Tensor Network Studies On Strong Correlated Systems

Strong correlated electronic systems have been the focus of interest for more than half a century,because of their intriguing properties,such as high tempera*-ture superconductivity,Mott metal-insulator transition,charge-spin separation,and quantum spin liquid,which are not observed in conventional metal or insu-lator materials.The Hubbard model is a minimal model for studying physical properties of strongly correlated electronic materials.Its physics is determined by the competi-tion between the kinetic energy of electrons and the on-site Coulomb interaction.This model captures many characteristic features of strongly correlated systems and serves as a paradigm for theoretical understanding of numerous phenomena in condensed matter physics,particularly the high-Tc superconductivity and the Mott metal-insulator transition.Despite its simplicity,exact results are available only from the Bethe Ansatz in one dimension and from Dynamical Mean-Field Theory(DMFT)in infinite dimensions.Density matrix renormalization group(DMRG)is a numerical method pro-posed by S.R.White.It is a powerful tool in solving one dimensional quantum system at zero temperature.However,due to the area law of entanglement en-tropy,DMRG can only deal with quasi one dimensional systems.Tensor network methods are proposed to solve two dimensional systems.It can both be applied to study statistic models and quantum systems.The thesis includes the five chapters.In Chapter 1,we introduces DMRG,different kinds of tensor network meth-ods and also the overlap with machine learning.In Chapter 2,optimal basis DMRG are proposed and applied to Hubbard model on two dimensional square lattice.In 1992,S.R.White proposed DMRG in real space,while in 1996 T.Xiang developed the momentum space DMRG and studies small onsite U/t Hubbard model.Besides,it has widely been used to calculate small molecule in quantum chemistry.In general,the entanglement entropy depends on the basis.We add the single particle basis transform to DMRG.The algorithm searches for the optimal single particle basis of Hubbard model on the fly.The numerical evidence shows great accuracy increase compared to real space DMRG.In the medium coupling region(U/t = 4?8),the accuracy increases by an order of magnitude and the entanglement entropy decreased by half.We calculate on different lattice size from 4 x 4 to 10 x 10 and analysis the optimal basis of the results.A detailed discussion of the single particle density matrix and mutual information is given.Besides,the algorithm is also applied to different doping cases.In Chapter 3,we study the duality of statistic models with tensor network representation.The local tensor gives the duality information,which helps to determine the critical point of statistic models.For 2D clock model,the ap-proximate self dual point is given by the equivalence of bond spectrum.The large q asymptotic behavior is studied.At self dual point,the bond spectrum is the eigenvector of the discrete Fourier transform(DFT)matrix.The harmonic oscillator Hamiltonian is symmetric between real and momentum space.The eigenstates are also the eigenvectors of DFT matrix.We construct a new model with the combination of eigenstates of harmonic oscillator.At Tc = 1,the model can represent all possible self-dual point of model with Zq symmetry.Villain model is one special case.In Chapter 4,we discuss the equivalence between restricted Boltzmann ma-chine(RBM)and tensor network states(TNS).RBM is a basic and important neural network in machine learning.The group lead by Troyer use RBM as wave function ansatz in variational Monte Carlo.The energy result is better compared to the best projected entangled pair state(PEPS),which is one kind of tensor network.All the RBM states has TNS representations.However,only a small tiny TNS can be mapped back to an RBM.The statistic model partition function or quantum wave function can be explicitly written into RBM form,including the Ising model,the cluster state and toric code model.With the equivalence,we find that the entanglement entropy of a local connected RBM obeys the area law.More importantly,the shift-RBM technique used by Troyer's group can cap-tures the volume law entanglement entropy with limited number of variational parameters.The equivalence also serves as a bridge between the two fields.The concepts and tools can be introduced into machine learning.By counting the bond dimension of corresponding TNS,we find that deep Boltzmann machine can capture more entanglement compared to RBM.In Chapter 5,we concludes the previous chapters and proposed some ideas on applications of optimal basis DMRG.We believe there will be more communi-cations between machine learning and tensor network methods.The volume law entanglement capability is a great advantage of RBM and inspires us to build more powerful wave function ansatz in both fields.