Tensor Network Algorithms From Bosonic System To Fermionic System

Strongly correlated quantum many-body systems pose some of the most exciting open problems in physics. Model systems such as the Hubbard model are thought to play an important role in the description of high-Tc superconductivity. Many attentions have been paid recently in condensed-matter physics with topological order, or are in symmetry protected or enriched topological phases. The rapid development in the field is accompanied by a deepened theoretical understanding of interacting quantum many-body systems.A particularly fresh approach to the problem is the theoretical programme based on the so called tensor network states, classes of physical states described by a rela-tively small number of parameters, which nevertheless capture natural ground states of interacting Hamiltonian models well.In this thesis, we accomplished three main works, including the development of Tensor Network methods and the application of the methods to the strongly correlated many-body systems:1. Developed a user-friendly package for the Tensor Network state methods.The tensor network states (TNS) methods have proven to be very powerful tools to investigate the strongcorrelated many-particle physics in one and two dimensions. The implementation of TNS methods depend heavily on the operations of tensors, including contraction, permutation, reshaping tensors, SVD and son on. Unfortunately, the most popular computer languages for scientific computation, such as Fortran and C/C++ do not have standard library for such operations, and therefore make the coding of TNS very tedious. We develop a Fortran2003 package that includes all kinds of basic tensor operations designed for TNS. It is user friendly and flexible for different forms of TNS,and greatly simplifies the coding of TNS methods.2. Extended the Boson PEPS method to Fermion PEPS methodIn the absence of exactly solvable models, accurate numerical simulations are es-sential in order to gain further insight into the physics of strongly correlated systems.While quantum Monte Carlo techniques are very powerful in simulating bosonic sys-tems, they suffer from the so-called negative sign problem in the case of fermionic and frustrated models. On the other hand, generic one-dimensional lattice systems can be accurately addressed with the density matrix renormalization group (DMRG) method,but this approach scales inefficiently with the lattice size in 2D systems. The PEPS method can in-principle solve the Fermion problem. We developed a simple method that can extend the Boson PEPS method to Fermion PEPS method.3. Apply TNS method to the extended Bose-Hubbard model on square lattice with frustrated tunneling.By using a state of art TNS method, we study the ground-state phase diagram of an extended Bose-Hubbard model on square lattice with frustrated next-nearest neigh-boring tunneling. In the hardcore limit, tunneling frustration stabilizes a peculiar half supersolid (HSS) phase with one sublattice being superfluid and the other Mott-Insulator away from half filling. A new phase separation regime composes of the HSS and su-perfluid phases is also identified. In the softcore case, the model show very rich phase diagrams above half filling, including three different types of supersolid phases depend-ing on the interaction parameters. The considered model provides a promising route to search for novel stable SS state induced by frustrated tunneling even in below half filling region, which can be implemented experimentally with dipolar atoms or molecules.