Exact Combinatorial Density of States for the Critical 1D Ising Model

TL;DR

This paper provides an exact combinatorial analysis of the 1D antiferromagnetic Ising model, revealing its spectral degeneracies, topological defects, and energy quantization, with implications for understanding quantum criticality.

Contribution

It introduces a novel exact framework for counting states and degeneracies in the 1D Ising model using topological and number-theoretic methods.

Findings

Degeneracies follow Fibonacci and Lucas sequences at ground state crossings.

Spectral gaps are identified near the fully polarized limit, forbidding certain energy levels.

Closed-form expressions for degeneracies at all energy levels are derived using transfer matrices.

Abstract

This work presents an exact microcanonical combinatorial analysis of the one-dimensional antiferromagnetic Ising model. At the primary ground-state level crossing $B/J=2$, degeneracies follow the Fibonacci and Lucas sequences for open chains and periodic rings, respectively. We extend this framework to the complete excitation spectrum, demonstrating that the density of states is constructed from topological defects governed by linear Diophantine equations and $p$-fold Fibonacci convolutions. Open boundaries act as fractional defects, densifying the chain spectrum into energy steps of $2J$, whereas the closed ring remains quantized in units of $4J$. Notably, this exact topological counting exposes non-trivial spectral gaps near the fully polarized limit, strictly forbidding the penultimate macroscopic energy levels in both topologies. Through the transfer matrix formalism, we derive exact closed-form expressions for the critical degeneracies at all energy levels. These results provide a rigorous analytical foundation for extracting exact residual entropies and exposing the intrinsic number-theoretic architecture of quantum critical manifolds.