Itinerant Magnetism

Itinerant magnetism develops from interacting, extended electronic states whose occupations and wavefunctions change with the magnetic order. In a magnetic metal, the same electronic states help form both the magnetization and the Fermi surface. Polarization splits bands; spatially modulated order mixes momenta; collective spin motion interacts with electron–hole excitations. A sufficiently strong reconstruction can also gap the entire Fermi surface, so an itinerant ordering mechanism need not leave a metal.

An itinerant magnet need not lack local moments on every time scale. A correlated metal can have strong atomic Hund correlations and a sizable short-time fluctuating moment alongside coherent low-energy quasiparticles. “Localized” and “itinerant” describe useful limits, often within the same material.

The Stoner Criterion develops the scalar uniform instability; the discussion here asks how band polarization, finite-wavevector response, and measured spectra distinguish material mechanisms.

Required background. Band Theory Overview supplies bands, occupations, and velocities, while Susceptibilities supplies the source-response and finite-wavevector instability logic.

Helpful background. Fermi Surface and Density of States support state counting; Ferromagnetism and Antiferromagnetism describe the resulting ordered phases.

The Magnetic Variable Is a Spin Density

For spin-$1/2$ electron fields, define the spin density measured in units of $\hbar$ per volume

$$\hat{\mathbf s}(\mathbf r) = \frac12 \hat\psi^\dagger_\alpha(\mathbf r) \boldsymbol\sigma_{\alpha\beta} \hat\psi_\beta(\mathbf r).$$

Its Fourier components are

$$\hat{\mathbf s}_{\mathbf q} = \int d^3r\, e^{-i\mathbf q\cdot\mathbf r} \hat{\mathbf s}(\mathbf r).$$

A uniform ferromagnet has

$$\left\langle \hat{\mathbf s}_{\mathbf 0} \right\rangle \ne \mathbf 0,$$

whereas an itinerant spin-density wave has

$$\left\langle \hat{\mathbf s}_{\mathbf Q} \right\rangle \ne \mathbf 0$$

at a nonzero ordering wavevector $\mathbf Q$.

The magnetic moment density is related to spin and orbital angular momentum through material-dependent $g$ tensors and spin–orbit coupling in the solid. Even in a nominally spin-only model, sign conventions require care because the electron’s magnetic moment is antiparallel to its spin. It is often cleaner to quote the spin polarization

$$\delta n = n_\uparrow-n_\downarrow$$

and state separately how it is converted to magnetization.

Itinerant electrons can change the local spin population by moving charge between orbitals and spin-resolved bands. This is additional amplitude freedom beyond a fixed local spin length. Fixed-length spin models still have longitudinal correlations, including multi-magnon fluctuations of the ordered component; they are not restricted to transverse response. The question is which degrees of freedom account for the observed longitudinal weight.

Exchange-Split Bands

A collinear band model

In a collinear mean-field description without spin–orbit mixing, write

$$\begin{aligned} \varepsilon_{n\mathbf k\uparrow} &= \varepsilon_{n\mathbf k} - \frac12\Delta_{n\mathbf k}, \\ \varepsilon_{n\mathbf k\downarrow} &= \varepsilon_{n\mathbf k} + \frac12\Delta_{n\mathbf k}. \end{aligned}$$

The labels $\uparrow$ and $\downarrow$ refer to a stated quantization axis. The exchange splitting $\Delta_{n\mathbf k}$ need not be constant across bands or momentum. In a multiorbital solid it is generally a matrix in orbital space and can mix with crystal-field and spin–orbit terms.

The spin polarization follows from the occupied states:

$$\delta n = \frac{1}{V} \sum_{n\mathbf k} \left[ f( \varepsilon_{n\mathbf k\uparrow} ) - f( \varepsilon_{n\mathbf k\downarrow} ) \right].$$

At fixed total electron number, exchange splitting also shifts the chemical potential. One cannot generally obtain the magnetic moment by splitting a density-of-states plot while leaving the Fermi level fixed.

Exchange splitting is not a Zeeman field

An externally applied Zeeman field is a controlled source. Exchange splitting is a self-consistent internal energy difference generated by interactions and polarization. In a simple scalar convention,

$$\Delta = I\,\delta n,$$

where $I$ is a Stoner interaction parameter. This equation is a mean-field closure, not a universal definition of $I$. Its numerical value depends on whether $\delta n$ is quoted per atom, per cell, or per volume and on whether the density of states is per spin or spin summed.

Exchange splitting can be large even when the ordered moment is modest if broad regions of momentum contribute with cancellations or if several orbitals polarize differently. Conversely, a spin-resolved photoemission splitting is not by itself proof of a bulk equilibrium ferromagnet; surface magnetism, domains, final-state effects, and matrix elements must be controlled.

Fermi volumes carry the polarization

For one spin-conserving, isotropic three-dimensional electron band at zero temperature,

$$n_\sigma = \frac{k_{F\sigma}^3}{6\pi^2}.$$

Hence

$$\delta n = \frac{ k_{F\uparrow}^3-k_{F\downarrow}^3 }{6\pi^2}.$$

This relation encodes the polarization in unequal occupied Fermi volumes for the stated band. In a crystal, completely filled bands also contribute to electron counting; with spin–orbit mixing, a spin-resolved Fermi volume alone no longer determines the spin expectation value. In a real crystal, several sheets, spin–orbit entanglement, and nontrivial band degeneracies replace this scalar picture, but quantum oscillations, angle-resolved photoemission, and first-principles calculations can still test the reconstructed Fermi surface.

Susceptibility as the Organizing Object

Couple a source field to the spin density through

$$\hat H_{\mathrm{src}} = - \sum_{\mathbf q} \mathbf h_{\mathbf q} \cdot \hat{\mathbf s}_{-\mathbf q}.$$

Since $\hat{\mathbf s}_{\mathbf q}$ is an extensive Fourier component and $\mathbf h_{\mathbf q}$ has units of energy, define the intensive response by $\delta\langle\hat s^a_{\mathbf q}\rangle/V=\sum_b\chi^{ab}(\mathbf q,\omega)h^b_{\mathbf q}$. For this minus-sign source coupling, the retarded susceptibility is

$$\chi^{ab}(\mathbf q,\omega) = + \frac{i}{\hbar V} \int_0^\infty dt\, e^{i(\omega+i0^+)t} \times \left\langle \left[ \hat s_{\mathbf q}^a(t), \hat s_{-\mathbf q}^b(0) \right] \right\rangle.$$

It answers three distinct questions:

1. At which wavevector is the paramagnet most susceptible?
2. Does an interaction drive a denominator to zero and create order?
3. In the ordered state, where do collective poles remain sharp relative to the electron–hole continuum?

Independent-particle kernel

For Bloch bands, the bare susceptibility has the schematic form

$$\chi_0^{ab}(\mathbf q,\omega) ={} \frac{1}{V} \sum_{\mathbf k,n,m} \times \frac{ f(\varepsilon_{m,\mathbf k+\mathbf q}) - f(\varepsilon_{n\mathbf k}) }{ \hbar\omega + \varepsilon_{n\mathbf k} - \varepsilon_{m,\mathbf k+\mathbf q} + i0^+ } \times \langle n\mathbf k\vert \hat s^a_{\mathbf q}\vert m,\mathbf k+\mathbf q\rangle \times \langle m,\mathbf k+\mathbf q\vert \hat s^b_{-\mathbf q}\vert n\mathbf k\rangle.$$

The matrix elements here use the one-particle Fourier spin vertices, $e^{\mp i\mathbf q\cdot\mathbf r}\sigma^{a,b}/2$; omitting the momentum-transfer phase while using full Bloch states would give the wrong selection rule. The occupation difference supplies phase space, the denominator supplies the electron–hole energy, and the matrix elements encode orbital and spin character. For an occupied $n$ and empty $m$, numerator and static denominator are both negative, giving a positive diagonal static response. A geometric nesting vector is therefore not sufficient by itself: the relevant states must have nonzero spin matrix elements and an interaction must reinforce that channel.

Interaction enhancement and its normalization

Let $\boldsymbol m$ collect intensive spin densities in the chosen spin/orbital channels, and define the induced source by

$$\boldsymbol h_{\mathrm{eff}}=\boldsymbol h+\mathsf K\boldsymbol m.$$

This definition fixes both the sign and normalization of the spin-channel vertex $\mathsf K$. Solving $\boldsymbol m=\boldsymbol\chi_0\boldsymbol h_{\mathrm{eff}}$ gives

$$\boldsymbol\chi= [\mathbb1-\boldsymbol\chi_0\mathsf K]^{-1}\boldsymbol\chi_0.$$

For $s^z=(n_\uparrow-n_\downarrow)/2$ and the earlier scalar splitting $\Delta=I\delta n$, the induced spin source is $2I m_z$. Thus $K=2I$, not $I$, in this spin-density convention. At zero temperature the unpolarized one-band response is $\chi_{0,s}=N(E_F)/2$ for a density of states per spin, so $K\chi_{0,s}=IN(E_F)$. The familiar scalar criterion is recovered without changing the definition of spin midway through the calculation.

The static equilibrium susceptibility is Hermitian and positive semidefinite for stable channels. With a Hermitian static vertex, inspect the Hermitian matrix

$$\mathsf A(\mathbf q)= \boldsymbol\chi_0^{1/2}(\mathbf q,0)\, \mathsf K\, \boldsymbol\chi_0^{1/2}(\mathbf q,0).$$

On the responsive subspace it has the same nonzero instability eigenvalues as $\boldsymbol\chi_0\mathsf K$. Its largest eigenvalue reaching one marks the loss of quadratic stability in this approximation. Following the tuning parameter identifies which wavevector reaches one first; a snapshot with several unstable channels does not determine that history or the final nonlinear ordered state.

The static qualification is essential. If total spin is conserved, a strictly uniform oscillating source cannot change its longitudinal expectation in an isolated sample: the corresponding retarded commutator is zero. The Pauli susceptibility instead describes equilibrium repopulation. Recover it through the thermodynamic static response, or the appropriate $\omega=0$ then $q\to0$ limit, with equilibration understood. It is not the $q=0$ finite-frequency response continued blindly to zero frequency.

The Scalar Stoner Limit

The Stoner Criterion article derives the scalar limit with explicit spin and density conventions. For one band, a momentum-independent interaction, and a paramagnetic density of states per spin $N(E_{\mathrm F})$, the uniform static susceptibility takes the mean-field form

$$\chi = \frac{\chi_{\mathrm P}}{ 1-I\,N(E_{\mathrm F}) }.$$

Here the scalar $\chi$ is the magnetic susceptibility $\partial M/\partial H$, obtained by restoring the magnetic-moment factors to the spin response above. If $N(E_{\mathrm F})$ is per spin and per volume in SI units, the noninteracting Pauli spin susceptibility is

$$\chi_{\mathrm P} = 2\mu_0\mu_B^2 N(E_{\mathrm F}).$$

This expression takes $g=2$ and describes the spin contribution only. More generally the prefactor is $2\mu_0(g\mu_B/2)^2$. Per-atom or per-cell densities of states require matching interaction and volume factors. The enhancement factor

$$\mathcal S := \frac{\chi}{\chi_{\mathrm P}} = \frac{1}{ 1-I\,N(E_{\mathrm F}) }$$

diverges at the scalar mean-field threshold

$$I\,N(E_{\mathrm F}) = 1.$$

The criterion captures the competition between the band-energy cost of unequal spin populations and the interaction-energy gain. It is useful because it connects magnetism to the Fermi-level density of states. It is incomplete because it suppresses momentum, orbital structure, self-energy corrections, vertex corrections, and critical spin fluctuations. A large noninteracting density of states is an invitation to test a magnetic instability, not a proof that the ground state is ferromagnetic.

Finite-Wavevector Spin-Density Waves

For a lattice with $N$ cells, use the same Fourier sign as above and define the spin-density-wave amplitude per cell

$$\mathbf m_{\mathbf Q} = \frac{1}{2N} \sum_{\mathbf k} \left\langle c^\dagger_{\mathbf k,\alpha} \boldsymbol\sigma_{\alpha\beta} c_{\mathbf k+\mathbf Q,\beta} \right\rangle.$$

This is $\langle\mathbf s_{\mathbf Q}\rangle/N$ in the lattice normalization. A real spin pattern includes the complex-conjugate component at $-\mathbf Q$. The order mixes states whose momenta differ by $\mathbf Q$. Commensurate order enlarges the magnetic unit cell and folds the zone. Genuinely incommensurate order has no finite magnetic primitive cell; a rational approximation or a momentum-space truncation must be stated.

For a collinear, one-orbital sector with $2\mathbf Q$ a chemical reciprocal vector, the two momenta close under scattering and the mean-field block is

$$\mathcal H_{\mathbf k} = \begin{pmatrix} \varepsilon_{\mathbf k} & \Delta_{\mathbf Q} \\ \Delta_{\mathbf Q}^{*} & \varepsilon_{\mathbf k+\mathbf Q} \end{pmatrix}.$$

The corresponding eigenvalues are

$$E_\pm(\mathbf k) ={} \frac{ \varepsilon_{\mathbf k} + \varepsilon_{\mathbf k+\mathbf Q} }{2} \pm \sqrt{ \left[ \frac{ \varepsilon_{\mathbf k} - \varepsilon_{\mathbf k+\mathbf Q} }{2} \right]^2 + \lvert\Delta_{\mathbf Q}\rvert^2 }.$$

For generic incommensurate collinear order, successive scatterings connect $\mathbf k+n\mathbf Q$; the displayed block is then only a justified two-patch approximation, not the full band problem. Perfect nesting is not required for order. Imperfectly gapped regions can leave reconstructed metallic pockets. Itinerant antiferromagnets are therefore often metals rather than Slater insulators.

Chromium is the classic material example: its incommensurate antiferromagnetism is described by a spin-density wave whose wavevector, Fermi-surface geometry, lattice strain, and accompanying charge modulation are experimentally linked. The example is instructive precisely because a simple nesting cartoon is not the whole material theory; orbital matrix elements, interactions, pressure, alloying, and fluctuations all matter.

Charge and Spin Density Waves compares reconstruction and collective motion in spatially modulated phases.

Collective Modes and the Stoner Continuum

Spin-flip electron–hole excitations

In a collinear spin-polarized metal, a transverse spin operator can remove an occupied majority-spin quasiparticle and create a minority-spin quasiparticle. The allowed continuum contains energies

$$\hbar\omega = \varepsilon_{m,\mathbf k+\mathbf q,\downarrow} - \varepsilon_{n,\mathbf k,\uparrow},$$

subject to

$$f( \varepsilon_{n,\mathbf k,\uparrow} ) = 1, \qquad f( \varepsilon_{m,\mathbf k+\mathbf q,\downarrow} ) = 0$$

at zero temperature. This set of spin-flip electron–hole states is the Stoner continuum.

The continuum is not one sharp dispersion. Its boundaries and spectral weight depend on all pairs of occupied and empty bands connected by $\mathbf q$, including their spin matrix elements.

A continuum boundary from two Fermi spheres

For a concrete three-dimensional example, take

$$\begin{aligned} \varepsilon_{\mathbf k\uparrow}&=Ak^2-\Delta/2,\\ \varepsilon_{\mathbf k\downarrow}&=Ak^2+\Delta/2,\\ A&=\frac{\hbar^2}{2m^*}. \end{aligned}$$

Choose $k_{F\uparrow}=1.2k_0$ and $k_{F\downarrow}=0.8k_0$, and use $E_0=Ak_0^2$. A common chemical potential then requires $\Delta=0.8E_0$ and $\mu=1.04E_0$. The unpolarized state at the same total density has $k_F=(1.12)^{1/3}k_0$ and $\mu_0\simeq1.0785E_0$. The change in chemical potential is part of conserving particle number.

A majority-to-minority spin flip has energy

$$E=\Delta+Aq^2+2Akq\cos\theta,$$

with $k\le k_{F\uparrow}$ and $|\mathbf k+\mathbf q|\ge k_{F\downarrow}$. In three dimensions its positive-energy support has boundaries

$$\begin{aligned} E_{\min}(q)&=\max\{0,\Delta+Aq^2-2Ak_{F\uparrow}q\},\\ E_{\max}(q)&=\Delta+Aq^2+2Ak_{F\uparrow}q. \end{aligned}$$

The lower edge reaches zero for $|k_{F\uparrow}-k_{F\downarrow}|\le q\le k_{F\uparrow}+k_{F\downarrow}$, where the two displaced Fermi surfaces intersect. “Reaches zero” means arbitrarily small positive excitation energies; it does not imply finite spectral weight at exactly zero energy. At $q=0$, this rigid-splitting model collapses to energy $\Delta$. These are continuum kinematics, not a calculation of a collective magnon pole.

The specified parabolic-band example conserves total density. Top: thick portions are occupied; the two Fermi radii are $0.8k_0$ and $1.2k_0$. Bottom: shaded spin-flip support, shown only up to $3E_0$. Its lower edge reaches zero from $q=0.4k_0$ to $q=2k_0$. Matrix elements determine the intensity inside this region; interactions determine whether an additional collective pole survives.

Collective transverse poles

Broken spin-rotation symmetry requires a low-energy transverse collective mode when spin–orbit coupling and external fields are neglected. In an itinerant ferromagnet, the susceptibility denominator can produce a magnon pole below the Stoner continuum. At small wavevector it can be sharp and quadratic:

$$\hbar\omega_{\mathrm m}(\mathbf q) \simeq Dq^2.$$

As the branch approaches the continuum, it can hybridize with electron–hole states, lose pole residue, and acquire Landau damping. A schematic pole condition is

$$\det \left[ \mathbb 1 - \boldsymbol\chi_0( \mathbf q,\omega ) \mathsf K \right] = 0.$$

The dissipative matrix is $\boldsymbol\chi_0^{\prime\prime}=(\boldsymbol\chi_0-\boldsymbol\chi_0^\dagger)/(2i)$ on the real-frequency axis. A pole generally broadens if it couples to a nonzero dissipative channel. For complex cross-channel matrices this is not an elementwise imaginary part. Kinematic overlap alone is insufficient when the relevant matrix element vanishes.

Longitudinal fluctuations

Longitudinal response changes the magnitude of the polarization. A fixed-length spin model has longitudinal multi-magnon correlations but lacks independent changes of the local spin representation through electronic occupation. In an itinerant magnet, those additional channels can affect thermodynamics, spectra, and the transition itself. Longitudinal weight often overlaps strongly with particle–hole continua and need not form a sharp amplitude mode.

Spin Waves and Magnons gives the material workflow for comparing collective branches, eigenvectors, cross sections, and linewidths. The Magnons article develops quantization and its limits.

Local-Moment and Itinerant Limits

The most useful comparison is operational:

Question	Local-moment limit	Itinerant limit
What carries magnetism?	moments identifiable before long-range order	spin polarization of extended quasiparticle states
What orders?	moment orientation, sometimes multipoles	momentum- and orbital-resolved spin density
What amplitude fluctuations are available?	fixed local spin length, but longitudinal multi-spin correlations	additional changes of electronic occupation and local moment amplitude
Natural low-energy model	spin Hamiltonian	interacting multiband electron Hamiltonian
Charge sector	may be insulating or metallic	often metallic; reconstruction can also fully gap the bands
Excitation continuum	multi-spin and fractional channels	spin-flip electron–hole channels
Above the transition	local moment can persist	polarization can weaken, but fluctuating moments can also persist
Parameter language	$S$, exchange tensors, anisotropy	bands, self-energies, vertices, $\chi(\mathbf q,\omega)$

No single measurement places a material permanently in one column.

Scale dependence

A metal can look local on femtosecond time scales and itinerant in its low-energy coherent response. Orbital-selective materials can have some orbitals forming robust moments while others remain mobile. Hund coupling can create a substantial instantaneous moment, while hybridization and intersite coherence determine how that moment orders.

Effective spin models can still work

If transverse collective modes are well separated from longitudinal and electron–hole continua, integrating out the itinerant electrons can produce an effective spin Hamiltonian over a limited energy range. Its exchange couplings may be long ranged, retarded, state dependent, and sensitive to the Fermi surface. Success of a low-energy spin-wave fit does not prove that the microscopic moments were rigid at all energies.

A metal is not automatically itinerant-magnetic

Local moments can coexist with a Fermi surface. Rare-earth metals, Kondo lattices above screening scales, and local-moment metals can conduct through electrons distinct from the moments. The diagnostic question is whether the electrons forming the relevant Fermi surface also form and reshape the magnetic order parameter.

Experimental Diagnostics

Spin-resolved band structure

Spin- and angle-resolved photoemission can resolve exchange-split occupied bands, while inverse photoemission and resonant spectroscopies access unoccupied states. Compare the observed splitting over momentum and orbital character rather than quoting one number.

Quantum oscillations and magnetotransport can reveal spin-split or reconstructed Fermi-surface sheets. Their interpretation requires care with magnetic breakdown, Zeeman splitting, spin–orbit coupling, domains, and field-induced transitions.

Ordered and fluctuating moments

Neutron diffraction measures the spatial Fourier components of the ordered magnetization density. Inelastic neutron scattering measures a moment correlation spectrum. Relating it to the dissipative susceptibility requires the equilibrium detailed-balance factor as well as form factors, polarization projection, and the stated moment normalization. A sharp low-$q$ magnon that broadens into a continuum is strong itinerant evidence when a band calculation reproduces the continuum kinematics and spectral weight.

X-ray magnetic circular dichroism can separate element-specific spin and orbital contributions subject to sum-rule assumptions. NMR, Mössbauer spectroscopy, and muon spin rotation access local fields and fluctuation time scales. These probes can establish that substantial local magnetism persists above an ordering transition even when the low-temperature phase has an itinerant Fermi surface.

Thermodynamics and the Rhodes–Wohlfarth ratio

Fit the high-temperature susceptibility only over a regime where a Curie–Weiss form is justified:

$$\chi(T) = \chi_0 + \frac{C}{T-\Theta}.$$

The effective moment $p_{\mathrm{eff}}$ is defined through the Curie constant. Rhodes and Wohlfarth introduce $p_c$ by

$$p_{\mathrm{eff}}^2 = p_c(p_c+2)$$

when moments are measured in units of $\mu_B$, and compare it with the low-temperature spontaneous moment $p_s$. A ratio

$$\frac{p_c}{p_s} \gg 1$$

is a classic signature of weak itinerant ferromagnetism: strong fluctuating response coexists with a small ordered moment.

This ratio is a diagnostic, not a binary theorem. Crystal-field moments, orbital contributions, Kondo screening, temperature-dependent backgrounds, unsaturated magnetization, and an inadequate Curie–Weiss window can all distort it.

The electronic heat-capacity coefficient and uniform susceptibility can also be combined into a Wilson ratio. Fermi-Liquid Theory Preview gives its normalization and interpretation. An enhanced Wilson ratio signals strong spin-channel interactions but does not by itself establish ordered itinerant magnetism.

Pressure, composition, and field

Weak itinerant magnets are often sensitive to small changes in lattice spacing, pressure, or composition because those changes reshape the density of states and susceptibility. This sensitivity is informative only with structural and disorder controls. A pressure-driven loss of magnetism can result from bandwidth changes, a first-order transition, competing order, or a change in electronic coherence.

First-Principles and Model Calculations

Spin-polarized density-functional calculations can predict:

• ordered moments and spin density;
• exchange-split band structures;
• magnetic anisotropy with spin–orbit coupling;
• relative energies of candidate magnetic structures;
• momentum-dependent susceptibilities in a chosen approximation.

Agreement with the ordered moment and one band splitting is not enough. A trustworthy comparison also checks Fermi-surface geometry, orbital character, mode energies, transition scales, and sensitivity to the exchange-correlation functional.

Fixed-spin-moment energy

A useful calculation constrains the total moment $M$ and evaluates

$$E_{\mathrm{FSM}}(M).$$

Near $M=0$,

$$E_{\mathrm{FSM}}(M) = E_0 + \frac{a_2}{2}M^2 + \frac{a_4}{4}M^4 + \cdots.$$

A negative $a_2$ signals a uniform mean-field instability in that computational approximation. Multiple minima can indicate metamagnetism or competing magnetic states. The result is still a zero-temperature electronic energy landscape; it does not automatically supply the finite-temperature transition or the fluctuation renormalization.

Dynamic susceptibility

Time-dependent density-functional theory, many-body perturbation theory, dynamical mean-field theory, and spin-channel RPA approximate the dynamic susceptibility differently. A response calculation must use a vertex consistent with its propagator to preserve spin conservation and the Goldstone mode. Artificial broadening, analytic continuation, and instrumental resolution must be separated from a predicted intrinsic width. Compare pole energies, spectral weight, and sum rules, not only a ground-state moment.

Spin Fluctuations Beyond Static Stoner Theory

Static Stoner mean field often predicts the existence and zero-temperature scale of polarization more successfully than it predicts finite-temperature behavior. It neglects the spatial and temporal spin fluctuations that reduce ordered moments, renormalize spectra, and commonly lower calculated transition temperatures.

Moriya’s spin-fluctuation and self-consistent-renormalization framework treats the spectrum of paramagnons as part of the thermodynamics. It explains why itinerant electrons can produce Curie–Weiss-like susceptibility and sizable fluctuating moments without becoming pre-existing rigid spins.

Near a magnetic quantum phase transition, gapless electron–hole excitations cannot always be treated as passive damping. They can alter the effective action of the order parameter and invalidate a purely local Landau expansion. In clean metallic ferromagnets, coupling to soft particle–hole modes can favor a first-order low-temperature transition or modulated phases; a continuous Hertz-type transition is not universal. Material disorder, dimensionality, spin–orbit coupling, and additional phases change the outcome.

Claims of “quantum critical itinerant ferromagnetism” should therefore identify:

• the tuning parameter and phase boundary;
• whether the transition is continuous or first order;
• disorder and residual-resistivity scales;
• the temperature window of any scaling law;
• alternative ordered or inhomogeneous phases;
• whether the measured exponents describe equilibrium criticality or a crossover.

Representative Material Regimes

Iron, cobalt, and nickel

Their spin-polarized Fermi surfaces and band structures establish a major itinerant component. Yet their finite-temperature behavior is not captured uniformly by a bare Stoner model. Iron retains more robust moment amplitude than nickel, while spin fluctuations substantially renormalize transition temperatures and spectra. Treating all three as identical “Stoner ferromagnets” loses the physics.

Chromium

Chromium is the canonical finite-$\mathbf Q$ itinerant antiferromagnet. Its incommensurate spin-density wave, Fermi-surface reconstruction, charge modulation, and pressure or alloy sensitivity make it a benchmark for susceptibility-driven order.

Weak itinerant ferromagnets

Materials such as ZrZn$_2$, Ni$_3$Al, and MnSi have small ordered moments and strong tunability. MnSi additionally has chiral spin–orbit interactions and helimagnetic order, so it is not a scalar Stoner ferromagnet. These systems are valuable precisely because pressure and field expose fluctuation and quantum-transition effects omitted by mean field.

Hund and multiorbital metals

Iron-based superconductors and related multiorbital compounds often show coherent reconstructed bands together with sizable local fluctuating moments. Orbital selectivity, Hund coupling, and nesting all contribute. The local-versus-itinerant question is therefore poorly posed unless an energy, time, orbital, and momentum scale is specified.

Exercises

1. Polarization from Fermi volumes

For one isotropic band at zero temperature, let

$$k_{F\uparrow} = 1.05\,\mathrm{nm}^{-1}, \qquad k_{F\downarrow} = 0.95\,\mathrm{nm}^{-1}.$$

Calculate $n_\uparrow-n_\downarrow$.

Solution

Using

$$n_\sigma = \frac{k_{F\sigma}^3}{6\pi^2},$$

gives

$$\begin{aligned} \delta n &= \frac{ (1.05)^3-(0.95)^3 }{6\pi^2} \,\mathrm{nm}^{-3} \\ &= \frac{ 1.157625-0.857375 }{6\pi^2} \,\mathrm{nm}^{-3} \\ &\simeq 5.07\times10^{-3} \,\mathrm{nm}^{-3}. \end{aligned}$$

The corresponding spin-only magnetic-moment magnitude is approximately $\mu_B\delta n$ for $g\simeq2$, subject to the stated spin labeling and electron-moment sign convention.

2. Which wavevector orders?

For a multiorbital paramagnet, the largest eigenvalue of

$$\boldsymbol\chi_0(\mathbf q,0) \mathsf K$$

is $0.78$ at $\mathbf q=0$ and $1.08$ at an incommensurate $\mathbf Q$. Which channel has lost local stability at this parameter value? What would be needed to determine which transition occurs first?

Solution

The finite-$\mathbf Q$ eigenvalue has crossed unity while the uniform eigenvalue has not. Thus the finite-wavevector spin-density-wave channel is locally unstable at this point. To identify which transition occurs first, follow the competing eigenvalues and free-energy minima as the parameter is tuned; the local criterion alone does not exclude a first-order transition elsewhere.

The unstable eigenvector identifies the orbital and spin combination in the linearized problem. Quoting only a total density of states would miss both the finite wavevector and the multiorbital structure.

3. A protected low-energy collective mode

In an ideal isotropic itinerant ferromagnet at zero field, a calculation produces a finite transverse gap at $\mathbf q=0$ even though spin–orbit coupling was omitted. What should be checked?

Solution

Exact spin-rotation symmetry and spontaneous ferromagnetic order require a gapless transverse Goldstone mode. Check:

1. self-consistency of the ordered mean field;
2. consistency between the one-particle self-energy and response vertex;
3. the sign and normalization of the transverse susceptibility;
4. numerical momentum and frequency resolution;
5. whether an explicit field or anisotropy entered inadvertently.

A response approximation that violates the relevant Ward identity can produce a spurious gap even when its ground-state polarization looks reasonable.

4. Rhodes–Wohlfarth diagnostic

A ferromagnetic metal has

$$p_{\mathrm{eff}} = 2.83, \qquad p_s = 1.00,$$

in units of $\mu_B$. Compute $p_c$ from

$$p_{\mathrm{eff}}^2 = p_c(p_c+2)$$

and find $p_c/p_s$.

Solution

Solving the quadratic equation gives the positive root

$$p_c = - 1 + \sqrt{ 1+p_{\mathrm{eff}}^2 }.$$

Since $2.83^2\simeq8.01$,

$$p_c \simeq -1+\sqrt{9.01} \simeq 2.00.$$

Therefore

$$\frac{p_c}{p_s} \simeq 2.$$

The ratio larger than one supports an intermediate or itinerant ferromagnetic interpretation, but it is not conclusive without checking the Curie–Weiss window, saturation, orbital moment, and other probes.

5. Collective mode entering a continuum

A transverse branch is resolution limited below $30\,\mathrm{meV}$, then broadens rapidly and loses integrated peak weight where a band calculation predicts spin-flip electron–hole states. Give the leading interpretation and two checks.

Solution

The leading interpretation is Landau damping: a collective transverse pole has entered the Stoner continuum and can decay into spin-flip electron–hole excitations.

Two essential checks are:

1. compare the measured onset in both momentum and energy with the matrix-element-weighted continuum, not only its geometric boundary;
2. verify that the missing pole weight is redistributed into magnetic continuum weight after background and resolution corrections.

Additional checks include temperature and polarization dependence, alternative phonon or orbital hybridization, and whether disorder broadening could mimic the onset.

References

• M. Brando, D. Belitz, F. M. Grosche, and T. R. Kirkpatrick, “Metallic Quantum Ferromagnets,” Reviews of Modern Physics 88, 025006 (2016), doi:10.1103/RevModPhys.88.025006.
• P. Dai, “Antiferromagnetic Order and Spin Dynamics in Iron-Based Superconductors,” Reviews of Modern Physics 87, 855–896 (2015), doi:10.1103/RevModPhys.87.855.
• E. Fawcett, “Spin-Density-Wave Antiferromagnetism in Chromium,” Reviews of Modern Physics 60, 209–283 (1988), doi:10.1103/RevModPhys.60.209.
• A. Georges, L. de’ Medici, and J. Mravlje, “Strong Correlations from Hund’s Coupling,” Annual Review of Condensed Matter Physics 4, 137–178 (2013), doi:10.1146/annurev-conmatphys-020911-125045.
• J. Kübler, Theory of Itinerant Electron Magnetism, 2nd ed. (Oxford University Press, 2021), doi:10.1093/oso/9780192895639.001.0001.
• T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism (Springer, 1985), doi:10.1007/978-3-642-82499-9.
• P. Rhodes and E. P. Wohlfarth, “The Effective Curie–Weiss Constant of Ferromagnetic Metals and Alloys,” Proceedings of the Royal Society A 273, 247–258 (1963), doi:10.1098/rspa.1963.0086.
• E. C. Stoner, “Collective Electron Ferromagnetism,” Proceedings of the Royal Society A 165, 372–414 (1938), doi:10.1098/rspa.1938.0066.