Projector formalism for kept and discarded spaces of matrix product states

TL;DR

This paper introduces a projector formalism and diagrammatic notation to explicitly characterize kept and discarded spaces in matrix product states, enhancing the formulation of MPS algorithms and enabling efficient computation of energy variances and excitations.

Contribution

It presents a novel projector formalism and diagrammatic notation for analyzing kept and discarded spaces in MPS, improving algorithmic capabilities.

Findings

Derived an explicit expression for the $n$-site energy variance.

Evaluated the energy variance numerically for a long-range hopping model.

Developed an efficient algorithm for low-lying $n$-site excitations.

Abstract

Any matrix product state $|\Psi\rangle$ has a set of associated kept and discarded spaces, needed for the description of $|\Psi\rangle$, and changes thereof, respectively. These induce a partition of the full Hilbert space of the system into mutually orthogonal spaces of irreducible $n$-site variations of $|\Psi\rangle$. Here, we introduce a convenient projector formalism and diagrammatic notation to characterize these $n$-site spaces explicitly. This greatly facilitates the formulation of MPS algorithms that explicitly or implicitly employ discarded spaces. As an illustration, we derive an explicit expression for the $n$-site energy variance and evaluate it numerically for a model with long-range hopping. We also describe an efficient algorithm for computing low-lying $n$-site excitations above a finite MPS ground state.