Topological invariants for 1-dimensional superconductors

Topological invariants for 1-dimensional superconductors Eddy Ardonne Jan Budich 1306.4459 1308.soon SPORE 13 2013-07-31

Intro: Transverse field Ising model H TFI = L 1 i=0 hσ z i + σ x i σ x i+1 σ s: Pauli matrices [σ x, σ y ]=2iσ z &cyclic Quantum phase transition at h = 1 between two ordered phases.

Intro: Transverse field Ising model H TFI = L 1 i=0 hσ z i + σ x i σ x i+1 σ s: Pauli matrices [σ x, σ y ]=2iσ z &cyclic Quantum phase transition at h = 1 between two ordered phases. The model is solved by means of a non-local Jordan-Wigner transformation Define fermionic levels: = 0 = 1 σi z =1 2c i c i σi z =( 1) F i c i = σi σ z + i c i = σi z σ i j<i j<i

Intro: Transverse field Ising model In terms of the fermions, the model reads H TFI = L 1 i=0 h(2c i c i 1) + L 2 ci c i+1 + c i c i+1 + c i+1c i + c i+1 c i i=0 ( 1) F c L 1 c 0 + c L 1 c 0 + c 0c L 1 + c 0 c L 1

Intro: Transverse field Ising model In terms of the fermions, the model reads H TFI = L 1 i=0 h(2c i c i 1) + L 2 ci c i+1 + c i c i+1 + c i+1c i + c i+1 c i i=0 ( 1) F c L 1 c 0 + c L 1 c 0 + c 0c L 1 + c 0 c L 1 Versatile model, which is a starting point for many generalizations: Kitaev: fermions themselves are the fundamental degrees of freedom: This gives Kitaev s Majorana chain Generalize the Majorana s to parafermions (Fendley; Shtengel, Alicea et.al) Realization of series of critical chains: Kitaev s 16-fold way

Advertisement: 1-d critical models 3 Consider n decoupled transverse field Ising models at the critical point: L 1 H (n) TFI = i=0 σ z i + σ x i σ x i+n spin-1/2 σ x i σ x i+1 σ z j L 1 1 0 2 i +1 i

Advertisement: 1-d critical models 3 Consider n decoupled transverse field Ising models at the critical point: L 1 H (n) TFI = i=0 σ z i + σ x i σ x i+n spin-1/2 σ x i σ x i+1 σ z j L 1 1 0 2 i +1 i Inspired by 2-d condensation transitions between topological phases, we add an appropriate boundary term, followed by a non-local spin transformation. We obtain a series of spin-1/2 chains, which are Jordan-Wigner solvable, and have critical so(n) 1 behaviour (realizing Kitaev s 16-fold way)! n 1 2 3 4 5 so(n) 1 Ising u(1) 4 su(2) 2 u(1) 2 u(1) 2 so(5) 1 Månsson, Lahtinen, Suorsa, EA PRB 13 & in preparation

Kitaev s Majorana chain In terms of the (polarized) fermions c i, Kitaev s model reads H Kitaev = i µ c i c i c i c i t c i c i+1 + c i+1 c i + ci c i+1 + c i+1 c i This model belongs to class D of the classification of free fermion systems In Bogoliubov de Gennes form, the model reads: Kitaev; Ryu et al.; c.f. Altland & Zirnbauer H Kitaev (k) = µ t cos(k) τ z + sin(k)τ y τ α : Pauli matrices in particle-hole space Anti-unitary particle-hole symmetry anti-commutes with the hamiltonian. C = τ x K, C 2 =1

Periodic table of topological phases Classification of topological phases of systems with time-reversal T, particlehole C, and/or chiral symmetry U CS = T C Kitaev; Ryu et al.; c.f. Altland & Zirnbauer Time reversal and particle-hole are anti-unitary, T = U T K,C = U C K satisfying [H, T ]=0, {H, C} =0 and T 2 = ±1, C 2 = ±1

Periodic table of topological phases Class T C U CS d =1 d =2 d =3 d =4 A 0 0 0 0 Z 0 Z AIII 0 0 1 Z 0 Z 0 BDI +1 +1 1 Z 0 0 0 D 0 +1 0 Z 2 Z 0 0 DIII 1 +1 1 Z 2 Z 2 Z 0 AII 1 0 0 0 Z 2 Z 2 Z CII 1 1 1 Z 0 Z 2 Z 2 C 0 1 0 0 Z 0 Z 2 CI +1 1 1 0 0 Z 0 AI +1 0 0 0 0 0 Z Calculating the Z 2 invariants is typically hard, and requires a dimensional extension in general (Budich & Trauzettel)

Outline Discussion of the Kitaev s Majorana chain, and its Imposing additional time reversal symmetry invariant. Construction of a topological invariant for 1D superconductors in class DIII Applying the invariant to a toy-model (p & s wave pairing and spin-orbit coupling) The effect of inversion symmetry: from D to DIII and back again... Z 2

Kitaev s Majorana chain To reveal the topological nature of Kitaev s chain H Kitaev = j µ c j c j c j c j t c j c j+1 + c j+1 c j + cj c j+1 + c j+1 c j we introduce Majorana fermions: γ 2j 1 = c j + c j, γ 2j = i(c j c j ) site j {γ i, γ j } =2δ i,j γ 2j 1 γ 2j

Kitaev s Majorana chain: two phases H Kitaev = i 2 (2µ)γ 2j 1 γ 2j +(t + )γ 2j γ 2j+1 +( t + )γ 2j 1 γ 2j+2 j For and, one finds a trivial phase: µ<0 t = =0 site j H Kitaev = iµ j γ 2j 1 γ 2j = µ j (2c j c j 1) c j fermion Majorana s from the same site are paired together! The ground state is completely empty (filled for µ>0)

Kitaev s Majorana chain: two phases H Kitaev = i 2 (2µ)γ 2j 1 γ 2j +(t + )γ 2j γ 2j+1 +( t + )γ 2j 1 γ 2j+2 j For t = > 0 and µ =0, one finds a non-trivial phase: H Kitaev = it j γ 2j γ 2j+1 = t j (2 c j c j 1)

Kitaev s Majorana chain: two phases H Kitaev = i 2 (2µ)γ 2j 1 γ 2j +(t + )γ 2j γ 2j+1 +( t + )γ 2j 1 γ 2j+2 j For t = > 0 and µ =0, one finds a non-trivial phase: site j site j+1 H Kitaev = it j γ 2j γ 2j+1 = t j (2 c j c j 1) c j fermion Majorana s from neighbouring sites are paired together! The Majorana s at the end are left unpaired. They combine into one, non-local fermionic mode with zero energy, which can be either filled or empty.

The pfaffian invariant Kitaev defines a Majorana number for open chains M = ±1: M = 1 if a Majorana bound state is present, M =+1if not To define a topological invariant, we need periodic boundary conditions! Consider the fermionic parity of the ground state of a hamiltonian H on a chain of length L: P H(L) By pairing up two systems in two different ways: L 1 L 2 L 1 L 2 γ 1 γ1 γ γ2 2 γ 1 γ1 γ γ2 2 one finds that P H(L 1 + L 2 ) = M(H)P H(L 1 ) P H(L 2 )

The pfaffian invariant Kitaev defines a Majorana number for open chains M = ±1: M = 1 if a Majorana bound state is present, M =+1if not To define a topological invariant, we need periodic boundary conditions! Consider the fermionic parity of the ground state of a hamiltonian H on a chain of length L: P H(L) By pairing up two systems in two different ways: L 1 L 2 L 1 L 2 γ 1 γ1 γ γ2 2 For systems of even length: M(H) =P H(L) γ 1 γ1 γ γ2 2 one finds that P H(L 1 + L 2 ) = M(H)P H(L 1 ) P H(L 2 )

The pfaffian invariant A general Majorana chain has the form: H = i 2 where A is a real, anti-symmetric matrix. γ l A l,m γ m l,m A can be brought into Jordan form with a real, orthogonal matrix W: A J = W AW T 0 λ = diag λ, λ 0 λ > 0 The parity of the ground state of the model A J is +1 (all fermion sites empty) W preserves the parity if det(w) = +1, and changes it for det(w) = -1

Relation to the Berry phase For Majorana chains with translation invariance, the invariant simplifies: M = sgn PfÃk=0PfÃk=π =det Wk=0 det W k=π

Relation to the Berry phase For Majorana chains with translation invariance, the invariant simplifies: M = sgn PfÃk=0PfÃk=π =det Wk=0 det W k=π In terms of the phases ϕ of the determinants, on has: M =( 1) ϕ 0 ϕ π π π Thus, M is a phase winding: ϕ 0 ϕ π = i k log det W (k) dk 0 A calculation of the Berry phase in the Bogoliubov de Gennes picture gives exactly the same result. Kitaev s pfaffian invariant can be seen as Berry phase (charge polarization)! Budich, EA

What are Majorana bound states good for? The topological phase can be found in nano-wires with SO interaction, deposited on an s-wave superconductor. Lytchin et al., Oreg et al. Several experimental groups have observed signatures consistent with the zero energy Majorana bound states. Mourik et al., Deng et al., Das et al. The Majorana bound states are non-abelian anyons, which might be used for topological quantum computation! How does one braid particles in one dimension? Use T-junctions! Alicea et al.

Can we introduce more degrees of freedom? If we take two spin copies of Kitaev s chain, one might be able to perform additional gates!

Can we introduce more degrees of freedom? If we take two spin copies of Kitaev s chain, one might be able to perform additional gates! Generically, a pair of Majorana bound states at the edge will hybridize, but if time-reversal symmetry is preserved, each half of the Kramers pair has nonabelian statistics. Liu et al.

1-D superconductors in class DIII In class DIII, we have both time reversal and particle-hole symmetry: T 2 = 1, C 2 =+1, represented via T = iσ y K, C = τ x K σ α : Pauli matrices in spin space τ α : Pauli matrices in particle-hole space Simple case: take a system h in class D, and construct its time-reversal conjugate (i.e., we have a spin-quantization axis) H = h 0 0 h Both copies have the same value of Kitaev s invariant M(h) =M(h )=±1

1-D topological invariant for class DIII Due to time-reversal symmetry, the Bloch bands come in pairs labeled by a Kramers index κ = I,II: u I α(k), u II α (k) At the time-reversal symmetric (real) momenta, their energies are equal (Kramers theorem).

1-D topological invariant for class DIII We consider the (log of the) Berry phase (i.e., the polarization) of the occupied band I: P I o = 1 2π 2π 0 dka I o(k) A I o(k) = iu I o(k) k u I o(k) Berry connection Time reversal symmetry relates the Berry connection of band I and II: A I o( k) =A II o (k) k χ α (k) It follows that and can only differ by an integer. P I o Po II Following Fu & Kane, 2006

1-D topological invariant for class DIII The polarization can be written as an integral over half the Brillouin zone Fu & Kane, 2006 Po I = 1 π Pf θo (π) A o (k)dk + i log 2π Pf θ o (0) 0 A o (k) =A I o(k)+a II o (k) θo (k) κ,κ = u κ o(k) T u κ o (k) In class AII, i.e., only time-reversal symmetry, Po I is not quantized. In class DIII, however, the additional particle-hole symmetry implies the quantization of, in half-integers! P I o The invariant in class DIII can finally be written as ν =e i2πp I o =e i π 0 dka o(k) Pf θ o (0) Pf θ o (π) = ±1

Example in class DIII As an example, we construct a non-trivial model in class DIII. We first take two time reversal conjugate copies of a Kitaev chain: H(k) = µ cos(k) σ 0 τ z + p sin(k) σ 0 τ y σ α : Pauli matrices in spin space τ α : Pauli matrices in particle-hole space We add a Rashba term, as well as an s-wave pairing, preserving both timereversal and particle-hole symmetry. There is no spin quantization axis! H(k) = µ cos(k) σ 0 τ z + p sin(k) σ 0 τ y + α sin(k) σ x τ z + s σ y τ y The topological phase competes with both the s-wave pairing, as well as the Rashba spin-orbit coupling.

Example in class DIII We can determine the phase diagram, by calculating the topological invariant for various parameters in the model. For µ = 0.5 and p =1.0, we obtain

Example in class DIII We can determine the phase diagram, by calculating the topological invariant for various parameters in the model. For µ = 0.5 and p =1.0, we obtain The purple region is the topolgically trivial phase, the yellow the non-trival phase. For large α > 1.0 and s =0, there is a metallic phase.

Example model in class DIII The general structure of phase diagram reveals that both the spin-orbit coupling as well as the s-wave paring compete with the topological phase, which can only occur in the region 1 <µ<1, as for the Kitaev chain.

DIII invariant with inversion symmetry In systems which also exhibit inversion symmetry, invariant simplifies. x x the topological Inversion symmetry is implemented by a unitary operator satisfying P inv H(k)P inv = H( k) and [P inv, T ]=0 The eigenvalues of P inv are ξ i = ±1. ν =e i2πp I o =e i π 0 dka o(k) Pf θ o (0) Pf θ o (π) P inv with P 2 = ±1 inv =1 The Berry connection can be related to the pfaffian of the matrix elements of, and be set to zero in an appropriate gauge. P inv T Following Fu & Kane, 2007

DIII invariant with inversion symmetry In systems which also exhibit inversion symmetry, invariant simplifies. x x the topological Inversion symmetry is implemented by a unitary operator satisfying P inv H(k)P inv = H( k) and [P inv, T ]=0 The eigenvalues of P inv are ξ i = ±1. ν =e i2πp I o =e i π 0 dka o(k) Pf θ o (0) Pf θ o (π) P inv with P 2 = ±1 The Berry connection can be related to the pfaffian of the matrix elements of, and be set to zero in an appropriate gauge. P inv T The matrix element of T, i.e. θ o (0) and θ o (π) are given in terms of the eigenvalues ξ i = ±1 Following Fu & Kane, 2007 inv =1

DIII invariant with inversion symmetry In the end of the day, the invariant is given by ν = α:occ ξ α (0)ξ α (π) where the product is over all occupied bands, such that each Kramers pair contribute once to the product (both partners have the same eigenvalue due to time-reversal symmetry). Inversion symmetry simplifies the invariant, only knowledge about the real momenta is necessary to determine if the system is in a topological phase!

D invariant with inversion symmetry The invariant in the DIII case with inversion: ν = α:occ ξ α (0)ξ α (π) where the product is over all occupied bands, such that each Kramers pair contribute once to the product (both partners have the same eigenvalue due to time-reversal symmetry). h 0 A D invariant is obtained by going to class DIII: H = 0 h and take the product over the bands of h, which is in class D, so the bands are not Kramers degenerate: ν = ξ α (0)ξ α (π) α:occ

Conclusion We reinterpreted Kitaev s Majorana invariant in terms of a Zak-Berry phase (charge polarization) We constructed the topological invariant of 1D superconductors in class DIII We used the invariant to study a toy-model with p- & s-wave pairing and spin-orbit coupling We considered the effect of inversion symmetry, simplifying the DIII invariant and giving a simple invariant for class D