(a) Shared empty place, second order: open chain of L = 400 places, J = 1.0, x1 = 100, g1 = g2 = 1, N_f = 1 Delta/J = 2.02: phi = |empty> (overlap 1.000000000000000), gap = 0.0201; xi = 1/arccosh(Delta/2J) = 7.07695; fitted decay length = 7.07695; max |M2/(4.14) - 1| over d = 1..160 = 5.7e-13 d : 1 5 10 20 40 80 M2 : -6.12e+00 -3.48e+00 -1.72e+00 -4.18e-01 -2.48e-02 -8.69e-05 d^6 |M2| : 6.12e+00 5.44e+04 1.72e+06 2.67e+07 1.01e+08 2.28e+07 Delta/J = 2.1: phi = |empty> (overlap 1.000000000000000), gap = 0.1001; xi = 1/arccosh(Delta/2J) = 3.17536; fitted decay length = 3.17536; max |M2/(4.14) - 1| over d = 1..88 = 2.2e-15 d : 1 5 10 20 40 80 M2 : -2.28e+00 -6.47e-01 -1.34e-01 -5.74e-03 -1.06e-05 -3.57e-11 d^6 |M2| : 2.28e+00 1.01e+04 1.34e+05 3.68e+05 4.33e+04 9.37e+00 Delta/J = 2.5: phi = |empty> (overlap 1.000000000000000), gap = 0.5001; xi = 1/arccosh(Delta/2J) = 1.44270; fitted decay length = 1.44270; max |M2/(4.14) - 1| over d = 1..40 = 2.4e-15 d : 1 5 10 20 40 80 M2 : -6.67e-01 -4.17e-02 -1.30e-03 -1.27e-06 -1.21e-12 -1.10e-24 d^6 |M2| : 6.67e-01 6.51e+02 1.30e+03 8.14e+01 4.97e-03 2.89e-13 Delta/J = 3.0: phi = |empty> (overlap 1.000000000000000), gap = 1.0001; xi = 1/arccosh(Delta/2J) = 1.03904; fitted decay length = 1.03904; max |M2/(4.14) - 1| over d = 1..29 = 2.9e-15 d : 1 5 10 20 40 80 M2 : -3.42e-01 -7.27e-03 -5.91e-05 -3.91e-09 -1.71e-17 -3.26e-34 d^6 |M2| : 3.42e-01 1.14e+02 5.91e+01 2.50e-01 7.00e-08 8.55e-23 (b) Shared empty place, exact M(d) = eps0(g,g) - eps0(g,0) - eps0(0,g) + eps0(0,0), N_f = 1, Delta/J = 3, L = 400, x1 = 100 limit = g1^2 g2^2 (A11 B22 + A22 B11) (fourth order, d -> infinity), A = K^{-1}, B = K^{-2} eps0(0,0) = 0.0e+00 g d: 1 10 40 160 limit(d=160) M(160)/limit - 1 0.05 -8.487e-04 1.347e-06 1.493e-06 1.493e-06 1.5000e-06 -4.45e-03 0.10 -3.332e-03 2.302e-05 2.358e-05 2.358e-05 2.4000e-05 -1.76e-02 0.20 -1.239e-02 3.564e-04 3.583e-04 3.583e-04 3.8400e-04 -6.68e-02 (c) Cell field: 18 cells (places {0,1}^18, dimension 262144), J = 1, x1 = 2 (cells 0..17), g1 = g2 = g; exact M by sparse diagonalization (eigsh) Delta/J = 3.0, g = 0.1 (|g1|+|g2| = 0.2 < (Delta-2J)/2 = 0.5): E0 = 0 (vacuum), xi = 1.03904; fitted decay length on 2 <= d <= 12: of M 1.03921, of M2 1.03902 d : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 M : -3.36e-03 -1.28e-03 -4.91e-04 -1.88e-04 -7.18e-05 -2.74e-05 -1.05e-05 -4.00e-06 -1.53e-06 -5.84e-07 -2.23e-07 -8.51e-08 -3.24e-08 -1.22e-08 M/M2 : 0.98566 0.98767 0.98889 0.98953 0.98983 0.98997 0.99003 0.99005 0.99007 0.99007 0.99007 0.99008 0.99012 0.99037 M/M2 over 1 <= d <= 14: min 0.98566, max 0.99037; over 4 <= d <= 14: spread 8.4e-04 Delta/J = 3.0, g = 0.2 (|g1|+|g2| = 0.4 < (Delta-2J)/2 = 0.5): E0 = 0 (vacuum), xi = 1.03904; fitted decay length on 2 <= d <= 12: of M 1.03974, of M2 1.03902 d : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 M : -1.29e-02 -4.96e-03 -1.90e-03 -7.28e-04 -2.79e-04 -1.06e-04 -4.07e-05 -1.55e-05 -5.93e-06 -2.27e-06 -8.66e-07 -3.31e-07 -1.26e-07 -4.73e-08 M/M2 : 0.94531 0.95254 0.95706 0.95941 0.96055 0.96106 0.96129 0.96140 0.96144 0.96146 0.96147 0.96149 0.96164 0.96257 M/M2 over 1 <= d <= 14: min 0.94531, max 0.96257; over 4 <= d <= 14: spread 3.2e-03 Delta/J = 2.5, g = 0.1 (|g1|+|g2| = 0.2 < (Delta-2J)/2 = 0.25): E0 = 0 (vacuum), xi = 1.44270; fitted decay length on 2 <= d <= 12: of M 1.44387, of M2 1.44223 d : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 M : -6.31e-03 -3.17e-03 -1.59e-03 -7.96e-04 -3.99e-04 -2.00e-04 -9.99e-05 -4.99e-05 -2.50e-05 -1.25e-05 -6.24e-06 -3.11e-06 -1.54e-06 -7.33e-07 M/M2 : 0.96196 0.96543 0.96864 0.97092 0.97238 0.97327 0.97379 0.97409 0.97426 0.97436 0.97444 0.97458 0.97503 0.97655 M/M2 over 1 <= d <= 14: min 0.96196, max 0.97655; over 4 <= d <= 14: spread 5.6e-03