Topological superconductors Ismail Achmed-Zade - PowerPoint PPT Presentation

Topological superconductors Ismail Achmed-Zade Arnold-Sommerfeld-Center December 14th, 2017 Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 1 / 24

Overview Superconductivity 1 BCS theory Topology 2 Topological Superconductor 3 Bulk-boundary Correspondence Experimental results 4 Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 2 / 24

What is a topological superconductor? Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 3 / 24

BCS theory Idea An electron pair interacts via a phonon k ′ + � � q � k − � q � q � � k ′ k Recall Phonons are the quanta of background lattice oscillations. Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 4 / 24

BCS theory Procedure Start with the Hamiltonian describing electrons coupling to the background lattice H = H 0 + H int . with q b † k a † H 0 = � q + � q � ω � q b � k ǫ � k a � � � � � k q a † q + b † H int = � k M � q a � k ( b � q ) . q ,� � � − � k + � Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 5 / 24

BCS theory Frohlichers Idea Using the operator b † � � � b � − � q q q a † S = + M � k a � k , q + � ǫ � k − ǫ � q − � ω � ǫ � k + ǫ � q − � ω � � q q k + � k + � we make a unitary transformation H ′ = e − S He S , which yields � ω q H ′ = H 0 + q | 2 a † k + q a † � | M � k ′ − q a k a k ′ ( ǫ k ′ − ǫ k ′ − q ) 2 − ( � ω q ) 2 k , k ′ , q Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 6 / 24

BCS theory Computation shows that electrons form Cooper pairs with opposite spin Bogoljubov approach � a k , ↑ a − k , ↓ � � = 0 Condensate of Cooper pairs Effective Hamiltonian � ǫ σ 1 ,σ 2 ( k ) a † H = k ,σ 1 a k ,σ 2 + 1 � V σ 1 ,...,σ 4 ( k , k ′ ) a † k ,σ 1 a † − k ,σ 2 a k ′ ,σ 3 a − k ′ ,σ 4 2 k ,σ 1 a k ,σ 2 + 1 � ǫ σ 1 ,σ 2 ( k ) a † � ∆ σ 1 ,σ 2 ( k ) a † k ,σ 1 a † ∼ − k ,σ 2 + h . c . 2 Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 7 / 24

A calculation for later Rewrite � ǫ σ 1 ,σ 2 ( k ) � � � ∆ σ 1 ,σ 2 ( k ) a k ,σ 2 H = ( a † k ,σ 1 , a − k ,σ 1 ) a † ∆ † − ǫ T σ 1 ,σ 2 ( k ) σ 1 ,σ 2 ( − k ) − k ,σ 2 In the 2D case we can assume normalized eigenstates of the form � cos(2 α ( k x , k y )) � | u ( k x , k y ) � = , sin(2 α ( k x , k y )) where α is a certain function of ǫ ( k ) and ∆( k ). Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 8 / 24

Bogoljubov quasi particles Particle-hole duality allows diagonalization of H , yielding Bogoljubov modes UHU † = E i γ † � k ,σ γ k ,σ , with = u k a k ↑ − v k a † γ k ↑ − k ↓ = u k a k ↓ + v k a † γ k ↓ − k ↑ . Important point: Possibility for Majorana excitations γ † 0 = γ 0 . Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 9 / 24

A little bit of math What is topology? Roughly topology is the study of shapes. It is one of the ’craziest’ parts of mathematics. We are going to study vector bundles over smooth manifolds and their characteristic classes, or rather the first Chern class c 1 and the first Chern c n � number 1 . Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 10 / 24

The Berry Connection Consider the Brillouin zone of an effectively 2D solid H k x , k y ψ ( k x , k y ) k y k x Berry Connection A ( � k ) = i � ψ ( � k ψ ( � � k ) |∇ � k ) � A − → A − ∇ φ F ij = ǫ ij ∂ i A j Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 11 / 24

The Berry Connection First Chern number ν = 1 � d A = 1 � � F ǫ ij ∂ i A j dk x dk y = 2 π ∈ Z . 2 π 2 π 2DBZ 2DBZ 2DBZ Appearance of field strength F Thm: c 1 ( L ) = F / 2 π For 2D insulators σ xy = − e 2 h ν Time reversal: A ( k ) �→ A ( − k ) implies F ( k ) �→ − F ( − k ) � � � F ( k ) �→ − F ( − k ) = − F ( k ) ⇒ ν = 0 . Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 12 / 24

Closer look at time reversal Kramers rule Consider a fermion in 2+1 dimensions T ψ ( t , x 1 , x 2 ) = i σ 1 ψ ( − t , x 1 , x 2 ) ⇒ T 2 = − 1 . Further by anti-unitarity of T , i.e. � u | v � = � Tv | Tu � � u | Tu � = � T 2 u | Tu � = −� u | Tu � . Thus � u | Tu � = 0 , and we get a 2-fold degenerate energy levels. Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 13 / 24

Closer look at time reversal Now we write a basis for our Hilbert space at momentum ( k x , k y ) H k x , k y = span � u 1 ( k ) , e i φ 1 Tu 1 ( − k ) , ... � . We obtain to different Berry connections per energy level A I n ( k ) and A II n ( k ). Then � � F I n + F II 2 πν = n = ν I + ν II = 0 , n by T symmetry. However ν I = − ν II ⇒ ( − 1) ν I = ( − 1) ν II ∈ Z 2 Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 14 / 24

Topological numbers γ k y = const k y k x 1D Example Take a one-dimensional slice in the 2D Brillouin zone of the SCD example at the beginning S 1 → S 1 � cos(2 α ( k x )) � �→ k x sin(2 α ( k x )) Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 15 / 24

Winding number This map gives a winding number as � π ω = 1 � ∂ k x (2 α ( k x )) ∼ sign( d ( k ))sign( ∂ k x ǫ ( k x )) 2 π − π k x : ǫ ( k x )=0 Because the path γ is homotopy equivalent to k y = const, the corresponding gapless mode has a flat energy dispersion. Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 16 / 24

Gapless modes on the boundary ν 1 ν 2 For gapped systems and E n < E F the following map is continuous BZ − → H � �→ ψ n ( � k ) k This implies that the Chern number is constant. How then do we model discontinuities? By allowing bands to intersect at the boundary. This gives rise to gapless modes. Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 17 / 24

Gapless modes on the boundary By allowing bands to intersect at the boundary. This gives rise to gapless modes. Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 18 / 24

1D quantum wire Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 19 / 24

Thank you Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 20 / 24

Boundary Theory The boundary of a 3+1 D topological superconductor admits a Majorana fermion � ψ i D ψ d 3 x , ¯ I Euclidean = Y where D αβ = ǫ αγ D γ β . The partition function is given by the Pfaffian Z ψ = Pf (D) . In the case of � � 0 D D = , −D T 0 we have Pf (D) = det D . Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 21 / 24

Kramers rule once more There is again a Kramers rule: If D χ = λχ, χ α = ( T χ ) α = ǫ αβ χ ∗ then ˜ β is also an eigenvector D ˜ χ = λ ˜ χ. Using [ D , T ] = 0 this implies  − λ i  λ i   ⇒ Pf (D) = Π ′ λ i  D = ⊕ i   − λ i  λ i Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 22 / 24

An anomaly? Fermionic anomalies do not affect the absolute value of Z ψ = | Z ψ | e i φ Time reversal sends Z ψ �→ ¯ Z ψ Z ψ ∈ R + implies no anomalies We would like to check the sign of Pf (D) = Π ′ λ i . Consider a family of these theories differing only by gauge transformations → apply index theory of the (lifted) Dirac operator (on the mapping torus) φ ( Y ) Y Y ( g 0 , A 0 ) φ · ( g 0 , A 0 ) It turns out that the index on a mapping torus of this form is always identically zero!! Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 23 / 24

But... If we regularize the Pfaffian and then we obtain a complex partition function Z ψ = | Z ψ | exp( i π 4 η ) . But considering the interaction with some bulk theory we can restore T invariance. The result however depends on the four manifold X , Z ψ = ( − 1) I X . I is the index of a Dirac operator on a manifold with boundary. Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 24 / 24