# Spontaneous magnetization of an ideal ferromagnet: Beyond Dyson's analysis

## Abstract

Using the low-energy effective field theory for magnons, we systematically evaluate the partition function of the O(3) ferromagnet up to three loops. Dyson, in his pioneering microscopic analysis of the Heisenberg model, showed that the spin-wave interaction starts manifesting itself in the low-temperature expansion of the spontaneous magnetization of an ideal ferromagnet only at order T{sup 4}. Although several authors tried to go beyond Dyson's result, to the best of our knowledge, a fully systematic and rigorous investigation of higher-order terms induced by the spin-wave interaction has never been achieved. As we demonstrate in the present paper, it is straightforward to evaluate the partition function of an ideal ferromagnet beyond Dyson's analysis, using effective Lagrangian techniques. In particular, we show that the next-to-leading contribution to the spontaneous magnetization resulting from the spin-wave interaction already sets in at order T{sup 9/2}--in contrast to all claims that have appeared before in the literature. Remarkably, the corresponding coefficient is completely determined by the leading-order effective Lagrangian and is thus independent of the anisotropies of the cubic lattice. We also consider even higher-order corrections and thereby solve--once and for all--the question of how the spin-wave interaction in an ideal ferromagnet manifests itself in themore »

- Authors:

- Facultad de Ciencias, Universidad de Colima, Bernal Diaz del Castillo 340, Colima C.P. 28045 (Mexico)

- Publication Date:

- OSTI Identifier:
- 21596844

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. B, Condensed Matter and Materials Physics

- Additional Journal Information:
- Journal Volume: 84; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevB.84.064414; (c) 2011 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1098-0121

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; ANISOTROPY; CORRECTIONS; CUBIC LATTICES; EXPANSION; FIELD THEORIES; HEISENBERG MODEL; INTERACTIONS; LAGRANGIAN FUNCTION; MAGNETIZATION; MAGNONS; SPIN WAVES; CRYSTAL LATTICES; CRYSTAL MODELS; CRYSTAL STRUCTURE; FUNCTIONS; MATHEMATICAL MODELS; QUASI PARTICLES

### Citation Formats

```
Hofmann, Christoph P.
```*Spontaneous magnetization of an ideal ferromagnet: Beyond Dyson's analysis*. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVB.84.064414.

```
Hofmann, Christoph P.
```*Spontaneous magnetization of an ideal ferromagnet: Beyond Dyson's analysis*. United States. doi:10.1103/PHYSREVB.84.064414.

```
Hofmann, Christoph P. Mon .
"Spontaneous magnetization of an ideal ferromagnet: Beyond Dyson's analysis". United States. doi:10.1103/PHYSREVB.84.064414.
```

```
@article{osti_21596844,
```

title = {Spontaneous magnetization of an ideal ferromagnet: Beyond Dyson's analysis},

author = {Hofmann, Christoph P.},

abstractNote = {Using the low-energy effective field theory for magnons, we systematically evaluate the partition function of the O(3) ferromagnet up to three loops. Dyson, in his pioneering microscopic analysis of the Heisenberg model, showed that the spin-wave interaction starts manifesting itself in the low-temperature expansion of the spontaneous magnetization of an ideal ferromagnet only at order T{sup 4}. Although several authors tried to go beyond Dyson's result, to the best of our knowledge, a fully systematic and rigorous investigation of higher-order terms induced by the spin-wave interaction has never been achieved. As we demonstrate in the present paper, it is straightforward to evaluate the partition function of an ideal ferromagnet beyond Dyson's analysis, using effective Lagrangian techniques. In particular, we show that the next-to-leading contribution to the spontaneous magnetization resulting from the spin-wave interaction already sets in at order T{sup 9/2}--in contrast to all claims that have appeared before in the literature. Remarkably, the corresponding coefficient is completely determined by the leading-order effective Lagrangian and is thus independent of the anisotropies of the cubic lattice. We also consider even higher-order corrections and thereby solve--once and for all--the question of how the spin-wave interaction in an ideal ferromagnet manifests itself in the spontaneous magnetization beyond the Dyson term.},

doi = {10.1103/PHYSREVB.84.064414},

journal = {Physical Review. B, Condensed Matter and Materials Physics},

issn = {1098-0121},

number = 6,

volume = 84,

place = {United States},

year = {2011},

month = {8}

}