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

Field-theoretical calculations predict that, at the upper critical dimension d_{c}=4, the finite-size scaling (FSS) behaviors of the Ising model would be modified by multiplicative logarithmic corrections with thermal and magnetic correction exponents (y[over ̂]_{t},y[over ̂]_{h})=(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 C_{1}∼L^{3}(lnL)^{y[over ̂]_{h}}, while the size of the second-largest cluster scales as C_{2}∼L^{3}(lnL)^{y[over ̂]_{h2}} with y[over ̂]_{h2}=-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 F_{1}∼L^{2}(lnL)^{y[over ̂]_{t}}, and the specific heat scales as c_{E}∼(lnL)^{2y[over ̂]_{t}}. This clarifies the long-standing open question that whether the specific heat for the critical Ising model at d_{c}=4 diverges logarithmically.