Probability distribution of magnetization in the one-dimensional Ising model: Effects of boundary conditions

TL;DR

This paper investigates how boundary conditions affect the probability distribution of magnetization in the 1D Ising model at low temperatures, providing exact scaling functions for various boundary types.

Contribution

It provides explicit calculations of finite-size scaling functions for magnetization distributions under different boundary conditions in the 1D Ising model.

Findings

Scaling functions depend on boundary conditions

Block magnetization distributions match free boundary results

Exact calculations at T -> 0 and large N

Abstract

Finite-size scaling functions are investigated both for the mean-square magnetization fluctuations and for the probability distribution of the magnetization in the one-dimensional Ising model. The scaling functions are evaluated in the limit of the temperature going to zero (T -> 0), the size of the system going to infinity (N -> oo) while N[1-tanh(J/k_BT)] is kept finite (J being the nearest neighbor coupling). Exact calculations using various boundary conditions (periodic, antiperiodic, free, block) demonstrate explicitly how the scaling functions depend on the boundary conditions. We also show that the block (small part of a large system) magnetization distribution results are identical to those obtained for free boundary conditions.