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Matrix product state recursion methods for strongly correlated quantum systems

  • Author(s): Tian, Yifan
  • Advisor(s): White, Steven R
  • et al.
Abstract

The main content of this dissertation consists of two projects that I studied during my graduate career. Among them, the first project has more successful results and thus will be the major content of this dissertation. Besides discussing the details of each project, this dissertation also introduces the background knowledge and fundamental techniques of this area.

The first project is "recursion methods for strongly correlated quantum system}, which is also the main project". In this project, we present a method for extrapolation of real-time dynamical correlation functions which can improve the capability of matrix product state methods to compute spectral functions. Unlike the widely used linear prediction method, which ignores the origin of the data being extrapolated, our recursion methods utilize a representation of the wavefunction in terms of an expansion of the same wavefunction and its translations at earlier times. This recursion method is exact for noninteracting Fermi system. Surprisingly, the recursion method is also more robust than linear prediction at large interaction strength. We test this method on the Hubbard two-leg ladder, and present more accurate results for the spectral function than previous studies.

The second project is "Mapping the Hubbard model to the t-J model using ground state unitary transformations". The effective low-energy models of the Hubbard model are usually derived from perturbation theory. Here we derive the effective model of the Hubbard model in spin space and t-J space using a unitary transformation from numerical optimization. We represent the Hamiltonian as Matrix product state(MPO) and represent the unitary transformation using gates according to tensor network methods. We obtain this unitary transformation by optimizing the unitary transformation between the ground state of the Hubbard model and the projection of the Hubbard model ground state into spin space and t-J space. The unitary transformation we get from numerical optimization yields effective models that are in line with perturbation theories. This numerical optimization method starting from ground state provides another approach to analyze effective low-energy models of strongly correlated electron systems.

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