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Progress of Theoretical Physics

On the Energy Sum Rule of the Ring Heisenberg Hamiltonian

K. Fujii, N. Genkawa

Published Jan 1, 1981• 0 citations
Publisher

Abstract

The sum rules for the energy eigenvalues of the ring Heisenberg Hamiltonian are discussed according to the classification of eigenstates through the total spin, the z-component of the total spin and the wave number. It is shown that the number of the eigenstates belonging to the energy sum obtained by our method substantially corresponds to one-n-th of that obtained by Van Vleck. With respect to the average energy defined as the quotient of the energy sum divided by the number of states, it is found that there exists a significant difference in its values between ours and Heisenberg's in the case of small n. The algebraic properties of the energy levels for atoms as many as eight are made clear by using the moments of the energy eigenvalues.

DOI
10.1143/ptp.65.154

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The sum rules for the energy eigenvalues of the ring Heisenberg Hamiltonian are discussed according to the classification of eigenstates through the total spin, the z-component of the total spin and the wave number. It is shown that the number of the eigenstates belonging to the energy sum obtained by our method substantially corresponds to one-n-th of that obtained by Van Vleck. With respect to the average energy defined as the quotient of the energy sum divided by the number of states, it is found that there exists a significant difference in its values between ours and Heisenberg's in the case of small n. The algebraic properties of the energy levels for atoms as many as eight are made clear by using the moments of the energy eigenvalues.
Employing the method of a symmetry lowering (form the permutation group Pn to the dihedral group Cnv), the eigenstates of the ring Heisenberg Hamiltonian H=-2J ∑ni=1Si·Si+1 are classified not only by total spin S and its z-component M, but also by wave number k=2 πL/n, where L= ±0, ±1, ±2, ∼, ±p for n=2p+1 or 2p. By the characters of irreducible representation of Pn and Cnv, we express the sum of the energy eigenvalues at fixed S, M and L, together with the number of the eigenstates contained in this sum. With respect to the spin configurations in the ring, it is revealed that there exist invariant subspaces under all symmetry operations of Cnv in the space of the spin-functions specified by M. By connecting this fact with the Wigner coefficients, the eigenfunctions are easily obtained.
According to quantum mechanics, the commutation property of the energy Hamiltonian with the momentum operator should give the definite values not only for energy but also for the momentum quantum levels. A difficulty provided by the standing-like boundary conditions of the electron gas is that the Hamiltonian eigenfunctions are different than eigenfunctions of the momentum operator. In results the electron momenta are obtained from the correspondence rule between the classical and quantum mechanics given by Landau and Lifshits. As a consequence the statistics of solutions representing not only the energy values but also the electron momenta should be taken into account. In the Heisenberg picture of quantum mechanics, the momenta are easily obtained because the electron oscillators are there directly considered. In fact, the Hamiltonian entering the Heisenberg method can be defined in two different ways each giving the set of the electron energies known from the Schr?dinger’s approach.
Sum rules for spin-1/21/21/2 quantum gases in well-defined-spin states: spin-independent interactions and spin-dependent external fields
Many-body eigenstates of spin-1/21/21/2 particles with defined total spins contain spin and spatial wavefunctions belonging to multidimensional irreducible representations of the symmetric group, unless the total spin has the maximal allowed value. Matrix elements in the basis of such eigenstates are analyzed for spin-dependent interactions with external fields and spin-independent ones between the particles. Analytical expressions are obtained for sums of the matrix elements and sums of their squared modules. The sum rules are applied to perturbative analysis of energy spectra.
Sums of matrix elements of spin-dependent two-body momentum-independent interactions and sums of their products are calculated analytically in the basis of many-body states with given total spin---the states built from spin and spatial wave functions belonging to multidimensional irreducible representations of the symmetric group, unless the total spin has the maximal allowed value. As in the first part of the series [V. A. Yurovsky, Phys. Rev. A 91, 053601 (2015)], the sum dependence on the many-body states is given by universal factors, which are independent of the Hamiltonians of noninteracting particles. The sum rules are applied to perturbative analysis of energy spectra and to calculation of two-body spin-dependent local correlations.
Analytical expressions are derived for sums of matrix elements and their squared moduli over many-body states with given total spin---the states built from spin and spatial wave functions belonging to multidimensional irreducible representations of the symmetric group, unless the total spin has the maximal allowed value. For spin-dependent one-body interactions with external fields and spin-independent two-body ones between the particles, the sum dependence on the many-body states is given by universal factors, which are independent of the interaction details and Hamiltonians of noninteracting particles. The sum rules are applied to perturbative analysis of energy spectra.
The algebraic structure of the hamiltonian of the many electron system is studied. Based upon the second quantization scheme, a purely algebraic-theoretical method is established in order to determine the total energies of the many electron system, where each energy eigenstate of one electron in the Hartree-Fock field of these electrons is occupied by a given number of electrons. This occupation number is either 0, 1 or 2, because of the value, 1/2, of the electron spin. Only the part of the hamiltonian related to the singly-occupied states are found to be algebraically important, in relation to which the so-called exchange algebra emerges. We state the general principle how to manipulate this algebra, and will work out in some cases of a few numbers of singly occupied states. In the case that the energy eigenstates of the Hartree-Fock field have degeneracies, the theory so far mentioned does not suffice, and more detailed and somewhat complicated study is required.