Itinerant Magnetism and Spin-Fluctuation Physics

In an itinerant magnet the same electrons form a Fermi surface and carry the magnetization. The Stoner criterion describes a weak-coupling instability of the paramagnetic Fermi sea; collective spin fluctuations then acquire damping from particle–hole excitations. This framework is distinct from a pre-existing lattice of fixed-length local moments, although real materials can interpolate between the limits.

Required background. Response Functions and Zero Sound supplies Fermi-liquid susceptibilities and collisionless collective modes; Coulomb Screening, Dielectric Response, and RPA supplies the bubble resummation. Helpful background. Pomeranchuk and Density-Wave Instabilities supplies competing Fermi-surface channels.

The Stoner instability

Take a contact repulsion $U n_\uparrow n_\downarrow$ and let $N(0)$ denote the density of states per spin at the Fermi energy. A small uniform magnetization density $m=n_\uparrow-n_\downarrow$ changes the energy as

$$\delta E=\frac{m^2}{4N(0)}-\frac U4m^2+O(m^4).$$

The paramagnet loses local stability when

$$U N(0)>1.$$

Equivalently, let $\Pi_0$ be the one-spin particle–hole bubble normalized by $\Pi_0(\mathbf0,0)=N(0)$. Up to the chosen magnetic-moment factor, the random-phase spin susceptibility is

$$\chi_s(\mathbf q,\omega) =\frac{2\Pi_0(\mathbf q,\omega)} {1-U\Pi_0(\mathbf q,\omega)},$$

so its uniform denominator is $1-UN(0)$. Other definitions move factors of two between the bubble, spin operator, and coupling; the denominator and the declared per-spin convention must be translated together. The familiar criterion was introduced as an itinerant-band mechanism by Stoner 1938, pp. 372–414.

The criterion marks a mean-field instability, not a quantitatively reliable Curie temperature or proof of a continuous transition. The density of states varies as the bands split, fluctuation corrections can be large, and nonanalytic fermionic responses can preempt the simplest Landau expansion.

Landau damping and paramagnons

Near a ferromagnetic instability in a clean metal, low-frequency particle–hole pairs give a schematic inverse susceptibility

$$\chi^{-1}(\mathbf q,i\omega_n) =r+cq^2+\gamma\frac{|\omega_n|}{v_Fq}+\cdots,$$

for $|\omega_n|\ll v_Fq$. Balancing $q^2$ against $|\omega|/q$ gives dynamical exponent $z=3$. Near a generic antiferromagnetic wave vector $\mathbf Q$ connecting portions of the Fermi surface,

$$\chi^{-1}(\mathbf Q+\mathbf q,i\omega_n) =r+cq^2+\gamma|\omega_n|+\cdots,$$

so the Gaussian balance gives $z=2$. The coefficient and even the functional form depend on Fermi-surface geometry, dimensionality, disorder, and whether $\mathbf Q$ connects hot points or extended nested regions. Hertz derived the order-parameter action by integrating out fermions Hertz 1976, §§ III–V.

The resulting broad collective excitation is a paramagnon. Far from the particle–hole continuum it may sharpen; inside the continuum it is damped rather than a stable magnon. The integrated spectral weight and momentum dependence distinguish this response from localized-spin waves, although mixed local-itinerant systems can show both.

Spin-fluctuation interactions

In a convention where positive $\chi_s$ is the spin response, exchanging a spin fluctuation generates the effective fermion interaction

$$V_{\mathrm{eff}}(q,\omega)\sim -g^2\chi_s(q,\omega) \,\boldsymbol\sigma_1\cdot\boldsymbol\sigma_2, \qquad g^2\chi_s>0.$$

Because $\boldsymbol\sigma_1\cdot\boldsymbol\sigma_2$ has eigenvalue $-3$ in a singlet and $+1$ in a triplet, this convention makes small-$q$ spin exchange repulsive in the singlet channel and attractive in the triplet channel. Ferromagnetic fluctuations can therefore favor odd-parity triplet structures. For antiferromagnetic fluctuations concentrated near $\mathbf Q$, a repulsive singlet kernel can instead contribute attractively to the projected gap equation when the singlet gap changes sign between $\mathbf k$ and $\mathbf k+\mathbf Q$. These are mechanisms within a model, not evidence that measured superconductivity is uniquely fluctuation mediated.

Local moments, itinerancy, and limits

The instantaneous moment $\langle\mathbf S_i^2\rangle$, ordered moment, Curie–Weiss susceptibility, bandwidth, and low-energy damping probe different time scales. A small ordered moment does not prove weak coupling; strong fluctuations can suppress order from sizable instantaneous moments. Conversely, a successful Stoner fit does not prove fixed-length moments are absent.

A controlled inference combines band-resolved $\chi_0$, absolute magnetic spectral weight, temperature and field evolution, and sensitivity to carrier density. RPA is controlled for weak effective interactions or appropriate large-degeneracy limits; close to a low-dimensional critical point, vertex corrections and fermion self-energy can invalidate it.

Exercises

1. A metal has per-spin $N(0)=2\,\mathrm{eV}^{-1}$ per cell and $U=0.4\,\mathrm{eV}$. Is the Stoner criterion met?

Solution

$UN(0)=0.8<1$, so the uniform paramagnet is stable within this Stoner approximation. The conclusion is model dependent and does not exclude a finite-wave-vector instability.

2. Obtain the Gaussian dynamical exponents for the two damping kernels above.

Solution

For the ferromagnet, scale $q^2\sim|\omega|/q$, hence $\omega\sim q^3$ and $z=3$. At generic antiferromagnetic $\mathbf Q$, $q^2\sim|\omega|$, hence $z=2$.

References

• Hertz, J. A. “Quantum Critical Phenomena.” Physical Review B 14 (1976): 1165–1184. DOI.
• Stoner, E. C. “Collective Electron Ferromagnetism.” Proceedings of the Royal Society A 165 (1938): 372–414. DOI.