Can electronic quantum criticality drive phonon-induced linear-in-temperature resistivity?

Abstract

Optical phonons naturally generate linear-in-$T$ resistivity in the high-temperature equipartition regime, but their finite gap prevents this mechanism from surviving to asymptotically low temperatures. Here we analyze whether proximity to an electronic quantum critical point can remove this obstruction by strongly softening an optical phonon. We first derive a model-independent criterion for such softened phonons to control low-temperature transport: besides reducing the renormalized optical gap, the Landau-damped phonon must acquire a dynamical exponent $z_p>d$, where $d$ is the spatial dimension of the phonon, so that a sufficiently large thermally occupied phase space survives as $T\to 0$. We then analyze a concrete mechanism in which the phonon couples nonlinearly to long-wavelength electronic collective modes near a $\vec{Q}=0$ quantum critical point, and apply it to the Ising-nematic case. Within a large-$N$ field theoretic formulation, the phonon softening is enhanced near criticality, but in the clean theory the resulting dynamics lies at or near the marginal boundary for asymptotic $T$-linear scattering. Including feedback from the softened phonon back onto the electronic critical sector further weakens the tendency toward robust low-temperature $T$-linear transport. Our results sharpen both the promise and the limitations of phonon-based explanations of strange-metal transport near electronic criticality.

Figures (5)

• Figure 1: Schematic consequences of phonon softening near an electronic quantum critical point. (a) As the system is tuned toward criticality ($r_1\rightarrow r_2\rightarrow r_3$), an optical phonon mode softens and its long-wavelength dispersion is renormalized, increasing the density of low-energy phonon states near the $\Gamma-$point, as shown in panel (b). (c) For a renormalized phonon dispersion $\omega_{{\boldsymbol{q}}}\sim q^{z_p}$, the largest thermally occupied momentum scales as $q(T)\sim T^{1/z_p}$, so the phase-space volume of thermally excited phonons scales as $[q(T)]^d\sim T^{d/z_p}$, where $d$ is the spatial dimension and $z_p$ is the phonon dynamical exponent. A larger $z_p$ therefore produces a larger low-energy phonon phase space. The solid and dashed curves illustrate the qualitatively different cases $z_p<d$ and $z_p>d$, where the characteristic thermal momentum scales are labelled by $q_1(T)$ and $q_2(T)$, respectively. In the low-temperature limit, $q_2(T)\gg q_1(T)$ in the scaling sense, implying that there is more phonon phase-space participating in scattering in the $z_p>d$ case, compared to the $z_p<d$ case.
• Figure 2: Crossover structure of the electron-phonon scattering rate $\Gamma_\text{ep}$ as a function of temperature $T$ and renormalized optical-phonon gap $\omega_D$. The qualitative behavior depends on whether the effective phonon dynamical exponent $z_p$ is smaller or larger than the critical value $z_c=d$, where $d$ is the spatial dimensionality of the phonon dispersion ($d=2$ or $3$ in the three scenarios discussed in the text). (a) For $1\le z_p\le z_c$, the scattering rate exhibits four distinct regimes, labeled (A,B,C,D). (b) For $z_p>z_c$, the regimes (B,C,D) merge into a single broadened equipartition-like regime (E), in which $\Gamma_\text{ep}$ becomes linear in $T$. The crossover scales $\omega_*$ and $\omega_\Lambda$ are defined in Eqs. \ref{['eq:omega_Lambda']} and \ref{['eq:omega_star']}, respectively.
• Figure 3: Momentum configuration for phonon-mediated umklapp scattering. The two circles denote the same FS repeated in adjacent Brillouin zones, separated by a reciprocal lattice vector ${\boldsymbol{b}}$. A fermion with momentum ${\boldsymbol{k}}$ on the right FS is equivalently described by $\tilde{{\boldsymbol{k}}}={\boldsymbol{k}}+{\boldsymbol{b}}$ on the neighboring FS, and can be scattered to ${\boldsymbol{k}}'=\tilde{{\boldsymbol{k}}}+{\boldsymbol{q}}$ by exchanging a phonon of momentum ${\boldsymbol{q}}$. The minimal phonon momentum needed to connect the two FSs is $\Delta_q$. The FS is drawn to be circular for simplicity.
• Figure 4: Renormalized optical-phonon dispersion near the $\Gamma$ point for $d_\text{el}=2$. The blue solid curve shows the tuned critical case $m_\varphi=0$ and $u=u_\text{max}$, where the phonon is fully softened and $\omega_{{\boldsymbol{q}}}^2\propto {\left|{\boldsymbol{q_\text{2D}}}\right|}$, consistent with the power counting. The other solid curves illustrate the effect of reducing $u$ below $u_\text{max}$ at criticality, which restores a finite optical gap. The dashed curves show the effect of moving away from the electronic quantum critical point at fixed coupling, producing a crossover from quadratic dispersion at the smallest momenta to the critical form at larger ${\left|{\boldsymbol{q_\text{2D}}}\right|}$.
• Figure 5: Momentum-space structure of $I({\boldsymbol{q}})$ in the different crossover regimes. The label $T$ denotes $I({\boldsymbol{q}})\sim T/\omega_{\boldsymbol{q}}^2$, while $T^2$ denotes $I({\boldsymbol{q}})\sim \gamma T^2/(\eta_{\boldsymbol{q}}\omega_{\boldsymbol{q}}^4)$. Panels (A1) and (A2) correspond to the Fermi-liquid regime with $\omega_D\gtrless \omega_*$, panels (B1) and (B2) to the generalized Bloch-Grüneisen regime with $\omega_D\gtrless \omega_*$, and panel (C) to the electron-phonon critical regime.