Sketch showing spins (up or down) on vertices of a square lattice, and defect variables (circled or blank) on the dual lattice formed by the square plaquettes.
A droplet of size . There are frozen boundary spins (black circles) and free bulk spins (white circles). The energy is determined by the state of the 36 defect variables that sit on the dual lattice. This energy can be divided into a boundary contribution that comes from the sites of the dual lattice marked with +, and a bulk contribution from the remaining sites. The partition sum is over the configurations of the bulk spins; these correspond to of the configurations of the defect variables. A given configuration of the defect variables contributes to the partition sum if and only if it satisfies the nine independent constraints set by the frozen boundary spins. For example, one such constraint is that the parity of the number of defects in the third row of the dual lattice is the same as the frozen spin combination , where the spins are identified in the figure.
(Top) Droplet entropy density vs for droplets of different sizes , with frozen boundary conditions enforcing even numbers of defects in every row or column, . The symbols are finite entropies calculated using the expressions of Appendix A. The entropy is extensive for . On approaching from above it deviates from the bulk entropy (solid line, ), and becomes subextensive for . (Bottom) Plot of total entropy against showing the (nonextensive) scaling predicted in (18).
Ground states of droplets with different boundary conditions. Spins are on vertices on the square lattice and are not shown. Excited plaquettes are marked with × symbols. (a) Boundary conditions enforce two rows and two columns with odd numbers of defects, so that and . There are two ground states that differ by flipping the two spins identified with black circles. (b) Boundary conditions enforce two rows with odd numbers of defects, but all columns with even numbers: and . There are distinct ground states of which two are shown: they differ only at the four marked spins.
Entropy as a function of for . The full line is the bulk entropy of the SPM. The open symbols are the entropies for frozen boundaries with different , as obtained from the finite-size partition functions of Appendix A. The + symbols give the entropy in the case where the boundary conditions are generated randomly.
Plots of the overlap function , averaged over boundary conditions at various temperatures. (Top) Average overlap as a function of temperature. (Middle) Collapse of this data as a function of (the dashed line is a guide to the eyes). (Bottom) Spatial dependence along the diagonal of the square droplet, showing strong correlations near the boundary.
Simulations of the SPM at for various system sizes and frozen boundary conditions. The relaxation time increases significantly as decreases through unity, but the long-time limit of would still appear to vanish since . We also show relaxation for and periodic boundary conditions. The relaxation can be well fitted by an exponential for large . At smaller the decay is slower than exponential.
Simulations of the SPM at for various system sizes and frozen boundary conditions. The long-time limit of increases from zero as becomes of order unity.
Sketch showing relation between square and rhomboid droplets . The spins are on the intersections of grids; is marked by a filled circle. The × symbol marks the plaquette interaction . Rotations of 120° leave both Hamiltonian and triangular lattices invariant. However, note that the boundaries of the droplets are not all equivalent, since the rhombus shape is not invariant under this rotation.
Illustration of Eq. (36) with . The product of the five spins marked by filled circles is given by the product of the 15 plaquette variables marked with crosses. Both the marked defects and the spins marked with circles (filled and empty) form parts of Sierpinski triangles: they are marked if and only if the relevant entry of Pascal’s triangle is odd.
Relaxation of finite systems at inverse temperature . Notice that the long-time limit of is changing, but that the relaxation time is approximately constant. Compare to Fig. 8, where the relaxation time increases sharply as increases.
Relaxation at different temperatures showing bulk results (circles) and two different values of . The actual system sizes used are (left) and 26; (middle) and 32; and (right) and 44. The dashed lines are guides to the eyes showing that the long-time limit scales as a function of only (there are weak deviations from scaling which we attribute to subleading corrections in and ).
Circular average of the normalized four-point dynamic correlator. The system size is with periodic boundaries, which is large enough to avoid finite-size effects. (Left panel) Three different temperatures, with times chosen so that . (Right panel) Collapse of the data of the left panel on rescaling of length as .
Data showing relaxation time for large systems as a function of inverse temperature, . The relaxation time is defined by . The error bars are smaller than the symbols shown. The dashed line is a fit to the form . Good fits are also possible with different coefficients for the quadratic term.
Article metrics loading...
Full text loading...