Study Of Entanglement Properties For Low Dimensional Quantum Many-body Systems

In this thesis,we study entanglement property in low dimensional quantum many-body systems,including bosonic system in 0 + 1 dimension,spin systems in 1 + 1 and 2 + 1 dimensions.In 0 + 1 dimensional bosonic system,entanglement depth is used to characterize the minimal number of particles in a system that are mutually entangled.For symmetric states,there is a dichotomy for entanglement depth:an N-particle symmetric pure state is either fully separable,or fully entangled-its entanglement depth is either 1 or N.We show that this dichotomy property for entangled symmetric states remains stable under non-symmetric noise.We propose an experimentally accessible method to detect entanglement depth in atomic ensembles based on a bound on the particle number population of Dicke states,and demonstrate that the entanglement depth of some Dicke states,for example the twin Fock state,is very stable even under a large arbitrary noise.Our observation can be applied to atomic Bose-Einstein condensates to infer that these systems can be highly entangled with the entanglement depth reaching system size,e.g.of several tens of thousands or more atoms.In 1 + 1 dimensional spin system,we focus on systems with symmetry protected topological(SPT)phases.We first study the geometry of reduced density matrices for states with SPT order.We observe ruled surface structures on the boundary of the convex set of low dimensional projections of the reduced density matrices.In order to signal the SPT order using ruled surfaces,it is important that we add a symmetry-breaking term to the boundary of the system-no ruled surface emerges in systems without boundary or when the added symmetry-breaking term represents a thermodynamic quantity.Although the ruled surfaces only appear in the thermodynamic limit where the groundstate degeneracy is exact,we analyze the precision of our numerical algorithm and show that a finite system calculation is sufficient to reveal the ruled surface structures.We then study the phase transition between different SPT phases.We carry out these studies for the two-legged spin-1 ladder system with D2×? symmetry group,where D2 is the discrete spin rotational symmetry and ? means interchain reflection symmetry.The system has one trivial phase and seven nontrivial SPT phases.We construct Hamiltonians to realize all these seven SPT phases and study the phase transitions between them.Our numerical results indicate that there exists no direct continuous transition between any two SPT phases.We interpret our results via the topological nonlinear sigma model effective field theory,and further conjecture that there exists in general no direct continuous transition between two SPT phases in one dimension if the symmetry group is discrete at all length scales.In 2 + 1 dimensional spin system,we apply symmetric tensor network state(TNS)to study the nearest neighbor spin-1/2 antiferromagnetic Heisenberg model on kagome lattice.We keep track of the global and gauge symmetries in the TNS update procedure and in tensor renormalization group(TRG)calculation.We use imaginary-time evolution to obtain the variational ground state,and then use symmetric TRG to compute the modular matrices.We find that the ground state is a gapped spin liquid with the Z2-topological order(or toric code type).The correlation length is about 10 unit cell.This line of studying entanglement properties for quantum many-body systems rang-ing from 0+1 dimension to 2 + 1 dimension will lead us to a deeper understanding of dimensionality,locality,entanglement and emergence.