arXiv 2405.040552024-05-07math-phmath.MP

有限图上 Bose-Hubbard 模型的多体量子 Gibbs 态展开

Expansion of the Many-body Quantum Gibbs State of the Bose-Hubbard Model on a Finite Graph

Zied Ammari, Shahnaz Farhat, Sören Petrat

AI总结：

针对有限图上 Bose-Hubbard 模型的平均场极限，本文证明多体 Gibbs 态关于逆温度的任意阶展开，并给出系数递归公式及前两项显式计算。

摘要：

我们考虑正温度下有限图上 Bose-Hubbard 模型的多体量子 Gibbs 态。我们用逆温度对相互作用进行标度，这对应于温度处于平均粒子数量级的平均场极限。对于该模型，已知当温度趋于无穷大时，多体 Gibbs 态收敛到离散非线性 Schrödinger 方程的 Gibbs 测度，即依据单体理论定义的 Gibbs 测度。本文中，我们通过证明以逆温度为小参数的多体 Gibbs 态任意阶展开来扩展这些结果。展开中的系数可利用递归公式计算为真空期望值，并且我们显式计算了前两个系数。

英文摘要：

We consider the many-body quantum Gibbs state for the Bose-Hubbard model on a finite graph at positive temperature. We scale the interaction with the inverse temperature, corresponding to a mean-field limit where the temperature is of the order of the average particle number. For this model it is known that the many-body Gibbs state converges, as temperature goes to infinity, to the Gibbs measure of a discrete nonlinear Schrödinger equation, i.e., a Gibbs measure defined in terms of a one-body theory. In this article we extend these results by proving an expansion to any order of the many-body Gibbs state with inverse temperature as a small parameter. The coefficients in the expansion can be calculated as vacuum expectation values using a recursive formula, and we compute the first two coefficients explicitly.

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