Polynomial degeneracy for the first \(m\) energy levels of the antiferromagnetic Ising model
type: Item
instance of: scholarly article
title: Polynomial degeneracy for the first \(m\) energy levels of the antiferromagnetic Ising model
zbMATH Open document ID: 1467.82016
DOI: 10.4171/AIHPD/101
author: Andrea Jiménez
published in: Annales de l'Institut Henri Poincaré D. Combinatorics, Physics and their Interactions (AIHPD)
publication date: 2021-06-09T00:00:00Z
review text: Summary: In this work, we continue our investigation on the antiferromagnetic Ising model on triangulations of closed Riemann surfaces. On the one hand, according to [\textit{R. Moessner} and \textit{A. P. Ramirez}, ``Geometrical frustration'', Phys. Today 59, No. 2, 24 (2006; \url{doi:10.1063/1.2186278})], the antiferromagnetic Ising model on triangulations exhibits geometrical frustration, a well-studied concept in condensed matter physics. Typical geometrically frustrated systems present an exponential ground state degeneracy. On the other hand, the dual graph of a triangulation of a closed Riemann surface is a cubic graph. Cubic bridgeless graphs have exponentially many perfect matchings [\textit{M. Chudnovsky} and \textit{P. Seymour}, Combinatorica 32, No. 4, 403--424 (2012; Zbl 1299.05263); \textit{L. Esperet} et al., Adv. Math. 227, No. 4, 1646--1664 (2011; Zbl 1223.05229)], which implies in the case of planar triangulations, an exponential ground state degeneracy. However, this phenomenon does not persist for triangulations of higher genus surfaces. A possible explanation for a geometrically frustrated system with a low ground state degeneracy is that exponentially many states exist at a low energy level. In this work, we constructively show that this explanation does not match with the behavior of all triangulations of closed Riemann surfaces. To be more specific, for each integer \(m \geq 1\), we construct a collection of triangulations \(\{T_n\}_{n > N(m)}\) of a fixed closed Riemann surface with the property that the degeneracy of each of the first \(m\) energy levels of \(T_n\) is a polynomial in the order \(n\) of \(T_n\).
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 82D40
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 7357094
zbMATH Keywords: antiferromagnetic Ising model
zbMATH Keywords: triangulations
zbMATH Keywords: ground state degeneracy
zbMATH Keywords: geometrical frustration
zbMATH Keywords: low energy level
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/aihpd/101
OpenAlex ID: W3133785503
description: scientific article; zbMATH DE number 7357094