Investigation Of Lanczos Optimization Method For Tensor Network State

One of the biggest challenges in condensed matter and statistical physics is the study of quantum many-body systems,especially the strongly correlated electron systems.Due to the large degrees and the complex interaction between particles in the system,there emerges extremely abundant phenomena.As the research progress of experiment and theory,people found a lot of new phenomena that traditional theory is not able to explain.Firstly,in high temperature super-conductor the electrons are strongly interacted and correlated.It would be very difficult to obtain the true physical picture using traditional perturbation theory and quantum field theory.Meanwhile one of the most commonly used numerical methods,quantum Menta Carlo,generally has the infamous minus sign problem for fermion systems that are not half filled.There are also other exotic quantum phases such as quantum Hall effect?QHE?that beyond traditional theory.There is no symmetry broken in such phases,which is surely not suitable for tradition-al symmetry breaking picture.These phases are now called topological ordered states.In recent years,tremendous progress has been achieved in the development of tensor network state?TNS?method,which have emerged as a powerful theoretical tool for investigating low-dimensional quantum lattice models.Traditional numer-ical renormalization group?NRG?method truncates physical degrees according to the energy.For systems in which the particles are only weakly interacted this may works well,however for strongly correlated systems,the electrons are strongly in-teracted and correlated with each other.It would be very difficult to determine which degrees are truly high energy ones.Thus we may truncate some higher energy degree which turns out to the lower one.The TNS approach uses completely different physical quantity,the entangle-ment.Generally the entanglement entropy of a random state would be propor-tional to the system size.Luckily the ground states are only weakly entangled states.For gapped systems,there is the famous entanglement area law which s-tates typically the entanglement entropy is proportional to the edge of the system.For critical system there is generally a logarithmic correction to the edge term.And TNS satisfy entanglement area law by construction,which makes it easily to catch the entanglement structure of the ground states.During the renor-malization calculation of TNS,the less entangled degrees are truncated.This truncation is equivalent to approximate the reduced density matrix.On the other hand,the entanglement is determined according to the Schmidt decomposition of the whole wave function,thus the entanglement truncation is determined by the global properties of the system.The most important technique issue for TNS methods is the optimization of the wave function and calculation of physical quantities.This would be quite easy and simple for one dimensional quantum systems due to the simple structure of 1D TNS,matrix product states?MPS?.There is no closed loop in MPS,so that the inner product can be calculated exactly,in fact all the properties of the wave function is determined by the transfer matrix constructed by the local tensor of MPS.The optimization of MPS is also simple.DMRG is the best optimization method to obtain the approximately best ground state wave function locally.It would be very different for two dimensional TNS,projected entangled pair states?PEPS?.The complex inner structure,i.e.the existence of loops,disables us calculate the inner product of PEPS wave functions exactly.The properties of PEPS are also determined by the transfer matrix constructed using the local tensors of the PEPS.However,in the two dimensional case,the transfer matrix is no longer a matrix,it's a matrix product operator?MPO?.Generally a MPO can not be diagonalized exactly.A lot of numerical methods,such as the bounday MPS and corner transfer matrix renormalization group,have been developed to approximately calculate the inner product.However the computation cost of these methods scales as D¹²,where D is the bond dimension of the local tensor of the TNS.The optimization of tensor networks is even much more complicated.The commonly used optimization methods include imaginary time evolution and energy functional variation.In these methods we need to iteratively calculate the transfer matrix a lot of times.This makes the total computation cost a very large coefficient times D¹².This restricts the bond dimension we could use.The accuracy relies on the bond dimension D very badly.However the computation and memory cost are also increasing quickly while enlarging D.This is the bottleneck of the current research of tensor network state method.We notice that in the traditional numerical calculation methods,boundary MPS for example,the local tensors of the wave function and its conjugate are combined together,which makes the bond dimension of the reduced tensor net-work to be D².We develop a new method which considers the local tensor of the wave function and its conjugate independently.So that we only need to calculate a tensor network of bound dimension D,rather than D².This method reduces the computation cost to some extent.However the computation cost is still very high,especially when we are trying to obtain the ground state wave function.To solve this problem,we propose a new approach of opimisation TNS.In this approach we use multiple TNS as our variational wave function which are generated by generalized Lanczos method.To be specific,we generate new many-body basis states of quantum lattice models us-ing TNS via Lanczos iteration.These basis states form a Krylov sub-space.After diagonalizing the effective Hamiltonian in this space,a better ground state wave function is obtained.The ground state energy keeps going down with Lanczos iterations.The ground-state wave function is represented as a linear superpo-sition composed from all the TNS generated by Lanczos iteration.In this way,the accuracy of the wave function in this approach would be higher not only via enlarging the bond dimension of the TNS,but more importantly also via gener-ating more TNS.So when the traditional TNS method not able to pursue better accuracy due the restriction of the bond dimension,we could stil use the wave function of traditional method as our initial wave function and obtain a better one by Lanczos optimization.This method is promising to solve the bottleneck of the current TNS methods.