Fig 1.
The image shows a system’s configuration, with blue circles representing + 1 spins and red circles symbolizing –1 spins.
Positive and negative interactions are represented by blue and red lines, respectively. Arrows indicate the local fields (in a pre-equilibrium state) calculated from Eq 3, with their lengths illustrating the field’s magnitude and their directions showing its orientation.
Fig 2.
Figure illustrates three extreme limits where paramagnetic, ferromagnetic, and spin-glass phases are expected.
Different types of spin-spin interactions that reduce the system’s energy and are more probable to be found inside the system are presented.
Fig 3.
Variance of local fields versus mean of local fields for (a) , (b)
.
The centroid of each cluster is marked by an ‘X’ and associated error bars estimated from the standard deviation of point-to-centroid distances. Red, green, and blue clusters represent the paramagnetic, spin-glass, and ferromagnetic phases, respectively. The standard deviation of each cluster centroid is represented by error bars in both x and y directions. (c): Variance of the local fields in the spin-glass phase (,
) as a function of lattice size. The dashed line represents the function
fitted to data.
Table 1.
The centroid position with associated standard deviations for each cluster at different system sizes.
Values are expressed as , where X and Y denote the centroid coordinates, and
and
represent the standard deviations along the x- and y-directions, respectively. Smaller error bars at larger N indicate improved stability and clustering accuracy.
Fig 4.
(a): These systems were simulated for (0,3] and
(0,3] without the prior knowledge about different phases, and the probability density function of points in feature space illustrates the dense areas as lighter color with a contour-plot.
The points stretch from the dense region corresponding to the spin glass phase (left top) to the dense region corresponding to the ferromagnetic phase (right bottom), representing the mixed phase region. (b): The dense areas are separated by removing the data less than threshold = 0.5 in the probability density function. (c): The centroid of each cluster is determined by the K-means algorithm.
Fig 5.
The probability density function of points in feature space of variance versus mean of the local field for (0,3] and
(0,3] displayed as a contour-plot.
The maxima of the function are marked by an ‘X’.
Fig 6.
Phase diagram: Dash lines indicate the boundaries of phase transition, obtained from theory, and the dash-dot line marks the AT line.
The color of each pixel shows the degree of similarity to one of three phases. Red points indicate similarity to the paramagnetic phase, green points to the spin-glass phase, and blue points to the ferromagnetic phase. The diagram is 30*30 pixels simulated for the lattice size N = 1024.