Hertz–Millis Theory and Landau Damping

Hertz–Millis theory integrates out metallic fermions to obtain a Landau-damped action for an order parameter. The construction predicts different dynamical exponents for clean ferromagnetic and generic antiferromagnetic transitions and often places the bosonic theory above its upper critical dimension. Its central assumption—that the remaining order-parameter functional is regular apart from the retained damping—is also its main point of failure.

Required background. Landau–Ginzburg–Wilson Quantum Criticality supplies order-parameter power counting; Polarization, Lindhard Functions, and the Particle–Hole Continuum supplies the nonanalytic fermion bubble. Helpful background. Itinerant Magnetism and Spin-Fluctuation Physics supplies the Stoner and paramagnon interpretation.

Integrating out the Fermi surface

Start from fermions coupled to an order parameter,

$$S=\int_k\psi_k^\dagger(-i\omega+\varepsilon_{\mathbf k})\psi_k +\frac12\int_q(r+cq^2)\phi_q\phi_{-q} +g\int_{k,q}\phi_q\psi_{k+q}^\dagger M\psi_k+\cdots .$$

Formally integrating the fermions gives

$$S_{\mathrm{eff}}[\phi]=S_\phi-\operatorname{Tr}\ln(G_0^{-1}+g\phi M).$$

The quadratic term contains the polarization $\Pi(\mathbf q,i\Omega_n)$. For a clean ferromagnetic channel at small $\mathbf q$,

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

valid for $|\Omega_n|\ll v_Fq$. Balancing terms gives $\Omega\sim q^3$, hence $z=3$. For a generic antiferromagnetic ordering vector $\mathbf Q$ that joins isolated hot spots,

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

giving $z=2$. Nesting, van Hove points, disorder, and extended hot regions change these kernels.

Hertz introduced this fermion-to-boson reduction Hertz 1976, §§ III–V; Millis developed its finite-temperature consequences Millis 1993, §§ II–IV.

Power counting and thermal mass

For a local quartic coupling, $[u]=4-(d+z)$. Thus a two-dimensional antiferromagnet has $d+z=4$ and marginal bosonic interactions, while a clean two-dimensional ferromagnet has $d+z=5$ and Gaussian leading exponents. Above the upper critical dimension, $u$ is RG-irrelevant but cannot be set to zero in the ordered phase.

At critical tuning it also generates a temperature-dependent mass. In the simplest Hertz–Millis regime,

$$\delta r(T)\sim u\,T^{(d+z-2)/z}$$

up to logarithms at marginal dimensions. The thermal phase boundary then scales with shift exponent

$$\psi=\frac{z}{d+z-2},$$

which need not equal $\nu z$. This is a direct example of a dangerously irrelevant coupling splitting the zero-temperature gap scale from the thermal ordering line.

Why integrating out gapless fermions can fail

The determinant generates every allowed bosonic vertex. Because the fermions are gapless on an extended Fermi surface, those vertices can be nonlocal and singular rather than analytic constants. In a clean itinerant ferromagnet, soft particle–hole modes generate nonanalytic momentum dependence and can drive the transition first order or toward modulated order; a purely local $\phi^4$ action misses that feedback Belitz, Kirkpatrick, and Vojta 2005, §§ III–IV.

At an antiferromagnetic transition, the hot fermions themselves lose quasiparticle coherence and feed back on the boson. Pairing and composite orders can become strongly enhanced. The bosonic quadratic damping remains useful, but it does not establish that all higher vertices are harmless or that a fermion-free theory controls the infrared.

The reduction is most credible when:

• the polarization is computed for the actual Fermi-surface geometry;
• generated vertices remain regular under the chosen scaling;
• fermion self-energy and vertex corrections are parametrically small;
• disorder and momentum relaxation are declared;
• no superconducting or first-order scale preempts the regime.

Retarded response and interpretation

Analytic continuation of $|\Omega_n|$ gives a retarded dissipative term with the sign required by positive spectral weight. For the antiferromagnet it is proportional to $-i\gamma\omega$ at low positive frequency; for the ferromagnet, $-i\gamma\omega/(v_Fq)$. One must continue from a spectral representation, not treat the absolute value as an analytic function.

A fitted $z=2$ or $z=3$ response supports a damping kinematics. It does not uniquely validate a local Hertz action or identify the ultimate fixed point. Momentum-resolved fermion spectra and higher-order bosonic observables test the omitted feedback.

Exercises

1. Find the effective dimension and quartic power counting for a $d=3$ antiferromagnetic metal.

Solution

$z=2$, so $d+z=5$. The quartic coupling has $[u]=4-5=-1$ and is irrelevant at the Gaussian fixed point, although it remains dangerous for the ordered phase and thermal mass.

2. Obtain the shift exponent for a $d=2$, $z=3$ ferromagnetic Hertz theory.

Solution

$\psi=z/(d+z-2)=3/(2+3-2)=1$. Thus the simplest theory predicts a thermal ordering scale linear in the tuning distance, subject to the known nonanalytic and first-order instabilities.

References

• Belitz, D., T. R. Kirkpatrick, and T. Vojta. “How Generic Scale Invariance Influences Quantum and Classical Phase Transitions.” Reviews of Modern Physics 77 (2005): 579–632. DOI.
• Hertz, J. A. “Quantum Critical Phenomena.” Physical Review B 14 (1976): 1165–1184. DOI.
• Millis, A. J. “Effect of a Nonzero Temperature on Quantum Critical Points in Itinerant Fermion Systems.” Physical Review B 48 (1993): 7183–7196. DOI.