Finite-size scaling behavior in trapped systems

TL;DR

This paper investigates finite-size scaling in two-dimensional Ising models under a spatially varying magnetic field, combining numerical transfer-matrix methods with conformal invariance and linear response theory to analyze different field regimes.

Contribution

It introduces a generalized finite-size scaling framework for trapped systems with variable magnetic fields, validated through numerical and theoretical analysis across multiple Ising models.

Findings

Agreement between theory and numerical magnetization profiles in low-field regime

Confirmation of $p$-dependent exponents in high-field correlation scaling

Extension of finite-size scaling concepts to trapped magnetic systems

Abstract

Numerical transfer-matrix methods are applied to two-dimensional Ising spin systems, in presence of a confining magnetic field which varies with distance $|{\vec x}|$ to a "trap center", proportionally to $(|{\vec x}|/\ell)^p$, $p>0$. On a strip geometry, the competition between the "trap size" $\ell$ and the strip width, $L$, is analysed in the context of a generalized finite-size scaling {\em ansatz}. In the low-field regime $\ell \gg L$, we use conformal-invariance concepts in conjunction with a linear-response approach to derive the appropriate ($p$-dependent) limit of the theory, which agrees very well with numerical results for magnetization profiles. For high fields $\ell \lesssim L$, correlation-length scaling data broadly confirms an existing picture of $p$-dependent characteristic exponents. Standard spin-1/2 and spin-1 Ising systems are considered, as well as the Blume-Capel model.