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arXiv:2005.01737

Perturbative PDF of the total magnetization of the 4D Ising model

Andrea Allais

Published May 4, 2020• 0 citations
arXivHTMLPDF

Abstract

We compute, at one loop in perturbation theory, the probability density function of the total magnetization $M$ of the Ising model on the 4-torus and the 4-sphere. We develop a single perturbative expansion that is valid in the symmetric phase as well as the broken symmetry phase, provided that the correlation length is large compared to the system size $L$. We find that, at the critical point, for large system size in lattice units, the PDF approaches $p(M)\sim \exp(-f(L) M^4)$. Consequently, the critical value of the Binder cumulant of the total magnetization is $U = 1 - \frac{4\,Γ(5/4)^2}{3\,Γ(3/4)^2}$. We validate our results by comparison with Monte Carlo simulation.

DOI
10.48550/arxiv.2005.01737

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field theory does not correctly describethe leading finite-size effects near the critical point of spin systems on a d-dimensional lattice withd > 4. We show that these finite-size effects require a description in terms of a lattice Hamiltonian.For n → ∞ and n = 1 explicit results are given for the susceptibility and for the Binder cumulant.They imply that recent analyses of Monte-Carlo results for the five-dimensional Ising model are notconclusive.PACS numbers: 05.70.Jk, 64.60.Ak, 75.40.Mg