Theoretical Studies Of Quantum Phase Transition In Two-dimensional Many-body Systems

Quantum information science is a new interdisciplinary subject combining quantum mechanics and information science.In recent years,quantum information science has not only brought new fire to traditional information science,but also brought new breakthroughs to all fields of physics.Many unique quantum resources were also created.Quantum many-body system is an effectice physical carrier to achieve quantum information.The understanding of quantum phase transitions and its controal have been an active research field of physics over many decades.However,recognizing the order parameters and the forms of symmetry breaking,a traditional approach to describe quantum phase transitions,as well as an exponential scale relationship between the number of particles and the Hilbert space have become two major obstacles to the exploration of quantum phase transitions in many-body systems.Fortunately,it has been shown that concepts from quantum information science can be widely exploited in the research of various quantum phase transitions without any prior knowledge of order parameters and symmetry breaking,providing a new perspective for the study of quantum phase transitions.In addition,with the development of computational methods,the density matrix renormalization group algorithm has become a powerful numerical tool for computing quantum many-body systems,partly solving the problem of space dimensions.Recently,a new strategy for reducing this complexity is to uses machine learning algorithms to solve physical problems.Specifically,it was suggested that quantum states represented by neural networks can potentially exhibit a volume-law of entanglement,which indicates a more powerful representation ability than tensor networks.In this paper,we first apply the the density matrix renormalization group method to study the quantum correlation and quantum phase transition in two-dimensional many-body systems.Among all the experimental platforms being developed for quantum many-body systems,the high-lying Rydberg atom trapped in optical lattices is a particularly promising one,therefore,the corresponding theoretical studies have practical applications.We investigate the multipartite entanglement,fidelity susceptibility,and quantum criticality of neutral atoms on a twodimensional square lattice,interacting via laser excitation to atom Rydberg states.It is found that the first derivative of residual entanglement with respect to detuning has peaks near the critical point,and corresponding critical behaviors are shown to obey conventional finite-sized scaling,from which we numerically determine the quantum critical point and the critical exponent of the associated correlation length.We also show that there is a sharp peak in the fidelity susceptibility near the critical point,and the critical exponent of the associated correlation length is obtained based on the finite size analysis.Next,the neutral atoms coupled to a highly excited Rydberg state on a two-dimensional triangular lattice are investigated by employing the density matrix renormalization group technique in the matrix product state form.The full ground-state phase diagram as a function of blockade radius and the detuning of the exciting laser is determined by the behavior of entanglement entropy.We find several quantum phases including stripe-ordered phase and symmetry-breaking density-wave-ordered phases featured with regular excitation patterns of different excitation densities ρ=1/3,1/4,and 1/7.In addition,aρ=2/3 ordered phase and an interesting " order-by-disorder”phase,which has been prepared experimentally,are also observed in this work.Our work provides an exploration of the possible quantum phases that can occur in the complex two-dimensional arrayed Rydberg systems,and thus could be a faithful theoretical guide for further experimental research.Another focus of this paper is to explore the different quantum phases and phase transitions of two-dimensional quantum many-body systems by using machine learning methods.Quantum spin liquid is a novel type of quantum state,in which any long-range magnetic order is suppressed by strong quantum fluctuation and is highly entangled even at absolute zero temperature.In addition,this exotic phase of matter is beyond Landau’s phase transition theory and not include any local order parameter or spontaneous symmetry breaking.We investigate the ability of the machine learning method based on the restricted Boltzmann machine in capturing physical quantities including the ground-state energy,spin-structure factor,magnetization,quantum coherence,and multipartite entanglement in the two-dimensional ferromagnetic spin liquids on a honeycomb lattice.It is found that the restricted Boltzmann machine can encode the many-body wavefunction quite well by reproducing accurate ground-state energy and structure factor.Further investigation on the behavior of multipartite entanglement indicates that the multipartite entanglement is richer in the gapless phase than the gapped spin-liquid phase,which suggests that the multipartite entanglement can characterize the spin-liquid phases.Additionally,we confirm the existence of a gapped non-Abelian topological phase in the spin liquids on a honeycomb lattice with a small magnetic field and determine the corresponding phase boundary by recognizing the change of the local magnetization and multipartite entanglement.