Density-matrix algorithms for quantum renormalization groups

TLDR

A formulation of numerical real‑space renormalization groups for quantum many‑body problems is presented, outlining several algorithms that exploit this framework. The authors present and demonstrate methods that retain the most significant eigenstates of a block density matrix—obtained by diagonalizing a larger lattice section—and test them on S=1/2 and S=1 Heisenberg chains. The approach yields far higher accuracy than standard methods, achieving at least 10⁻⁹ precision for S=1 Heisenberg chain energies, and is broadly applicable to one‑dimensional quantum lattice systems, providing a wide variety of static properties.

Abstract

A formulation of numerical real-space renormalization groups for quantum many-body problems is presented and several algorithms utilizing this formulation are outlined. The methods are presented and demonstrated using S=1/2 and S=1 Heisenberg chains as test cases. The key idea of the formulation is that rather than keep the lowest-lying eigenstates of the Hamiltonian in forming a new effective Hamiltonian of a block of sites, one should keep the most significant eigenstates of the block density matrix, obtained from diagonalizing the Hamiltonian of a larger section of the lattice which includes the block. This approach is much more accurate than the standard approach; for example, energies for the S=1 Heisenberg chain can be obtained to an accuracy of at least ${10}^{\mathrm{\ensuremath{-}}9}$. The method can be applied to almost any one-dimensional quantum lattice system, and can provide a wide variety of static properties.