Real-space correlation functions of nearest-neighbour spin ice

<p>Real-space correlation functions&nbsp;<span class="math-tex">\(\langle \sigma_i\sigma_j\rangle\)</span> of the nearest-neighbour spin ice Hamiltonian <span class="math-tex">\(H = J \sum_{\langle ij\rangle} \sigma_i\sigma_j\)</span> on the pyrochlore lattice, from Monte Carlo simulations on 128&times;128&times;128 cubic unit cells.</p> <p><strong>File formats</strong></p> <p>The file name indicates the temperature (in units of J) where the measurement was taken.</p> <p>Files with extension .dat contain the correlation functions as double-precision real floats.<br> Files with extension .err contain the standard errors as single-precision real floats.</p> <p>Both file types contain an array of shape (4, 4, 4, 128, 128, 128). The meaning of the indices (from major to minor):</p> <ul> <li>sublattice index &mu;</li> <li>sublattice index &nu;</li> <li>index &lambda; of FCC lattice point in the cubic unit cell</li> <li>offset of cubic lattice cells</li> </ul> <p>Entry [&mu;, &nu;, &lambda;, x, y, z] corresponds to the correlator <span class="math-tex">\(\langle \sigma(\vec R + \vec r^\mathrm{FCC}_\lambda + \vec r_\mu)\sigma(\vec r_\nu)\rangle\)</span>, where <span class="math-tex">\(\vec R = [xyz] \)</span>, the FCC index &lambda; corresponds to the lattice points <span class="math-tex">\(\vec r^\mathrm{FCC}_0 = [000], \vec r^\mathrm{FCC}_1=[011]/2, \vec r^\mathrm{FCC}_2=[101]/2,\vec r^\mathrm{FCC}_3 = [110]/2\)</span>, and the sublattice&nbsp; indices correspond to the site offsets <span class="math-tex">\(r_0 = [111]/8, r_1=[1\bar1\bar1]/8, r_2=[\bar11\bar1]/8,r_3=[\bar1\bar1 1]/8\)</span> from the nearest FCC lattice point. Coordinates x,y,z run between 0 and 127 in periodic boundary conditions.</p> <p>The utility loader.py loads the files, shapes them in the correct array format, and extracts individual correlators.</p> <p><strong>Details of data generation</strong></p> <p>We performed Monte Carlo simulations of nearest-neighbour spin ice using the efficient loop-string algorithm introduced in <a href="https://doi.org/10.1103/PhysRevB.90.220406">Phys. Rev. B 90, 220406(R)</a>. For nonzero temperatures, correlations between different strings were excluded, which effectively averages all spin configurations compatible with a given loop graph. We ran 32 independent Markov chains at each temperature point and obtained 4096 Monte Carlo samples in each. The reported standard errors are the error on the mean of the 32 Markov chains.</p>