Tensor Network Approach To The Topological Phases And Phase Transitions In The Low-dimensional Strongly Correlated Quantum Systems

The quantum topological phases in low-dimensional strongly correlated systems are attractive in condensed matter physics,the quasi-particle excitations in the topological phases can be used to perform topological quantum computation.As the phases of matter beyond the Ginzburg-Landau-Wilson paradigm,there is a new kind of order—topological order in the topological phases of matter.It has been demonstrated that the topological order originates from quantum many-body entanglement in the ground state wavefunc-tions.In order to microscopically describe quantum many-body entanglement,a novel tool called tensor network state method has been invented.In this dissertation,we start from the tensor network state representations of ground state wavefunctions,characterize the topological phases and study their topological phase transitions.To realize topological quantum computation,a class of one-dimensional quantum many-body systems—the parafermion systems have been proposed.In the edge states of their topological phases,there exist zero modes that can be used to perform topologi-cal quantum computation.However,in previous studies,the topological phases in these systems haven’t been characterized from the perspective of quantum many-body entangle-ment.Therefore,we systematically construct the matrix product states(one-dimensional tensor network states)for the ground state wavefunctions of topological and symmetry protected topological phases of parafermion systems for the first time.From these matrix product states,we find that the topological order originates from the gauge symmetry in the entanglement degrees of freedom of the matrix product states,different gauge symme-tries and different topological phases are one-to-one correspondence.We further study the situation with extra imposed symmetry,and find that the symmetry protected topological order gives rise to degeneracy in the full entanglement spectrum.These results show how topological order emerges from quantum many-body entanglement from a novel view-point,deepen our understanding of the topological phases,and present the new methods of detecting topological order from the ground state wavefunctions.According to the universal features of the 2-dimensional bosonic topologically or-dered phases,the mathematics describing underlying topological order has been found.Due to that there is no local order parameter in the topologically ordered phases,a unified framework describing topological phase transitions is still lacking.To study topologi-cal phase transitions,we start from the ground state wavefunctions of the 2-dimensional Abelian topological states.We construct a tensor network state of a deformed wavefunc-tion with one tuning parameter,it includes two distinct topologically ordered phases and a spontaneous symmetry breaking phase,as well as the phase transitions among them.We also find the quantum-classical mapping between the norm of deformed wavefunction and the partition function of the classical eight-vertex/Ashkin-Teller model,and exactly determine the effective field theories describing the quantum critical points.Moreover,we also study the topological phase transitions of a 2-dimensional non-abelian topological state—the Fibonacci topological state.Taking advantages of the quan-tum duality,we construct a deform Fibonacci topological state with the string tension and the dual string tension,which correspond to two tuning parameters.And we map the norm of the deformed wavefunction into the partition function of a two-coupled(3+√5)/2-state Potts model.This statistical model is unusual in the sense that the degrees of freedom are irrational numbers and the non-local Boltzmann weights are negative.Using some spe-cial techniques,we also derive the tensor network state representation of the Fibonacci topological state.We obtain the global phase diagram of the deformed wavefunction and investigate the properties of the phase transition lines with the help of tensor network algorithms.We propose the new ideas for studying quantum topological phase transitions.Our results display the deep connection between topological phase transitions and statistical physics,it also inspires us to construct the theoretical framework of generic topological phase transitions.