Applications Of Quantum Entanglement In Tensor Networks

Strongly correlated quantum many-body physics is one of the most challenging and fundamental directions in modern physics.In such systems,traditional perturbation theory fails,and the exact diagonalization also encounters the Exponential Wall problem.Tensor-network states,together with the related tensor network methods,can precisely compute the ground state and low-energy excited states of local gapped systems.Recently,non-Hermitian physics is undergoing a revolution driven by the breakthrough in experiments and theory.To utilize tensor networks to study the properties of general non-Hermitian models,it is necessary to first judge whether the tensor-network state has sufficient representation capacity through the entanglement properties of non-Hermitian quantum states.Although their topological properties are well understood,many fundamental questions about entanglement properties remain unclear.On the other hand,the difficult optimization problem of tensor network algorithms restricts their promotion and application.The first half of this thesis utilizes the correlation matrix technique to systematically investigate the properties of entanglement in non-Hermitian free fermionic systems.First,we discuss the basic methods and concepts such as the diagonalization of the nonHermitian free fermion models,the way of defining the ground states,the definitions of the density operator and the reduced density operator,and so on.Then,for the biorthogonal density operator defined by the left and right ground states,we give detailed proof for the establishment of the correlation matrix technique in calculating the reduced density operator and summarize the six major steps to calculate the reduced density operator.For any one-band models,under the thermodynamic limit and consider the size of the subsystem l>>1,we use the Fisher-Hartwig theorem to prove that the entanglement entropy satisfies logarithmically corrected area law.Besides,the number of gapless modes Nf and the effective central charge c strictly satisfy the relationship c=Nf/2.For the more general one-dimensional,quasi-one-dimensional and two-dimensional cases,the results of numerical calculations confirm that the above relationship still holds,leading to the conclusion that the entanglement properties are only determined by the structure of the Fermi surface.Furthermore,we demonstrate that the main contribution of entanglement entropy comes from those entangled pairs across the boundary.These results demonstrate that it is feasible to study the properties of non-Hermitian models via tensor networks.The second half of this thesis is devoted to solving some long-standing optimization problems in tensor networks.For tensor-network states which are loop-less structures and therefore have canonical forms,the entanglement between their environment and local tensors will be precisely described by bond matrice.Also,the bond matrice will be completely determined by the local tensors,so the energy of the tensor-network states is a functional of the local tensor.Considering these facts,we propose a variational algorithm to optimize the tensors by minimizing the energy functional and realize this algorithm with automatic differentiation.Taking the antiferromagnetic Heisenberg model defined on the Bethe lattice and the triangular Husimi lattice at z=4 as the test examples,the projected entangled pair states(PEPS)and projected entangled simplex states(PESS)representing the ground states are updated respectively.We find that the results of variational optimization are consistent with those of the simple update algorithm when τ≈0.Unlike the imaginary-time evolution,this algorithm provides an optimization scheme without Trotter error and completes the optimization of tensornetwork states with loop-less structures directly.For the tensor network state with loop structures,the full update algorithm accurately calculates the entanglement between the environment and the local tensors,but the process of optimizing and updating the local tensors is not stable.With the help of automatic differentiation,we improve the unstable optimization process here.The antiferromagnetic Heisenberg models defined on the square lattice and the kagome lattice are used as the test models,and the PEPS and PESS corresponding to the respective ground states are optimized and updated.Compared with the results of the full update algorithm,we confirm the stability and accuracy of this improved algorithm and greatly improves the instability problem in full update algorithm.When contracting a 2D tensor network,the loop optimization for tensor network renormalization(LoopTNR)can effectively remove the short-range entanglement degrees of freedom that are irrelevant to the results.In the original scheme,the key step of tensor decomposition and low-rank approximation are accomplished through the variational optimization of periodic matrix product states,which is cumbersome and runs the risk of falling into local optima.We improve this variational optimization process using automatic differentiation and take the 2D Ising model as a benchmark model.Comparing the free energy at and away from the phase transition temperature with the original LoopTNR results,the validity of the improvement is confirmed.The results show that we can achieve LoopTNR more directly,stably,and precisely with the automatic differentiation.