论文标题
哈伯德模型通过随机演算和吉尔萨诺夫转换的动力学
The Dynamics of the Hubbard Model through Stochastic Calculus and Girsanov Transformation
论文作者
论文摘要
作为典型的量子许多身体问题，我们考虑了Bose-Hubbard模型中密度矩阵元素的时间演变。对于任意初始状态，可以从SDE或随机微分方程系统获得这些数量。对于此SDE系统，可以应用Girsanov转换。这具有从初始状态中的所有信息移入转换系统的平均场部分。在G = Un固定的大n极限中，转换系统的扩散部分消失了，结果，确切的量子动力学由ODE系统给出，该系统证明是时间依赖于时间依赖的离散Gross Pitaevskii方程。对于两个站点的玻色 - 哈伯德模型，GP方程还原为数学摆和粒子不平衡等于振荡性的摆速度，或者可以具有滚动，然后对应于自我捕获或绝缘相。作为一种副产品，我们还发现具有四分之一双井潜力的数学摆的等效性。崩溃和复兴是一个更微妙的现象，为了看到这些现象是SDE系统的扩散部分，或者必须考虑量子校正。这可以通过近似和崩溃来完成，并且可以通过数值和分析计算复制复兴。由于菲涅耳或维也纳扩散过程的期望值，我们可以完全以这种方式编写密度矩阵元素，可以从抛物线二阶PDE中获得，因此我们还获得了各种精确的PDE表示。该论文的目的是为量子制定许多身体系统的有效计算方案而编写，因此形式主义是通用的，适用于任意维度，任意跳跃矩阵以及适当调整的量子。
As a typical quantum many body problem, we consider the time evolution of density matrix elements in the Bose-Hubbard model. For an arbitrary initial state, these quantities can be obtained from an SDE or stochastic differential equation system. To this SDE system, a Girsanov transformation can be applied. This has the effect that all the information from the initial state moves into the drift part, into the mean field part, of the transformed system. In the large N limit with g=UN fixed, the diffusive part of the transformed system vanishes and as a result, the exact quantum dynamics is given by an ODE system which turns out to be the time dependent discrete Gross Pitaevskii equation. For the two site Bose-Hubbard model, the GP equation reduces to the mathematical pendulum and the particle imbalance is equal to the velocity of that pendulum which is either oscillatory or it can have rollovers which then corresponds to the self trapping or insulating phase. As a by-product, we also find an equivalence of the mathematical pendulum with a quartic double well potential. Collapse and revivals are a more subtle phenomenom, in order to see these the diffusive part of the SDE system or quantum corrections have to be taken into account. This can be done with an approximation and collapse and revivals can be reproduced, numerically and also through an analytic calculation. Since expectation values of Fresnel or Wiener diffusion processes, we write the density matrix elements exactly in this way, can be obtained from parabolic second order PDEs, we also obtain various exact PDE representations. The paper has been written with the goal to come up with an efficient calculation scheme for quantum many body systems and as such the formalism is generic and applies to arbitrary dimension, arbitrary hopping matrices and, with suitable adjustments, to fermionic models.
