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## Effective Hamiltonians of the pyrochlore antiferromagnet

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**Author**

Hizi, Uzi

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**Abstract**

The pyrochlore lattice Heisenberg antiferromagnet is a highly frustrated model,
and possesses, classically, a macroscopic continuous ground state degeneracy. We
study the semiclassical limit of large spin length S and examine the effect of quantum
fluctuations on the energy within various theories. In each of these theories,
we focus on deriving an effective Hamiltonian, as a function of a small number
of degrees of freedom. The effective Hamiltonian gives us a simple formula for
calculating the energy and facilitates the search for a unique ground state among
the large number of classical ground states.
First, we consider the harmonic spin-wave theory, in which we keep only the
lowest order (in 1/S) correction to the classical Hamiltonian. We perform a detailed
analysis of the harmonic order spin-wave modes and, using a real-space loop
expansion, produce an effective Hamiltonian, in which the degrees of freedom are
Ising variables representing products of the classical spin directions around loops
in the lattice. We find a family of exactly degenerate collinear ground states, related
by gaugelike Z2 transformations and provide bounds for the zero-temperature
entropy.
We carry the spin-wave calculation to the next ?anharmonic? order in the 1/S
expansion, utilizing a a self-consistent variational Hamiltonian approach, equivalent
to Hartree-Fock approximation. We find that the harmonic degeneracy is
broken, but there remain a large number of seemingly degenerate ground states.
We develop an alternative approximation, employing the widely used, but not
well controlled generalization of the SU(2) ?= Sp(1) theory to Sp(N), in the limit of
infinite N. We develop an effective Hamiltonian for this mean-field theory, using
an analytical loop-expansion. We find that in this case, the ground state of the
large-N theory cannot possibly be the physical ground state in the limit S ? 1,
since it is not a harmonic spin-wave ground states. Nonetheless, when restricted to
the manifold harmonic spin-wave ground states, both the anharmonic spin-waves
and the large-N theory result in similar effective Hamiltonian.
We further demonstrate that the harmonic theory can readily be applied to determine
the harmonic-order ground state manifolds of the Heisenberg Hamiltonian
on related lattices, and to field-induced collinear magnetization plateau states.

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**Date Issued**

2006-06-22#####
**Subject**

frustrated magnetism; pyrochlore; spin-waves

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**Type**

dissertation or thesis