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:

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 (vacuum) or 1 ($\tau$ particle), with the constraint that no two adjacent $\tau$ particles can fuse to vacuum (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
basis_pbc = anyon_basis(N, true, measure_class=:Fibo)

# Generate Ising anyon basis  
basis_ising = anyon_basis(N, true, measure_class=:IsingX)

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

Both for state and density matrix in specific symmetry sector: anyon_rdm(N, subsystems, state, k=0)

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