[Paper Review] The story of magnetism: from Heisenberg, Slater, and Stoner to Van Vleck, and the issues of exchange and correlation

Navinder Singh|arXiv (Cornell University)|Jul 30, 2018

Geomagnetism and Paleomagnetism Studies9 references4 citations

TL;DR

This paper traces the historical development of magnetism theories from classical to quantum mechanics, emphasizing the foundational role of Van Vleck's 'middle-road' theory in unifying itinerant and localized electron models. It explains how exchange and correlation effects were resolved through advances like the Slater-Stoner model, the Heisenberg model, and ultimately the Self-Consistent-Renormalization (SCR) theory, which incorporates spin fluctuations and strong correlations beyond mean-field approximations.

ABSTRACT

This article is devoted to the development of the central ideas in the field of magnetism. The presentation is semi-technical in nature and it roughly follows the chronological order. The key contributions of Van Vleck, Dorfman, Pauli, Heisenberg, and Landau are presented. Then the advent of the field of itinerant electron magnetism starting with the investigations of Bloch and Wigner, and more successful formulation by Slater and Stoner is presented. The physical basis of the Slater-Stoner theory is discussed and its problems are summarized. Then, an overview of the debate between itinerant electron view of Stoner and localized electron view of Heisenberg is presented. Connected with this debate are the issues of exchange interactions. The issues related to the origin of exchange interaction in Stoner model are discussed. We review the "middle-road" theory of van Vleck and Hurwitz--the very first theory which takes into account the electron correlation effects in the itinerant model. We close our presentation with the discussion of the very important issue of strong electron correlation in the itinerant picture.

Motivation & Objective

• To resolve the paradox between classical paramagnetism (Langevin) and the Bohr-van Leeuwen theorem, which predicts zero magnetization in classical statistical mechanics.
• To trace the evolution of theoretical magnetism from pre-quantum models to quantum treatments, highlighting Van Vleck's pivotal role in introducing electron correlation into itinerant electron systems.
• To examine the debate between itinerant (Stoner) and localized (Heisenberg) electron models, and to analyze the role of exchange interactions and correlation effects in resolving this divide.
• To establish the theoretical foundation for modern treatments of strong electron correlation in transition metals, culminating in the SCR theory as a key advancement beyond mean-field approximations.

Proposed method

• Analyzing the Bohr-van Leeuwen theorem to show the failure of classical statistical mechanics in explaining magnetism due to phase-space invariance under vector potential transformation.
• Applying quantum statistical mechanics to paramagnetism in real gases, as pioneered by Van Vleck, to derive quantum corrections to classical magnetization.
• Using the ligand field theory to explain magnetic anisotropy and crystal field splitting in d-electron salts, linking electronic structure to magnetic behavior.
• Formulating the Slater-Stoner model for itinerant electron magnetism, based on the Stoner criterion for ferromagnetism in metals.
• Introducing the concept of indirect s-d exchange via the Vonsovsky-Zener model to explain ferromagnetism in transition metals.
• Extending the analysis to non-mean-field treatments via the Friedel-Alexander-Anderson-Moriya theory and the SCR theory, which includes spin fluctuation renormalization and strong correlation effects.

Experimental results

Research questions

• RQ1 Why does classical statistical mechanics fail to predict magnetization in the presence of an external magnetic field, as formalized in the Bohr-van Leeuwen theorem?
• RQ2 How did Van Vleck’s 'middle-road' theory provide the first systematic treatment of electron correlation in itinerant electron systems?
• RQ3 What is the physical origin of the Slater curve, and how does it predict the sign of exchange interaction in transition metals?
• RQ4 Why did the localized Heisenberg model fail to describe iron-group metals, and how did the itinerant Stoner model resolve this?
• RQ5 How do spin fluctuations and strong correlation effects beyond mean-field theory, as captured in the SCR theory, improve the description of magnetic order in transition metals?

Key findings

• The Bohr-van Leeuwen theorem proves that classical statistical mechanics cannot account for magnetism, as the partition function becomes independent of the vector potential, leading to zero magnetization.
• Van Vleck's quantum statistical treatment of paramagnetism in real gases provided the first quantum mechanical foundation for understanding electron correlation in magnetic systems.
• The ligand field theory, developed by Van Vleck, explained the magnetic anisotropy and energy splitting in d-electron salts, crucial for understanding transition metal compounds.
• The Slater-Stoner model successfully described itinerant electron ferromagnetism but failed to account for strong correlation effects, necessitating further theoretical development.
• The SCR theory, developed by Moriya and Kawabata, provided a non-perturbative, self-consistent treatment of spin fluctuations and correlation effects, going beyond Hartree-Fock and RPA approximations.
• The debate between itinerant and localized models was resolved in favor of the itinerant picture for iron-group metals, but only after incorporating correlation effects via advanced theories like SCR.