Hierarchical mean-field theory in quantum statistical mechanics:  A bosonic example

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.