Finite-size scaling at infinite-order phase transitions

Abstract

For systems with infinite-order phase transitions, in which an order parameter smoothly becomes nonzero, a new observable for finite-size scaling analysis is suggested. By construction this new observable has the favourable property of diverging at the critical point. Focussing on the example of the F -model we compare the analysis of this observable with that of another observable, which is also derived from the order parameter but does not diverge, as well as that of the associated susceptibility. We discuss the difficulties that arise in the finite-size scaling analysis of such systems. In particular we show that one may reach incorrect conclusions from large-system size extrapolations of observables that are not known to diverge at the critical point. Our work suggests that one should base finite-size scaling analyses for infinite-order phase transitions only on observables that are guaranteed to diverge.

Numerical study of the F model with domain-wall boundaries

“…Previously we found behavioural similarities between d ln P 0 /dβ and the susceptibility χ of the staggered polarization for PBCs [3]. Since there are no known analytical expressions for the asymptotic behaviour of P 0 for DW-BCs we fall back on the leading corrections known for PBCs [59].…”

“…[2]. This choice was also used in our previous work [3]. The same thermodynamic behaviour is obtained for 'free' and (conjecturally) 'Néel' boundary conditions, where the arrows on the external edges are respectively left free or fixed to alternate [6,10].…”

“…In this work we prefer speed over sample accuracy as this allows us to investigate much larger systems, thus improving the reliability of our subsequent analysis of the thermodynamic limit. Rather than CFTP we thus use the full lattice multi-cluster algorithm [54], as in [3], with a reported dynamic exponent z = 0.005 ± 0.022 for PBCs [55], so that the correlation time can be considered independent of system size in practice. The accuracy of our simulations is checked in Sec.…”

“…On the other hand it causes the model's thermodynamics to depend on the choice of boundary conditions used at the intermediate analysis for finite size [6][7][8]. (In fact, this phenomenon in the context of graphene [9] originally motivated [3] and the present work.) PBCs are commonly employed and are compatible with the translational invariance that is present for infinite systems.…”

Experimental measures of topological sector fluctuations in the F-model

“…Exact solutions by Lieb [2] and Sutherland [3] in 1967 and Baxter [4] in 1973 later placed it the small class of exactly solved models in statistical mechanics [5]. Theoretical interest in the F-model has long been sustained by its relevance to surface roughening (BCSOS model [6]), to quantum many body problems [7], and as an exemplar of an infinite-order transition [8][9][10].…”