How to define quantum mean-field solvable Hamiltonians using Lie algebras

TL;DR

This paper establishes a Lie-algebraic framework to define and identify quantum Hamiltonians that are exactly solvable within mean-field theories, including higher-order fermionic Hamiltonians and those with complex eigenstate structures.

Contribution

It introduces a general criterion for mean-field solvability of quantum Hamiltonians using Lie algebras, applicable to various particle types and Hamiltonian complexities.

Findings

Identifies mean-field solvable Hamiltonians beyond quadratic order.

Reveals some Hamiltonians require different quasi-particle rotations per eigenstate.

Provides a Lie-algebraic criterion for mean-field solvability.

Abstract

Necessary and sufficient conditions for quantum Hamiltonians to be exactly solvable within mean-field theories have not been formulated so far. To resolve this problem, first, we define what mean-field theory is, independently of a Hamiltonian realization in a particular set of operators. Second, using a Lie-algebraic framework we formulate a criterion for a Hamiltonian to be mean-field solvable. The criterion is applicable for both distinguishable and indistinguishable particle cases. For the electronic Hamiltonians, our approach reveals the existence of mean-field solvable Hamiltonians of higher fermionic operator powers than quadratic. Some of the mean-field solvable Hamiltonians require different sets of quasi-particle rotations for different eigenstates, which reflects a more complicated structure of such Hamiltonians.