IQOQI UNIVERSITY OF INNSBRUCK AUSTRIAN ACADEMY OF SCIENCES

Introduction to the Physics of Anyons with Majorana Fermions as an Example

M. Baranov

Institute for Quantum Optics and Quantum Information, Center for Quantum Physics, University of Innsbruck

Anyon Physics of Ultracold Atomic Gases Kaiserslautern 10 – 15 December 2018 Lecture 1: Appearance of Anyons

Lecture 2: Majorana Fermions as an example of non-Abelian Anyons Lecture 1: Appearance of Anyons

Exchange and statistics

Why dimension 2?

Braid group. Abelian and no-Abelian anyons

Formal introduction of anyons: Fusion of anyons and anyon Hilbert space Exchange and statistics Particle exchange and statistics

Behavior of the state (wave function) Statistics under the exchange of two identical (quasi)particles

Exchange of two (quasi)particles

1 2

      (r1,r2 ,)  (r2 ,r1,)  ? (r1,r2 ,) Properties of many-body wave functions:

     (R1, R2 ,;r1,r2 ,)  Ri positions of quasiparticles  rj positions of particles

 has to be single-valued with respect to particle coordinates rj  but not necessarily with respect to quasiparticle coordinates Ri Exchange as adiabatic dynamical evolution General statements:

1. Adiabatic theorem: States in a (possibly degenerate) energy subspace separated from others by a gap remain in the subspace when the system is changed adiabatically without closing the gap.

2. Change under adiabatic transport = combination of Berry’s phase/matrix and transformation of instantaneous energy eigenstate (explicit monodromy)

3. Change under adiabatic transport is invariant, but Berry’s phase/matrix and eigenstate transformation depend on choice of gauge (and can be shifted from one to the other) Exchange as adiabatic dynamical evolution

For a unique ground state  (single-valued) 2 1 separated by a gap from excited states

i 1   e  ,     dt E(t)  time time t  dynamical phase d 1 2 Berry phase   i  dt  |    (path)  dt g 1 2 (single-valued w.f.)

 g (path) geometrical phase

 (topology of the path) statistical angle - of interest! i Exchange statistics   e  Why dimension 2?  Constraint on the exchange R12 relative position 2 exchange ˆ double exchange ˆ    R R12 , f

2 12, f 1 time time t

1  2 R12,i

 R 0 12,i

  R  R 12, f 12,i   R12, f  R12,i

Is  ˆ 2 an identity? 3D case: ˆ 2 1 (identity!) C The contour C can be deformed    0 R12,i to a point R12  const ( R12,i )  R (i.e., to the case when nothing happens) 12 , f  without crossing the origin R12  0

0

Conclusion: in 3D only bosons (   0 ) or fermions (    )     2D case: ˆ 2 1 C The contour C cannot be deformed    0 R12,i to a point R12  const ( R12,i )  R12, f (i.e., to the case when nothing happens)  without crossing the origin R12  0

0

Conclusion: in 2D more possibilities, not only bosons or fermions Braid group. Abelian and non-Abelian anyons Particle exchange in 2D: Braid group (for N particles)   Trajectories that wind around starting from initial positions   R1,, RN to final positions R 1 ,  , R N (the same set – identical particles)

generated by ˆi defining relations (braiding of particles i and i+1)

ˆiˆ j  ˆ jˆi for i  j  2     Ri1 Ri Ri1 Ri2

ˆiˆi1ˆi  ˆi1ˆiˆi1 

time time t      Ri1 Ri Ri1 Ri2

Note: 1. Braid group is infinite dimensional ( ˆ 2  1 !) in contrast to finite-dimensional permutation group (p ˆ 2  1 )

2. Braid group is non-Abelian ˆiˆi1  ˆi1ˆi Representations of the braid group: statistics of particles

Elements of the braid group (trajectories of particles)

representation

Changes of the states under the evolution (particle statistics)

1. One-dimensional representations: unique (ground) state

2. Higher-dimensional representations: degenerate (ground) state 1. One-dimensional (Abelian) representations

Unique (ground) state 

Transformation under braiding operation ˆ  ˆ ei 

with arbitrary  - Abelian anyons

Examples: 1. bosons (   0 ) and fermions (   )

2. quasiholes in the at FQHE Laughlin state  1/ M    / M Example 2. quasiholes in the FQHE Laughlin state  1/ M  R. Laughlin, 1983 Trial wave function for N fermions at positions ri with n quasiholes at positions R  : z j  (x j  iy j ) / lB Z  (X   iY ) / lB Choice 1 n N 1 2 1 2 n 1 Z n N N  z 4M   4 j M 1 M j1  1 (Z , zi )  (Z  Z  ) e (Z  zi )(zk  zl ) e M    1 i1 kl

normalization quasihole Laughlin w.f.

N d 2 z | (Z , z ) |2 1 O(e Z Z )  i 1  i i1 M

e  B Berry’s phase = Aharonov-Bohm phase (geometric)  g (path)  of a charge encircling flux of M c q  e / M  B  BA D. Arovas, J.R. Schrieffer,F.Wilczek,1984 Exchanging two quasiholes give a phase of    / M R. Laughlin, 1987 from eigenstate transformation (explicit monodromy) B. Blok, X.G. Wen, 1992 Example 2. quasiholes in the FQHE Laughlin state  1/ M

Choice 2 (different “gauge”: single-valued )

n N 1 2 1 2 n 1 Z n N N  z 4M   4 j M 1 M j1  1 (Z , zi )   Z  Z  e (Z  zi )(zk  zl ) e M    1 i1 kl

Eigenstate transformation (analytic continuation) = trivial

Berry’s phase = Aharonov-Bohm phase + statistical angle    / M 2. Higher-dimensional representations excited states Degenerate ground state  ,  1,, g with an orthonormal basis  gap 

  Transformation under braiding operation ˆ

ˆ   U (ˆ)   d with matrix 푈 휎 = 푃푒푥푝(푖 푑푡푚 )ℳ (mˆ )  i  |    dt  Berry matrix explicit monodromy of the w.f.

Particles are non-Abelian anyons if U (ˆ1) U (ˆ 2 )  U (ˆ 2 ) U (ˆ1) for at least two  ˆ 1 and ˆ 2 (do not commute!)

Examples: Ising anyons (Majorana fermions, 휈 = 5 / 2 qH-state), Fibonacci anyons Conditions for non-Abelian anyons:

Robust degeneracy of the ground state:

The degeneracy cannot be lifted by local perturbations (which are needed, i.e., for braiding)

Degenerate ground states cannot be distinguished by local measurements

  Vloc    C

Braiding of identical particles changes state within the degenerate manifold, but should not be visible for local observer Nonlocal measurements: parity measurements (for Majorana fermions), etc.

Require topological states of matter with topological degeneracy and long-range entanglement Comment: In real world (finite systems, etc.): GS degeneracy is lifted

excited states

gap 

  ~   

  Condition on the time of operations:  T   

slow enough to be adiabatic

fast enough to NOT resolve the GS manifold Formal introduction of Anyons:

Fusion rules and Hilbert space Set of particles (anyons) 1, a, b, c 1 - vacuum

Fusion of anyons

Fusion of two Anyons a , b :

how do they behave as a combined object seen from distances much large than the separation between them r  l

? a r l “Observer” b Fusion rules

a c r l “Observer” b a b

c c - integers ab   Nab c Nab c c

Non-Abelian anyons if c for some a and b  Nab 1 c ( c for simplicity) Hilbert space – fusion chains Nab 1 (no creation and annihilations operators!)

ℋ푛 : n anyons, 푎 1 , ⋯ , 푎 푛 where 푎 1 , ⋯ , 푎 푛 − 1 fuse into 푎 푛 (⋯ (푎1× 푎2) × 푎3) × ⋯ ) × 푎푛−1 = 푎푛

Basis vectors

푎2 푎3 푎4 푎푛−2 푎푛−1 ⋯ 푎1 푎푛 푒1 푒2 푒푛−3

푎1 and 푎 2 fuse into 푒 1 , 푒1 and 푎 3 into 푒 2 , …, 푒푛−3 and 푎 푛 − 1 into 푒 푛 uniquely specified by the intermediate fusion outcomes 푒1, ⋯ , 푒푛−3

dim ℋ = 푁푒1 푁푒푛 Dimension 푛 푎1푎2⋯ 푒푛−3푎푛−1 푒1…푒푛−3 Associativity of braiding: 푎 × 푏 × 푐 = 푎 × 푏 × 푐 = 푑

a b c a b c

e abc e'  Fd ee' e' d d a,b fuse to e; b,c fuse to e’; e,c fuse to d e’,a fuse to d

F-matrix (basis change) guaranties the invariance of the Hilbert space construction: Different fusion ordering is equivalent to the change of the basis vectors Exchange properties of anyons (in a given fusion channel)

a b a b

ab  Rc

c c

R-matrix (braiding in a given fusion channel) Consistency conditions for F- and R-matrices

a b c a b c b

e  F abc e' ab  d ee'  Rc e' d d c c

F- and R-matrices satisfy the pentagon and hexagon consistency relations Pentagon relation

12c a34 234 1e4 123 F5 da F5 cb  Fd ce F5 db Fb ea e Hexagon relation

231 1b 123 13 213 12 F4 cd R4 F4 ba  Rc F4 ca Ra b Set of anyons with F- and R-matrices satisfying the pentagon and hexagon equations completely determine an Anyon model.

If no solution exists the hypothetical set of anyons and fusion rule are incompatible with local quantum physics.

Alternative approach: topological field theories Examples:

1: Fibonacci anyons 1,

Fusion rules:   1 1{1,}  {1,}

2: Ising anyons 1,  ,

 - non-Abelian anyon  - fermion

Fusion rules:   1       1 1{1, , }  {1, , }

(more in the next lecture) Hilbert space for 2 N Majorana fields  m ( m  1 ,  , 2 N ) Fusion rules:   1       1 1{1, , }  {1, , }

1. Split in N pairs 훾2푗−1, 훾2푗

2. Specify fusion channels ( 1 ,  ) j for each pair ( j 1,, N ) The fusion tree is now uniquely define. The resulting “charge” is either 1 or  depending on the number 푛 휓 of the  fusion outputs:

1 for even 푛휓 for odd 푛휓

훾1 훾2 훾3 훾4 훾5 훾6 훾2푁−1 훾 … 2푁 or or or or or or or

Equivalent to the Hilbert space for fermionic modes End of the Lecture 1

Thank you for your attention!