Numerical studies of correlated electron systems

本研究的目的是发展和研究数值方法来解决相关电子问题。我们将考虑Hubbard型哈密顿量来描述相关电子系统的低能物理，如过渡金属氧化物和有机化合物。随机方法的困扰的符号问题，导致所需的数值工作的指数增加的系统大小和逆温度的函数。我们的目标是开发和研究近似方案，以规避这个符号问题。我们将研究的方法之间的共同点是，它们都能够以最小的代价再现鞍点（或平均场）结果。在某种意义上，它们可以被看作是考虑平均场解周围量子涨落的一种系统方法。更具体地说，我们将集中讨论三种算法。i）从任意晶格拓扑和带填充的哈伯德模型出发，我们可以通过将费米子味的数量从2增加到N来推广它。作为N的增长值和推广性质的函数，我们将流到给定的鞍点或平均场解。作为N的函数，符号问题变得越来越不严重，从而使我们能够在越来越大的晶格上进行模拟。在这个框架内，我们将调查条纹相位和d-密度波状态超出平均场近似。ii）最近引入了新的算法来数值求解中间维的Hubbard型模型。我们将集中在两个方法：路径积分重整化群方法（PIRG）以及约束路径量子蒙特卡罗算法（CPQMC）。这两种方法都没有符号问题，但都有自己的集合近似。特别是，PIRG方法已被广泛用于研究中维的Mott金属-绝缘体转变。我们的目标是进一步发展和调查这些方法的可靠性。

The aim of this research proposal is to develop and investigate numerical approaches to tackle the correlated electron problem. We will consider Hubbard type hamiltonians to describe the low energy physics of correlated electron systems such as transition metal oxides and organic compounds. Stochastic approaches are plagued by the sign problem which results in an exponential increase of the required numerical effort as a function of system size and inverse temperature. Our goal is to develop and investigate approximate schemes to circumvent this sign problem. The common point between the methods we will investigate are that they are all capable - at a minimal cost - to reproduce saddle point (or mean-field) results. They may in a certain sense, be seen as a systematic way of taking into account quantum fluctuations around the mean-field solution. To be more specific, we will concentrate on three algorithms. i) Starting from the Hubbard model on arbitrary lattice topologies and band-fillings we can generalize it by enhancing the number of fermion flavors from two to N. As a function of growing values of N and the nature of the generalization, we will flow to a given saddle point or mean-field solution. As a function of N the sign problem becomes less and less severe thus allowing us to carry out simulations on increasingly large lattices. Within this framework we will investigate both stripes phases and d-density wave states beyond the mean-field approximation. ii) New algorithms have recently been introduced to solve nummerically Hubbard type models in intermediate dimensions. We will concentrate on two methods: the Path Integral Renormalization Group approach (PIRG) as well as the constrained path quantum Monte Carlo algorithm (CPQMC). Both methods are free of the sign problem but come with their own set approximations. In particular, the PIRG method has been extensively used to investigate the Mott metal-insulator transition in intermediate dimensions. Our aim is to further develop and investigate reliability of those approaches.