Numerical Methods for Quantum Monte Carlo Simulations of the Hubbard Model ∗

One of the core problems in materials science is how the interactions between electrons in a solid give rise to properties like ∗This work was partially supported by the National Science Foundation under Grant 0313390, and Department of Energy, Office of Science, SciDAC grant DEFC02 06ER25793. Wenbin Chen was also supported in part by the China Basic Research Program under the grant 2005CB321701. 2 Bai, Chen, Scalettar, Yamazaki magnetism, superconductivity, and metal-insulator transitions? Our ability to solve this central question in quantum statistical mechanics numerically is presently limited to systems of a few hundred electrons. While simulations at this scale have taught us a considerable amount about certain classes of materials, they have very significant limitations, especially for recently discovered materials which have mesoscopic magnetic and charge order. In this paper, we begin with an introduction to the Hubbard model and quantum Monte Carlo simulations. The Hubbard model is a simple and effective model that has successfully captured many of the qualitative features of materials, such as transition metal monoxides, and high temperature superconductors. Because of its voluminous contents, we are not be able to cover all topics in detail; instead we focus on explaining basic ideas, concepts and methodology of quantum Monte Carlo simulation and leave various part for further study. Parts of this paper are our recent work on numerical linear algebra methods for quantum Monte Carlo simulations. 1 Hubbard model and QMC simulations The Hubbard model is a fundamental model to study one of the core problems in materials science: How do the interactions between electrons in a solid give rise to properties like magnetism, superconductivity, and metal-insulator transitions? In this lecture, we introduce the Hubbard model and outline quantum Monte Carlo (QMC) simulations to study many-electron systems. Subsequent lectures will describe computational kernels of the QMC simulations. 1.1 Hubbard model The two-dimensional Hubbard model [8, 9] we shall study is defined by the Hamiltonian: H = HK +Hμ +HV , (1.1) where HK , Hμ and HV stand for kinetic energy, chemical energy and potential energy, respectively, and are defined as HK = −t ∑ 〈i,j〉,σ (ciσcjσ + c † jσciσ), Hμ = −μ ∑