Tensor Network States Algorithm For Two-dimensional Quantum Many-body Systems

Developing efficient algorithms to simulate strongly correlated quantum many-body systems is in the center of the modern condensed matter physics.In the context of strongly interacting systems,where the conventional perturbation theory fails,re-vealing their physical nature is mainly dependent on the numerical simulation methods,such as,exact diagonalization,quantum Monte Carlo(QMC)method and density ma-trix renormalization group(DMRG).These numerical methods have been widely used in studying strongly correlated quantum systems and have achieved great success.How-ever,developing new efficient algorithms is still urgent,because of the limitations of the previous methods:e.g.,ED encounters the so-called "Exponential Wall";QMC suf-fers from the notorious sign problem for fermionic and frustrated systems;and DMRG is limited to 1D or quasi-1D systems and does not work well for higher dimension sys-tems.Recently,inspired by the insight of quantum entanglement in the perspective of quantum information theory,the algorithms based on the tensor network states(TNS),particularly,matrix product states(MPS)and projected entangled pair states(PEPS),which is a natural extension of MPS to higher dimensions that satisfies both area law and size consistency,have been proved to be powerful simulation methods to exploit the strongly correlated systems.Based on MPS representation people build a complete theory to describe 1D systems.But for the 2D systems,there are still some difficulties hindering the power of the simulation due to the complexity of PEPS and our limited computing capability.We hope to develop an efficient method to optimize PEPS wave functions aiming to make PEPS applicable for some complicated sysmtes.Based on TNS representation for 2D systmes,the thesis describe some methods developed by us to optimize the ground states of quantum many-body systems,and it is composed of two main parts.The first part describes how to use replica exchang molecular dynamics(REMD)method to escape the local minimum when optimizing TNS wave funtions.When using TNS as variational wave functions to obtain the ground states,a very key step is how to optimize TNS efficiently to avoid being trapped into local minimum.We developed the REMD method for the optimization TNS.Treating tensor elements as generalized coordinates,we can map the optimization problem with respect to energy function to a classical mechanic system.Before the optimization starts,we set a series of different temperatures.According to the energy function which is the potential energy of the classical system,we start from a random state and simulate the systems by molecular dynamics(MD)method.At last we will get a series of sulutions at different temper-atures and the zero temperuature sulution is what we need.During the MD procees,replica exchage method is adopted to heip the system exscape from the local minimum.In the second part,we introduce an efficent optimization method of projected en-tangled pair states(PEPS)to solve 2D systems.PEPS is a good representation of the ground state of 2D systems.But because of compuational complexity,the practical application of PEPS is limited to small bond dimension D of PEPS,which may not correctly capture the physics of some complicted systems.We demonstrate a gradient optimization method combing Monte Carlo sampling technique can be used optimize PEPS wave function efficiently.We first obtain a rough ground state by simple update imaginary time evolution,then use energy gradients with respect to tensor elements to optimizate the rough states further to get an accurate groud state.When calculating energies and gradients Monte Carlo sampling technique is adopted.Compared with pre-vious methods,our method has a very low computational scaling,and is more accurate to optimize PEPS,which has great potential to solve some long standing problems.