A large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered. The dependence of the finite-volume magnetizationMper(h, L) in cubes of sizeLdunder periodic boundary conditions on the external fieldh is analyzed. For the case where two phases coexist at the infinite-volume transition pointht, we prove that, independent of the details of the model, the finite-volume magnetization per lattice site behaves likeMper(ht)+M tanhMLd(h-ht) withMper(h) denoting the infinite-volume magnetization and M=1/2Mper(ht+0)-Mper(ht-0). Introducing the finite-size transition pointhm(L) as the point where the finite-volume susceptibility attains the maximum, we show that, in the case of asymmetric field-driven transitions, its shift isht-hm(L)=O(L-2d), in contrast to claims in the literature. Starting from the obvious observation that the number of stable phases has a local maximum at the transition point, we propose a new way of determining the pointhtfrom finite-size data with a shift that is exponentially small inL. Finally, the finite-size effects are discussed also in the case where more than two phases coexist.
%0 Journal Article
%1 Borgs1990Rigorous
%A Borgs, Christian
%A Kotecký, Roman
%D 1990
%J Journal of Statistical Physics
%K scaling critical-phenomena finite-size phase-transitions
%N 1
%P 79--119
%R 10.1007/bf01013955
%T A rigorous theory of finite-size scaling at first-order phase transitions
%U http://dx.doi.org/10.1007/bf01013955
%V 61
%X A large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered. The dependence of the finite-volume magnetizationMper(h, L) in cubes of sizeLdunder periodic boundary conditions on the external fieldh is analyzed. For the case where two phases coexist at the infinite-volume transition pointht, we prove that, independent of the details of the model, the finite-volume magnetization per lattice site behaves likeMper(ht)+M tanhMLd(h-ht) withMper(h) denoting the infinite-volume magnetization and M=1/2Mper(ht+0)-Mper(ht-0). Introducing the finite-size transition pointhm(L) as the point where the finite-volume susceptibility attains the maximum, we show that, in the case of asymmetric field-driven transitions, its shift isht-hm(L)=O(L-2d), in contrast to claims in the literature. Starting from the obvious observation that the number of stable phases has a local maximum at the transition point, we propose a new way of determining the pointhtfrom finite-size data with a shift that is exponentially small inL. Finally, the finite-size effects are discussed also in the case where more than two phases coexist.
@article{Borgs1990Rigorous,
abstract = {{A large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered. The dependence of the finite-volume magnetizationMper(h, L) in cubes of sizeLdunder periodic boundary conditions on the external fieldh is analyzed. For the case where two phases coexist at the infinite-volume transition pointht, we prove that, independent of the details of the model, the finite-volume magnetization per lattice site behaves likeMper(ht)+M tanh[MLd(h-ht)] withMper(h) denoting the infinite-volume magnetization and M=1/2[Mper(ht+0)-Mper(ht-0)]. Introducing the finite-size transition pointhm(L) as the point where the finite-volume susceptibility attains the maximum, we show that, in the case of asymmetric field-driven transitions, its shift isht-hm(L)=O(L-2d), in contrast to claims in the literature. Starting from the obvious observation that the number of stable phases has a local maximum at the transition point, we propose a new way of determining the pointhtfrom finite-size data with a shift that is exponentially small inL. Finally, the finite-size effects are discussed also in the case where more than two phases coexist.}},
added-at = {2019-06-10T14:53:09.000+0200},
author = {Borgs, Christian and Koteck\'{y}, Roman},
biburl = {https://www.bibsonomy.org/bibtex/2518705fc025764c84de48e4c97f40491/nonancourt},
citeulike-article-id = {3612691},
citeulike-linkout-0 = {http://dx.doi.org/10.1007/bf01013955},
citeulike-linkout-1 = {http://www.springerlink.com/content/pl1r7jg2r04217n9},
day = 1,
doi = {10.1007/bf01013955},
interhash = {20727ce2928b66d122131c69e7801bfa},
intrahash = {518705fc025764c84de48e4c97f40491},
journal = {Journal of Statistical Physics},
keywords = {scaling critical-phenomena finite-size phase-transitions},
month = oct,
number = 1,
pages = {79--119},
posted-at = {2009-06-29 15:56:38},
priority = {2},
timestamp = {2019-08-01T16:20:05.000+0200},
title = {{A rigorous theory of finite-size scaling at first-order phase transitions}},
url = {http://dx.doi.org/10.1007/bf01013955},
volume = 61,
year = 1990
}