Extending the code from two dimensions to three dimensions is trivial and the chance for mistakes is rather small. However, we have tested the code at least at µ = 0 using the results from previous work. Here we compare results for χ obtained using the worm algorithm with that obtained using the microcanonical improved Metropolis (MM) algorithm and the available results in the literature using the cluster method [28]. The comparison is shown in Table VII. The reason for us to choose β = 0 . 45421 is because this is known to be the critical value of the coupling where the theory undergoes a phase transition from a normal phase to a superfluid phase. At the critical coupling we expect χ ∝ L $^{γ/ν}$. A fit of our data to this form yields the value of γ/ν = 1 . 99 as expected from [28], and is shown in figure 10. TABLE VI: Checks of the observables with exact solution on 2 × 2 lattices. 28 β χ χ MC ρ ρ MC κ κ MC ρ$_{s}$ ρ MC s µ = 0 . 0 0 . 1 1 . 2207 1 . 2206(1) 0 2(2) × 10 − 5 0 . 01005 0 . 01001(4) 0 . 01005 0 . 01008(4) 0 . 2 1 . 4831 1 . 4830(2) 0 − 5(4) × 10 − 5 0 . 04064 0 . 04063(8) 0 . 04064 0 . 04076(8) 0 . 5 2 . 3838 2 . 3835(4) 0 1(9) × 10 − 5 0 . 2526 0 . 2527(2) 0 . 2526 0 . 2526(2) 1 . 0 3 . 2730 3 . 2733(4) 0 2(2) × 10 − 4 0 . 7796 0 . 7796(4) 0 . 7796 0 . 7794(4) 5 . 0 3 . 8728 3 . 8733(3) 0 1(1) × 10 − 3 4 . 809 4 . 806(4) 4 . 809 4 . 813(4) µ = 0 . 5 0 . 1 1 . 2362 1 . 2361(1) 0 . 00590 0 . 00592(2) 0 . 01559 0 . 01560(5) 0 . 01007 0 . 01013(4) 0 . 2 1 . 5188 1 . 5188(2) 0 . 02374 0 . 02378(5) 0 . 0640 0 . 0640(1) 0 . 04100 0 . 04105(7) 0 . 5 2 . 4692 2 . 4693(4) 0 . 1429 0 . 1430(1) 0 . 4341 0 . 4344(3) 0 . 2581 0 . 2581(2) 1 . 0 3 . 3245 3 . 3247(4) 0 . 4190 0 . 4193(2) 1 . 6553 1 . 6564(10) 0 . 7867 0 . 7861(4) µ = 1 . 0 0 . 1 1 . 2861 1 . 2861(2) 0 . 01809 0 . 01812(4) 0 . 03851 0 . 03857(8) 0 . 01018 0 . 01024(4) 0 . 2 1 . 6316 1 . 6317(2) 0 . 07160 0 . 07162(8) 0 . 1637 0 . 1638(2) 0 . 04241 0 . 04242(8) 0 . 5 2 . 6899 2 . 6900(4) 0 . 3865 0 . 3868(2) 1 . 2530 1 . 2539(8) 0 . 2747 0 . 2748(2) 1 . 0 3 . 4317 3 . 4317(4) 1 . 0053 1 . 0055(3) 5 . 487 5 . 488(3) 0 . 8032 0 . 8033(4)