# Mutual cost of two held sources, leading order in 1/S, cubic torus (units as stated per block) (5.8) with kappa_h = J S, kappa_a = h_p against direct inverse (L=6): max error 4.88e-14 ## (1) Transverse sources: separation part J E_r/(g1 g2 (m1perp.m2perp)) single objects: -eps1 eps2 ghat(r); cells (x, x+e2), + on the first object: -eps1 eps2 sum ghat ghat_inf(0) = 0.2527310 (gamma_3 = 0.2527310) r eps1eps2 single: J E_r/(g1g2) 4 pi r ghat_inf(r) cells: J E_r/(g1g2) pi r sum ghat_inf torus check: ghat_L(r)-ghat_L(8) for L=16, 32, 64 vs ghat_inf(r)-ghat_inf(8) 2 +1 -0.042889 1.0779 -0.157642 0.9905 0.029359 0.032566 0.032862 0.032901 3 -1 0.027545 1.0384 0.106137 1.0003 0.014233 0.017247 0.017521 0.017557 4 +1 -0.020289 1.0198 -0.079621 1.0005 0.007303 0.010028 0.010269 0.010301 5 -1 0.016101 1.0117 0.063679 1.0003 0.003573 0.005889 0.006087 0.006113 6 +1 -0.013365 1.0077 -0.053058 1.0001 0.001456 0.003213 0.003358 0.003377 7 -1 0.011430 1.0055 0.045475 1.0001 0.000347 0.001353 0.001432 0.001442 8 +1 -0.009988 1.0041 -0.039790 1.0000 0.000000 0.000000 0.000000 0.000000 ## Continuum coefficients along n, single: J S E/(g1 g2) = -eps1 eps2 kappa_par / r^3, kappa_par = sqrt(2z)/(32 pi^3) = 0.0034913 held spin (defect), r >> l_b: S E/(J sig1 sig2) = +eps1 eps2 lambda / r^5, lambda = 0.0209480 changed links, J E/(d1 d2 S) * r^7: parallel to r: eps=+1 -0.0674991 (formula -0.0674991), eps=-1 -0.0325858 (formula -0.0325858) perpendicular: eps=+1 -0.0064008 (formula -0.0064008), eps=-1 -0.0122197 (formula -0.0122197) ## (2) Quadratic sources, lattice (zero modes left out); ratio = lattice / continuum columns: r, eps1eps2, along single [J S E/(g1g2), ratio], along cell [.., ratio vs 4x], defect [S E/(J s1 s2), ratio], link par [J E/(d1d2S), ratio], link perp [.., ratio] L = 32 2 +1 -4.624e-04 1.060 -1.491e-03 0.854 5.671e-04 0.866 -1.274e-03 2.417 -1.174e-04 2.347 3 -1 1.198e-04 0.927 4.142e-04 0.801 -1.211e-04 1.405 -2.430e-05 1.631 -1.119e-05 2.002 4 +1 -3.922e-05 0.719 -1.480e-04 0.678 1.949e-05 0.952 -6.880e-06 1.670 -6.874e-07 1.759 5 -1 1.640e-05 0.587 6.200e-05 0.555 -6.361e-06 0.949 -5.446e-07 1.306 -2.119e-07 1.355 6 +1 -7.170e-06 0.444 -2.790e-05 0.432 2.055e-06 0.763 -3.006e-07 1.247 -2.761e-08 1.207 7 -1 3.351e-06 0.329 1.293e-05 0.318 -8.651e-07 0.694 -4.524e-08 1.143 -1.714e-08 1.155 8 +1 -1.510e-06 0.221 -5.918e-06 0.217 3.521e-07 0.551 -3.768e-08 1.171 -3.017e-09 0.988 9 -1 6.659e-07 0.139 2.578e-06 0.135 -1.646e-07 0.464 -7.892e-09 1.158 -2.875e-09 1.125 10 +1 -2.585e-07 0.074 -1.014e-06 0.073 7.468e-08 0.356 -8.418e-09 1.247 -5.211e-10 0.814 11 -1 8.809e-08 0.034 3.369e-07 0.032 -3.849e-08 0.296 -2.283e-09 1.366 -7.612e-10 1.214 12 +1 -2.467e-08 0.012 -9.778e-08 0.012 2.099e-08 0.249 -2.906e-09 1.543 -1.184e-10 0.663 L = 64 2 +1 -4.970e-04 1.139 -1.616e-03 0.926 5.756e-04 0.879 -1.274e-03 2.416 -1.174e-04 2.348 3 -1 1.393e-04 1.077 4.876e-04 0.943 -1.251e-04 1.451 -2.430e-05 1.631 -1.119e-05 2.002 4 +1 -5.146e-05 0.943 -1.957e-04 0.897 2.119e-05 1.036 -6.868e-06 1.667 -6.922e-07 1.772 5 -1 2.483e-05 0.889 9.493e-05 0.850 -7.344e-06 1.096 -5.434e-07 1.303 -2.117e-07 1.353 6 +1 -1.312e-05 0.812 -5.144e-05 0.796 2.620e-06 0.972 -2.969e-07 1.231 -2.873e-08 1.256 7 -1 7.695e-06 0.756 3.007e-05 0.738 -1.231e-06 0.987 -4.446e-08 1.124 -1.699e-08 1.145 8 +1 -4.679e-06 0.686 -1.850e-05 0.678 5.840e-07 0.914 -3.586e-08 1.114 -3.397e-09 1.113 9 -1 2.998e-06 0.626 1.181e-05 0.616 -3.205e-07 0.904 -7.274e-09 1.068 -2.759e-09 1.080 10 +1 -1.950e-06 0.559 -7.737e-06 0.554 1.761e-07 0.841 -7.248e-09 1.074 -6.713e-10 1.049 11 -1 1.306e-06 0.498 5.163e-06 0.492 -1.058e-07 0.814 -1.750e-09 1.047 -6.603e-10 1.053 12 +1 -8.771e-07 0.434 -3.487e-06 0.431 6.334e-08 0.752 -1.997e-09 1.060 -1.788e-10 1.001 L = 128 2 +1 -5.071e-04 1.162 -1.653e-03 0.947 5.768e-04 0.881 -1.274e-03 2.416 -1.174e-04 2.348 3 -1 1.453e-04 1.124 5.105e-04 0.987 -1.256e-04 1.457 -2.430e-05 1.631 -1.119e-05 2.002 4 +1 -5.554e-05 1.018 -2.116e-04 0.970 2.146e-05 1.049 -6.867e-06 1.667 -6.925e-07 1.773 5 -1 2.784e-05 0.997 1.068e-04 0.956 -7.510e-06 1.120 -5.434e-07 1.303 -2.117e-07 1.353 6 +1 -1.545e-05 0.956 -6.063e-05 0.938 2.724e-06 1.011 -2.967e-07 1.230 -2.881e-08 1.260 7 -1 9.546e-06 0.938 3.740e-05 0.919 -1.304e-06 1.046 -4.445e-08 1.123 -1.699e-08 1.145 8 +1 -6.180e-06 0.906 -2.447e-05 0.897 6.354e-07 0.994 -3.578e-08 1.112 -3.429e-09 1.123 9 -1 4.239e-06 0.885 1.674e-05 0.874 -3.590e-07 1.012 -7.265e-09 1.066 -2.757e-09 1.079 10 +1 -2.985e-06 0.855 -1.186e-05 0.849 2.049e-07 0.978 -7.205e-09 1.067 -6.865e-10 1.072 11 -1 2.178e-06 0.830 8.638e-06 0.823 -1.282e-07 0.986 -1.743e-09 1.042 -6.589e-10 1.051 12 +1 -1.616e-06 0.800 -6.433e-06 0.796 8.072e-08 0.959 -1.970e-09 1.046 -1.869e-10 1.046 ratio extrapolated to L -> infinity: Aitken from L = 32, 64, 128 (geometric finite-size error); the L = 128 value where the change from 64 to 128 is below 1e-3 or not geometric 2 1.172 0.956 0.881 2.416 2.348 3 1.144 1.007 1.459 1.631 2.002 4 1.055 1.007 1.051 1.667 1.773 5 1.057 1.016 1.125 1.303 1.353 6 1.047 1.029 1.020 1.230 1.260 7 1.073 1.053 1.060 1.123 1.145 8 1.104 1.095 1.017 1.112 1.124 9 1.180 1.170 1.047 1.066 1.079 10 1.322 1.318 1.032 1.067 1.075 11 1.672 1.675 1.071 1.042 1.051 12 3.173 3.225 1.103 1.046 1.053