Sparse Diagonalization

The Hamiltonian matrix in quantum mechanics for systems with local interactions are sparse, meaning most of their entries are zero. Storing and manipulating dense matrices of size $N \times N$ requires $O(N^2)$ memory and $O(N^3)$ computational time for diagonalization. For large $N$, this becomes infeasible. The Lanczos method is designed to take advantage of this sparsity, requiring only matrix-vector products rather than explicit matrix storage or manipulation.

The Lanczos method is a powerful algorithm for finding a few extremal eigenvalues and their corresponding eigenvectors of large, sparse matrices. It is particularly useful for quantum lattices, where systems are often described by high-dimensional matrices that are too large to handle with dense matrix techniques. The Lanczos method exploits the sparsity of these matrices to efficiently compute the desired eigenvalues and eigenvectors.

The implementation of the sparse diagonalization in ALPS is also discussed.