Reducing Memory Cost of Exact Diagonalization using Singular Value Decomposition

We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramatically reduced memory requirements, {\em without restricting to variational ansatzes}. The lattice of size $N$ is partitioned into two subclusters. At each iteration the Lanczos vector is projected into two sets of $n_{{\rm svd}}$ smaller subcluster vectors using singular value decomposition. For low entanglement entropy $S_{ee}$, (satisfied by short range Hamiltonians), the truncation error is expected to vanish as $\exp(-n_{{\rm svd}}^{1/S_{ee}})$. Convergence is tested for the Heisenberg model on Kagom'e clusters of 24, 30 and 36 sites, with no lattice symmetries exploited, using less than 15GB of dynamical memory. Generalization of the Lanczos-SVD algorithm to multiple partitioning is discussed, and comparisons to other techniques are given.

💡 Research Summary

The paper introduces a memory‑efficient variant of the Lanczos exact‑diagonalization (ED) method by incorporating singular‑value decomposition (SVD) into each Lanczos iteration. The central idea is to split the lattice of N sites into two equal subclusters, treat the many‑body wavefunction as a matrix connecting the two halves, and compress this matrix by retaining only the n_svd largest singular values. After each application of the Hamiltonian, the resulting state is orthogonalized within each subcluster, the inter‑cluster coupling matrix C is formed, and an SVD of C is performed. The new wavefunction is then truncated to the top n_svd singular components, thereby preventing the exponential growth of the Lanczos basis and keeping the memory footprint proportional to n_svd·2^{N/2} rather than 2^{N}.

The authors analyze the truncation error ε in terms of the bipartite entanglement entropy S_ee. Assuming the singular‑value spectrum follows a power‑law density ρ_p(s) ∝ s^p (analogous to a classical gas with constant specific heat), they derive an asymptotic error estimate ε ∼ n_svd^{−S_ee} exp