Finite-size-scaling analysis of the XY universality class between two and three dimensions: An application of Novotny's transfer-matrix method

TL;DR

This study uses Novotny's transfer-matrix method to analyze the XY universality class's critical behavior between two and three dimensions, revealing a smooth interpolation of critical exponents consistent with analytical predictions.

Contribution

It extends Novotny's finite-size-scaling approach to the XY universality class in fractional dimensions, providing new insights into criticality between 2D and 3D.

Findings

The correlation-length critical exponent er(d) varies smoothly between 2D and 3D limits.

Simulation results align with analytical 1/N-expansion predictions.

Methodology adapts transfer-matrix techniques for fractional dimensions.

Abstract

Based on Novotny's transfer-matrix method, we simulated the (stacked) triangular Ising antiferromagnet embedded in the space with the dimensions variable in the range 2 \le d \le 3. Our aim is to investigate the criticality of the XY universality class for 2 \le d \le 3. For that purpose, we employed an extended version of the finite-size-scaling analysis developed by Novotny, who utilized this scheme to survey the Ising criticality (ferromagnet) for 1 \le d \le 3. Diagonalizing the transfer matrix for the system sizes N up to N=17, we calculated the $d$-dependent correlation-length critical exponent \nu(d). Our simulation result \nu(d) appears to interpolate smoothly the known two limiting cases, namely, the KT and d=3 XY universality classes, and the intermediate behavior bears close resemblance to that of the analytical formula via the 1/N-expansion technique. Methodological details including the modifications specific to the present model are reported.