A novel general mapping for bosonic and fermionic operators in Fock space

TL;DR

This paper introduces a new, general method for efficiently computing the action of bosonic and fermionic operators on state vectors in Fock space, avoiding matrix representations and enabling scalable simulations.

Contribution

The authors propose a novel enumeration-based approach to represent configurations, simplifying calculations of operator actions on state vectors in second quantization.

Findings

Significantly reduces computational complexity for operator actions.

Enables simulation of larger many-particle systems.

Provides explicit formulas for various fermionic and bosonic systems.

Abstract

In this paper we provide a novel and general way to construct the result of the action of any bosonic or fermionic operator represented in second quantized form on a state vector, without resorting to the matrix representation of operators and even to its elements. The new approach is based on our proposal to compactly enumerate the configurations (i.e., determinants for fermions, permanents for bosons) which are the elements of the state vector. This extremely simplifies the calculation of the action of an operator on a state vector. The computations of statical properties and of the evolution dynamics of a system become much more efficient and applications to systems made of more particles become feasible. Explicit formulations are given for spin-polarized fermionic systems and spinless bosonic systems, as well as to general (two-component) fermionic systems, two-component bosonic systems, and mixtures thereof.