An exact limit for the Peierls distortion energy in an antiferromagnetic chain by means of a renormalization procedure

Abstract

A renormalization procedure gives a rigorous upper bound for the groundstate energy per spin for a Peierls-distorted antiferromagnetic chain with Heisenberg interaction.

Renormalization techniques for quantum spin systems. Ground-state energies

“…For the rigid lattice (w0 -+ oo) the method of this subsection may be easily improved. Here one finds eo = 0 and the corresponding value of ~equals: -36/23 = -1.5652, whereas, even in the rather simple approximation of subsection 2.1 a better value is found: of us/v2 [19].…”

“…The ground-state energy per spin~*(~)[19].The renormalization gives a change of sign of the distortion parameter: Fixed points of the RG transformation are given by the equation F(~)= -~which for~ 0 has the solutions~= 0, 1. The stable point W'i = 1 is reached, with sufficient accuracy, after a reasonable number of transformations, for all values~~0) e in the interval [0, 1.5].…”

“…So the cylindrical symmetry is conserved in the projection. For the evaluation of H~'~1õ ne has to know the projection of the H~°, already given in (19), as well as the projection of the interaction terms H~J, which may be derived in a particular example on the basis of (21). The general result has the form [16]…”

“…Here only the overall symmetry of the scaled Hamiltonian~as a consequence of the same symmetry of~is derived. Now the renormalized constant part of Hamiltonian H~'~" may be found combining the results in (19) and (22)…”

An exact upper bound for the ground-state energy of the triangular antiferromagnetic lattice by means of a renormalization procedure

“…techniques have offered new possibilities to calculate approximate values for the ground-state energy of antiferromagnetic spin i systems [ 1, 21 or to give a rigorous upper bound for this energy [3]. In this letter we give an upper bound per spin for the triangular lattice with an antiferromagnetic Heisenberg interaction between nearest neighbours.…”