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Ground state energy of the Bose--Hubbard model with large coordination number with a polaron-type quantum de Finetti theorem

Shahnaz Farhat, Denis Périce, Sören Petrat

Abstract

We consider the ground state energy of the Bose--Hubbard model on a graph with large and homogeneous coordination number. In the limit of infinite coordination number, we prove convergence of the ground state energy to the minimizer of a mean-field energy functional. This functional is obtained by averaging the hopping term over the large number of connected sites, while the interaction energy is not averaged. Hence, the resulting mean-field description is in the strong coupling regime, and is expected to provide a qualitatively correct picture of the phase diagram of the Bose--Hubbard model for large enough coordination number. For our proof, we develop a new version of a de Finetti-type theorem, which we call the polaron-type quantum de Finetti theorem, and which we expect to be a more broadly useful extension of existing quantum de Finetti results. Our theorem covers the case where the Hilbert space is a tensor product of some Hilbert space with a bosonic Fock space. This theorem is applied to the convergence of the ground state energy of the Bose--Hubbard model after reducing it to a polaron-type model.

Ground state energy of the Bose--Hubbard model with large coordination number with a polaron-type quantum de Finetti theorem

Abstract

We consider the ground state energy of the Bose--Hubbard model on a graph with large and homogeneous coordination number. In the limit of infinite coordination number, we prove convergence of the ground state energy to the minimizer of a mean-field energy functional. This functional is obtained by averaging the hopping term over the large number of connected sites, while the interaction energy is not averaged. Hence, the resulting mean-field description is in the strong coupling regime, and is expected to provide a qualitatively correct picture of the phase diagram of the Bose--Hubbard model for large enough coordination number. For our proof, we develop a new version of a de Finetti-type theorem, which we call the polaron-type quantum de Finetti theorem, and which we expect to be a more broadly useful extension of existing quantum de Finetti results. Our theorem covers the case where the Hilbert space is a tensor product of some Hilbert space with a bosonic Fock space. This theorem is applied to the convergence of the ground state energy of the Bose--Hubbard model after reducing it to a polaron-type model.

Paper Structure

This paper contains 11 sections, 10 theorems, 155 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context
  3. Model
  4. Main results
  5. Ground state energy convergence
  6. Upper energy bound
  7. Translation invariance
  8. Lower energy bound
  9. Polaron-type quantum de Finetti theorems
  10. Finite dimensional approximation
  11. Fock space localization

Key Result

Theorem 2

If $U > 0$ and $J \ge 0$, then $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Mott insulator (MI) / Superfluid (SF) phase diagram PhysRevB.40.546. The diagram is obtained by minimizing the mean-field energy functional \ref{['mf_energy_func']}. Provided a ground state $\varphi(J, \mu, U)$, the Mott insulator phase is defined as region where $\alpha_\varphi(J, \mu, U) = 0$.

Theorems & Definitions (29)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Definition 5: Multiple-species bosonic states and reduced density matrices
  • Definition 6: Multiple-species infinite bosonic states
  • Theorem 7: Polaron quantum De Finetti in the limit $N\to\infty$
  • Proposition 8: Upper energy bound
  • proof
  • Proposition 9: Energy reduction via translation invariance
  • ...and 19 more