[Paper Review] Current conservation and ratio rules in magnetic metals with Coulomb repulsion

Kosuke Odagiri|arXiv (Cornell University)|Dec 28, 2011

Magnetic Properties of Alloys4 references3 citations

TL;DR

This paper derives exact all-order ratio rules for magnetic metals with Coulomb repulsion by applying Ward–Takahashi identities to spin-current conservation in ferromagnetic and antiferromagnetic phases. It shows that the ratio of densities of states governs magnetic stability, with ferromagnetism requiring a rising density of states and antiferromagnetism a concave one, and confirms these predictions with experimental and theoretical consistency at zero temperature.

ABSTRACT

From general considerations of spin-symmetry breaking associated with (anti-)ferromagnetism in metallic systems with Coulomb repulsion, we obtain interesting and simple all-order rules involving the ratios of the densities of states. These are exact for ferromagnetism under reasonable conditions, and nearly exact for anti-ferromagnetism. In the case of ferromagnetism, the comparison with the available experimental and theoretical numbers yields favourable results.

Motivation & Objective

• To establish exact all-order relations between electronic densities of states and magnetic order parameters in metallic systems with Coulomb repulsion.
• To investigate how spin-symmetry breaking in ferromagnetic and antiferromagnetic phases leads to universal ratio rules through current conservation.
• To test the validity of these rules against experimental and theoretical data, particularly in ferromagnetic metals.
• To analyze the role of radiative corrections from magnons in destabilizing long-range antiferromagnetic order at finite temperatures.
• To extend Gribov's approach to axial-current conservation in high-energy physics to the context of condensed matter magnetic metals.

Proposed method

• Uses Dyson–Schwinger equations and Ward–Takahashi identities derived from spin-current conservation in SU(2) spin-symmetry-broken phases.
• Applies a generalized Coulomb interaction (including photon exchange) without specifying its form, as it does not affect the ratio rules.
• Derives the ferromagnetic ratio rule via U(1) current mixing and diagrammatic techniques, showing exact relations between densities of states.
• Constructs Feynman rules for spin currents, magnons, and Higgs-boson self-energy to compute two-point functions and tadpole contributions.
• Performs tadpole cancellation at zero and finite temperature, incorporating Fermi-Dirac statistics and Bose-Einstein distributions.
• Evaluates finite-temperature corrections to the magnetization, showing a T^{3/2} dependence consistent with known results.

Experimental results

Research questions

• RQ1 What exact all-order relations exist between the densities of states and magnetic order parameters in ferromagnetic metals with Coulomb repulsion?
• RQ2 How do Ward–Takahashi identities and current conservation constrain the form of the ratio rules in spin-symmetry-broken phases?
• RQ3 To what extent are the derived ratio rules exact or approximate in the case of antiferromagnetism compared to ferromagnetism?
• RQ4 How do radiative corrections from magnons affect the stability of long-range antiferromagnetic order at finite temperatures?
• RQ5 Can the derived ratio rules be tested against experimental data, and what measurable parameters are involved?

Key findings

• An exact ratio rule is derived for ferromagnetism that relates the densities of states of spin-up and spin-down electrons, with the condition that the density of states must rise with energy for ferromagnetism to be stable.
• For antiferromagnetism, the ratio rule involves the bare spin exchange energy and requires the density of states to be concave, indicating a different stability condition.
• The finite-temperature analysis shows that the magnetization decreases as T^{3/2}, consistent with established results, due to bosonic tadpole contributions from magnons.
• Tadpole cancellation at finite temperature requires the fermionic and bosonic contributions to exactly balance, leading to a measurable dependence on ΔE, Δh, and mϕ.
• Radiative corrections from magnons at finite temperature destabilize long-range antiferromagnetic order, suggesting that genuine long-range order is not permitted in metallic antiferromagnets.
• The framework is general and nonperturbative, allowing direct comparison with experimental data through measurable parameters such as the density of states, exchange energy, and magnon mass.