Logarithmic finite-size scaling of the self-avoiding walk at four dimensions

The n-vector spin model, which includes the self-avoiding walk (SAW) as a special case for the n -> 0 limit, has an upper critical dimensionality at four spatial dimensions (4D). We simulate the SAW on 4D hypercubic lattices with periodic boundary conditions by an irreversible Berretti-Sokal algorithm up to linear size L = 768. From an unwrapped end-to-end distance, we obtain the critical fugacity as z(c) = 0.147 622 380(2), improving over the existing result z(c) = 0.147 622 3(1) by 50 times. Such a precisely estimated critical point enables us to perform a systematic study of the finite-size scaling of 4D SAW for various quantities. Our data indicate that near z(c), the scaling behavior of the free energy simultaneously contains a scaling term from the Gaussian fixed point and the other accounting for multiplicative logarithmic corrections. In particular, it is clearly observed that the critical magnetic susceptibility and the specific heat logarithmically diverge as chi similar to L-2(ln L)(2 (y) over caph) and C similar to (ln L)(2 (y) over capt), and the logarithmic exponents are determined as (y) over cap (h) = 0.251(2) and (y) over cap (t) = 0.25(3), in excellent agreement with the field theoretical prediction (y) over cap (h) = (y) over cap (t) = 1/4. Our results provide a strong support for the recently conjectured finite-size scaling form for the O(n) universality classes at 4D.

Automated summary

The study focuses on the critical properties of the self-avoiding walk model in four spatial dimensions, obtaining a more precise critical fugacity and observing scaling behavior near the critical point, which includes terms from the Gaussian fixed point and multiplicative logarithmic corrections. The results provide strong support for the conjectured finite-size scaling form for the O(n) universality classes at 4D.