Theory of non-Abelian statistics: fusion space of topo. exc. What - PowerPoint PPT Presentation

Theory of non-Abelian statistics: fusion space of topo. exc. What are the most general properties of the topological excitations? can be boson, can be fermion, can be semion, ... Consider a state with quasiparticles | i 1 , i 2 , i 3 , · · · � at � x 1 , � x 2 , � x 3 , · · · , which is a gapped ground state of H + δ H trap x 1 ) + δ H trap x 2 ) + δ H trap ( � ( � ( � x 3 ) + · · · i 1 i 2 i 3 • The ground state subspace of the above Hamiltonian is the fusion space V F ( i 1 , i 2 , i 3 , · · · ) of the quasiparticles i 1 , i 2 , i 3 , · · · . • We assume the above ground state degeneracy is stable arbitary purterbations around � x 1 , � x 2 , � x 3 , · · · and the traped quasiparticles are said to be simple . • If the ground state subspace is not stable against any perturbations δ H ( � x 1 ) near � x 1 , then the quasiparticle i 1 at � x 1 is composite . • If i 1 is composite, we can add δ H ( � x 1 ) to split the ground state subspace: V F ( i 1 , i 2 , i 3 , · · · ) → V F ( j 1 , i 2 , i 3 , · · · ) ⊕ V F ( k 1 , i 2 , i 3 , · · · ) ⊕ · · · We denote i 1 = j 1 ⊕ k 1 ⊕ · · · . Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category

Fusion algebra of (non-Abelian) topological excitations • For simple i , j , if we view ( i , j ) as one particle, it may correspond to a composite particle: k V F (˜ V F ( i , j , l 1 , l 2 , · · · ) = ⊕ ˜ k , l 1 , l 2 , · · · ) N ij k =1 V F = ⊕ k ⊕ k k ( k , l 1 , l 2 , · · · ) α ij α ij (k , ..) 2 i ⊗ j = ⊕ k N ij (k , ..) k k → the fusion algebra . (i,j,...) 1 Associativity : = � = � ( i ⊗ j ) ⊗ k = i ⊗ ( j ⊗ k ) = ⊕ l N ijk N ijk m N ij n N jk m N mk n N in l l , l l l Quantum dimension and vector space fractionalization: • In general, we cannot view V F ( i , j , k , · · · ) as V ( i ) ⊗ V ( j ) ⊗ V ( k ) ⊗ · · · , and dim[ V F ( i , i , i , · · · )] � = d n i , d i ∈ Z . Quasiparticle i may carry fractional degree freedom. dim[ V F ( i , i , · · · , i )] = � m 2 · · · N m n − 2 i = ( N i ) n − 1 m i N ii m 1 N m 1 i ∼ d n 1 i 1 i where the matrix ( N i ) jk = N ji k , and d i the largest eigenvalue of N i . • d i is called the quantum dimension of the quasiparticle i . Abelian particle → d i = 1. Non-Abelian particle → d i � = 1. Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category

Relation between fusion spaces and the F -matrix • Two di ff erent ways to fuse i , j , k → l : i j k i j k F α in V F ( i , j , k , · · · ) = ⊕ m ⊕ N ij m n ij m =1 V F l m m ( m , k , · · · ) α m mk jk α ij α ij α l α n l l N mk = ⊕ m ⊕ N ij =1 V F m =1 ⊕ l ⊕ m l , m ( l , · · · ) α ij α ij α mk m ; α mk l l = ⊕ l {| l ; α ij m , α mk , m � } ⊗ V F ( l , · · · ) l V F ( i , j , k , · · · ) = ⊕ n ⊕ N jk n =1 V F n n ( i , n , · · · ) α jk α jk = ⊕ n ⊕ N jk N in l =1 V F n =1 ⊕ l ⊕ l , n ( l , · · · ) n l α jk α in α jk n ; α in = ⊕ l {| l ; α jk n , α in l , n � } ⊗ V F ( l , · · · ) l , m � = � ijk ; m , α ij m , α mk • | l ; α jk | l ; α jk n , α in n , α in l , n � l F l n , α jk n , α in l ; n , α jk n , α in l where F ijk is an unitary matrix. l Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category

η δ χ χ δ ε φ γ δ α α β χ γ φ γ κ η ϕ β α δ α β χ φ γ Consistent conditions for F ijk ; m αβ and UFC l ; n χδ i j k l i j k l Two di ff erent ways of fusion and are related via m q n s p p two di ff erent paths of F-moves: i j k l i j k l i j k l � � � � � � q , δ , � F mkl ; n βχ q , δ , � ; s , φ , γ F mkl ; n βχ F ijq ; m α� = � = � Φ Φ p ; s φγ Φ m m q q , n p ; q δ� p ; q δ� s p p p � i j k l � � i j k l � � i j k l � t , η , ϕ F ijk ; m αβ t , η , ϕ ; s , κ , γ F ijk ; m αβ F itl ; n ϕχ = � = � t t Φ Φ p ; s κγ Φ m n ; t ηϕ n ; t ηϕ s n n p p p   i j k l t , η , κ ; ϕ ; s , κ , γ ; q , δ , φ F ijk ; m αβ p ; s κγ F jkl ; t ηκ F itl ; n ϕχ = � s ; q δφ Φ q n ; t ηϕ  .  s p The two paths should lead to the same unitary trans.: � � F ijk ; m αβ p ; s κγ F jkl ; t ηκ F mkl ; n βχ F ijq ; m α� F itl ; n ϕχ = n ; t ηϕ s ; q δφ p ; q δ� p ; s φγ t , η , ϕ , κ � Such a set of non-linear algebraic equations is the famous pentagon identity. Moore-Seiberg 89 N ij k , F ijk ; m αβ → Unitary fusion category (UFC) l ; n χδ Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category

UFC and topological quasiparticles in di ff erent dimensions • Topological excitations in 1+1D are described/classi fi ed by (non-Abelian) UFC. i j k Consider topological excitations described by an arbitary UFC, can we realize them via a 1+1D lattice model? • Topological excitations in 2+1D (and beyond) are described by Abelian (symmetric) UFC: N ij k = N ji k . i j j i k In higher dimension, topological excitations also have non-trivial braiding properties. Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category

β β α α γ Braiding and R-matrix i j i j • Two ways to fuse: R V F ( i , j , · · · ) = ⊕ k , α ˜ V F α ( k , · · · ) = ⊕ k {| k ; α � � } ⊗ V F ( k , · · · ) k k V F ( i , j , · · · ) = ⊕ k , β V F β ( k , · · · ) = ⊕ k {| k ; β � } ⊗ V F ( k , · · · ) • | k , α � � = � β R ij ; α k ; β | k , β � where R ij ; α k ; β is an unitray matrix. • Relation to the spin θ i = e i 2 π s i of the particle: j i 2 π rotation of ( i , j ) = 2 π rotation of k j i j i 2 π rotation of ( i , j ) = 2 π rotation R R of i and j and exchange i , j twice θ i θ j R ij ; γ k ; β R ji ; β k ; α = θ k δ γα k k k Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category

β α φ λ η δ λ γ γ ε α α χ δ Consistent conditions for R ij ; α k ; β and UMTC j i k j j i k i k R F p n p l l l F R j i k j i k j j i k i k R F m n m l l m l l Hexagon identity: � p ; � F ikj ; p �λ F kij ; p φλ l ; m αγ R mk ; γ F ijk ; m αβ R ik ; φ l ; n ηδ R jk ; η n ; χ = l ; β l ; n χδ m αβ N ij k , F ijk ; m αβ , R ij ; α k ; β → Unitary modular tensor category (UMTC) l ; n χδ which describes non-Abelian statistics of 2+1D topo. excitations. Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category

Boundary of topological order → gravitational anomaly • Boundary of (some) topologically ordered states is gapless Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category

Boundary of topological order → gravitational anomaly • Boundary of (some) topologically ordered states is gapless • Boundary of topologically ordered states has gravitational anomaly Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category

Boundary of topological order → gravitational anomaly • Boundary of (some) topologically ordered states is gapless • Boundary of topologically ordered states has gravitational anomaly There is an one-to-one correspondence between effective d -dimensional topological theory Topologically orders and d − 1 -dimensional with ordered gravitational anomalies gravitational state anomaly Example 1 (gapless): • 1+1D chiral fermion L = i ( ψ † ∂ t ψ − ψ † ∂ x ψ ) → � ( k ) = vk . Gravitational anomalous, cannot appear as low energy e ff ective theory of any well-de fi nded local 1+1D lattice model. • But the above chiral fermion theory cannot appear as low energy e ff ective theory for the boundary of a 2+1D topologically ordered state – the ν = 1 IQH state (which has no topological excitations ). • The same bulk → many di ff erent boundary of the same gravitational anomaly, e.g. 3 edge modes ( v 1 k , − v 2 k , v 3 k ) Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category