Logarithmic finite-size scaling of the four-dimensional Ising model

Field-theoretical calculations predict that, at the upper critical dimension dc = 4, the finite-size scaling (FSS) behaviors of the Ising model would be modified by multiplicative logarithmic corrections with thermal and magnetic correction exponents (yt, yh) = (1/6, 1/4). Using high-efficient cluster algorithms and the lifted worm algorithm, we present a systematic study to the FSS of the four-dimensional Ising model at criticality in the Fortuin-Kasteleyn (FK) bond and loop representations. In the FK representation, the size of the largest cluster is observed to scale as C1 similar to L3(ln L)yh, while the size of the second-largest cluster scales as C2 similar to L3(ln L)yh2 with yh2 = -1/4 a new correction exponent not yet predicted from field theory. In the loop representation, we observe that the size of the largest loop cluster scales as F1 similar to L2(ln L)yt, and the specific heat scales as cE similar to (ln L)2yt. This clarifies the long-standing open question that whether the specific heat for the critical Ising model at dc = 4 diverges logarithmically.