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Journal of Xinxiang University

Compared Study in the Energy Spectra of Ground-state and Excited-state in Small Size Heisenberg Chain with XXZ Quantum Model Presented by Haldane

Xiuxia Zhang

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

In this paper,the energy spectrum of the ground-state and the excited-state in the model of small size one-dimensional Heisenberg finite chains(n20) was discussed.The Complete Diagonal and Lanczos Iterative method of Hamiltonian matrix are actualized by TITPACK program which designed by Hidetoshi Nishimori.But the finite chain revised method is mainly actualized by MATLAB program.All programs would be given in this paper.The investigative object was small size one-dimensional Heisenberg finite chains(n20),the TITPACK program was used to calculate the eigenvector of ground-state and excited-state|Ψ0〉、|Ψ1〉、|Ψ2〉、|Ψ3〉,the energy eigenvalue of ground-state and excited-state λ0 、λ1、λ2、λ3,the variation relation between and the correlated function of Sx 、Sz and the system Hamilton parameter.Using the finite chain revised method,the error was 0.000 047 2 between calculated and theoretic ground-state eigenvalue.The absolute value of ground-state energy square was inversed ratio the chain length,such as E(n)=-0.443 1-0.848 4/n2,through analyze the result of the program and find the law between the result and the quantity,in the small size one-dimensional Heisenberg finite chains,S=Aexp(-d/B) in the XXZ model.It is completely difference with XXZ Quantum model presented by Haldane.Only some energy spectra were intersected.

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In this paper,the energy spectrum of the ground-state and the excited-state in the model of small size one-dimensional Heisenberg finite chains(n20) was discussed.The Complete Diagonal and Lanczos Iterative method of Hamiltonian matrix are actualized by TITPACK program which designed by Hidetoshi Nishimori.But the finite chain revised method is mainly actualized by MATLAB program.All programs would be given in this paper.The investigative object was small size one-dimensional Heisenberg finite chains(n20),the TITPACK program was used to calculate the eigenvector of ground-state and excited-state|Ψ0〉、|Ψ1〉、|Ψ2〉、|Ψ3〉,the energy eigenvalue of ground-state and excited-state λ0 、λ1、λ2、λ3,the variation relation between and the correlated function of Sx 、Sz and the system Hamilton parameter.Using the finite chain revised method,the error was 0.000 047 2 between calculated and theoretic ground-state eigenvalue.The absolute value of ground-state energy square was inversed ratio the chain length,such as E(n)=-0.443 1-0.848 4/n2,through analyze the result of the program and find the law between the result and the quantity,in the small size one-dimensional Heisenberg finite chains,S=Aexp(-d/B) in the XXZ model.It is completely difference with XXZ Quantum model presented by Haldane.Only some energy spectra were intersected.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
A new method of determining the spectra of simple quantum-mechanical spin Hamiltonians by direct Hamiltonian-matrix diagonalization is presented. The method is illustrated by applying it to the isotropic one-dimensional Heisenberg antiferromagnet, for which calculations were performed for longer spin chains than by previous direct Hamiltonian-matrix diagonalizations. For spin s=1/2 it was possible to accurately determine the ground- and first-excited-state energies for chains of length N=32 and 64. For s=1 the longest chain for which a similar computation was performed was N=32; the singlet-triplet Haldane gap is found to be 0.421\ifmmode\pm\else\textpm\fi{}0.005 which is consistent with earlier Monte Carlo results. Long-spin-chain calculations are performed iteratively by solving smaller-chain problems, and constructing from their eigenvalues, and eigenstates, the longer-spin-chain basis and Hamiltonian matrices. The computations were performed on a small computer, which suggests that much larger problems could be handled with a parallel-processing supercomputer.
The one-dimensional Heisenberg antiferromagnet of S = 1 spins is studied by numericaldiagonalization method based on the Lanczos algorithm. The Haldane gap of this system isnumerically estimated from finite-size clusters under the twisted boundary condition up tothe 30-site system. The finite-size results are analyzed by a convergence-acceleration technique; the multistep procedure of the convergence acceleration is successfully carried out.The present estimate for the gap magnitude is more precisely than the previously reportedvalue from a Lanczos-diagonalization study. It also agrees well with the result reported ina study based on the density matrix renormalization group calculation. The present studyclarifies that Lanczos-diagonalization calculations can sufficiently contribute for preciselyestimating an energy gap although this method can treat only small clusters.
We carry out finite-size extrapolations of numerical-diagonalization data of the S =1 Heisenberg chain having a nonzero energy gap between the unique singlet ground state and the first excited state, namely the Haldane gap. Very precise estimates of the ground-state energy per site E g /N = −1.4014840447(39) and the staggered component of the magnetic structure factor S π = 3.864356(31) at T =0 are successfully obtained from the finite-size data of system sizes up to N = 24 under the twisted boundary condition by the sequence interval squeeze method, which was applied to a precise estimation of the Haldane gap by Nakano and Terai [J. Phys. Soc. Jpn. 78 (2009) 014003]. The present estimates are compared with other estimates in previous studies from various methods including the quantum Monte Carlo simulation and the density matrix renormalization group calculation.
The method of de Vega and Woynarovitch (1985) is used to calculate finite-size corrections to the ground-state energy in different sectors for the XXZ Heisenberg chain. Finite-size scaling amplitudes and correction-to-scaling exponents in the critical region are derived. Using conformal invariance, a scaling dimension x=( pi - gamma )/2 pi is extracted corresponding to the electric field operator in the 8-vertex model: this confirms a conjecture of Baxter and Kelland (1974). Finite-size scaling properties near the Kosterlitz-Thouless critical point Delta =-1 are discussed.
In this thesis, we analyze how the DMRG method works and use it on the one-dimensional (1D) antiferromagnetic Heisenberg Chain with open boundaries. We explicitly demonstrate how DMRG works for both the spin 1/2 and the spin 1 model and write programs for them in Python. A detailed process of the 1D spin 1/2 Heisenberg chain for four sites is shown to get exact values for its ground state energy values and to show how each step of the DMRG process works. We then compare the DMRG method to the analytically exact values and get very precise results. The DMRG method uses what is called the density matrix, formed by the eigenvector of the system we want to approximate. The exact ground state eigenvalue is obtained for N=14 sites and comparing that value with the DMRG method with only the 40 (D=40) most important states kept of the density matrix we get a difference from the exact result by about 3.5*10^(-13) after using four sweeps with the finite DMRG method, calculated in a small fraction of the time. We calculate the ground state eigenvalues for bigger systems up to N=50 and see the ground state energy per site E_0/N approach the Bethe ansatz result E_0/N=1/4-ln(2) in the thermodynamic limit (when N goes to infinity). The same approach is used for the spin 1 chain and we write a program, computing ground state eigenvalues ranges of N up to 50 and here in the thermodynamic limit E_0/N approaches approximately -1.401485(2). Finally we use DMRG to target the lowest five eigenstates of both the spin 1/2 and spin 1 systems to visualize the energy gap from the ground state to the first excited energy level in the thermodynamic limit. We get the energy gap approaching 0 for the spin 1/2 model and approaching the Haldane gap delta E =0.4105(1) for the spin 1 chain with open boundaries with a fourfold degeneracy of the ground state.