Raz Carmi

The project is written for MATLAB 5.2 .

In order to run the simulation write: start

The files are: start.m start.mat simulate.m
picabst.m picabst.mat picsett.m picsett.mat picresu.m
picresu.mat picalgo.m picalgo.mat texabst.m texsett.m
texresu.m texalgo.m

__Abstract:__

Spontaneous formation of topological defects in non-equilibrium phase
transitions was extensively studied in the past few years. This issue is
relevant to the early universe theories, superfluids 4He and 3He, liquid
crystals, superconductors and Bose-Einstein condensation. A theory of bubble
collision of domains having different values of the phase belonging to
the complex order parameter can help to predict the net spontaneous flux
which should be measured in a rapid superconducting phase transition. The
simulation is based on a 2D hexagonal lattice where each point represents
a domain of order parameter with random phase and constant amplitude. The
matrix element (a point in the lattice) has a real value (the phase). Between
each three points a quantized vortex, antivortex or homogeneous site may
be created according to the three random phases and a condition called
''the relaxed geodesic rule'' (described by an appropriate function with
independent random variable). The geodesic rule is the assumption of minimal
gradient and the ''relaxing'' is the probability for deviation from this
rule. The program should find the random array of vortices and antivortices,
and the net flux (which is a measurable quantity in a real experiment).
The relaxing strength can be changed in different simulations and the influence
on the net flux and on the total vortex density can be investigated.

__Parameter setting:__

There are four setting options:

1. Matrix dimension: determines the size of the square matrix
for the simulation.

A real superconductor sample is typically
analog to a matrix of the order of

magnitude of 1000 to 10,000 square but it
also depends on the rapidness of the

quench. A simulation with such a matrix will
execute for a long while.

Here, we are limited to smaller systems in
order to show the behavior of the results

in a reasonable period of time.

2. No. of quenches: determines the number of independent simulation
runs which are

included in the statistics. Each quench is
analog to a cooldoun of the system from

the normal to the superconducting state and
then counting the spontaneous

vortices. Where this number is higher the
accuracy of the statistics is better, but it

takes longer to complete the simulation.

3. Max. relaxing and No. of relaxing points: determine the maximal
relaxing strength

and the number of the relaxing points which
will appear in the comparison

between the net and the total flux. The scale
for the relaxing strength is chosen to

be equal to 1 for "very high" relaxing. The
first result to appear in the graph

(relaxing=0) is always the reference point:
net flux = total flux =1 (as a relative

value).

__Results:__

The main result of the simulations is that the increasing of the net
flux (vortices minus antivortices) is much sensitive to the relaxed geodesic
rule than the increasing of the total flux. The maximal theoretical statistical
value of the net flux is the square root of the total flux (similar to
the standard deviation in a random walk). Here we can see that even when
the relaxing strength is not too high the net flux is close to about 1/10
of the maximal value or even higher (providing the matrix dimension is
not too small). This result has an experimental implication for superconductors.

The graphs show:

1. A sample of the vortex array (40x40). When the relaxing strength
start growing the density is higher and

more double vortices/antivortices appear.
Too many of these objects is not physically reasonable since

they are unstable in a real system.

2. Distribution Histograms of the net and the total flux, containing
the number of quenches defined in the

setting. The average of the net flux is always around
zero. The distribution of the total flux is very close to

its average.

3. A comparison between the net flux and the total flux, both relative
to their values without relaxing. Here,

the net flux is actually the standard deviation
of the distribution (red). The total flux is the average of the

distribution (blue).

__Algorithm:__

The important aspect of the algorithm is how to define properly the
relaxed geodesic rule, while maintaining the correct physical meaning.
The main steps are:

1. Each matrix element of the square array has a random value between
0 to 2pi, represents the phase of the

complex order parameter.

2. The interactions between the matrix elements are based on a structure
of hexagonal lattice, because a

vortex may be created between 3 different phase
values. That is, if the sum of the phase differences

equals +-2pi.

3. In the simple geodesic rule the phase difference is always minimized.
If the geodesic rule is relaxed, there

is some probability, for example, that a phase difference
of pi/2 will be considered as -3pi/2.

4. Here, the program calculates the sum of the 3 phase difference values
around each contact point in the

lattice and checks if it is equal to 0 (homogeneous
site), +-1 (vortex or antivortex), +-2 or more (double

vortex/antivortex). Before the summation each phase
difference is "relaxed" with a probability determined

by the relaxing strength parameter.

5. For a physical meaning a non-symmetric probability is declared (in
some mathematical steps). For

example, if there is originally a vortex between
three phase regions, the probability for homogeneous site

is higher than for a double vortex (in an appropriate
way), since in a real system a double vortex is more

costly in energy.