Finite-size Scaling Behavior In High Dimensions

Finite-size scaling(FSS)is a fundamental theory describing the asymptotic approach of finite-size systems to the thermodynamic limits near continuous phase transition points.Below the upper critical dimension dc,FSS has been widely accepted and proven to be a powerful tool for extracting critical exponents from finite-size systems.However,compared to FSS in low dimensions,FSS above the upper critical dimension is surprisingly subtle,which has been a long-standing debate and deeply depends on the boundary conditions.Based on these debates,this thesis studies the high-dimensional FSS behavior of a series of statistical physical models under periodic boundary conditions.We observe very rich physical phenomena and extract a simple physical scenario,where the FSS behaviors on high-dimensional tori are simultaneously governed by the complete graph(CG)asymptotics and the Gaussian fixed-point(GFP)asymptotics.Moreover,we propose that the Ising model has simultaneous two upper critical dimensions under the Fortuin-Kateleyn(FK)geometric representation.Based on Monte Carlo simulations and renormalization group analysis,we provide a simple physical picture that the FSS behavior of the Ising model on periodic boundary conditions in d>dc=4 is simultaneously dominated by the CG asymptotics and the GFP asymptotics.Additionally,We conjecture the form of the free energy density function,which successfully explains existing experimental results.Furthermore,based on extensive simulations from d=4 to 7,insights from renormalization group theory,rigorous solutions and numerical results on the CG,we conjecture that the Ising model in the FK representation has two upper critical dimensions(dc=4,dp=6).We found that as long as d>dc,there are simultaneous two configuration sectors,two lengthscale behavior,and two scaling windows.When 4<d<6,the scaling behavior is governed by the GFP asymptotics and CG-Ising asymptotics.While for d≥6,the scaling behavior is controlled by the CG-Ising asymptotics and high-dimensional percolation asymptotics,which includes the CG-percolation asymptotics and GFP asymptotics.Moreover,we extend the physical picture of high-dimensional FSS to the upper critical dimension dc,and verify the conjectured form of the free energy density by simulating the self-avoiding walk(SAW)model at large scales.We observe the logarithmic divergence of the specific heat firstly,consistent with the predictions of field theory,and refute some numerical results.The structure of this paper is as follows.In Chapter 1,we introduce some basic knowledge.In Chapter 2,we present the research results of the Ising model on the CG(d→∞)and observe strong percolation effects in the FK representation,which can be well understood from the perspective of the loop representation via the loopcluster algorithm.In Chapter 3,we study the FSS behavior of the FK Ising model on a five-dimensional lattice with periodic boundaries condition,which is simultaneously dominated by the CG asymptotic and GFP asymptotic.In Chapter 4,we present the research results on high-dimensional tori(d≥5),showing that there are simultaneous two upper critical dimensions(dc=4,dp=6)in the FK representation.In Chapter 5,we extend the scenario above dc to dc and study the FSS behavior of the self-avoiding random walk(SAW)on a four-dimensional torus.In Chapter 6,we introduce the FSS behavior of the percolation model on a seven-dimensional torus.In Chapter 7,we will summarize the entire paper and look forward to the future,proposing questions that require further research.