Transport near the Ising-nematic quantum critical point of metals in two dimensions

TL;DR

The paper investigates dc transport in two-dimensional metals near an Ising-nematic quantum critical point, where strong coupling renders conventional quasiparticle pictures insufficient. Using the memory-matrix formalism, it decomposes momentum relaxation into disorder-induced channels (random-field, forward, large-angle, and $2k_F$ backscattering) and computes their distinct temperature dependences, revealing a dominant random-field contribution that diverges as $T\to0$. A self-consistent analysis of nematic fluctuations at $T>0$ shows the boson mass scales as $m^2(T)\sim T\ln(1/T)$ near the QCP, with broader crossovers to $z=1$ dynamics at higher temperatures. Together, these results establish a framework for non-Fermi-liquid transport in strongly interacting metals and predict near-linear or divergent $T$-dependent resistivity regimes controlled by impurity structure and nematic fluctuations. The work also highlights connections to holographic approaches and underscores the need to treat momentum relaxation and order-parameter dynamics on equal footing in quantum critical metals.

Abstract

We consider two-dimensional metals near a Pomeranchuk instability which breaks 90$^\circ$ lattice rotation symmetry. Such metals realize strongly-coupled non-Fermi liquids with critical fluctuations of an Ising-nematic order. At low temperatures, impurity scattering provides the dominant source of momentum relaxation, and hence a non-zero electrical resistivity. We use the memory matrix method to compute the resistivity of this non-Fermi liquid to second order in the impurity potential, without assuming the existence of quasiparticles. Impurity scattering in the $d$-wave channel acts as a random "field" on the Ising-nematic order. We find contributions to the resistivity with a nearly linear temperature dependence, along with more singular terms; the most singular is the random-field contribution which diverges in the limit of zero temperature.

Figures (9)

• Figure 1: Schematic of the resistivity, $\rho (T)$, due to scattering off a random field $h_0$. The computations are perturbative in $h_0$, and break down at small enough $T$.
• Figure 2: Feynman graphs for the polarizability $\Pi_0$ and the $\Gamma_i$. The full lines are fermions, and the dashed lines are $\phi$ propagators.
• Figure 3: Corrections to the $\phi$ self energy at order $1/N_f$.
• Figure 4: Values of $m^2 (T)$ obtained by solving Eq. (\ref{['meqn']}) for $A=B=U=E=1$ and $N_f = 2$. The values of $\widetilde{s}$ range from $\widetilde{s}=-0.15$ (bottom) to $\widetilde{s}=0.075$ (top) in steps of 0.025. The quantum-critical value, $\widetilde{s}=0$, is the black line. For $\widetilde{s} <0$, the values of $m(T)$ become exponentially small at low $T$: this is an artifact of the approximations made in obtaining Eq. (\ref{['meqn']}). The proper solution has $m(T)$ vanish at a non-zero temperature $T=T_I (s)$ sketched in Fig. \ref{['fig:phasediag']}, corresponding to an Ising phase transition below which there is long-range Ising-nematic order.
• Figure 5: Phase diagram in the vicinity of the Ising-nematic quantum critical point at $s=s_c$ and $T=0$, deduced from Eq. (\ref{['meqn']}). The full line at $T=T_I$ is an Ising phase transition to a metal with long-range Ising-nematic order, $\langle \phi \rangle \neq 0$. The dashed lines are crossovers. The boundary of the Fermi liquid scales as $T \sim \widetilde{s}^{3/2}$. The boundaries of the quantum-critical region, and $T_I$, scale as $T \sim |\widetilde{s}|/\ln (1/|\widetilde{s}|)$. In the Fermi liquid region the leading temperature dependence of $m^2 (T)$ scales as $T^2$. In the quantum critical region, and in the intermediate region between the two dashed lines, the leading temperature dependence of $m^2 (T)$ scales linearly with $T$ up to logarithms, and this influences the $T$ dependence of all observables.