Zeroes in the Complex Beta Plane Of 2D Ising Block Spin Boltzmannians

TL;DR

This paper analyzes the distribution of zeroes of effective Boltzmannian polynomials in the complex Beta plane for the 2D Ising model, revealing insights into the regularity and potential pathologies of block spin transformations.

Contribution

It computes and examines the zeroes of effective Boltzmannians for 2D Ising model block spins using transfer matrix techniques, highlighting their distribution and implications.

Findings

Zeroes reflect the regularity of block spin transformations.

Spurious zeroes approach the positive real axis at large Beta.

Potential link between zeroes and renormalization group pathologies.

Abstract

Effective Boltzmannians in the sense of the block spin renormalization group are computed for the 2D Ising model. The blocking is done with majority and Kadanoff rules for blocks of size 2 by 2. Transfer matrix techniques allow the determination of the effective Boltzmannians as polynomials in u=exp(4 Beta) for lattices of up to 4 by 4 blocks. The zeroes of these polynomials are computed for all non-equivalent block spin configurations. Their distribution in the complex Beta plane reflects the regularity structure of the block spin transformation. In the case of the Kadanoff rule spurious zeroes approach the positive real Beta axis at large values of Beta. They might be related to the renormalization group pathologies discussed in the literature.