Tensor Network Algorithms And Their Applications In Dynamics Of Quantum Many-body Systems

Strongly correlated many-body physics,which gave rise to many physical systems of interest such as high temperature superconductor and spin-liquid systems,lies in the core of condensed matter physics.Analytically and numerically,these systems with strong correlations are hard to study,since their Hilbert space grows exponentially with the number of particles in the system.However,in the recent 20 years,based on a deeper understanding of the most important properties in quantum systems ? correlation and entanglement,and also the rapid development of numerical tools such as Density Matrix Renormalization Group,Tensor Network States and Time Dependent Variational Principle,we can finally study the most concerned topics in many-body physics—the properties of ground state and low-energy states,and quantum dynamics.In this thesis,we accomplished three main works,starting from the development of Tensor Network to applying Tensor Network on quantum dynamics:1.Based on the definition of geometry of relative entropy,by applying replica-exchange Monte Carlo algorithm,we developed a unified numerical solution for all kinds of correlationsIn order to study different quantum systems,all kinds of definitions for entangle-ment and correlations have been introduced.While these definitions work well in cer-tain areas,a unified measurement for all kinds of correlations is still needed to compare one type of correlation to another.In order to solve this problem,based on the defini-tion of geometry of relative entropy,we transform all measurements of correlations into minimization problems in certain Hilbert space.By applying replica-exchange Monte Carlo algorithm,we are able to achieve efficient solutions to all kinds of correlations defined by relative entropy.This method acts as a complement to the analytical solution which is usually hard to achieve,and paves way for the unified definition of correlations.2.Combine Size Consistency and Area Law as a new set of principles for Tensor Network constructionFor a gapped Hamiltonian,take A as a part of its ground state,the entanglement between A and the rest of the state is determined by the surface of A.This principle,which is called Area Law,is believed to be the explanation of DMRG's success in 1-D systems and also the gold standard for construction of Tensor Networks.However,we realized that Area Law is not the only standard for Tensor Network.It only describes the entanglement,but not the energy additivity.By combining Size Consistency and Area Law together,we build a new set of principles for Tensor Network construction that is more comprehensive.3.Study Kibble-Zurek Mechanism under long-ranged correlation with Ma-trix Product State and Time Dependent Variational PrincipleRecently,on one hand,the rapid progresses of experimental technologies for ion trap have made it possible to study dynamics of quantum many-body systems in exper-iments.On the other hand,by choosing suitable Tensor Network and applying Time Dependent Variational Principle,we could reliably simulate the real-time evolution of 1-D systems with up to 100 sites for more than 500 unit time.Based on these develop-ments,we study Kibble-Zurek Mechanism under long-ranged correlation and provide a new view for Kibble-Zurek Mechanism in quantum realm.