Non-Abelian Statistics as a Berry Phase in the Honeycomb Lattice Model New J. Phys. 11 (2009) 093027 arXiv:0901.3674 Ville Lahtinen and Jiannis K. Pachos DAQIST workshop, 18 September 2009, Maynooth
Anyons in an exactly solvable model Kitaev's honeycomb lattice model: A.Y. Kitaev, Annals of Physics , 321:2, 2006 An exactly solvable 2D spin model on a honeycomb lattice Known to support non-Abelian Ising anyons based on Chern number and CFT arguments
Anyons in an exactly solvable model Kitaev's honeycomb lattice model: A.Y. Kitaev, Annals of Physics , 321:2, 2006 An exactly solvable 2D spin model on a honeycomb lattice exactly solvable exactly solvable Known to support non-Abelian Ising anyons based on Chern number and CFT arguments We can do more: Demonstrate the fusion rules from the spectrum Calculate the non-Abelian statistics from the eigenstates Understand the non-Abelian behavior microscopically Provide methods and predictions for future experiments
Anyons in an exactly solvable model To demonstrate the Ising anyons, one needs to demonstrate: (1) Ising fusion rules 1 or Ψ σ x σ = 1 + Ψ DONE! Ψ x σ = σ VL et al., Ann. Phys. 323, 9 (2008) Ψ x Ψ = 1 σ σ
Anyons in an exactly solvable model To demonstrate the Ising anyons, one needs to demonstrate: (1) Ising fusion rules 1 or Ψ σ x σ = 1 + Ψ DONE! Ψ x σ = σ VL et al., Ann. Phys. 323, 9 (2008) Ψ x Ψ = 1 σ σ (2) Statistics ? R 2 = e -iπ/4 ( ) 0 1 σ σ R 2 = e -iπ/4 1 0
The honeycomb lattice model The Hamiltonian
The honeycomb lattice model The Hamiltonian z Anisotropic nearest neighbour couplings y x
The honeycomb lattice model The Hamiltonian Effective magnetic field gives z next-to-nearest interaction y x
The honeycomb lattice model The Hamiltonian z Represent Pauli operators by Majorana fermions Represent Pauli operators by Majorana fermions y x
The honeycomb lattice model The Hamiltonian +1 +1 +1 +1 +1 +1 Represent Pauli operators by Majorana fermions +1 -1 +1 Represent Pauli operators by Majorana fermions +1 +1 -1 +1 +1 +1 -1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 Static background Z 2 gauge field living on every link +1 +1 Fixing u fixes the gauge and the physical sector
The honeycomb lattice model The Hamiltonian +1 +1 +1 +1 +1 +1 Represent Pauli operators by Majorana fermions +1 -1 +1 Represent Pauli operators by Majorana fermions +1 +1 -1 +1 +1 +1 -1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 The physical sectors are labeled by the plaquette +1 +1 operators (Wilson loops): Eigenvalue w p = -1: a vortex at plaquette p
The honeycomb lattice model The Hamiltonian Represent Pauli operators by Majorana fermions Represent Pauli operators by Majorana fermions The physical sectors are labeled by the plaquette operators (Wilson loops): Eigenvalue w p = -1: a vortex at plaquette p
The honeycomb lattice model To solve the model: Choose the system size and fix the boundary conditions (we work on torus). Represent Pauli operators by Majorana fermions Represent Pauli operators by Majorana fermions The physical sectors are labeled by the plaquette operators (Wilson loops): Eigenvalue w p = -1: a vortex at plaquette p
The honeycomb lattice model To solve the model: Choose the system size and fix the boundary conditions (we work on torus). Fix the physical sector (the vortex configuration) by fixing the gauge field u. Represent Pauli operators by Majorana fermions Represent Pauli operators by Majorana fermions The physical sectors are labeled by the plaquette operators (Wilson loops): Eigenvalue w p = -1: a vortex at plaquette p
The honeycomb lattice model To solve the model: Choose the system size and fix the boundary conditions (we work on torus). Fix the physical sector (the vortex configuration) by fixing the gauge field u. Represent Pauli operators by Majorana fermions Represent Pauli operators by Majorana fermions Choose the parameters J and K on all links (we consider the non-Abelian phase with globally J x =J y =J z and K > 0). The physical sectors are labeled by the plaquette operators (Wilson loops): Eigenvalue w p = -1: a vortex at plaquette p
The honeycomb lattice model To solve the model: Choose the system size and fix the boundary conditions (we work on torus). Fix the physical sector (the vortex configuration) by fixing the gauge field u. Represent Pauli operators by Majorana fermions Represent Pauli operators by Majorana fermions Choose the parameters J and K on all links (we consider the non-Abelian phase with globally J x =J y =J z and K > 0). Dump the Hamiltonian into a number cruncher. The physical sectors are labeled by the plaquette operators (Wilson loops): Eigenvalue w p = -1: a vortex at plaquette p
The honeycomb lattice model To solve the model: Choose the system size and fix the boundary conditions (we work on torus). Fix the physical sector (the vortex configuration) by fixing the gauge field u. Represent Pauli operators by Majorana fermions Represent Pauli operators by Majorana fermions Choose the parameters J and K on all links (we consider the non-Abelian phase with globally J x =J y =J z and K > 0). Dump the Hamiltonian into a number cruncher. Wait. The physical sectors are labeled by the plaquette operators (Wilson loops): Eigenvalue w p = -1: a vortex at plaquette p
The honeycomb lattice model To solve the model: Choose the system size and fix the boundary conditions (we work on torus). Fix the physical sector (the vortex configuration) by fixing the gauge field u. Represent Pauli operators by Majorana fermions Represent Pauli operators by Majorana fermions Choose the parameters J and K on all links (we consider the non-Abelian phase with globally J x =J y =J z and K > 0). Dump the Hamiltonian into a number cruncher. Wait. ... while waiting figure out what to do with the The physical sectors are labeled by the plaquette spectrum and eigenvectors .. operators (Wilson loops): Eigenvalue w p = -1: a vortex at plaquette p
The honeycomb lattice model To solve the model: Choose the system size and fix the boundary conditions (we work on torus). Fix the physical sector (the vortex configuration) by fixing the gauge field u. Represent Pauli operators by Majorana fermions Represent Pauli operators by Majorana fermions Choose the parameters J and K on all links (we consider the non-Abelian phase with globally J x =J y =J z and K > 0). Dump the Hamiltonian into a number cruncher. Wait. ... while waiting figure out what to do with the The physical sectors are labeled by the plaquette spectrum and eigenvectors .. operators (Wilson loops): ... go walk through Australia on Google street view .. Eigenvalue w p = -1: a vortex at plaquette p
The honeycomb lattice model To solve the model: Choose the system size and fix the boundary conditions (we work on torus). Fix the physical sector (the vortex configuration) by fixing the gauge field u. Represent Pauli operators by Majorana fermions Represent Pauli operators by Majorana fermions Choose the parameters J and K on all links (we consider the non-Abelian phase with globally J x =J y =J z and K > 0). Dump the Hamiltonian into a number cruncher. Wait. ... while waiting figure out what to do with the The physical sectors are labeled by the plaquette spectrum and eigenvectors .. operators (Wilson loops): ... go walk through Australia on Google street view .. Eigenvalue w p = -1: a vortex at plaquette p Get the results, change the parameters and repeat.
Vortex transport How to simulate the transport of vortices? The parameters J ij and K ikj appear in the Hamiltonian always paired with the local gauge fields u ij .
Vortex transport How to simulate the transport of vortices? The parameters J ij and K ikj appear in the Hamiltonian always paired with the local gauge fields u ij . Assume local control on link ( ij )
Vortex transport How to simulate the transport of vortices? The parameters J ij and K ikj appear in the Hamiltonian always paired with the local gauge fields u ij . Assume local control on link ( ij ) J ij , K ikj 0 The vortex occupies both plaquettes
Vortex transport How to simulate the transport of vortices? The parameters J ij and K ikj appear in the Hamiltonian always paired with the local gauge fields u ij . Assume local control on link ( ij ) J ij , K ikj -J ij , -K ikj Equivalent to changing u ij -> - u ij .
Zero modes and fusion rules Mode spectrum Full low-energy spectrum
Zero modes and fusion rules Mode spectrum Full low-energy spectrum “continuum of states” gap
Zero modes and fusion rules Mode spectrum Full low-energy spectrum
Zero modes and fusion rules 2 n -fold degeneracy for 2n vortices Mode spectrum Full low-energy spectrum Zero mode - energy converges exponentially to E=0
Zero modes and fusion rules 2 n -fold degeneracy for 2n vortices Interactions lift degeneracy when vortices nearby each other Mode spectrum Full low-energy spectrum Zero mode energy oscillates when vortices nearby.
Zero modes and fusion rules 2 n -fold degeneracy for 2n vortices Interactions lift degeneracy when vortices nearby each other Mode spectrum Full low-energy spectrum
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