Tensor Network Algorithm Studies On Dynamical Correlation Function Of Strong Correlated Systems

Condensed matter physics is the main branch of physical research at present.It has successfully explained the physical properties of many materials.However,many newly emerging strong correlated electronic materials have not yet been well understood.It is necessary to study such systems including experiments,theories and calculations.Physics is an experiment based subject,and the basic mechanism of most of the experiments is to show the interaction between the measuring medium and the target material by detecting the feedback of the measuring medium.The interaction between the medium and the material corresponds to the dynamical process of a material for some kind of excitation,which can be expressed by the dynamical correlation functions.It is of great significance to understand the mechanism of the physical properties of materials by constructing the theoretical model and using the computational method to solve the dynamical correlation function of the model.Therefore,the dynamical correlation function is very important in the field of strong correlated systems.It can be observed reliably through various physical experimental method.However,it is difficult to calculate the dynamical correlation function accurately and efficiently in the field of computational physics,especially in the field of tensor renormalization group.The purpose of this paper is to further promote and improve the numerical method of tensor renormalization group for dynamical correlation function.Tensor renormalization group is a new numerical method for strongly correlated systems,and its wave function is represent by tensor network state.It can well describe the ground state of the physical model which satisfied the area law.This paper summarizes the mature algorithm of the tensor renormalization group in the one-dimensional case: the density matrix renormalization group(DMRG)method.For the ground state of one dimensional system that satisfies the area law,DMRG method can be used to solve with the matrix product state(MPS)representation of wave function.However,the excited state wave functions needed to be involved in the dynamical properties of the calculation,can not usually be guaranteed to satisfy the area law,and can not be accurately expressed in MPS.Therefore,the existing DMRG based dynamical calculation methods have obvious shortcomings.This paper compares the mainstream DMRG based dynamical algorithm:the iterative algorithm represented by continued fraction method(CF),the core idea is to extend the fractional part of the dynamical correlation function to the approximate of continuous fraction.The calculation process is starting at the ground state,and the dynamical correlation function of the whole spectrum of the target system is obtained through the gradual iteration process.Because this method starts from the ground state,it can get a more accurate low frequency excitation spectrum,but due to the error will accumulate with the iterative process,it is difficult to accurately calculate the middle and high frequency result.The direct method represented by the correction vector method(CV)is used to obtain the sparse linear equations by the deformation of the Green’s function formula,and then the dynamical correlation function is obtained directly.Because there is no error accumulation similar with iterative process in CF method,this method can get very high accuracy result of dynamical correlation function.However,this method can only get the corresponding data of one momentum and one frequency.The cost of calculations required to calculate the total spectrum is too much.At the same time,the condition number for solving the equations is very large.In order to get the high precision results,the calculation cost is very large.Therefore,this method is difficult to apply to the calculation of large scale problem because of the calculation cost.The method of time evolution based on the timedependent density matrix renormalization group(tDMRG),evolve the physical final state from the physical initial state with time evolution,and then the dynamical correlation function is obtained by the Fourier transform.Therefore,this method can get relatively accurate high frequency dynamical correlation function which corresponding to short time evolution.But the low frequency result is very difficult to calculate,because the information of low frequency dynamics needs to evolve for a long time,and the entanglement entropy corresponding to the physical state increases with time,the band dimension required by DMRG increases rapidly,so the rapid increase in computation cost makes it often difficult to get the exact long intertemporal evolution results in the actual calculation.Chebyshev polynomials are the best uniform approximation polynomials of finite interval functions.It has a very important theoretical status and application foundation in the field of function approximation.With the idea of using the Chebyshev polynomials to expand the dynamical correlation function,the calculation method of the Chebyshev matrix product state approach(CheMPS)is proposed.The calculation results which are superior to other methods can be obtained.The calculation cost is less than that of the CF method.When the accuracy of the calculation results is very high(similar to the CV method),it can well solve the one-dimensional antiferromagnetic Heisenberg model as an example.However,because the CheMPS method is still an iterative algorithm,the MPS representation of the wave function is similar to that of the CF method,so the actual calculation can not strictly satisfy the Chebyshev iterative relationship.The approximate error of MPS in the calculation process will accumulate with the iterative process and make the result deviate from the actual model,so it still needs to be further studied and improvement.In order to solve the above problems,we propose a method named Reorthonormalization of Chebyshev tensor network states dynamical method(ReCheTNS),and its MPS form Reorthonormalization of Chebyshev matrix product states dynamical method(ReCheMPS),which is no longer directly use the MPS which deviates from the formula for iterative calculation.Instead,a series of MPS obtained by orthogonalization is used as an orthonormal basis vector to stretch into the effective Hilbert space.Thus,the Hamiltonian operator is represented effectively,and the results of the dynamical correlation function are calculated,effectively reducing the MPS approximation error influence on the results.In this paper,one dimensional XY model and one dimensional antiferromagnetic Heisenberg model are calculated by ReCheMPS method,and compared with the results of strict result and CheMPS methods.All data are agreement with strict results,and the reliability and accuracy of the ReCheMPS method are verified better than the CheMPS method.The results of the ReCheMPS method depend only on the effective Hamiltonian produced by the Chebyshev vector in the form of MPS,which contain more effective information,so the accuracy of the calculation results is significantly higher than that of the CheMPS method.At the same time,the resolution of the calculated results no longer depends on the Chebyshev expansion series.The effective Hamiltonian can be diagonalization and obtained from the effective Hilbert space and there is no more approximation to the results.Therefore,the result has arbitrary high resolution for the δ function form and can be selected according to the actual needs of broadening.The extra computation of the ReCheMPS method is very small compared with other iterative methods.The calculation of the effective Hilbert space is only required to calculate the MPS inner product.The corresponding computation is much smaller than the iterative variational solution for the corresponding Chebyshev vector MPS.The process of obtaining the Hamiltonian and getting the dynamical results after the effective Hilbert space is only several times of small matrix product,so the proportion of the extra computation in the ReCheMPS method is about a few percentage points.In summary,compared with the CheMPS method,the ReCheMPS method greatly improves the accuracy and resolution of the result with only small increase in calculation cost,has considerable practical value.The DMRG based dynamical calculation methods and the ReCheMPS methods proposed in this paper are all numerical methods for computing physical systems with finite size.In practice,it is often necessary to extrapolate the results of dynamical correlation function under thermodynamic limit.(1)The most commonly used extrapolation idea is to get the results of different size systems,and to extrapolate the infinite limit physical results through the method of function extrapolation.This method requires high accuracy and large size calculation results,which is difficult to satisfy in actual calculation.(2)In addition,the CheMPS method tries to get a more smooth curve by using the finite size dynamical results to broaden and fuzzy finite size discrete energy levels,and then estimates the infinite result of the thermodynamic limit.The results resolution of this method has broken so that the result of the thermodynamic limit is also deformed.The selection of the width of the widening parameters is difficult to be quantified accurately.The results of the calculation of different sizes usually need to be selected artificially in order to obtain a consistent thermodynamic limit.(3)The high resolution dynamical results can be calculated at the first,and then use the δfunction and weight to fit the results,and the results of the thermodynamic limit can be obtained by combining the results of multiple system size.This method relies on the accuracy of the calculated results,and it is still acceptable for more accurate low frequency spectral results in the iterative algorithm.If the results are not accurate enough,the method is invalid.(4)In order to improve the above method,Smooth estimation method of finite size extrapolation is proposed.This method is combined with the ReCheMPS method to obtain the approximate of the dynamical correlation function in the thermodynamic limit.The smoothing estimation method is based on the diagonalization of effective Hamiltonian obtained by ReCheMPS method.Firstly,the main peak position in the result is selected,and then the noise and degeneracy caused by numerical calculation is eliminated to get the final result.The smoothing estimation method avoids the shortcomings of the traditional numerical extrapolation error.The results of the different system sizes obtained by the effective use of the existing data and the application of the smoothing estimation method are all consistent with the strict results of the infinite system.Therefore,the smoothing estimation method can be of great help to the theoretical analysis of real material properties.Finally,the ReCheMPS method is extended to higher dimensional and more general physical models.It shows us that the ReCheMPS method can also be used to calculate two-dimensional or three-dimensional systems,and can be widely applied to spin,fermion or boson model,also can use more general tensor network state method to expand and calculate.In a sentence,the ReCheMPS method is quite general.