Topological Superconductivity

Materials Theory

6th MaNEP winter school, Saas Fee, January 2015 Uppsala

Open PD and PhD positions!

Ångström laboratory Contents • Introduction to superconductivity – BCS theory – BdG formulation • Unconventional superconductivity • Topological matter • Topological superconductivity – Chiral superconductors – “Spinless” superconductors – Majorana fermions Introduction to Superconductivity

What is it?

How do we describe it? What is Superconductivity? • What is superconductivity? – Electric transport without resistance – Meissner effect

Superconductor

Magnet History of Superconductivity

Superconductivity in Hg:

Heike Kammerlingh Onnes

Tc Superconductors

Department of Energy Nobel Prizes How? • But how do the electrons move without resistance? – The electrons are all in a coherent quantum state with a fixed phase (condensate) i' = 0e

– One of the few macroscopic manifestations of QM – Spontaneous gauge symmetry breaking – Higgs mechanism (discovered by P. W. Anderson before Higgs et al.) Higgs Mechanism • In the Standard Model: – Interactions with the Higgs field (Higgs bosons) give masses to the W+,- and Z bosons due to electroweak (gauge) symmetry breaking

• In superconductors: – The superconducting pairs give mass to the electromagnetic field (photons) due to spontaneous symmetry breaking of the gauge symmetry (fixed phase on Ψ) – Explored by P.W. Anderson for superconductivity 2 years before Higgs et al.

Meissner effect = The “Higgs mass” of the electromagnetic field expels it from the superconductor But Wait! • All electrons in one state: a superconducting condensate – Pauli exclusion principle??

How do electrons end up in the same state?

Statistics • Fermions (half-integer spin): – Electrons, quarks, neutrinos – Only one electron per quantum state • Bosons (integer spin): – Photons, gauge, and Higgs boson – As many particles as you wish in one quantum state (Bose-Einstein condensate, BEC)

Maybe two electrons can form an electron pair!? Sort of, but not the whole story…. Electron Pairing • Electrons are negatively charged, how can they possibly pair? – In fact, they can pair pretty easily (at least at very, very low temperatures)

• The only instability of a Fermi Liquid is to attractive interactions – Any attractive interaction destroys the Fermi Liquid and creates electron pairs: Cooper pairs Attractive Interaction • Where is attractive interaction coming from?

Normal state At Tc In the superconducting state

Electron-lattice vibration (phonon) interactions gives an effective attractive potential between two electrons Conventional Superconductors

• Superconductivity from electron-phonon interactions – Superconductivity in “normal” metals

– Low temperatures, Tc < 40 K BCS Theory • How do the Cooper pairs condense (superconduct)?

• Bardeen-Cooper-Schrieffer (BCS) theory: – Condensation of Cooper pairs (with a fermionic wave function) – Truly many-body state (not a two-body bound state condensing) (Higher temperatures than BECs) Cooper Pairs

Pair of electrons above the Fermi surface (FS) with no net momentum:

Fourier transform ( ):

spatial property of the Cooper pair attractive For the energy of a Cooper pair is < 0

! FS is unstable to Cooper pair formation for any V<0 (see e.g. Tinkham: Introduction to superconductivity) BCS Wave Function

Coherent state of Cooper pairs (ansatz):

with

• Ill-defined number of particles:

phase of order parameter: fixed (but same energy for all phases)

• Breaks gauge invariance: BCS Hamiltonian

Pairing Hamiltonian:

kinetic (band) energy

We can determine uk, vk in ψBCS by minimizing <ψBCS|H-µN|ψBCS> but it is a bit clumsy (see e.g. Tinkham: Introduction to superconductivity)

Instead we use mean-field theory with the order parameter

pair amplitude at k (off-diagonal long-range order)

Mean-Field Treatment

ignore fluctuations (mean-field approximation)

Set the order parameter

Matrix Formulation (BdG)

Define the Nambu spinor

TRS: ε = ε k k constant

Bogliubov -de Gennes (BdG) formulation: 2x2 matrix problem ! Solve by finding eigenvalues and vectors

! ( )

Eigenstates = Quasiparticles

QP energy (eigenvalue):

Bogoliubov QP operators (eigenvector): tranformation

(Same coefficients as

in the BCS wave fcn)

Diagonal Hamiltonian:

QP excitations Condensation energy

Quasiparticle Properties

• Excitations out of the condensate, always positive energy

Band structure Density of states (DOS)

P. Coleman: Introduction to Many Body Physics Quasiparticle Properties II

• Excitation

creates electron with momentum k and spin up, destroys electron with momentum –k and spin down • At E = 0: ! QP is equal parts electron and hole

! Can become its own antiparticle (Majorana fermion)

• annihilates ground state puts 1 electron in state –k, and prevents a Cooper pair at

momentum k

Additional formulas

January 8, 2015

u 2 + v 2 =1 | k| | k|

Fk = c k ck h # "i

u 2 + v 2 =1 | k| | k|

1 u 2 = v 2 = | k| | k| 2

ak BCS =0 "| i † † † † a k BCS = c k (uk0 + vk0 ck c k ) 0 #| i # 0" 0# | i k =k Y06

E /kB T 1 a† ak =(1+e k0 ) h k0 0 i † † † † † † Vk,k0 ck c k c k0 ck0 = Vk,k0 [(ck c k Fk)+Fk][(c k0 ck0 Fk0 )+Fk0 " # # " " # # " ⇡ Xk,k0 Self-ConsistentXk,k0 Equation † † † † (c c F )(cc F ) Vk,k0 (Fk0 ck c k + Fkc k0 ck0 FkFk0 ] † † † " # # " Xk,k0 Having found the diagonal Hamiltonian we can determine the superconducting order parameter self-consistently: = V F = V c c k k,k0 k0 kE,k0/kB Tk0 1k0 a† a =(1+e k0 h )# "i k = Vk,k c k ck = Vk,k uk v⇤ a† akk k+0a† ak 1 0 0 0 0 0 k0 kk0h0 00 i k0 k0 0 h # "i h X" " X# # i k,k0 k,k0 X X Fermi-Dirac E /kB T 1 a† ak =(1+1 e k0 )k distribution k0 0 0 h k i=(c†c† F †V)(kcc,k0 F ) 2 Ek k 0 1 X0

(c c F )(cc F ) 1 † † † = V F k=0 V c c k =k Vkk,k,k0 k0 tanh[Ek,kk0/(2k0BTk)]0 0 0 h # "i 2 k0 Ek0 k0 Xk0 X X

k = Vk,k Fk = Vk,k c k ck 0 0 0 0# 0" 1 h k0 i At T =k0 0: = k0 V X Hk =(X k) kzˆ,k0 2⇥ · Ek0 Xk0 1 = V k0 k 2 k,k0 E 1 k0 k0 = H = Vvk0 ( ktanh[) zˆE /(2k T )] k ~XkF,k0 k0 B 2 E⇥k0 · Xk0

1 k0 k = Vk,k0 tanh[Ek0 /(2kBT )] 2 Ek k UHc⇢(=(E0 F)=1k) zˆ X0 ⇥ ·

H =( k) zˆ ⇥ · HVpot= ~=vFV((r)k) zˆ ⇥ ·

H = ~vF ( k) zˆ ⇥ 1 · EUc⇢(EF )=1 res / V

Uc⇢(EF )=1 Vpot = V (r) !1

V = V (r) pot 1 Eres2 / V 1 E res / V V !1

V !12

2 Summary BCS Theory

BCS wave function: BCS = (uk + vkck† c† k ) 0 | i " # | i k Y Cooper pair

Hamiltonian with kinetic energy εk and a pairing potential Vkl: Quasiparticle energy: E = "2 + 2 k k | k| Energy gap = q order parameter

Quasiparticles:

Order parameter (Ψ before): Introduction to Superconductivity

What is it? A charged superfluid of Cooper pairs (2 electrons) with fermionic character

Cooper pairs formed by effective attractive interaction

How do we describe it? BCS theory for the condensation

BdG matrix formalism Unconventional Superconductivity

What is unconventional?

How do we handle unconventional superconductivity? Conventional Superconductors

• Superconductivity from electron-phonon interactions – Superconductivity in “normal” metals

– Low temperatures, Tc < 40 K 1 l – Isotropic pairing ( k = V kl = Δ) 2 El Xl Superconducting Symmetries Superconducting order parameter is fermionic i.e. odd under particle exchange:

(k)= ( k) ↵ ↵ i' ↵(k)=e ⌘(k)↵ orbital spin

! η even function in k

! η odd function in k Spatial Symmetries

y • Conventional superconductors: x – Spin-singlet, s-wave (η k-independent)

y • Cuprate (high-Tc) superconductors: 2 2 – Spin-singlet, d-wave (η = kx - ky )

UNCONVENTIONAL • Exotic superconductors:

3 i – Spin-triplet, p-wave ( He, Sr2RuO4?) x – Topological “spinless” superconductors kx + ik y with Majorana fermions i Superconducting Symmetries

Vk,k’ (and the band structure) determines the pairing symmetry, but it often very hard to determine • Lattice fluctations: spin-singlet s-wave • Antiferromagnetic spin fluctuations: spin-singlet d-wave (extended s-wave) • Ferromagnetic spin fluctuations: spin-triplet p-wave • Strong on-site repulsion (Heisenberg interaction): spin-singlet d-wave

Is there a way to determine the possible pairing symmetries in a material without knowledge of Vk,k’?

Yes, we can do a general symmetry group analysis See e.g. Sigrist and Ueda, RMP 63, 239 (1991) General Pairing Hamiltonian

General pairing Hamiltonian:

Mean-field order parameter:

E /kB T 1 a† ak =(1+e k0 ) h k0 0 i

(c†c† F †)(cc F )

k = Vk,k Fk = Vk,k c k ck 0 0 0 h 0# 0"i Xk0 Xk0

E /kB T 1 a† ak =(1+e k0 ) h k0 0 i 1 k0 k = Vk,k0 2 Ek /kBET 1 (c ca† aFk )(cc=(1+F ) e 0 )k0 † †k0 †0 k0 h i E /kB T 1 a† ak =(1+X e k0 ) h k0 0 i

k = Vk,k Fk (c=†c† VFk†,k)(ccc k Fck) 1 0 0 k 0 h 0# 0"i XkMatrix0 (c†cXk† 0 F †0)(Formulationcc F ) k = Vk,k0 tanh[ Ek0 /(2kBT )] 2 Ek0 k0 1 k k == Vk,k FVk = 0 Vk,k c k ck T Xk 0 k0,k0 0 h 0# 0"i 4-componentk notation2 Ek k0(Nambu): k =X0 Vkk,0k Fk =X0 Vk,k c k ck X0 0 0 h 0# 0"i Xk0 Xk0 1 1 k0 ˆ k0 = V "k(k=)tanh[0 E V/k(2,kk(kT )]) ˜k 2 k,k0 E 2 k0 0 EB ! = a† k0 1 kk0 ak kXk0 ˆ = k0 V 0 H k †(k)X k",k(0 k)0 2 Ek ✓ 0 ◆ Xk0 1 ˆ ˜ "(k)0 (kk)0 =akk†= Vk,k0 tanh[akEk0 /(2kBT )] H ˆ2 (k) "(Ekk) 1† k 0 0 ψ even ✓ X0 k0 ◆ Spin-singletk = pairing:Vk,k 0 tanh[Ek0 /(2kBT )] 2 Ek0 function of k H =(k0 k) zˆ X ˆ (k)[ck† c† k "(kck†)⇥c0† k ](·k) ˜ = "a† # " # a H k ˆ (k) "(k) k ✓ "(†k) ˆ (k)0 ◆ ˜ 0 = ak† ak H ˆ †(k) "(k)0 dz(k)[ck† c† ✓k + ck† c† k ] ◆ " # " # Spin-triplet pairing: (k)[ck† c † k ck† c† k ] H = ~vF" (# k" ) #zˆ d vectorial odd (k)[ck† c† k ⇥ck† c† ·k ] " # " # function of k [ dx(k)+idy(k)]ck† c† k " " dz(k)[ck† c† k + ck† c† k ] " # " # [d (k)+id (k)]c† c† mz =x 0: dz(ky)[ck† ck† k k+ ck† c† k ] " # # # " # Uc⇢(EF )=1 mz = 1: [ dx(k)+idy(k)]ck† c† k " " H =( k) zˆ [ dx(⇥k)+·idy(k)]ck† c† k " " [dx(k)+idy(k)]ck† c† k # # [d (k)+id (k)]c† c† H = ~vFx ( k) y zˆ k k Vpot ⇥= V· (r#) # H =( k) zˆ ⇥ · 2

H =( k) zˆ ⇥ ·

H = ~vF ( k) zˆ ⇥1 · Eres H = ~v/F (V k) zˆ 2 ⇥ ·

2 2 E /kB T 1 a† ak =(1+e k0 ) h k0 0 i

E /kB T 1 a† ak =(1+e k0 ) h k0 0 i (c†c† F †)(cc F )

(c†c† F †)(cc F ) k = Vk,k Fk = Vk,k c k ck 0 0 0 h 0# 0"i Xk0 Xk0 k = Vk,k Fk = Vk,k c k ck 0 0 0 h 0# 0"i k0 1 k0 X X k0 k = Vk,k0 2 Ek0 Xk0 1 k0 k = Vk,k0 2 Ek0 1 k0 X k0 k = Vk,k0 tanh[Ek0 /(2kBT )] 2 Ek0 Xk0 1 k0 k = Vk,k0 tanh[Ek0 /(2kBT )] 2 Ek0 k0 X "(k)0 ˆ (k) ˜ = a† a H k ˆ (k) "(k) k ✓ † 0 ◆ "(k)0 ˆ (k) ˜ = a† a H k ˆ (k) "(k) k ✓ † 0 ◆ (k)[ck† c† k ck† c† k ] " # " #

(k)[ck† c† k ck† c† k ] " # " # dz(k)[ck† c† k + ck† c† k ] " # " #

dz(k)[ck† c† k + ck† c† k ] " # " # [ dx(k)+idy(k)]ck† c† k " "

[ dx(k)+idy(k)]ck† c† k " " [dx(k)+idy(k)]ck† c† k # #

p 2 2 Ek = "(k) + d(k) q(k) | 2 | ± | Finite q = non-unitary p Self-consistency: 2

(unitary states)

Linearize close to Tc:

Linear eigenvalue equation

Solution

• The largest eigenvalue gives Tc • The eigenfunction spaces (Δ) form a basis of an irreducible representation (irrep) of the symmetry group of the equation (V)

! Possible SC symmetries belong to irreps of the symmetry group of H ! SC state always breaks U(1) but can also break – Crystal lattice symmetry – Spin-rotation symmetry – Time-reversal symmetry

Below Tc: – 2nd SC transition from same V ! new Δ belongs to same irrep – 2nd SC transition from other V

Basis gap functions: D4h

• D4h = tetragonal symmetry (cuprates with kz = 0)

Spin-singlet s-wave, extended s-wave d(x2-y2)-wave d(xy)-wave

Spin-triplet

p(x)- and p(y)-wave degenerate

Sigrist and Ueda, RMP 63, 239 (1991) Basis gap functions: D6h

• D6h = hexagonal symmetry (graphene, Bi2Se3 TIs with kz = 0,)

Spin-singlet s-wave, extended s-wave

d(x2-y2)-wave and d(xy)-wave degenerate Spin-triplet

Sigrist and Ueda, RMP 63, 239 (1991) Multiple Order Parameters The superconducting state especially interesting if it has multiple components

• Two-dimensional irreps gives Δ1+iΔ2 – singlet d(x2-y2)+id(xy)-wave for hexagonal lattices (graphene?)

– triplet (mz = 0) p(x)+ip(y)-wave for square lattices (Sr2RuO4) Break time-reversal symmetry (TRS) Topological superconductors

• Subdominant pairing – singlet d(x2-y2)+is-wave (cuprates with small s-wave component) Breaks TRS but not a topological superconductor Unconventional Superconductivity

What is unconventional? Any SC state which is does not have spin-singlet s-wave order

How do we handle unconventional superconductivity? Generalized formalism using a 4 x 4 BdG equation