Hyperoctahedral Chen calculus for effective Hamiltonians

TL;DR

This paper introduces a hyperoctahedral group algebra framework for computing effective Hamiltonians, providing new algebraic tools and series expansions relevant to quantum physics and chemistry applications.

Contribution

It develops a novel algebraic structure based on hyperoctahedral groups for effective Hamiltonian calculations, extending existing algebraic methods.

Findings

Effective Hamiltonian computations are best modeled by hyperoctahedral group algebras.

The adiabatic evolution operator admits a Picard-type series and exponential expansion.

New algebraic structures facilitate applications in quantum physics and chemistry.

Abstract

The algebraic structure of iterated integrals has been encoded by Chen. Formally, it identifies with the shuffle and Lie calculus of Lyndon, Ree and Sch\"utzenberger. It is mostly incorporated in the modern theory of free Lie algebras. Here, we tackle the problem of unraveling the algebraic structure of computations of effective Hamiltonians. This is an important subject in view of applications to chemistry, solid state physics, quantum field theory or engineering. We show, among others, that the correct framework for these computations is provided by the hyperoctahedral group algebras. We define several structures on these algebras and give various applications. For example, we show that the adiabatic evolution operator (in the time-dependent interaction representation of an effective Hamiltonian) can be written naturally as a Picard-type series and has a natural exponential expansion.