Hierarchical Mean-Field Theories in Quantum Statistical Mechanics

TL;DR

This paper introduces a hierarchical mean-field framework for analyzing phase coexistence and competition in quantum systems, enabling the study of complex phase diagrams and critical points with an approximate but comprehensive approach.

Contribution

The paper develops a hierarchical mean-field method that captures all competing thermodynamic phases simultaneously in quantum statistical mechanics.

Findings

Determined the phase diagram of a bosonic lattice model.

Identified quantum critical points involving competing orders.

Demonstrated the method's effectiveness in complex phase analysis.

Abstract

We present a theoretical framework and a calculational scheme to study the coexistence and competition of thermodynamic phases in quantum statistical mechanics. The crux of the method is the realization that the microscopic Hamiltonian, modeling the system, can always be written in a hierarchical operator language that unveils all symmetry generators of the problem and, thus, possible thermodynamic phases. In general one cannot compute the thermodynamic or zero-temperature properties exactly and an approximate scheme named ``hierarchical mean-field approach'' is introduced. This approach treats all possible competing orders on an equal footing. We illustrate the methodology by determining the phase diagram and quantum critical point of a bosonic lattice model which displays coexistence and competition between antiferromagnetism and superfluidity.