Unconventional Pairings and Nodal Topology in Inversion Symmetry
Transkript
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