Corner Transfer Matrix approach to the Yang-Lee Singularity in the 2D Ising Model

The goal of this work is to investigate the complex magnetic field analytic continuation of the 2D Ising model near the Yang-Lee singularity. The exact location of the singularity is not known precisely, and one of the results produced is an improved quality estimate of its location \xi₀ \approx i0.1893506. Another primary result is the numerical verification of the scaling function conjecture by Fonseca \& Zamolodchikov in \cite{fonseca2003ising} regarding the structure of the free energy scaling function near the Yang-Lee singularity, as well as proposed logarithmic operators in the Ising free energy at order H². In accomplishing the above results, this thesis starts by detailing the slice of the Ising model's history pertinent to this work. For an established model such as the Ising model it would be impractical to cover every branch and generalisation, so this work focuses on the standard 2D Ising model with magnetic field in both the square and triangular cases. The relevant historical developments covered focus on the Ising scaling theory, general renormalisation arguments and conformal field theory (and the more modern logarithmic extensions) used to underpin our current understanding of the model. This thesis then details the theory and implementation details for the main numerical technique used: the Corner Transfer Matrix Renormalisation Group (CTMRG) method. The Ising susceptibility data obtained in \cite{orrick2001susceptibility,chan2011ising} plays a critical role in providing numerical evidence for the extended logarithmic operators proposed in this work. These operators are predicted by the theory of a logarithmic minimal model $\mathcal L\mathcal M(3,4)$ for a conformal field theory of central charge $c=1/2$ \cite{kogan1998origin} which describes the Ising model at the critical point. The operators appear as the result of the integer difference in conformal weight between the identity $[I]$ and energy [ε] conformal families for the $c=1/2$ CFT \cite{pearce2006logarithmic}, and are also proposed from the theory of low $Q$ Potts model \cite{nivesvivat2021logarithmic,VASSEUR2014435} (a generalisation of the Ising model). These extended logarithmic operators are used in conjunction with numerical data collected through the Corner Transfer Matrix Renormalisation Group method to obtain a high precision estimate of the free energy scaling function. CTMRG is a powerful numerical technique with high convergence, compatible with studying the Ising model in the complex realm through an extension developed in this work. The scaling function has been characterised extensively near the thermodynamic critical point in prior work \cite{mangazeev2010scaling,fonseca2003ising,barouch1973zero,tracy1973neutron,wu1976spin} with highly accurate expansion coefficients up to order \xi⁸ and less accurate estimates for terms as high as order \xi¹⁴. This work aims to provide numerical verification for the conjectured form of the scaling function given in \cite{fonseca2003ising} for the region near the Yang-Lee singularity, and to act as a condensed resource for those tackling the vast body of knowledge associated with the Ising model.

Author dc:creator

• Hagan, Bryte

Language dc:language.iso

en_AU

Handle dc:identifier.uri

http://hdl.handle.net/1885/307430

OAI identifier oai:identifier

oai:openresearch-repository.anu.edu.au:1885/307430