The He II Theory Preserving the Symmetry of the Initial Hamiltonian of the System

TL;DR

This paper introduces a new microscopic He II theory that preserves the initial Hamiltonian's symmetry, providing a more accurate description of the ground state and excited states, especially in complex geometries.

Contribution

The paper presents a novel method for constructing the atom number operator in He II theory, differing from Bogoliubov's approach by maintaining operator non-commutativity.

Findings

Predictions align with Bogoliubov theory in simple geometries.

Differences arise in complex geometries due to non-commuting operators.

The new theory impacts calculations of helium flow through tunnel transitions.

Abstract

The article suggests a method for the construction of the number of atoms preserving microscopic He II theory. The suggested theory can provide the ground state wave function (WF) as an expansion in series by small parameters. In addition, errors in the definition of the WFs of the excited states turn out to be vanishingly small while the system size increases. Predictions of the proposed theory and of the Bogoliubov theory are identical if helium occupies a simply connected volume. However, there are differences in the general case. These differences are due to the fact that the operator of the occupation number of the zero momentum state does not commute with the creation and annihilation operators of phonons. This contradicts the Bogoliubov assumption that the creation and annihilation operators of atoms in the zero momentum state can be replaced by a c-number. The latter, as is known, should commute with any operator. This should lead to a difference in the results of calculation of helium flow through the tunnel transition. Therefore, the suggested theory is not equivalent to Bogoliubov theory.