Particle Number Conservation and Block Structures in Matrix Product States

TL;DR

This paper explores how particle number conservation manifests as block sparsity in matrix product states and operators, providing methods for efficient numerical algorithms in quantum physics.

Contribution

It introduces a general framework linking particle number conservation to block structures in matrix product states and operators, with explicit constructions and algorithmic implications.

Findings

Block sparsity characterizes eigenvectors of particle number operators.

Block structures enable efficient operations and rank truncation.

Explicit constructions of particle operators in matrix product form.

Abstract

The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. This is shown to generalize to other classes of operators. Imposing block sparsity yields a scheme for conserving the particle number that is commonly used in applications in physics. Operations on such block structures, their rank truncation, and implications for numerical algorithms are discussed. Explicit and rank-reduced matrix product operator representations of one- and two-particle operators are constructed that operate only on the non-zero blocks of matrix product states.