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Basis Functions

This section covers the fundamental basis generation and manipulation functions for anyon chains.

Anyon Basis Generation

The anyon_basis function generates the computational basis states for various anyon models in different symmetry sector:

Fibonacci Anyons

For Fibonacci anyons, where quantum dimension is $\phi = \frac{1+\sqrt{5}}{2}$, the basis consists of configurations that satisfy the fusion constraints. Each site can be in state 0 ($\tau$ particle) or 1 (vacuum), with the constraint that no two adjacent $1$ particles can fuse (forbidden configurations 11are excluded). If you are familiar with PXP model, you will quickly get it.

Ising Anyons

For Ising anyons (Majorana fermions), where quantum dimension $d=\sqrt{2}$, the basis includes all possible $\sigma$ and 1 configurations, representing the two types of Ising anyons.

Nomral Spin

quantum dimension $d=2$.

Symmetry

The symmetry you can choose nowadays is:

  • Translational symmetry $T$
  • Topological symmetry $Y$

Potential symmetry:

  • Inversion symmetry $I$

Usage Examples

using FibonacciChain

# Generate Fibonacci basis for 6 sites with PBC
N = 6
model_fibo = AnyonModel(FibonacciAnyon(), N; pbc=true)
basis_pbc = anyon_basis(model_fibo)

# Generate Ising anyon basis  
model_ising = AnyonModel(IsingAnyon(), N; pbc=true)
basis_ising = anyon_basis(model_ising)

# Generate basis in momentum sector k=0
basis_k0, rep_dict = anyon_basis(model_fibo, 0)

Hamiltonian Construction

Now we support

  • Ising interaction: $H = -\sum_i Z_iZ_{i+1} - \sum_i X_i$
  • Ferromagnetic Fibonacci anyon interaction: $H = -\sum_i \Pi_{i}^0$
  • Antiferromagnetic Fibonacci anyon interaction: $H = -\sum_i \Pi_{i}^1$

Topological Symmetries

The topological symmetry operations are crucial for understanding the anyonic nature of the system. They encode how the fusion outcomes transform under topological operations.

Reduced Density Matrices

For a given model and subsystems, compute reduced density matrices:

  • anyon_rdm(model, subsystems, state)
  • Sector version: anyon_rdm_sec(model, subsystems, state, k)

And we also have disjoint_rdm for two parallel chains with two different anyon types.