Quantum phase transitions of metals in two spatial dimensions: II. Spin density wave order

TL;DR

This work reexamines the quantum critical SDW transition in two-dimensional metals using the Abanov-Chubukov spin-fermion model and a field-theoretic $1/N$ RG framework. It demonstrates that Hertz-Millis theory fails in $d=2$ due to nonlocal fermion-induced interactions that renormalize the boson sector, yielding a dynamical exponent $z$ away from 2 and non-Fermi-liquid behavior at hot spots. The analysis reveals breakdown of naive large-$N$ counting from infrared singularities and UV-enhanced diagrams, including a log-squared enhancement of both pairing and density/bond-order vertices, indicating strong competition with superconductivity and Ising-nematic/VBS tendencies. The results emphasize intricate coupling between critical bosons and a Fermi surface in 2D and motivate exploration of alternative RG schemes for a controlled description. Overall, the paper advances understanding of quantum criticality in metals by uncovering novel scaling, topology-based diagram enhancements, and intertwined instabilities near SDW criticality.

Abstract

We present a field-theoretic renormalization group analysis of Abanov and Chubukov's model of the spin density wave transition in two dimensional metals. We identify the independent field scale and coupling constant renormalizations in a local field theory, and argue that the damping constant of spin density wave fluctuations tracks the renormalization of the local couplings. The divergences at two-loop order overdetermine the renormalization constants, and are shown to be consistent with our renormalization scheme. We describe the physical consequences of our renormalization group equations, including the breakdown of Fermi liquid behavior near the "hot spots" on the Fermi surface. In particular, we find that the dynamical critical exponent z receives corrections to its mean-field value z = 2. At higher orders in the loop expansion, we find infrared singularities similar to those found by S.-S. Lee for the problem of a Fermi surface coupled to a gauge field. A treatment of these singularities implies that an expansion in 1/N, (where N is the number of fermion flavors) fails for the present problem. We also discuss the renormalization of the pairing vertex, and find an enhancement which scales as logarithm-squared of the energy scale. A similar enhancement is also found for a modulated bond order which is locally an Ising-nematic order.

Figures (23)

• Figure 1: Square lattice Brillouin zone showing the Fermi surface appropriate to the cuprates. The filled circles are the hot spots connected by the SDW wavevector $\vec{Q} = (\pi,\pi)$. The locations of the continuum fermion fields $\psi_1^\ell$ and $\psi_2^\ell$ is indicated.
• Figure 2: Configuration of the $\ell=1$ pair of hot spots, with the momenta of the fermion fields measured from the common hot spot at $\vec{k}=0$, indicated by the filled circle. The Fermi velocities $\vec{v}_{1,2}$ of the $\psi_{1,2}$ fermions are indicated.
• Figure 3: Modification of the Fermi surfaces in Fig. \ref{['fig:fermions']} by SDW order with $\langle \phi \rangle \neq 0$. The full lines are the Fermi surfaces, and the white, light shaded, and dark shaded regions denote momenta where 0, 1, and 2 of the bands are occupied. The upper and lower lines are boundaries of hole and electron pockets respectively.
• Figure 4: The boson self-energy at $N = \infty$. The full lines represent the $\psi_{1,2}$ fermions, and the dashed lines represent the boson $\phi^a$.
• Figure 5: The leading contribution to the fermion self-energy.