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Iteration Methods For Approximating The Lowest Order Energy Eigenstate of A Given Symmetry For One- and Two-Dimensional Systems

Chad E. Junkermeier

Published Jan 1, 2003• 1 citation
OpenAlex

Abstract

Using the idea that a quantum mechanical system drops to its ground state as its temperature goes to absolute zero several operators are devised to enable the approximation of the lowest order energy eigenstate of a given symmetry; as well as an approximation to the energy eigenvalue of the same order.

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•Jan 1, 2026•0 citations
Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.
Significance The ground state of a quantum mechanical system is the lowest-energy eigenstate of the Hamiltonian. In isolation, it persists unchanged forever, with symmetries dictated by those of the Hamiltonian. But near-eigenstates of broken symmetry can persist for long times, even on the scale of human measurement. The appearance of broken symmetries of the electron density or spin density in a density functional calculation can reveal strong correlations among the electrons that are present in a symmetry-unbroken wavefunction. Symmetry breaking can arise when a wave-like fluctuation drops to zero frequency. The presented examples are the stretched hydrogen molecule, antiferromagnetism in solids, and the static charge-density wave in a low-density jellium, which is shown quantitatively to be a zero-frequency plasma wave.
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.