Numerical studies of correlated electron systems

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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.