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Checkerboard Lattice Gauge Transformations

Understanding sublattice transformations and their role in the Ising model

What is a Checkerboard Lattice?

The checkerboard lattice is a way of visualizing the 2D square lattice as two interleaved sublattices, often called "even" and "odd" sites. This is useful for certain transformations and algorithms, especially in the context of antiferromagnetic interactions and cluster updates. Note that we use this on a bipartite lattice, such as the square lattice with an even number of sites per side.

For a site (i, j):
- Even sublattice: (i + j) % 2 == 0
- Odd sublattice: (i + j) % 2 == 1

Gauge Transformations in the Ising Model

Gauge transformations are changes of variables that leave the physical properties of the system invariant. In the Ising model, a common transformation is to flip all spins on one sublattice, which can map an antiferromagnetic system to a ferromagnetic one.

s'_{i,j} = η_{i,j} · s_{i,j},
where η_{i,j} = (-1)^{i+j}

This transformation is especially useful for cluster algorithms, allowing them to treat antiferromagnetic couplings as ferromagnetic by flipping the sign of the interaction on one sublattice.