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

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Introduction to the Physics of Anyons with Majorana Fermions As an Example 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 t dynamical phase d 1 2 Berry phase i dt | (path) 1 2 dt g (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 1 2 R12, f R12 , f time t R12,i 1 2 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 deformedR12 , f R12,i to a point R12 const ( R12,i ) (i.e., to the case when nothing happens) without crossing the origin R12 0 0 0 Conclusion: in 3D only bosons ( 0 ) or fermions ( ) 2D case: ˆ 2 1 C The contour C cannot be deformed 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 0 Conclusion: in 2D more possibilities, not only bosons or fermions Braid group. Abelian and non-Abelian anyons Particle exchangeˆ 2 in1 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) ˆiˆ j ˆ jˆi for i j 2 Ri1 Ri Ri1 Ri2 ˆiˆi1ˆi ˆi1ˆiˆi1 time t Ri1 Ri Ri1 Ri2 Note: 1. Braid group is infinite dimensional ( !) in contrast to finite-dimensional permutation group (p ˆ 2 1 ) 2. Braid group is non-Abelian ˆiˆi1 ˆi1ˆi 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 j1 1 (Z , zi ) (Z Z ) e (Z zi )(zk zl ) e M 1 i1 kl normalization quasihole Laughlin w.f. N d 2 z | (Z , z ) |2 1 O(e Z Z ) i 1 i i1 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 j1 1 (Z , zi ) Z Z e (Z zi )(zk zl ) e M 1 i1 kl 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 c “Observer” a b c c - integers ab Nab c Nab c c a r Non-Abelian anyons if c for some a and b l Nab 1 c b ( 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 b c e abc e' Fd ee' e' d d a,b fuse to e; b,c fuse to e’; a 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 ab Rc a b c c R-matrix (braiding in a given fusion channel) Consistency conditions for F- and R-matrices a b c b c e abc e' Fd ee' e' ab d d Rc a b 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. 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 1 or or or or or1 1{1, , } {1, , } Equivalent to the Hilbert space for fermionic modes End of the Lecture 1 Thank you for your attention!.
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