arXiv 0904.30752009-04cond-mat.quant-gascond-mat.othercond-mat.stat-mech

Bethe格点上Bose-哈伯德模型的精确解

Exact solution of the Bose-Hubbard model on the Bethe lattice

Guilhem Semerjian, Marco Tarzia, Francesco Zamponi

备注：: 27 pages, 16 figures, minor corrections
发表信息：: Phys. Rev. B 80, 014524 (2009)
DOI：: 10.1103/PhysRevB.80.014524

摘要：

量子Bethe晶格模型在热力学极限下的精确解相当于求解一个泛函自洽方程。本文给出了Bethe格点上玻色-哈伯德模型的两种等价形式的方程。第一种方法基于相干态路径积分，在大连通性极限中导致Fisher等人[Phys. Rev. B { \bf 40}，546（1989）]在首阶的平均场处理，并导致玻色子动力学平均场论作为第一修正，如最近由Byczuk和Vollhardt [Phys. Rev. B { \bf 77}，235106（2008）]。我们得到了另一种形式的方程使用的占领数表示，可以很容易地解决了任意的数值精度，任何有限的连接。因此，我们计算模型的超流和莫特绝缘体阶段之间的过渡线，随着热力学可观的空间和虚时间依赖的关联函数。贝特晶格的有限连通性相对于无限连通的对应晶格而言，包含了更丰富的物理内容：晶格中位置之间的距离概念得以保留，并且玻色子在莫特绝缘体相中仍然是弱移动的。Bethe格结构可以被看作是有限维模型的一个近似。我们的预测和量子蒙特卡罗模拟的结果在二维和三维之间确实表现出定量合理的协议。

英文摘要：

The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent state path integral, leads in the large connectivity limit to the mean field treatment of Fisher et al. [Phys. Rev. B {\bf 40}, 546 (1989)] at the leading order, and to the bosonic Dynamical Mean Field Theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B {\bf 77}, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.

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