Quantum Entanglement and Topological Order Ashvin Vishwanath UC - PowerPoint PPT Presentation

Quantum Entanglement and Topological Order Ashvin Vishwanath UC Berkeley Ari Turner M. Oshikawa Tarun Grover Yi Zhang Berkeley-> Amsterdam ISSP Berkeley->KITP Stanford

OUTLINE • Part 1: Introduction – Topological Phases, Topological entanglement entropy. – Model wave-functions. • Part 2: Topological Entropy of nontrivial bipartitions. – Ground state dependence and Minimum Entropy States. – Application: Kagome spin liquid in DMRG. • Part 3: Quasi particle statistics (modular S-Matrix) from Ground State Wave-functions. Ref: Zhang, Grover, Turner, Oshikawa, AV: arXiv:1111.2342

Conventional (Landau) Phases Distinguished by spontaneous symmetry breaking. Can be diagnosed in the ground state wave-function by a local order parameter. • Magnets (broken spin • Solid (broken • Superfluids ψ symmetry) M translation) Solid In contrast –topological phases…

Topological Phases Integer topological phases Fractional topological phases • Integer Quantum Hall & • Fractional Quantum Hall Topological insulators • Gapped spin liquids • Haldane (AKLT) S=1 phase • Interacting analogs in Topological Order: D=2,3 (Kitaev, Chen-Gu-Wen, Lu&AV) 1. Fractional statistics excitations (anyons). Non-trivial surface states 2. Topological degeneracy on closed manifolds. How to tell – given ground state wave-function(s)? Entanglement as topological `order parameter’.

Topological Order – Example 1 • Laughlin state ( ν=1/2 bosons) [`Chiral spin Liquid’] 2 𝑓 − 𝑨 𝑗 2    Ψ {𝑨 𝑗 } = 𝑨 𝑗 − 𝑨 2   𝑗 ( r , r , r ) ( r , r , r ) 𝑘  1 2 N C 1 1 2 N 𝑗<𝑘 Lattice Version • Ground State Degeneracy (N=2 g ): 𝑗 N=1 N=2 s is a semion Quasiparticle Types: { 1 , s }. # = Torus degeneracy

Topological Order – Example 2 Ising (Z 2 ) Electrodynamics • Here E=0,1; B=0, π  E   (mod 2) (E field loops do not end) 0 Ψ = • Degeneracy on torus=4. • Degeneracy on cylinder=2 (no edge states)

Topological Order – Example 2 Z 2 Quantum Spin Liquids Ψ = P G ( Ψ BCS ) RVB spin liquid: (Anderson ’73). Effective Theory: Z 2 Gauge Theory. Recently, a number of candidates in numerics with no conventional order. Definitive test: identify topological order. Square J1_J2 Kagome (Yan et al) Honeycomb Hubbard (Jiang et al, Wang et al) (Meng et al)

Entanglement Entropy • Schmidt Decomposition: Ψ = 𝑞 𝑗 𝐵𝑗 ⊗ 𝐶𝑗 𝑗 Entanglement Entropy (von-Neumann): 𝑇 𝐵 = − 𝑞 𝑗 log 𝑞 𝑗 𝑗 Note: 1) 𝑇 𝐵 = 𝑇 𝐶 2) Strong sub-additivity (for von-Neumann entropy)        S S S S S S S 0 A B C BC AC AB ABC

Topological Entanglement Entropy • Gapped Phase with topological order. – Smooth boundary, circumference L A : A   Topological Entanglement Entropy S aL - A (Levin-Wen;Kitaev-Preskill) ϒ =Log D . (D : total quantum dimension). A B Abelian phases: 𝐸 = √{𝑈𝑝𝑠𝑣𝑡 𝐸𝑓𝑕𝑓𝑜𝑓𝑠𝑏𝑑𝑧} Z 2 gauge theory: ϒ =Log 2 Constraint on boundary – no gauge charges inside. Lowers Entropy by 1 bit of information.

Entanglement Entropy of Gapped Phases • Trivial Gapped Phase: – Entanglement entropy: sum of local contributions. Curvature Expansion (smooth boundary) :       (    2 2  X F a a a a ) 0 1 2 4 l A Z 2 symmetry of Entanglement Entropy: 1/ κ S A = S B AND κ→ - κ . So a 1 =0 B No constant in 2D for trivial phase.      2  S dl [ a a ] A 0 2 A    2  a L 0 A L A Grover, Turner, AV: PRB 84, 195120 (2011)

Extracting Topological Entanglement Entropy • Smooth partition • General Partition boundary on lattice? A B C OR A B         - S S S S S S S A B C BC AC AB ABC • Problem: (LevinWen;Preskill Kitaev)   `topological entanglement Strong subadditivity implies: 0 Identical result with Renyi entropy entropy’ depends on ground state. (Dong et al, Zhang et al) How does this work with generic states?

Topological Entropy of Lattice Wavefunctions ϒ – From MonteCarlo Evaluation of Gutzwiller Projected Lattice wave- functions . 𝒇 −𝑻 𝟑 = 〈𝑻𝑿𝑩𝑸 𝑩 〉 . (Y. Zhang, T. Grover, AV Phys. Rev. B 2011.) 0.42 ± 0.14 Good agreement for chiral spin liquid. Z2 not yet in thermodynamic limit(?) Alternate approach to diagnosing topological order: Entanglement spectrum (Li and Haldane, Bernevig et al.). Closely related to edge states Does not diagnose Z2 SL Cannot calculate with Monte Carlo.

Part 2: Ground State Dependence of Topological Entropy

Topological Entanglement in Nontrivial bipartitions • Nontrivial bipartition - entanglement cut is not contractible. Can `sense’ degenerate ground states. • Result from Chern-Simons field theory: (Dong et al.) • Abelian topological phase with N ground states on torus. There is a special basis of ground states for a cut, such that: 𝑂 ( 𝑞 𝑜 = 𝑑 𝑜 2 ) Ψ = • 𝑑 𝑜 |𝜚 𝑜 〉 𝑜=1 𝟐 𝑶 𝛿 = 2𝛿 0 − 𝒒 𝒐 𝐦𝐩𝐡 𝒐=𝟐 𝒒 𝒐 Topological entropy in general reduced . 0 ≤ 𝛿 ≤ 2𝛿 0 For the special states 𝜚 𝑜 , equal to usual value ( 𝛿 = 2𝛿 0 =2log D). These Minimum Entropy States correspond to quasiparticles in cycle of the torus

Eg. Z 2 Spin Liquid on a Cylinder • Degenerate sectors: even and odd 𝑓 E winding around cylinder. Minimum Entropy States: 𝑝 0, 𝜌 = ( 𝑓 ± |𝑝〉)/√2 A B 0〉, |𝜌 • The minimum entropy states ( 𝛿 = log 2) are `vison ’ states – magnetic flux through the cylinder that entanglement surface can measure. • State 𝑓 has 𝛿 = 0. Cancellation from: Ψ = 𝐵, 𝑓𝑤𝑓𝑜 𝐶, 𝑓𝑤𝑓𝑜 + 𝐵, 𝑝𝑒𝑒 𝐶, 𝑝𝑒𝑒 √2

Application: DMRG on Kagome Antiferromagnet A B Depenbrock,McCulloch, Schollwoeck Jiang, Wang, Balents: (arxiv:1205:4858). Log base 2 arXiv:1205.4289 • Topological entanglement entropy found by extrapolation within 1% of log 2 . • Minimum entropy state is selected by DMRG (low entaglement). • Possible reason why only one ground state seen.

Ground State Dependence of Entanglement Entropy    • Chiral spin liquid on Torus: 2   ( r , r , r ) ( r , r , r )  1 2 N C 1 1 2 N – Degenerate ground states from changing boundary  C conditions on Slater det.  1        cos sin 1 2 Trivial Bipartition: No ground state dependence.

Ground State Dependence of Entanglement Entropy • Chiral spin liquid on Torus: – Degenerate ground states from changing boundary  C conditions on Slater det.  1        cos sin 1 2 Non trivial Bipartition: Ground state dependence!

Ground State Dependence of Topological Entropy from Strong Sub-additivity • Strong subadditivity: 𝑇 𝐵𝐶𝐷 + 𝑇 𝐶 − 𝑇 𝐵𝐶 − 𝑇 𝐶𝐷 ≤ 0 𝜹 𝟐 𝜹 𝟏 𝜹 𝟑 Obtain `uncertainty’ relation: 𝜹 𝟐 + 𝜹 𝟑 ≤ 𝟑𝜹 𝟏 Naïve result, 𝛿 1 = 𝛿 2 = 2𝛿 0 𝑑𝑏𝑜𝑜𝑜𝑝𝑢 hold from general quantum information requirement. True even without topological field theory .

Part 3: Mutual Statistics from Entanglement • Relate minimum entropy states along independent torus cuts. (modular transformation: S matrix) 𝜚′ 1 = 𝑻 𝜚 1 𝜚 2 𝜚′ 2 𝑁𝐹𝑇: 𝜚 1 , 𝜚 2 𝑁𝐹𝑇: 𝜚′ 1 , 𝜚′ 2 S encodes quasiparticle braiding statisitics: Chiral Spin Liquid: e s 𝑇 ab b 𝑏 𝑇 = 1 1 1 e −1 1 √2 s

Statistics from Entanglement – Chiral Spin Liquid S 1   1.09 0.89 Semion Statistics!       0.89 1.0 9 2 Wavefunction `knows’ about semion exciations; Zhang, Grover, Turner, Oshikawa, AV (2011).

Conclusions • Entanglement of non-trivial partitions can be used to define `quasiparticle ’ like states, and extract their statistics. • Useful to distinguish two phases with same D. (eg. Z 2 and doubled chiral spin liquid, no edge states) Less prone to errors. • Can topological entanglement entropy constrain new types of topological order (eg D=3)? • Experimental measurement? Need nonlocal probe.