Restricted Spin Set Lattice Models A route to Topological Order

Transcription

1 Restricted Spin Set Lattice Models A route to Topological Order R. Zach Lamberty with Stefanos Papanikolaou and Christopher L. Henley Supported by NSF grant DMR and an NSF GRF APS March Meeting 2012

2 } Our Definition: A state is Topologically Ordered if it possess an entropically degenerate ensemble (in the thermodynamic limit) which cannot be distinguished by local measurements, and cannot be reached by local moves Topological Order Quantum Topological Order } A new knob in the model connects critical models to topologically ordered models Critical Topologically Ordered

3 } Our Definition: A state is Topologically Ordered if it possess an entropically degenerate ensemble (in the thermodynamic limit) which cannot be distinguished by local measurements, and cannot be reached by local moves Topological Order Quantum Topological Order } A new knob in the model connects critical models to topologically ordered models Critical Topologically Ordered

4 } Group } Lattice (Abelian or Non-Abelian) } Periodic Boundary } Directed Spins on edges } Plaquette Constraint } Sectors = loop product } Allowed spin set } Made of conjugacy classes (related by symmetry) Plaquette Product Sector Labels } Non-trivial correlations } Mediated interactions

5 } Group } Lattice (Abelian or Non-Abelian) } Periodic Boundary } Directed Spins on edges } Plaquette Constraint } Sectors = loop product } Allowed spin set } Made of conjugacy classes (related by symmetry) Plaquette Product Sector Labels } Non-trivial correlations } Mediated interactions

6 } Group } Lattice (Abelian or Non-Abelian) } Periodic Boundary } Directed Spins on edges } Plaquette Constraint } Sectors = loop product } Allowed spin set } Made of conjugacy classes (related by symmetry) Plaquette Product Sector Labels } Non-trivial correlations } Mediated interactions

7 } Group } Lattice (Abelian or Non-Abelian) } Periodic Boundary } Directed Spins on edges } Plaquette Constraint } Sectors = loop product } Allowed spin set } Made of conjugacy classes (related by symmetry) Plaquette Product Sector Labels } Non-trivial correlations } Mediated interactions

8 } Group } Lattice (Abelian or Non-Abelian) } Periodic Boundary } Directed Spins on edges } Plaquette Constraint } Sectors = loop product } Allowed spin set } Made of conjugacy classes (related by symmetry) Plaquette Product Sector Labels } Non-trivial correlations } Mediated interactions

9 } Group } Lattice (Abelian or Non-Abelian) } Periodic Boundary } Directed Spins on edges } Plaquette Constraint } Sectors = loop product } Allowed spin set } Made of conjugacy classes (related by symmetry) Plaquette Product Sector Labels } Non-trivial correlations } Mediated interactions

10 Local Local

11 Local Local

12 Global Global

13 Global Global

14 Global Global

15 Global Global

16 Global Global

17 Global Global

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22 Dimer Model

23 Sector Probabilities Sector Probabilities } Q1: How do we determine a model has Topological Order? } A1: Use sector probabilities as statistical probes } Constrained random walks of defects can change sectors } Sample configurations Sector Sector

24 Sector Probabilities Sector Probabilities } Q1: How do we determine a model has Topological Order? } A1: Use sector probabilities as statistical probes } Constrained random walks of defects can change sectors } Sample configurations Sector Defect Walk Sector

25 Sector Probabilities Sector Probabilities Dimer Model Topologically Ordered Increasing Abelian

26 Sector Probabilities Sector Probabilities Six-Vertex Model Topologically Ordered Increasing Abelian

27 Defect Interactions } Q2: Do we see interactions between defects? Defect Interactions } A2: Restricted spin set non-trivial correlations Distribution of distances from one defect to the other Entropic cost

28 Defect Interactions } Q2: Do we see interactions between defects? Defect Interactions } A2: Restricted spin set non-trivial correlations Distribution of distances from one defect to the other Entropic cost

29 Defect Interactions } The reduced spin set can lead to mediated interactions between defects Critical : TO: Defect Interactions

30 } We have both Abelian and Non-Abelian Classical models with Topological Order } Convergent sector probabilities are a useful diagnostic of Topological Order } Restricting spin degrees of freedom can tune within a family of models from one which is critical to one which is topologically ordered

31 Braiding Features of the Model Bonus Slide Braiding

32 Braiding Features of the Model Bonus Slide Braiding

33 Braiding Features of the Model Bonus Slide Braiding

34 Braiding Features of the Model Bonus Slide Braiding

35 Braiding Features of the Model Bonus Slide Braiding