Quantum critical behavior in itinerant electron systems: Eliashberg theory and instability of a ferromagnetic quantum critical point

We consider the problem of fermions interacting with gapless long-wavelength collective bosonic modes. The theory describes, among other cases, a ferromagnetic quantum-critical point (QCP) and a QCP towards nematic ordering. We construct a controllable expansion at the QCP in two steps: we first create a non-Fermi-liquid “zero-order” Eliashberg-type theory, and then demonstrate that the residual interaction effects are small. We prove that this approach is justified under two conditions: the interaction should be smaller than the fermionic bandwidth, and either the band mass $m_B$ should be much smaller than $m=k_F∕v_F$, or the number of fermionic flavors $N$ should be large. For an SU(2) symmetric ferromagnetic QCP, we find that the Eliashberg theory itself includes a set of singular renormalizations which can be understood as a consequence of an effective long-range dynamic interaction between quasiparticles, generated by the Landau damping term. These singular renormalizations give rise to a negative nonanalytic $q^{3∕2}$ correction to the static spin susceptibility, and destroy a ferromagnetic QCP. We demonstrate that this effect can be understood in the framework of the ${\varphi }^4$ theory of quantum criticality. We also show that the nonanalytic $q^{3∕2}$ correction to the bosonic propagator is specific to the SU(2) symmetric case. For systems with a scalar order parameter, the $q^{3∕2}$ contributions from individual diagrams cancel out in the full expression of the susceptibility, and the QCP remains stable.

Figure 1

(a) Polarization bubble. (b) One-loop fermionic self-energy. (c) Two-loop fermionic self-energy.

Figure 2

Three-leg vertices: (a) zero external momentum and frequency; (b) finite momentum; (c) generic vertex.

Figure 3

(a) Cooper pairing vertex. (b) Four-leg vertex.

Figure 4

“${\varphi }^4$” and “${\varphi }^3$” type diagram.

Figure 5

Corrections to the polarization bubble from diagrams with one and two extra bosonic lines.

Figure 6

Integration contour: the hatched region stands for a branch cut, and the cross for a pole.

Figure 7

Three-loop contribution to the fermionic self-energy.

Figure 8

The ten two-loop diagrams for the charge susceptibility.