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A physicist associated with the Stoner criterion for itinerant ferromagnetism.

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.

The spin-independent term cancels in the signed sum. The linearized stationary equation becomesThe even powers vanish because the equation is odd under $M\mapsto -M$. Equivalently, the mean-field action has curvatureThus the unpolarized stationary point becomes unstable whenAt low temperature, the derivative of the Fermi distribution samples a narrow window around $\mu$, so ${\chi }_0\simeq \nu (\mu )$ for a smooth density of states. Measuring energy from the Fermi level gives $U_c\nu (0)\simeq 1$, the Stoner criterion in the spin-summed convention fixed above. This identifies local instability of the paramagnetic state; the detailed transition order also depends on higher terms in the action.

In the isotropic ordered phase of itinerant ferromagnetism, continuous spin-rotation symmetry leaves a degenerate direction of magnetization. Its transverse fluctuations produce one gapless, quadratically dispersing mean-field itinerant ferromagnetic Goldstone mode, $E(\mathbf{q})\propto q^2$. The two broken spin generators form a canonical pair, as for the type-B Goldstone boson of the localized ferromagnet. The system also remains metallic, with exchange-split Fermi surfaces and low-energy spin-conserving particle-hole excitations. Longitudinal-amplitude fluctuations and the spin-flip continuum are distinct from the long-wavelength magnon; their energy and damping scales depend on the band structure. Spin anisotropy, if added, could gap the transverse mode, but it is absent from the stated Hamiltonian.

Exchange interactions arise from quantum electronic motion, Coulomb interactions and fermionic antisymmetry. A localized-spin description can be written $H={\sum }_{i<j}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. In this explicit plus-sign convention $J<0$ favors parallel spins and $J>0$ favors antiparallel spins; many texts put a minus sign into the Hamiltonian and reverse that convention.

Direct exchange involves overlapping orbitals and their Coulomb exchange matrix elements. The Pauli exclusion principle correlates spin symmetry with spatial symmetry, changing the electronic energy. Orbital occupation and competing kinetic processes determine which spin state is favored; the preference is not universally ferromagnetic.

Superexchange couples moments through virtual hopping, often via a nonmagnetic ligand. For two half-filled orbitals in the simple repulsive one-band model, the antiferromagnetic coupling is $J\simeq 4|t{|}^2/U$: virtual double occupation lowers the antiparallel-spin state more effectively. Orbital geometry and intermediate-state interactions can change the sign in other configurations. Real charge transport can remain insulating while this magnetic coupling acts.

Double exchange instead uses actual carrier motion between mixed-valence sites. Local Hund's coupling ties the mobile electron's spin to a core moment. Parallel neighboring cores allow a larger effective hopping amplitude and a lower carrier kinetic energy, favoring ferromagnetism. The original model explicitly connected conduction and magnetic order in manganese perovskites.^[ref]

Conduction-electron-mediated exchange couples localized moments through a metal's electronic spin response. Its strength and sign depend on separation and the Fermi surface; in a simple three-dimensional asymptotic model the range factor oscillates as $\cos (2k_{F}r)/r^3$. The original nuclear-moment calculation was motivated by resonance-line broadening in silver.^[ref] In an itinerant metal, an alternative description compares the kinetic-energy cost of unequal spin populations with the exchange-energy gain; the Stoner criterion expresses when spontaneous polarization is favorable.

The resulting order also depends on the lattice and competing couplings. In ferromagnetism, moments align to give a nonzero moment within a domain. In antiferromagnetism, opposing sublattices cancel their moments. In ferrimagnetism, oppositely ordered sublattices have unequal moments and leave a net one. A bulk zero magnetization alone cannot distinguish antiferromagnetic order from a disordered state or canceling magnetic domains.

The manganese monoxide neutron diffraction experiment makes that distinction experimentally. A monochromatic neutron beam scatters from powdered material, and a counter records angular intensity. Comparing the same compound at room temperature and $80\mathrm{K}$ reveals additional low-temperature magnetic reflections, indexed by a larger magnetic unit cell, while nuclear reflections persist. That spatial periodicity is evidence of alternating order even though the net moment cancels.^[ref]

Exchange interaction arises from Coulomb interaction together with the antisymmetry required for identical electrons. Different spin states require different spatial symmetry and therefore different pair correlations and Coulomb energy; exchange is not the classical magnetic dipole force between spins. For electrons in distinct atomic orbitals, positive direct exchange favors parallel spins, underlying Hund's rules and the formation of local moments.

In a solid with localized moments, a common convention is $H=-J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. Then $J>0$ favors ferromagnetism and $J<0$ favors antiferromagnetism. Superexchange through a ligand and other virtual hopping processes can produce either effective sign depending on orbital occupancy and geometry; half-filled orbitals often give antiferromagnetic coupling. For itinerant electrons, the Stoner criterion describes a competition between exchange energy gain and the kinetic cost of splitting the spin populations.

Experiments distinguish these descriptions through both static and dynamic response. Magnetic neutron diffraction locates magnetic Bragg peaks and therefore the ordering wavevector; an antiferromagnet can have such peaks despite zero bulk moment. Inelastic neutron scattering measures spin-wave dispersion, from which an exchange scale can be fitted. A Curie-Weiss susceptibility intercept gives complementary information about dominant coupling, but its sign alone does not establish the actual ordered structure.

The Hubbard model combines band motion and local repulsion:Hopping favors itinerant states, while positive $U$ penalizes simultaneous opposite-spin occupation on one site. In a uniform mean-field treatment, a spin imbalance reduces the interaction cost but raises the band-filling energy. The Stoner criterion for instability of the unpolarized metal is $U{g}_{\uparrow }(E_F)>1$ when $g_{\uparrow }$ is the density of states per site, per energy, for one spin. Using a total two-spin density of states changes the displayed numerical convention.

The ordered mean-field state has exchange-split spin bands and a spontaneous magnetization, with the kinetic and interaction costs balancing at equilibrium. This is a model of itinerant ferromagnetism, not a guarantee that every repulsive Hubbard system becomes ferromagnetic. Filling, lattice structure and competing order matter: near half filling on a bipartite lattice, antiferromagnetism and a Mott insulator can be favored instead. The approximation also omits important spin fluctuations and strong-correlation effects.

Electrons contribute magnetic response through both spin and orbital motion. In an ordinary weakly interacting metal, a field splits the two spin energies. Only states near the Fermi energy can change occupancy because of the Pauli exclusion principle, producing Pauli paramagnetism. For a total two-spin density of states $g(E_F)$, the simple result is ${\chi }_P={\mu }_0{\mu }_B^{2}g(E_F)$. Orbital cyclotron motion gives Landau diamagnetism; for an isotropic parabolic free-electron band it contributes $-{\chi }_P/3$. Filled shells can also supply a diamagnetic background.

Interactions can favor parallel spin occupancy. In an itinerant description, the exchange-energy gain competes with the kinetic-energy cost of redistributing electrons between spin bands. The Stoner criterion describes when an unpolarized state becomes unstable, with the precise factor in $Ig(E_F)$ depending on whether the density of states counts one spin or both. This mechanism can yield spontaneous ferromagnetism and magnetic domains. Alternatively, sufficiently localized moments in a metal can interact through conduction electrons. A high density of states alone is therefore not a complete microscopic explanation of a material's magnetism; band structure, electron interactions and temperature matter.

Why degenerate perturbation theory is needed. If several unperturbed states have equal or nearly equal energy, a perturbation can mix them at leading order. In that subspace, diagonalize the perturbation matrix rather than using nondegenerate formulas with small energy denominators. The resulting eigenvectors identify the combinations selected by the coupling, and the eigenvalues give the leading splitting. Other remote states can then be included perturbatively.

For the linear combination of atomic orbitals description of molecular hydrogen, equivalent atomic $1s$ states on the two nuclei supply the nearly degenerate starting functions. In an orthonormal two-state model, equal diagonal energy $E_a$ and real hopping $t$ give $E_a\pm t$ and symmetric/antisymmetric combinations. For actual overlapping atomic functions with overlap $S$ and Hamiltonian coupling $h$, one instead solves a generalized eigenvalue problem:The bonding state has enhanced density between the nuclei and the antibonding state a separating node. Two opposite-spin electrons occupying the bonding molecular orbital describe the simplest closed-shell hydrogen-molecule bond. Electron-electron interaction and nuclear repulsion are needed to calculate a realistic total binding energy; the two-state splitting alone is not the full molecular energy.

In the nearly-free electron model, the analogous degenerate states are plane waves $|\mathbf{k}⟩$ and $|\mathbf{k}-\mathbf{G}⟩$ at a Bragg scattering. A Fourier coefficient $U_{\mathbf{G}}$ of the periodic potential couples them. At exact degeneracy $E_B$, diagonalizing their two-by-two matrix givesThe associated standing waves place electronic density differently relative to the ionic potential. These are the same gap-opening combinations illustrated in Q1(b). The labels of the two starting basis states survive as mixed orbital or plane-wave character, rather than independent crossing eigenstates.

Origin of broad and narrow bands. Spatially extended s/p-derived orbitals have substantial overlap and therefore large hopping amplitudes and broad bands. More localized d orbitals overlap less strongly and produce narrower bands. A multiorbital tight-binding calculation assigns the on-site orbital energies and hopping matrices, forms Bloch sums, and diagonalizes the resulting matrix at each wavevector. Different hopping ranges and orbital symmetry determine the bandwidths and allowed interband coupling. The two-band description retains one broad combination and one nearly flat d-derived combination; a real transition metal contains several d orbitals and more complicated dispersions.

Hybridized eigenvalues and eigenstates. Let $|{\psi }_{\mathbf{k}}⟩=a|{\varphi }_{\mathbf{k}}⟩+b|{\chi }_{\mathbf{k}}⟩$, in an orthonormal basis. Hermitian operator fixes the reverse coupling to be $V^{\ast }$. Projection onto the two components givesSolving this quadratic yields the band hybridization dispersionFor nonzero $V$, one eigenvector representation is $(a,b)\propto (V,E-E_1)$, followed by normalization. Thus the wavefunctions exchange broad-band and d-band character through the avoided crossing; the levels are equal mixtures at the uncoupled crossing.

At $V=0$, the parabola crosses the horizontal level at $k=\pm k_0$, with $k_0=\sqrt{2m{E}_d}/\hslash$. For small nonzero $|V|$, the curves avoid crossing with minimum direct separation $2|V|$. The lower branch follows the parabola below the crossing and flattens toward $E_d$ above it; the upper does the opposite. The lower branch satisfies $E_{-}<E_d$ everywhere and tends to $E_d$ from below as $k\to \infty$, while the upper branch always lies above $E_d$. This is the perfectly flat-band continuum approximation, with zone-boundary intersections excluded as requested.

Fixed particle number and the larger Fermi sphere. Denote the uncoupled broad-band Fermi radius by $k_0$ to distinguish it from the new one $K$. With two spin states, the broad-band number density at $V=0$ is $k_0^3/(3{\pi }^2)$. The stipulated equal population in the flat sector makes the total density twice that value. At zero temperature and any nonzero hybridization, all occupied states fit into the monotonic lower branch below $E_d$, while the upper branch is empty. ThereforeThis is the stated factor increase. It uses fixed total particle number, equal spin counting, an exactly flat unhybridized band and enough available states below any Brillouin-zone restriction. In generic pairs of dispersive bands both hybridized pockets can remain occupied; then Fermi-volume conservation in two hybridized electron bands fixes a sum of volumes rather than this one radius. The exactly flat band is degenerate at $V=0$, so its initial occupation must be specified, as it is here.

Fermi-velocity suppression. Set $\Delta (k)=E_1(k)-E_d$. The exact derivative of the lower branch isAt $k=K$, $E_1(K)=2^{2/3}{E}_d$ and $\Delta (K)=(2^{2/3}-1)E_d>0$. For $|V|\ll \Delta (K)$, the square bracket divided by two is $|V{|}^2/\Delta (K{)}^2+O(|V{|}^4/\Delta (K{)}^4)$. Since group velocity is ${\hslash }^{-1}dE/dk$ and the broad-band reference velocity is $v_F^{(1)}=\hslash k_0/m$,so the flat-band hybridization Fermi-velocity suppression isThe reference velocity is that of the original broad-band Fermi sphere, not the broad-band slope at the larger $K$; omitting the radius change loses the factor $2^{1/3}$.

The low-temperature qualification becomes more stringent as $V\to 0$: the new chemical potential approaches $E_d$ with $E_d-\mu \simeq |V{|}^2/\Delta (K)$, and the relevant thermal energy must be small compared with this hybridization scale. At fixed nonzero temperature the singular flat-band redistribution is not automatically resolved for arbitrarily small coupling. A small but finite narrow-band dispersion can also change the occupation topology.

Density of states and magnetism. For an isotropic three-dimensional branch with spin degeneracy two,A small Fermi velocity therefore produces a large density of states. The localized, partially filled d-derived bands in iron, cobalt and nickel place many states close to the Fermi level; broad s/p-derived conduction bands in copper, silver and gold coexist with filled d bands lying below it. This explains the substantially larger Fermi-level density of states of the former group within the band picture. An exact factor of ten requires the actual band parameters and is not fixed by the two-band toy model.

A high density of states lowers the kinetic-energy cost of spin polarization. A repulsive exchange interaction can then satisfy the Stoner criterion and stabilize ferromagnetism. With a single-spin density of states $g_s$ and an on-site interaction normalization, the criterion is $U{g}_s(E_F)>1$; with spin-summed $g=2g_s$, it is $Ug(E_F)>2$. The large d-state contribution favors this instability in Fe, Co and Ni. The noble metals do not satisfy the corresponding instability condition in the same ordinary state; this does not imply that they have no weak magnetic response whatsoever.

The energy favoring spin polarization must compete with the increase in Fermi kinetic energy. Assume the usual three-dimensional zero-temperature parabolic conduction band at fixed $N$ and $V$. Write the spin polarization as $\zeta =(N_{\uparrow }-N_{\downarrow })/N$, so $N_{\uparrow ,\downarrow }=N(1\pm \zeta )/2$. The unpolarized Fermi energy isThe two spin populations have Fermi energies $E_F(1\pm \zeta {)}^{2/3}$. Summing their individual $3E_{F,\sigma }/5$ mean kinetic energies givesThe specified interaction is an energy per electron, $U(1-{\zeta }^2)/4$. Thus the energy change per electron near the unpolarized state isThe quadratic coefficient changes sign at the Stoner threshold for an interaction normalized per electron,The positive quartic term gives a continuous onset in this model. More globally, $f^{\prime \prime }(\zeta )=(E_F/3)[(1+\zeta {)}^{-1/3}+(1-\zeta {)}^{-1/3}]-U/2$. Convexity of $x^{-1/3}$ makes this nonnegative for $U\le 4E_F/3$, so no lower polarized minimum appears before that threshold.

At fixed number density $N/V$, the critical $U$ is independent of the total sample size. A comparison restricted to a fully polarized state gives a different numerical coefficient, $U>\frac{12}{5}{E}_F(2^{2/3}-1)$, but the same requested density scaling; it misses the partially polarized onset. One must also distinguish this $U$, with units of energy, from a contact coupling multiplying an interaction-energy density $g{n}_{\uparrow }{n}_{\downarrow }$: here $g=U/(N/V)$. Importing a differently normalized Stoner criterion would introduce an erroneous extra density factor.

Specify the spin counting first. Let $g_s=g_s(E_F)$ denote the unpolarized single-spin density of states per atom per energy, and put $\delta n=n_{\uparrow }-n_{\downarrow }$. At fixed total occupation $n$, the two Hubbard model mean-field energies can be writtenThe common shift $Un/2$ can be absorbed into the chemical potential. Lowering the up-spin band by $\Delta$ adds $g_s\Delta$ carriers, while raising the down-spin band removes the same number. ThusIf $n_a$ is the atom number density, the magnetization is $M=n_a{\mu }_B\delta n$. The Stoner-enhanced magnetic susceptibility in SI volume units is thereforeHere ${\chi }_P$ is the noninteracting Pauli paramagnetism result. If the paper's $g(E_F)$ counts both spins, $g=2g_s$ and the same result isA susceptibility per atom omits $n_a$. Stating the spin and volume conventions avoids an otherwise spurious factor of two or a dimensional mismatch.

The Stoner criterion is $U{g}_s(E_F)>1$, equivalently $Ug(E_F)>2$ with spin-summed $g$. To see the instability, transfer $\eta$ electrons per atom from one spin band to the other. The band-filling cost is ${\eta }^2/g_s$, whereas the interaction energy changes by $-U{\eta }^2$. Their sum $(1/g_s-U){\eta }^2$ becomes negative at the threshold. The unpolarized state then ceases to be a local minimum, and nonlinear band filling determines the spontaneous ferromagnetism. The negative continuation of the displayed linear susceptibility beyond the threshold is not a stable paramagnetic response.

The constant term in the spin sum cancels. Linearizing the saddle point equation givesThe paramagnetic auxiliary-field curvature isConsequently the Stoner criterion for the loss of local paramagnetic stability isThe printed $\nu (0)$ uses energy measured from the Fermi level. For a smooth density of states, $-n_F^{\prime }$ becomes a narrow unit-area peak at $\mu$; a singular density of states or appreciable temperature requires the integral form. The curvature establishes local instability, not the order or location of every possible thermodynamic transition.

With exact spin-rotation symmetry, itinerant ferromagnetism selects a magnetization direction. The two broken transverse generators form one quadratic type-B Goldstone boson, a long-wave ferromagnetic magnon with $E_q\sim D{q}^2$. Longitudinal amplitude motion is not symmetry-protected to be gapless. In a metallic ordered state, particle-hole excitations remain near each surviving exchange-split Fermi surface; these are distinct from the collective magnon. The itinerant spin-flip particle-hole continuum starts at exchange scale $2|M|$ at $q=0$ in this simple mean-field spectrum, while its finite-$q$ boundaries differ. Spin anisotropy, spin-orbit coupling or an external field can gap the transverse mode; none is present in the stated model.

For a spin-degenerate band, the total density of states includes both spin species and is twice the single-spin density of states. Per lattice site, $\nu (ϵ)=2N^{-1}{\sum }_k\delta (ϵ-{ϵ}_k)$. Using a separate explicit spin sum then requires a compensating factor one half when converting momentum sums. The normalization matters in the Stoner criterion.

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.

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 $U{g}_s<1$.