PBCLattice2D

include/spatial/pbc_lattice.hpp provides num::PBCLattice2D, a small struct that precomputes the four periodic-boundary neighbor index arrays for an \(N \times N\) square lattice.

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Problem

Periodic square-lattice algorithms repeatedly access the four nearest neighbors of each site. Precomputed neighbor arrays replace modulo operations inside sweeps and cluster traversals.

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Routine Reference

struct num::PBCLattice2D {
    int N;
    std::vector<int> up, dn, lt, rt;   // N*N each
 
    explicit PBCLattice2D(int N);
};

Construction is \(O(N^2)\) and done once; subsequent lookups are direct array reads.

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Index Layout

Flat row-major layout: site \((row, col)\) has flat index \(i = row \cdot N + col\).

\[ \texttt{up}[i] = ((row - 1 + N) \bmod N) \cdot N + col \]

\[ \texttt{dn}[i] = ((row + 1) \bmod N) \cdot N + col \]

\[ \texttt{lt}[i] = row \cdot N + (col - 1 + N) \bmod N \]

\[ \texttt{rt}[i] = row \cdot N + (col + 1) \bmod N \]

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Example

num::PBCLattice2D nbr(N);
 
// Metropolis sweep -- neighbor sum with no modulo arithmetic
real ns = spins[nbr.up[i]] + spins[nbr.dn[i]]
        + spins[nbr.lt[i]] + spins[nbr.rt[i]];
 
// BFS cluster detection
num::connected_components(N*N,
    [&](int i) { return spins[i] < 0.0; },
    [&](int i, auto&& visit) {
        visit(nbr.up[i]); visit(nbr.dn[i]);
        visit(nbr.lt[i]); visit(nbr.rt[i]);
    });

Used by: Ising IsingLattice::sweep, IsingLattice::sweep_umbrella.