DC resistivity at the onset of spin density wave order in two-dimensional metals

TL;DR

This work analyzes DC transport at a spin-density-wave quantum critical point in two-dimensional metals, where strong interactions couple to the entire Fermi surface. It adopts a momentum-relaxation framework in which fast momentum-conserving scattering sets local equilibration and DC resistivity is controlled by weak perturbations that relax momentum, analyzed via the memory-matrix formalism. The central finding is the separation of disorder effects into two channels: long-wavelength random-mass disorder yields a linear-in-$T$ resistivity with logarithmic corrections in the Hertz $d=2,z=2$ limit (with a crossover to a $z=1$ regime exhibiting a higher power of $T$), while short-wavelength disorder produces a residual resistivity and a separate linear-$T$ term from vertex corrections. The results illuminate how bosonic critical fluctuations plus disorder determine transport near SDW order and offer experimental and theoretical avenues for extending to other 2D density-wave transitions; the framework is applicable to iron-based superconductors near SDW criticality and suggests careful control of long-wavelength disorder to modulate the linear-$T$ resistivity.

Abstract

The theory for the onset of spin density wave order in a metal in two dimensions flows to strong coupling, with strong interactions not only at the `hot spots', but on the entire Fermi surface. We advocate the computation of DC transport in a regime where there is rapid relaxation to local equilibrium around the Fermi surface by processes which conserve total momentum. The DC resistivity is then controlled by weaker perturbations which do not conserve momentum. We consider variations in the local position of the quantum critical point, induced by long-wavelength disorder, and find a contribution to the resistivity which is linear in temperature (up to logarithmic corrections) at low temperature. Scattering of fermions between hot spots, by short-wavelength disorder, leads to a residual resistivity and a correction which is linear in temperature.

Figures (5)

• Figure 1: (a) The two pockets of fermions separated by the SDW ordering wavevector ${\bm K}=(\pi,\pi)$. (b) The resulting pair of Fermi surfaces after shifting the pocket centered at $(\pi,\pi)$ to $(0,0)$ intersect at 4 hot spots as shown.
• Figure 2: Resummation of graphs to obtain the Green's function for $\phi_\mu\phi_\mu$. The diamonds denote $\phi_\mu\phi_\mu$ operators and the circles denote the quartic interaction. The wavy lines represent the vector boson propagators.
• Figure 3: Temperature driven crossover in the scaling of the random-mass contribution to $\rho_{xx}(T)$ from $T$ to $T^4$ as $T$ is increased. Here, $\gamma=1$, $\epsilon=1$ and the momentum cutoff $\Lambda=100$.
• Figure 4: Graphs for the contribution to $G_{\dot{P}_x,\dot{P}_x}(i\Omega)$ due to inter hot spot scattering. The vertices provide factors of $Q^{ij}V_0+Q^{ij}V_0\Gamma_0$. The solid lines are fermion propagators and the wavy lines are vector boson propagators. The dotted lines carry internal momentum and the external bosonic Matsubara frequency $i\Omega$, and have propagators equal to $1$. The first graph in the series of graphs in $(a)$ is the free fermion contribution. The subsequent graphs represent the corrections due to renormalization of the fermion propagators at one loop, but evaluate to $0$ due to factors of $\int d\xi/(i\omega-\xi)^m=0$, $m\in\mathbb{Z}$ and $m\geq2$. The graph in $(b)$ is the simplest vertex correction. Here too, for the same reason, further graphs of the same type but with self-energy rainbows on the fermion propagators also evaluate to $0$.
• Figure 5: Numerical solution (solid) of Eq. \ref{['eq:rt2']}, and $\gamma T+\epsilon T^2$ (dashed), for $\epsilon=1$ and $\gamma=1$.