Quantum Phase Transition And The Ground-state Fidelity From Tensor Network Representations

The approximation ground-state wave functions of the one/two-dimensionalquantum systems in the thermodynamic limit are obatined by employing the infinitetensor net-work algorithm. The bifurcation in ground-state fidelity per lattice site of spinquantum system has been studied in this thesis, from which the transiton point and thesymmetry of systems are obtained exactly. Meanwhile, the local order parameter isextracted from the reduced density martix. Some physical observables of theone-dimensional quantum has been also obtained. The continuous phase transitions ofthe spin quantum systems in the thermodynamic limit are better understood.The quantum models have been studied in this thesis include: one-dimensionalspin-1XXZD model, one-dimensional spin-1/2two-spin and three-spin interactioncompeting model, one-dimensional spin-1/2the nearest neighbor and the next nearestneighbor interaction Heisenberg model, one-dimensional spin-1/2antiferromagnetic andferromagnetic alternating interaction Heisenberg model, two-dimensional Ising model.The approximation ground-state wave functions for these quantum models have beenobtained by employing the infinite tensor net-work algorithm: infinite matrix productstate for one spatial dimension; infinite project entanglement-pair states for two spatialdimesion. Physical observables of the quantum models measured in this thesis include:von Neumann entropy, local order parameter, the ground-state fidelity per lattice site,scaling relation between von Neumann entropy of critical point and truncationdimensions, divergence exponent for correlation length of critical point with finitetruncation dimensions for one spatial dimension, the ground-state fidelity per lattice sitefor two spatial dimension quantum Ising model, from which we can confirm that theresults measured in the corresponding quantum models.On the other hand, the infinte tensor network algorithm is also employed to studythe spin quantum system with Kosterlitz-Thouless phase transition, which transferredfrom gapped system to gapless system, for the first time. The spin quantum systemsexisting Kosterlitz-Thouless phase transitions include one-dimensional spin-1XXZDmodel, one-dimensional spin-1/2the nearest neighbor and the next nearest neighborinteraction model. When the infinite tensor network algorithm is used to conductnumerical simulation, introduce of finite truncation dimension may cause that theone-dimensional quantum system in the thermodynamic limit appears continuous spontaneous symmetry breaking, which is in contradiction with the Mermin-Wegnertheorem. To deal with this contradiction, we propose two concepts, i.e., fakespontaneous symmetry breaking and pseudo-order paeameter. The computational resultsreveal that the fake spontaneous symmetry breaking and pseudo-order parameter willvanish as the truncation dimension being infinite. In this thesis, we have studiedone-dimensional spin-1/2two-spin and three-spin interaction competing model,one-dimensional spin-1/2the nearest neighbor and the next nearest neighbor interactionHeisenberg model which has the frustration in spin configuration, and one-dimensionalspin-1/2antiferromagnetic and ferromagnetic alternating interaction Heisenberg model,which is resort to the translation three-site and four-site invariance of the infinitetensor-net work algorithm, respectively. The central charge of the Haldane-Isingtransition of one-dimensional spin-1/2antiferromagnetic and ferromagnetic alternatinginteraction Heisenberg model is obtained for the first time. The two-dimensionalquantum Ising model is an important model on the theory of two-dimensional strongcorrelation system. The two-dimensional infinite tensor network algorithm is extandedto two-dimensional situation, in which the ground-state fidelity per lattice site for thetwo-dimensional quantum Ising model is calculated and graphed.Besides, the ground-state degenerate degree for one-dimensional spin-1/2two-spinand three-spin interaction competing model is analysed by using the infinite tensornet-work algorithm. The bifurcation in the ground state fidelity per lattice site is causedby the (fake) spontaneous symmetry breaking in quantum system. The bifurcation canbe used to distinguish the degenerate states in the (fake) spontaneous symmetrybreaking phase, and the catastrophe point of the bifurcation is the critical point of thesystem as the truncation dimension to be infinite.Finally, we can safely draw the conclusion that the ground-state fidelity per latticesite is able to characterize quantum phase transitions, which provides a powefulunifying framework for understanding of the complex quantum systems, regardless ofwhat type of internal order and the dimension of the quantum system are present inquantum many-body states.