Mean-Field Dynamics of the Bose-Hubbard Model in High Dimension
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912281146687488 |
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| author | Farhat, Shahnaz Périce, Denis Petrat, Sören |
| author_facet | Farhat, Shahnaz Périce, Denis Petrat, Sören |
| contents | The Bose-Hubbard model effectively describes bosons on a lattice with on-site interactions and nearest-neighbour hopping, serving as a foundational framework for understanding strong particle interactions and the superfluid to Mott insulator transition. This paper aims to rigorously establish the validity of a mean-field approximation for the dynamics of quantum systems in high dimension, using the Bose-Hubbard model on a square lattice as a case study. We prove a trace norm estimate between the one-lattice-site reduced density of the Schrödinger dynamics and the mean-field dynamics in the limit of large dimension. Here, the mean-field approximation is in the hopping amplitude and not in the interaction, leading to a very rich and non-trivial mean-field equation. This mean-field equation does not only describe the condensate, as is the case when the mean-field description comes from a large particle number limit averaging out the interaction, but it allows for a phase transition to a Mott insulator since it contains the full non-trivial interaction. Our work is a rigorous justification of a simple case of the highly successful dynamical mean-field theory (DMFT) for bosons, which somewhat surprisingly yields many qualitatively correct results in three dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05304 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean-Field Dynamics of the Bose-Hubbard Model in High Dimension Farhat, Shahnaz Périce, Denis Petrat, Sören Mathematical Physics Quantum Gases The Bose-Hubbard model effectively describes bosons on a lattice with on-site interactions and nearest-neighbour hopping, serving as a foundational framework for understanding strong particle interactions and the superfluid to Mott insulator transition. This paper aims to rigorously establish the validity of a mean-field approximation for the dynamics of quantum systems in high dimension, using the Bose-Hubbard model on a square lattice as a case study. We prove a trace norm estimate between the one-lattice-site reduced density of the Schrödinger dynamics and the mean-field dynamics in the limit of large dimension. Here, the mean-field approximation is in the hopping amplitude and not in the interaction, leading to a very rich and non-trivial mean-field equation. This mean-field equation does not only describe the condensate, as is the case when the mean-field description comes from a large particle number limit averaging out the interaction, but it allows for a phase transition to a Mott insulator since it contains the full non-trivial interaction. Our work is a rigorous justification of a simple case of the highly successful dynamical mean-field theory (DMFT) for bosons, which somewhat surprisingly yields many qualitatively correct results in three dimensions. |
| title | Mean-Field Dynamics of the Bose-Hubbard Model in High Dimension |
| topic | Mathematical Physics Quantum Gases |
| url | https://arxiv.org/abs/2501.05304 |