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(48MB mpg file) This is a movie of the evolution of domain walls with temperature in a two-dimensional spin glass.
There are 256x256 Ising spins and the temperature is reduced from 0.5 to 0.025, with uniform steps per each frame. The
correlations between neighboring spins are computed for periodic and antiperiodic (twisted in the x-direction) boundary
conditions. Comparing the two sets of correlation functions gives the domain wall. As the temperature is lowered, the
domain wall remains relatively unchanged and then rapidly shifts. This change portrays temperature chaos (see Bray and
Moore; Fisher and Huse), that is, the extreme sensitivity of the thermodynamic state to temperature. At very low temperatures,
one large domain wall dominates. The grey scale is logarithmic, i.e., dark regions indicate a large (nearly unit) change in
the correlation function with boundary condition, while light gray areas are areas that are only very weakly sensitive to
the boundaries.
[Huse, Middleton, Thomas,
arxiv.org:1103.1946 and
arxiv.org:1012.3444.]
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Coarsening in combinatorial algorithms.
Coarsening is familiar in physical systems, where a disordered system
establishes a phase by expanding volumes of uniform phase. Here, the
coarsening occurs during the application of an algorithm borrowed from
computer science and applied to finding ground states in disordered
magnets. The two rows show the coarsening of an auxiliary variable, the
height, in the push-relabel algorithm, during the deterministic search
for the solution. In the cited paper, connections are made between the
dynamics of the algorithms and the nature of the ground states. In
particular, the algorithms, though quite rapid, slow down near critical
points.
[Middleton,
cond-mat/0104185.]
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Droplets in disordered systems. These are low energy excitations above
the ground state configuration - the regions indicate where the excited
state differs from the ground state.
The picture on the left is a sample droplet (in a two dimensional
elastic medium) constrained to contain
the origin of a sample, but unconstrained in scale (except by sample size.)
The measurements of the indicated radii show that the droplets are "compact"
in the sense that their areas grow like the square of their linear dimension.
The picture on the right is of droplets excited by bulk perturbation of
the bond strengths in a two dimensional spin glass.
[Middleton,
cond-mat/0007375.]
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Introduction of domain walls by the expansion of sample
size in a 2-dimensional spin glass. The ground state is computed for two
system sizes (L and L'), with the disorder in the common region identical.
The solid lines indicate the relative domain walls introduced and the
window w is where changes in the ground state are measured.
[Middleton,
cond-mat/9904285.]
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Visualization of disorder-induced
defects in a two-dimensional elastic medium at zero temperature.
The ends of the strings represent +/- defect pairs. The defect
density depends
on the strength of the disorder. The strings themselves indicate where the
strain relative to the defect-free state
is localized: in contrast with defects in the pure 2D XY model, where
the elastic strain is spread throughout space, disorder leads to confinement
of the strain associated with the defects to a line.
[Middleton,
cond-mat/9807374.]
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Configuration of heights in a 3-dimensional elastic medium, with scalar
displacements, subject to quenched disorder. The black and white cubes indicate
displacements up and down from the average position (transparent cubes, which are
not visible!) The simulations on this model indicate a logarithmic roughness and
elastic costs which support some details
of the theoretical arguments
(cond-mat/9610146)
in favor of the existence of a Bragg glass in three
dimensions
[McNamara, Middleton, Zeng,
cond-mat/9905058.]
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Defect wall in the above model (3D elastic medium) generated by twisted
boundary conditions. Note that the domain wall is fractal, quite rough, and
though its surface area is relatively small, it is intersected by any vertical
plane.
[McNamara, Middleton, Zeng,
cond-mat/9905058.] (142 KB postscript)
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Not a simulation, but a figure from a proof.
In the referenced paper, it is shown that determining the barriers to the
motion of interfaces is generally an NP-hard problem (computationally
intractable, in practice), even though finding the low-lying states can be
done rapidly (polynomial time)
[Middleton, PRE 59, 2571 (1999);
cond-mat/9902203.]
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Visualization of the
application of an algorithm to find ground state of lines
in a random potential: shown is the exact ground state for 100
lines in a 300x300 area potential. [
Zeng, Middleton, and Shapir, Physical Review Letters 77, 3204 (1996).]
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Mincut algorithm applied to condensed matter
physics. The application
here is to find the ground state of an interface in the
random bond Ising model. [Reference:
A. Middleton, Phys. Rev. E 52, R3337 (1995).]
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Currents in a 128x128 array of mesoscopic dots, with
underlying charge disorder.
The top picture shows the time-averaged current just above
the threshold voltage for conduction, while the lower picture
shows the currents for a somewhat higher voltage.
The color scale is logarithmic in the current; the red dots
carry the most current, the blue dots the least.
[Reference:
A. Middleton, N. Wingreen, Phys. Rev. Lett. 71, 3198 (1993).]
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Avalanches generated by thermal excitation in a model
of charge-density waves.
[Middleton, Physical Review B 45, 9465 (1992).]
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