Some remarks on the coherent-state variational approach to nonlinear boson models

TL;DR

This paper compares different mean-field variational schemes for nonlinear many-boson systems, analyzing their algebraic structures, relationships, and applications, with a focus on coherent-state approaches and their properties.

Contribution

It provides a detailed comparison of Gutzwiller-like, number-preserving, and Glauber-like variational states, revealing their algebraic relations and connections to coherent states in bosonic models.

Findings

States |Z> can be expressed as superpositions of |ξ> states.

|ξ> states are shown to be SU(M) coherent states.

Derived Hamiltonian structures for different variational schemes.

Abstract

The mean-field pictures based on the standard time-dependent variational approach have widely been used in the study of nonlinear many-boson systems such as the Bose-Hubbard model. The mean-field schemes relevant to Gutzwiller-like trial states $|F>$, number-preserving states $|\xi >$ and Glauber-like trial states $|Z>$ are compared to evidence the specific properties of such schemes. After deriving the Hamiltonian picture relevant to $|Z>$ from that based on $|F>$, the latter is shown to exhibit a Poisson algebra equipped with a Weyl-Heisenberg subalgebra which preludes to the $|Z>$-based picture. Then states $|Z>$ are shown to be a superposition of $\cal N$-boson states $|\xi>$ and the similarities/differences of the $|Z>$-based and $|\xi>$-based pictures are discussed. Finally, after proving that the simple, symmetric state $|\xi>$ indeed corresponds to a SU(M) coherent state, a dual version of states $|Z>$ and $|\xi>$ in terms of momentum-mode operators is discussed together with some applications.