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Quantum entanglement, topological order, and tensor category theory Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Local unitary trans. defines gapped quantum phases Two gapped states, |Ψ(0)� and |Ψ(1)� (or more precisely, two ground state subspaces), are in the same phase iff they are related � through a local unitary (LU)� evolution �1 � � |Ψ(1)� = P e − i 0 dg H(g ) |Ψ(0)� � where H(g ) = i Oi (g ) and Oi (g ) are local hermitian operators. Hastings, Wen 05; Bravyi, Hastings, Michalakis 10 • LU evolution =�local unitary transformation: � �1 |Ψ(1)� = P e − i T 0 dg H(g ) |Ψ(0)� = |Ψ(0)� ground−state subspace Δ −>finite gap ε −> 0 • The local unitary transformations define an equivalence relation: Two gapped states related by a local unitary transformation are in the same phase. A gapped quantum phase is an equivalence class of local unitary transformations – a conjecture. Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category A gapped quantum liquid phase: • A gapped quantum phase: ψN1 ψN2 ψN3 ψN4 HN1 , HN2 , HN3 , HN4 , · · · LU LU LU LU HN� 1 , HN� 2 , HN� 3 , HN� 4 , · · · ψN’1 ψN’2 ψN’3 ψN’4 Ni+1 = sNi , s ∼ 2 gLU gLU gLU • A gapped quantum liquid phase: ψN1 ψN2 ψN3 ψN4 HN1 , HN2 , HN3 , HN4 , · · · LU LU LU LU HN� 1 , HN� 2 , HN� 3 , HN� 4 , · · · ψN’1 ψN’2 ψN’3 ψN’4 Nk+1 = sNk , s ∼ 2 Generalized local unitary (gLU) trans. Nk ground−state subspace Δ −>finite gap ε −> 0 Nk+1 Nk gLU LU • 3+1D toric code model → a 3+1D gaped quantum liquid. • Stacking 2+1D FQH states and Haah cubic model → gapped quantum state, but not liquids. Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Bosonic/fermionic gapped quantum phases Both local bosonic and fermionic systems have the following local property: Vtot = ⊗i Vi |Ψ(1)� = |Ψ(0)� • Bosonic gapped phases�are the equivalent classes of LU transformation: LU = Uijk , which acts within Vi ⊗ Vj ⊗ Vk , and [Uijk , Ui � j � k � ] = 0, e.g. Uijk = e i(bi bj bk† +h.c.) • Fermionic gapped phases � aref the equivalent classes of fermionic LU transformation: fLU = Uijk , which does not act within Vi ⊗ Vj ⊗ Vk , but f , Uf [Uijk i �j �k � ] = 0, e.g. f Uijk =e i(ci cj ck† ck +h.c.) Gapped quantum liquids for bosons and fermions have very different mathematical structures Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category LU trans. defines long-range entanglement (ie topo. order) For gapped systems with no symmetry: • According to Landau theory, no symmetry to break → all systems belong to one trivial phase Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category LU trans. defines long-range entanglement (ie topo. order) For gapped systems with no symmetry: • According to Landau theory, no symmetry to break → all systems belong to one trivial phase • Thinking about entanglement: Chen-Gu-Wen 2010 - There are long range entangled (LRE) states - There are short range entangled (SRE) states |LRE� �= |product state� = |SRE� local unitary transformation LRE state SRE product state Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category LU trans. defines long-range entanglement (ie topo. order) For gapped systems with no symmetry: • According to Landau theory, no symmetry to break → all systems belong to one trivial phase • Thinking about entanglement: Chen-Gu-Wen 2010 - There are long range entangled (LRE) states → many phases - There are short range entangled (SRE) states → one phase |LRE� �= local unitary transformation LRE state SRE product state |product state� = |SRE� g2 local unitary transformation SRE SRE product product state state topological order LRE 1 LRE 2 local unitary transformation LRE 1 LRE 2 SRE phase transition g1 • All SRE states belong to the same trivial phase • LRE states can belong to many different phases = different patterns of long-range entanglements defined by the LU trans. = different topological orders Wen 1989 → A classification by tensor category theory Levin-Wen 05, Chen-Gu-Wen 2010 Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → more entangled Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = (| ↑� + | ↓�) ⊗ (| ↑� + | ↓�) = |x� ⊗ |x� → unentangled Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = (| ↑� + | ↓�) ⊗ (| ↑� + | ↓�) = |x� ⊗ |x� → unentangled • = | ↓� ⊗ | ↑� ⊗ | ↓� ⊗ | ↑� ⊗ | ↓�... → unentangled Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = (| ↑� + | ↓�) ⊗ (| ↑� + | ↓�) = |x� ⊗ |x� → unentangled • • = | ↓� ⊗ | ↑� ⊗ | ↓� ⊗ | ↑� ⊗ | ↓�... → unentangled = (| ↓↑� − | ↑↓�) ⊗ (| ↓↑� − | ↑↓�) ⊗ ... → short-range entangled (SRE) entangled Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = (| ↑� + | ↓�) ⊗ (| ↑� + | ↓�) = |x� ⊗ |x� → unentangled • = | ↓� ⊗ | ↑� ⊗ | ↓� ⊗ | ↑� ⊗ | ↓�... → unentangled = (| ↓↑� − | ↑↓�) ⊗ (| ↓↑� − | ↑↓�) ⊗ ... → short-range entangled (SRE) entangled � � � = |0�x1 ⊗ |1�x2 ⊗ |0�x3 ... • Crystal order: |Φcrystal � = � • = direct-product state → unentangled state (classical) Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = (| ↑� + | ↓�) ⊗ (| ↑� + | ↓�) = |x� ⊗ |x� → unentangled • = | ↓� ⊗ | ↑� ⊗ | ↓� ⊗ | ↑� ⊗ | ↓�... → unentangled = (| ↓↑� − | ↑↓�) ⊗ (| ↓↑� − | ↑↓�) ⊗ ... → short-range entangled (SRE) entangled � � � = |0�x1 ⊗ |1�x2 ⊗ |0�x3 ... • Crystal order: |Φcrystal � = � • = direct-product state → unentangled state (classical) • Particle condensation (superfluid) � � � � |ΦSF � = all conf. � Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = (| ↑� + | ↓�) ⊗ (| ↑� + | ↓�) = |x� ⊗ |x� → unentangled • = | ↓� ⊗ | ↑� ⊗ | ↓� ⊗ | ↑� ⊗ | ↓�... → unentangled = (| ↓↑� − | ↑↓�) ⊗ (| ↓↑� − | ↑↓�) ⊗ ... → short-range entangled (SRE) entangled � � � = |0�x1 ⊗ |1�x2 ⊗ |0�x3 ... • Crystal order: |Φcrystal � = � • = direct-product state → unentangled state (classical) • Particle condensation (superfluid) � � � � = (|0�x1 + |1�x1 + ..) ⊗ (|0�x2 + |1�x2 + ..)... |ΦSF � = all conf. � = direct-product state → unentangled state (classical) - Superfluid, as an exemplary quantum state of matter, is actually very classical and unquantum from entanglement point of view. Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Scramble the phase: local rule → global dancing pattern ΦSF ({z1 , ..., zN }) = 1 → unentangled product state • Local dancing rules of a FQH liquid: (1) every electron dances around clock-wise (ΦFQH only depends on z = x + iy ) (2) takes exactly three steps to go around any others (ΦFQH ’s phase change 6π) � → Global dancing pattern ΦFQH ({z1 , ..., zN }) = (zi − zj )3 • A general theory of multi-layer Abelian FQH state: � � 1 � I I KII I J KIJ − 4 i,I |ziI |2 (zi − zj ) (zi − zj ) e I ;i<j I <J;i,j Low energy effective theory is the K -matrix Chern-Simons theory L = K4πIJ aI daJ • An integer number of edge modes c = dim(K ) = number of layers. Even K -matrix classifies all 2+1D Abelian topological order (but not one-to-one) Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category • A systematic theory of single-layer non-Ableian FQH state: Pattern of zeros Sa : a-electron cluster has a relative angular momentum Sa Wen-Wang 08 S2 = 3 S2 = 1 S3 = 9 S3 = 5 S4 = 18 S4 = 10 ν=1/3 Laughlin ν=1/2 Pfaffian • Local dancing rules are enforced by Hamiltonian to lower energy. • Only certain sequences Sa correspond to valid FQH states. Which? • Different POZ Sa give rise to different topological properties • A fractional number of edge modes c �= integer. Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Sum over a subset of product states To make topological order, we need to sum over many different product states, but we should not sum over everything. � all spin config. | ↑↓ ..� = | →→ ..� Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Sum over a subset of product states To make topological order, we need to sum over many different product states, but we should not sum over everything. � all spin config. | ↑↓ ..� = | →→ ..� • sum over a subset of spin config.: � � � � 2 |ΦZloops �= � � � � # of loops � |ΦDS � = (−) � loops • Can the above wavefunction be the ground states of local Hamiltonians? Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category Sum over a subset: local rule → global dancing pattern 2D 3D • Local dancing rules of a string liquid: (1) Dance � while � holding � hands � (no�open ends) � � = Φstr , Φstr = Φstr (2) Φstr � � → Global dancing pattern Φstr =1 • Local dancing rules of another string liquid: (1) Dance � while � holding � hands � (no�open ends) � � = Φstr , Φstr = −Φstr (2) Φstr � � → Global dancing pattern Φstr = (−)# of loops � � • Two string-net condensations → two topological orders: Levin-Wen 05 Z2 topo. order Sachdev Read 91, Wen 91 and double-semion topo. order. Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category