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Progress of Theoretical Physics

Gel'fand Levitan Method for Quantum Spin Systems

Masatoshi Imada

Published Aug 1, 1982• 2 citations
Publisher

Abstract

The quantum Gel'fand Levitan method is formulated for the Ising-like Heisenberg Ising spin chain. The solution of the Gel'fand Levitan equation makes it possible to express the spin operators in terms of the creation and the annihilation operators of Bethe's eigenstates. The spin operators are expressed by the normal product expansion and the lowest order term is explicitly calculated.

DOI
10.1143/ptp.68.527

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The quantum Gel'fand Levitan method is formulated for the Ising-like Heisenberg Ising spin chain. The solution of the Gel'fand Levitan equation makes it possible to express the spin operators in terms of the creation and the annihilation operators of Bethe's eigenstates. The spin operators are expressed by the normal product expansion and the lowest order term is explicitly calculated.
The quantum generalization of the Gelfand-Levitan method is presented for the nonlinear Schr\"odinger model. The basic dispersion relation for operator Jost functions is derived, and the Heisenberg field operator is expressed in terms of scattering-data operators. Construction of Green's functions in the zero-density vacuum is discussed. The four-point function is explicitly calculated from the expression for the field operator and compared with the result of a direct Feynman graph summation. In addition it is proved for any number of particles that the Hamiltonian eigenstates constructed from the quantized scattering data are identical with those previously obtained by means of Bethe's ansatz.
In the present thesis we perform a step towards understanding the effec- tiveness of Hamiltonian perturbation theory in quantum dynamics, which is made possible by the Hamiltonian framing of quantum mechanics. To such a purpose, we focus our attention on quantum spin chains, namely on the isotropic Heisenberg ferromagnet, which is well known to be equivalent, via the Jordan-Wigner transformation, to a system of interacting fermions, and is also known to be exactly integrable through the Bethe ansatz. The interaction term of the Hamiltonian, of order four in fermionic creation and annihilation operators, is treated as a perturbation in the regime of small excitations: small number of reversed spins or, equivalently, small number of fermions per site. The calculation of the first order normal form Hamiltonian amounts to erasing all the nonresonant interaction terms, and allows to draw interesting conclusions on the dynamics of the system, e.g. to determine ap- proximately conserved quantities. Moreover, we get a good correspondence between our perturbative energy spectrum and the exact one computed via the Bethe ansatz, the approximation being of course the more accurate the smaller is the number of fermions per site. The present method, here tested in an exactly solvable case, can be applied to any nonintegrable system of weakly interacting fermions (and/or bosons).
We give an infinite volume formulation of quantum inverse scattering method for Heisenberg XXZ model, by making use of Jost solutions of difference equations satisfying prescribed asymptotic boundary conditions. We find operator identities among the Jost solutions which enable us to obtain Gel'fand-Levitan equations more easily than any previous formulations. By solving an integral equation, we express scattering data operators in terms of solely reflection coefficient operators, which are creation and annihilation operators of magnons and their bound states. We also see that when expressed in terms of them, Hamiltonian and other integrals of motion are diagonalized in free bosonic forms so that magnon and their bound states constitute a complete set of bases of Fock space.