the light of OPs, gap and energy
Nodal Topology of the mixed state
=
Nodal Topology of the pure ESP triplet state
- Point (Dirac) node at k=0
- No closed line nodes
- No angular line nodes
k→∞
k=0
μ decides on the topology of the OP
Observable consequences of the nodes in energy
k
0
k1
CLN
Observable consequences of the nodes in energy
k
0
k1
k2
Comparison of the point and the closed line nodes in the energy
Point node: Cv~ T3
}
CLN Cv~ T
Energy point and closed line nodes show universal scaling behaviour of
Cv(T)~ Tn and ρ(E) ~ En-1 .
Comparison of different types of nodes
Topological superconductivity
(using Particle-hole symmetry at work)
Almost like a fully gapped superconductivity (except E=0 surface solutions)
Fully gapped in the bulk (a significant singlet solution present)
Gappless superconductivity on the surface (pure triplet SC)
Zero energy mode
(Majorana BS)
E
Bulk Excitation Band
e

-e
GS
Low energy
mixed or singlet
Dominant GS
Majorana doublet
(due to Time reversal invariance at k=0)
topologically unprotected
Helical superconductor (TRS)
2 spinless Majorana BS's
Chiral superconductor (TRSB)
OR
No Majorana BS here
because of spin mixing
Topologically controllable Exciton Condensates (TEC)
- Exciton condensates formed by two topological surface states
A
μ
e
μ
h
Topological SC in doped QSHI's
- gating V
AB
- doping
B
Thank You