Ferromagnetic Fermi Liquid Theory

• Ferromagnetic Fermi Liquid Theory is a microscopic framework capturing low-energy spin and collective dynamics in itinerant ferromagnets through Landau Fermi liquid principles.
• It incorporates spontaneous magnetization and explicit quasiparticle interactions via Landau parameters to model exchange stiffness, spin-wave dispersion, and spin-current dynamics.
• The theory predicts instabilities in isotropic ferromagnets with short-range interactions and underpins advanced renormalization group and soft-mode field theory extensions.

Ferromagnetic Fermi Liquid Theory (FFLT) is the microscopic theoretical framework describing low-energy spin, magnetization, and collective dynamics in metallic ferromagnets within the Landau Fermi liquid paradigm. FFLT extends the phenomenology of paramagnetic Fermi liquids by incorporating spontaneous spin polarization, explicit quasiparticle interactions, and their consequences for collective excitations and stability criteria. Central to FFLT are the roles of Landau parameters in the antisymmetric (spin) channel, the theory of spin-current dynamics, the construction of collective-mode spectra, and the characterization of stability and instabilities in itinerant ferromagnets.

1. Fundamental Equations of Ferromagnetic Fermi Liquid Theory

FFLT is formulated in terms of two coupled macroscopic variables: the magnetization density $\vec{M}$ and spin-current density $\vec{J}_i$. In the absence of spin-orbit coupling and for small deviations about a uniform ground-state magnetization $M_0\hat{y}$, the Onsager–Leggett-type dynamical equations are (Saslow et al., 2024):

$$\partial_t \vec{M} + \partial_i \vec{J}_i = -\gamma \vec{M} \times \vec{B} - \frac{\vec{M}}{\tau_M}$$

$$\partial_t \vec{J}_i + G\,\partial_i \vec{M} = \vec{J}_i \times (\vec{B} + A\vec{M}) - \frac{\vec{J}_i}{\tau_J}$$

• $\gamma$ is the gyromagnetic ratio.
• $G\,\partial_i \vec{M}$: vector spin pressure ("exchange-stiffness term").
• $\vec{J}_i \times A\vec{M}$: mean-field precession of spin current about the local magnetization.
• $-\vec{M}/\tau_M$ and $-\vec{J}_i/\tau_J$: phenomenological longitudinal and transverse damping.

The coefficients $G$ (spin stiffness) and $A$ (internal molecular-field strength) are determined in terms of microscopic Landau quasiparticle interactions, while $\tau_M$ and $\tau_J$ depend on collision integrals.

2. Landau Parameters and Microscopic Structure

The key microscopic building blocks are the Landau parameters extracted from the quasiparticle interaction function $f^{s,a}(\cos\theta)$, decomposed into symmetric ("s") and antisymmetric ("a") spin channels and expanded in Legendre harmonics:

$$f^{s,a}(\cos \theta) = \sum_{\ell=0}^{\infty} f_\ell^{s,a} P_\ell(\cos\theta)$$

$F_\ell^{s,a} = N(0)\,f_\ell^{s,a}$

Here, $F_0^a$ (Stoner parameter) governs the spin susceptibility and ferromagnetic instability criterion; $F_1^a$ controls the current–magnetization coupling and spin-current dynamics (Mineev, 2011, Saslow et al., 2024, Belitz et al., 2011).

The equilibrium spin-current prefactor $u$ and the explicit forms of $G$ and $A$ in terms of Fermi velocities $v_{F\uparrow,\downarrow}$, densities of states $N_{\uparrow,\downarrow}$, and antisymmetric Landau parameters are given by:

$$u = v_{F\uparrow}(1+F_1^a N_\downarrow/N(0)) + v_{F\downarrow}(1+F_1^a N_\uparrow/N(0))$$

$$G = u^{-1}\left\{ v_{F\uparrow}^2(1+F_1^a N_\downarrow/N(0)) + v_{F\downarrow}^2(1+F_1^a N_\uparrow/N(0)) + 2f_0^a[v_{F\uparrow}N_\uparrow + v_{F\downarrow}N_\downarrow] \right\}$$

$A = \frac{4u}{\hbar}(f_0^a-f_1^a/3)$

In the paramagnetic limit, these reduce to versions used in Leggett’s theory of liquid $^3$He (Saslow et al., 2024). The static susceptibility is given by

$$\chi = \frac{(g\mu_B)^2\,[N_\uparrow^{-1}+N_\downarrow^{-1}]+4\,f_0^a}{4}$$

which, in the paramagnet, simplifies to $\chi = \chi_0/(1 + F_0^a)$.

3. Collective Modes: Spin Waves and Spin-Current Oscillations

Linearization of the dynamical equations yields two principal transverse collective modes (Saslow et al., 2024, Mineev, 2011):

• Exchange (spin-wave) mode:

$\omega \approx \gamma H + D k^2 - i/\tau_J$

with $D = G/M_0$ the spin-wave stiffness. For $B=0$, it is gapless.

• Spin-current (zero-sound–like) mode:

$\omega \approx \omega_{\text{mf}} \pm i\,\delta$

with $\omega_{\text{mf}} = 4u\lambda/\hbar$ and $\lambda = f_0^a - f_1^a/3$. This mode involves oscillations of the spin current about the static magnetization.

The spin-wave dispersion for the hydrodynamic regime is (Mineev, 2011):

$\omega(q) = \omega_L + (D'' - i D')q^2$

where

$$D'' = D_0 / (\gamma H),\quad D' = D_0 / (\gamma H)^2\tau$$

and $D_0 = v_F^2(1 + F_0^a)(1 + F_1^a/3)/3$.

4. Instability and Limitations of Homogeneous Isotropic Ferromagnetism

A core result of FFLT in systems with short-range repulsion is the inherent instability of the homogeneous ferromagnetic phase against transverse, finite-$q$ fluctuations (Mineev, 2011). This is reflected in the sign of the dissipative component $D'$. In ferromagnets with $F_0^a < -1$, $D' < 0$, and transverse inhomogeneous deviations of magnetization grow exponentially in time:

$$\delta M_\perp(q, t) \sim e^{i(q \cdot r - D'' q^2 t)} e^{|D'| q^2 t}$$

Thus, collisions in the isotropic Stoner model inject rather than damp such excitations. The root cause is the interplay of Landau parameters and the finite quasiparticle lifetimes. Such an instability is not present in paramagnets ($F_0^a > -1$), for which $D'>0$ and dissipation damps fluctuations. This finding renders the isotropic, short-range Stoner ferromagnet intrinsically unstable—contrasted with the apparent stability of real metallic ferromagnets, which are stabilized by long-range exchange, crystalline anisotropy, spin–orbit coupling, and other effects not captured within FFLT with only $F_0^a$, $F_1^a$ (Mineev, 2011).

A plausible implication is that attempts to realize Stoner ferromagnetism in cold-atom gases with tunable short-range repulsion are fundamentally hampered by this instability, in agreement with experiments where molecular (pairing) instabilities dominate over magnetism.

5. Soft-Mode Field Theories and Renormalization Group Extensions

FFLT can be reformulated in terms of effective soft-mode field theories as developed by Belitz and Kirkpatrick (Belitz et al., 2011). In this approach, the collective spin excitations arise as massless (Goldstone) modes associated with spontaneous spin-rotation symmetry breaking in the Fermi liquid. The construction employs matrix-valued soft-field variables in the spin-triplet, $\ell=0$ channel; after integration over massive modes, the effective action reduces in the static and uniform limit to a Landau free-energy expansion:

$$\Delta F[M] = \frac{1}{2}(1+F_0^a)M^2 + \frac{1}{4}u_s M^4 + \cdots - H \cdot M$$

The ferromagnetic instability occurs when $F_0^a \to -1$. The static susceptibility and equation of state within this formalism coincide with standard FFLT results. Furthermore, the field-theoretic formulation allows one to:

• Systematically include fluctuations via a loop expansion (controlled by $G \sim 1/v_F$),
• Obtain explicit RG flow equations (e.g., $dg/d\ell = (d-1)g - A_d g^2 + \ldots$ with $g = N_0\gamma$),
• Identify and analyze the behavior near the ferromagnetic quantum critical point,
• Extend the theory to non-$s$-wave order parameters and complex soft-mode couplings.

This suggests that the soft-mode approach not only encapsulates traditional FFLT and its predictions for susceptibilities and Goldstone modes but also generalizes it to contexts requiring controlled RG analyses and the systematic treatment of fluctuations beyond mean field.

6. Relation to Paramagnetic Fermi Liquid Theory and Experimental Signatures

In the paramagnetic limit ($M_0 \to 0$), FFLT reduces exactly to the Leggett theory for the spin dynamics of normal $^3$He, with:

$$G \to G_P = v_F^2(1+F_1^a/3)(1+F_0^a),\quad A \to A_P = (4v_F(1+F_1^a/3)/\hbar)(F_0^a - F_1^a/3)$$

Here, no spontaneous magnetization emerges unless $F_0^a = -1$ (the Pomeranchuk–Stoner instability criterion) (Saslow et al., 2024, Belitz et al., 2011).

Experimental probes—such as inelastic neutron scattering and Brillouin light scattering—measure the spin-wave stiffness $D$, allowing extraction of combinations of $F_0^a$ and $F_1^a$. Measurements of dynamic susceptibilities and resonance linewidths provide access to the relaxation times $\tau_M, \tau_J$ and thus to dissipative transport parameters.

7. Broader Context, Generalizations, and Open Directions

While FFLT provides a comprehensive account of spin and spin-current dynamics in isotropic, itinerant ferromagnets with short-range interactions, its limitations must be recognized. Real materials stabilize ferromagnetism via mechanisms (long-range interactions, band structure, anisotropy) absent in the idealized FFLT. The theory forms the backbone for more advanced soft-mode analyses, RG studies, and the inclusion of multiple Landau parameters and symmetry channels. It continues to be an essential tool in exploring universal aspects of quantum phase transitions and collective phenomena in correlated electron systems (Saslow et al., 2024, Belitz et al., 2011, Mineev, 2011).