Itinerant ferromagnetism (${ϵ}_{k\sigma }={ϵ}_k-\sigma M$)

Ferromagnetic spin polarization carried by mobile electrons rather than a separate fixed-spin lattice. A homogeneous spin field splits the electron bands and competes with a positive field cost. Mean-field instability is governed by the Stoner criterion; the ordered metal supports transverse spin waves as well as fermionic particle-hole excitations.

Spin reversal in exchange-split bands creates an electron-hole excitation with energy ${ϵ}_{k+q}-{ϵ}_k+2M$ for a majority-to-minority flip in the simple uniform model with $M>0$. Its allowed continuum depends on occupied initial and empty final states. At $q=0$ all such allowed transitions have exchange energy $2M$; finite momentum changes the thresholds. A collective ferromagnetic magnon can remain gapless below this continuum when spin rotation is exact. The continuum is not the same excitation as a longitudinal amplitude mode.

An isotropic spin-polarized metal has a degenerate magnetization direction and a long-wavelength transverse magnon with quadratic dispersion. Nonzero spin density pairs the two transverse broken generators into one type-B Goldstone boson. Exchange-split Fermi surfaces also support fermionic low-energy excitations; the magnon is distinct from longitudinal amplitude motion and the spin-flip continuum.

In the homogeneous spin-vector mean-field scheme with field cost $M^2/(2U)$ and spin-summed density of states per site, the paramagnetic curvature is proportional to $U^{-1}-\int \nu (ϵ)[-n_F^{\prime }(ϵ)]dϵ$. It changes sign at $U{\chi }_0=1$. For a smooth low-temperature density of states, ${\chi }_0\simeq \nu (\mu )$, giving the familiar Fermi-level criterion. Interaction and spin-counting conventions must be specified before comparing coefficients.

For a uniform exchange field with cost $N{M}^2/(2U)$ and spin-band energies ${ϵ}_k-\sigma M$, the grand potential is even in M. Its curvature at zero field is $N[U^{-1}-{\chi }_0]$, with ${\chi }_0=\int \nu (-n_F^{\prime })dϵ$ and spin-summed density of states per site. Negative curvature proves local paramagnetic instability. It does not by itself exclude a first-order transition before that instability, nor determine the stable finite-field amplitude.

For a three-dimensional zero-temperature parabolic Fermi gas, let $\zeta =(N_{\uparrow }-N_{\downarrow })/N$ and $E_F={\hslash }^2(3{\pi }^{2}N/V{)}^{2/3}/(2m)$. Suppose the interaction energy per electron is $U{N}_{\uparrow }{N}_{\downarrow }/N^2=U(1-{\zeta }^2)/4$. The total energy per electron isIts Taylor expansion has quadratic coefficient $E_F/3-U/4$ and positive quartic coefficient $E_F/81$. Hence the unpolarized state loses stability at $U_c=4E_F/3\propto (N/V{)}^{2/3}$. This $U$ has units of energy. It differs from a contact coupling $g$ defined by interaction-energy density $g{n}_{\uparrow }{n}_{\downarrow }$, for which $g=U/n$ and $g_c=4E_F/(3n)$. Normalization must be fixed before comparing critical couplings. For $U\le 4E_F/3$, convexity of $x^{-1/3}$ implies $f^{\prime \prime }(\zeta )=(E_F/3)[(1+\zeta {)}^{-1/3}+(1-\zeta {)}^{-1/3}]-U/2\ge 0$, so the unpolarized minimum is a global minimum; a first-order polarized minimum does not pre-empt this threshold in the stated model.

Let $g_s(E_F)$ be the single-spin density of states per site per energy, and let the on-site Hubbard model repulsion have energy $U$. For a weak uniform magnetic field $H$, the half-splitting of the two spin bands is $\Delta =U\delta n/2+{\mu }_0{\mu }_{B}H$, where $\delta n=n_{\uparrow }-n_{\downarrow }$. Constant density of states gives $\delta n=2g_s\Delta$. ThereforeThis is the response per site; multiplying by the site number density gives the dimensionless SI volume magnetic susceptibility. With spin-summed density of states $g=2g_s$, the denominator is $1-Ug/2$. Its vanishing is the mean-field Stoner criterion. The unpolarized linear-response expression is stable only for $U{g}_s<1$.

For an interaction-energy density $U{n}_{\uparrow }{n}_{\downarrow }$, let $n_{\uparrow }=n/2+\delta$, $n_{\downarrow }=n/2-\delta$ at fixed total carrier density. If $g_s(E_F)$ is the unpolarized single-spin density of states per unit volume, the kinetic-energy cost of the two occupation shifts is ${\delta }^2/g_s(E_F)$, whereas the interaction changes by $-U{\delta }^2$. ThereforeFor the three-dimensional free electron model, $g_{\mathrm{tot}}(E_F)=3n/(2E_F)$ and $U_c=4E_F/(3n)$. Both the interaction normalization and the spin counting in the density of states must be specified when quoting a Stoner criterion.

Integrating free fermions in a constant auxiliary spin field gives $S_{\mathrm{eff}}=\beta N{M}^2/(2U)-{\sum }_{k\sigma }\ln [1+e^{-\beta ({ϵ}_k-\mu -\sigma M)}]$. Differentiation gives the self-consistent spin polarization. The extensive action is a mean-field restriction of the full auxiliary-field functional integral, not an exact omission of all spatial/time fluctuations.