Critical exponents for the long-range Ising chain using a transfer matrix approach

TL;DR

This paper investigates the critical behavior of the long-range Ising chain with decaying interactions using an efficient transfer matrix method, providing new estimates for critical exponents through finite size scaling.

Contribution

It introduces a transfer matrix approach for long-range Ising chains, enabling precise calculation of critical exponents for large systems with long-range interactions.

Findings

Critical exponents for the correlation length are estimated.

The transfer matrix method reduces computational complexity.

Finite Range Scaling hypothesis is validated for these systems.

Abstract

The critical behavior of the Ising chain with long-range ferromagnetic interactions decaying with distance $r^\alpha$, $1<\alpha<2$, is investigated using a numerically efficient transfer matrix (TM) method. Finite size approximations to the infinite chain are considered, in which both the number of spins and the number of interaction constants can be independently increased. Systems with interactions between spins up to 18 sites apart and up to 2500 spins in the chain are considered. We obtain data for the critical exponents $\nu$ associated with the correlation length based on the Finite Range Scaling (FRS) hypothesis. FRS expressions require the evaluation of derivatives of the thermodynamical properties, which are obtained with the help of analytical recurrence expressions obtained within the TM framework. The Van den Broeck extrapolation procedure is applied in order to estimate the convergence of the exponents. The TM procedure reduces the dimension of the matrices and circumvents several numerical matrix operations.