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The Bogoliubov-Bose-Hubbard model: existence of minimizers and absence of quantum phase transition

Norbert Mokrzański, Marcin Napiórkowski

TL;DR

The BBH framework rigorously analyzes a Bogoliubov-inspired variational model for the Bose-Hubbard lattice by introducing the BBH functional $\mathcal{F}(\gamma,\alpha,\rho_0)$ and proving the existence of minimizers in both grand canonical and canonical ensembles. Through convexity, restricted minimization, and Euler–Lagrange analysis, the authors establish a thermally driven phase transition between a superfluid (with $\alpha\neq0$ and $\rho_0$ possibly positive) and a Mott-insulator phase (with $\alpha\equiv0$ and $\rho_0=0$) at $T>0$, described by the monotone function $J(T,\theta)$. At zero temperature, for $\mu>0$ the system remains superfluid, implying no quantum phase transition with respect to $U$; the canonical case yields the same qualitative picture with a fixed density constraint. The results provide a rigorous phase diagram and demonstrate that thermal effects, not quantum fluctuations, drive the transition in this BBH setting, offering a solid bridge between Bogoliubov theory and lattice boson physics with clear implications for condensation and symmetry breaking in lattice models.

Abstract

We consider a variational approach to the Bose-Hubbard model based on Bogoliubov theory. We introduce the grand canonical and canonical free energy functionals for which we prove the existence of minimizers. By analyzing their structure we show the existence of a thermally driven phase transition by showing that the system is superfluid at sufficiently low temperatures and insulating at high temperatures. In particular, we show that this model does not exhibit a quantum phase transition.

The Bogoliubov-Bose-Hubbard model: existence of minimizers and absence of quantum phase transition

TL;DR

The BBH framework rigorously analyzes a Bogoliubov-inspired variational model for the Bose-Hubbard lattice by introducing the BBH functional and proving the existence of minimizers in both grand canonical and canonical ensembles. Through convexity, restricted minimization, and Euler–Lagrange analysis, the authors establish a thermally driven phase transition between a superfluid (with and possibly positive) and a Mott-insulator phase (with and ) at , described by the monotone function . At zero temperature, for the system remains superfluid, implying no quantum phase transition with respect to ; the canonical case yields the same qualitative picture with a fixed density constraint. The results provide a rigorous phase diagram and demonstrate that thermal effects, not quantum fluctuations, drive the transition in this BBH setting, offering a solid bridge between Bogoliubov theory and lattice boson physics with clear implications for condensation and symmetry breaking in lattice models.

Abstract

We consider a variational approach to the Bose-Hubbard model based on Bogoliubov theory. We introduce the grand canonical and canonical free energy functionals for which we prove the existence of minimizers. By analyzing their structure we show the existence of a thermally driven phase transition by showing that the system is superfluid at sufficiently low temperatures and insulating at high temperatures. In particular, we show that this model does not exhibit a quantum phase transition.

Paper Structure

This paper contains 16 sections, 33 theorems, 334 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Grand canonical approach.
  4. Canonical approach.
  5. Existence of minimizers
  6. Boundedness from below
  7. Convexity and double minimization
  8. Restricted problem and the existence of restricted minimizers
  9. Variational derivatives
  10. Positive temperature case
  11. Non-zero condensate density case
  12. Zero condensate density case
  13. Zero temperature case
  14. The phase transition and phase diagram
  15. Proof of the theorem in the canonical setting
  16. ...and 1 more sections

Key Result

Theorem 1

For any $U > 0$, $\mu \in \mathbb{R}$ and $T \ge 0$ there exists a minimizer $(\gamma^{\min}, \alpha^{\min}, \rho_0^{\min})\in\mathcal{D}$ of the BBH functional BBH. If $\mu\leq 0$, then the minimizer is unique. Furthermore, for every minimizer the following equivalence holds: $\rho_0^{\min} = 0$ if

Figures (1)

  • Figure 1: Qualitative grand canonical phase diagrams of the model. The boundaries of the unknown area (marked as $"?"$) are given by the relations $J(T,0) = \frac{\mu}{2U}$ and $J(T,4\mu + 2U) = \frac{\mu}{2U}$.

Theorems & Definitions (67)

  • Theorem 1
  • Definition
  • Theorem 2
  • Remark 2.1
  • Theorem 3
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • ...and 57 more