198 research outputs found
Fate of partial order on the trillium and distorted windmill lattices
The classical Heisenberg model on the trillium and distorted windmill
lattices exhibits a degenerate ground state within large- theory, where the
degenerate wavevectors form a surface and line, in 3-dimensional space,
respectively. We name such states partially ordered to represent the existence
of long-range order along the direction normal to these degenerate manifolds.
We investigate the effects of thermal fluctuations using Monte Carlo (MC)
methods, and find a first order transition to a magnetically ordered state for
both cases. We further show that the ordering on the distorted windmill lattice
is due to order by disorder, while the ground state of the trillium lattice is
unique. Despite these different routes to the realization of low temperature
ordered phases, the static structure factors obtained by large- theory and
MC simulations for each lattice show quantitative agreement in the cooperative
paramagnetic regime at finite temperatures. This suggests that a remnant of the
characteristic angle-dependent spin correlations of partial order remains above
the transition temperatures for both lattices. The possible relevance of these
results to -Mn, CeIrSi, and MnSi is discussed.Comment: 10 pages, 9 figures, published versio
A discrete history of the Lorentzian path integral
In these lecture notes, I describe the motivation behind a recent formulation
of a non-perturbative gravitational path integral for Lorentzian (instead of
the usual Euclidean) space-times, and give a pedagogical introduction to its
main features. At the regularized, discrete level this approach solves the
problems of (i) having a well-defined Wick rotation, (ii) possessing a
coordinate-invariant cutoff, and (iii) leading to_convergent_ sums over
geometries. Although little is known as yet about the existence and nature of
an underlying continuum theory of quantum gravity in four dimensions, there are
already a number of beautiful results in d=2 and d=3 where continuum limits
have been found. They include an explicit example of the inequivalence of the
Euclidean and Lorentzian path integrals, a non-perturbative mechanism for the
cancellation of the conformal factor, and the discovery that causality can act
as an effective regulator of quantum geometry.Comment: 38 pages, 16 figures, typos corrected, some comments and references
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