Criticality in Alternating Layered Ising Models : I. Effects of connectivity and proximity

TL;DR

This study investigates how connectivity and proximity in alternating layered Ising models influence specific heat behavior and critical temperatures, revealing finite-size scaling and decay effects related to strip widths and boundaries.

Contribution

It provides a detailed numerical analysis of the effects of connectivity and proximity on critical phenomena in layered Ising models, extending understanding of finite-size scaling and boundary effects.

Findings

Finite-size scaling holds near critical points with increasing strip widths.

Enhancement of specific heat is dominated by boundary decay factors.

Decay of enhancement near critical points is slower, involving logarithmic factors.

Abstract

The specific heats of exactly solvable alternating layered planar Ising models with strips of width $m_1$ lattice spacings and ``strong'' couplings $J_1$ sandwiched between strips of width $m_2$ and ``weak'' coupling $J_2$, have been studied numerically to investigate the effects of connectivity and proximity. We find that the enhancements of the specific heats of the strong layers and of the overall or `bulk' critical temperature, $T_c(J_1,J_2;m_1,m_2)$, arising from the collective effects reflect the observations of Gasparini and coworkers in experiments on confined superfluid helium. Explicitly, we demonstrate that finite-size scaling holds in the vicinity of the upper limiting critical point $T_{1c}$ ($\propto J_1/k_B$) and close to the corresponding lower critical limit $T_{2c}$ ($\propto J_2/k_B$) when $m_1$ and $m_2$ increase. However, the residual {\it enhancement}, defined via appropriate subtractions of leading contributions from the total specific heat, is dominated (away from $T_{1c}$ and $T_{2c}$) by a decay factor $1/(m_1+m_2)$ arising from the {\it seams} (or boundaries) separating the strips; close to $T_{1c}$ and $T_{2c}$ the decay is slower by a factor $\ln m_1$ and $\ln m_2$, respectively.