Microscopic origin of the nemato-elastic coupling and dynamics of hybridized collective nematic-phonon excitations

Abstract

Electronically-driven nematic order breaks the rotational symmetry of a system, e.g., through a Pomeranchuk instability of the Fermi surface, with a concomitant distortion of the lattice. As a result, in a metal, the nematic collective mode interacts with two different sets of gapless excitations: the particle-hole excitations of the metal and the lattice fluctuations that become soft at the induced structural transition, namely, the transverse acoustic phonons. However, the \textit{dynamics} of these hybridized collective modes formed by the transverse acoustic phonons and the metallic electronic-nematic fluctuations has remained largely unexplored. Here we address this problem by developing a formalism in which the nemato-elastic coupling is obtained microscopically from the direct coupling between electrons and transverse acoustic phonons enabled by impurities present in the crystal. We then demonstrate the emergence of hybrid nemato-elastic modes that mix the characteristics of the transverse phonons and of the nematic fluctuations. Near the nematic quantum critical point in a metal, two massless modes emerge with intertwined dynamic behaviors, implying that neither mode dominates the response of the system. We systematically study the non-trivial dependence of these collective modes on the longitudinal and transverse momenta, revealing a rich landscape of underdamped and overdamped modes as the proximity to the quantum critical point and the strength of the electron-phonon coupling are changed. Since dynamics play an important role for determining superconducting instabilities, our results have implications for the study of pairing mediated by electronic nematic fluctuations.

Figures (11)

• Figure 1: Schematic electronic-nemato-elastic system. The nematic (i.e., quadrupolar) fluctuations of the Fermi surface allows the formation of underdamped particle-hole excitations with momentum slightly off the diagonal directions as the QCP is approached Oganesyan01. Further away from the diagonal directions, fluctuations become overdamped and rapidly decay into the particle-hole continuum. The transverse phonons soften precisely along the diagonal directions and the directional dependence of the nematic fluctuations implies that the two degrees of freedom strongly hybridize. As a result of this hybridization, the nematic mode becomes damped even at the QCP, while a new mixed massless coherent mode emerges.
• Figure 2: Nemato-elastic modes without low-energy electronic excitations. (a) Uncoupled nematic and acoustic phononic dispersions near an electronic nematic critical point located at $r=0$. The mass of the nematic mode reaches zero at the critical point, while the phonons are unaffected. This is illustrated schematically on the right; the Fermi surface distorts at the nematic transition while the lattice remains tetragonal. (b) Coupled nematic and phononic degrees of freedom near the electronic nematic transition. In this case, the nematic fluctuations do not go soft near the transition. Instead, the velocity of the phonon vanishes, as is expected at a structural transition, and the phonon dispersion becomes quadratic [see inset in (b)]. In this case, the Fermi surface distortion is accompanied by a tetragonal-to-orthorhombic lattice distortion.
• Figure 3: Evolution of the nematic modes uncoupled from the lattice. Nematic dispersion for $v_n/v_F=0.60$ and $\lambda_{\rm nem}/v_F = 0.25$ as the nematic QCP is approached. (a)--(c) Angular-dependence and (d)--(f) $q$-dependence of the dispersion at $\theta_{\mathbf{q}}=\tfrac{\pi}{4}$ for different values of $x$ approaching the QCP. The real (a) and imaginary (b) parts of the angular dependence have been normalized by $\omega_{0,\mathrm{re}}$ and $\omega_{0,\mathrm{im}}$, respectively, where $\omega_{0,\mathrm{re}}$ is the value of the dispersion at $\theta_{\mathbf{q}}=\tfrac{\pi}{4}$ and $q=q_0$ [see Eq. \ref{['eq:omega_sol_3']}] and $\omega_{0,\mathrm{im}}$ is the imaginary part of the dispersion at the angle where the real part vanishes and the imaginary part peaks. The angular dependence of the ratio between the real and the (absolute value of the) imaginary part is shown in (c), plotted on a log-scale. Note that the width of the region where the real part is finite scales with $(v_n^2 q^2 + x)^{1/4}$, and thus narrows with $\sqrt{q}$ at the QCP. The $q$-dependence for fixed $\theta_{\mathbf{q}}=\tfrac{\pi}{4}$ in (d)--(f) is shown on a log-log scale to highlight the different regimes discussed in the text. For finite $x$, the dependence is linear in small $q$, while at the QCP the real part becomes quadratic and the imaginary part becomes cubic in $q$. Consequently, the ratio, shown in (f), tends to a constant for finite $x$ but diverges as $q\rightarrow 0$ at the QCP.
• Figure 4: Imaginary part of the renormalized nematic propagator. (a)--(c) depicts the nematic spectral function, $\Im \chi(s)$, for $x=0$ while (d)--(f) corresponds to the case $x=10^{-4}$ as a function of $s= \omega / q_0$ with $q_0 = 10^{-4}$ for $\lambda_{\rm ph}/v_s=0$ [(a) and (d)], $\lambda_{\rm ph}/v_s=0.01$ [(b) and (e)], and $\lambda_{\rm ph}/v_s=0.04$ [(c) and (f)]. As $\lambda_{\rm ph}/v_s$ becomes finite, the single nematic mode splits into two which are separated by a near-complete suppression of spectral weight.
• Figure 5: Imaginary part of the renormalized phonon propagator. (a)--(c) depicts the phonon spectral function, $\Im \mathcal{D}(s)$, for $x=0$ while (d)--(f) corresponds to the case $x=10^{-4}$ as a function of $s= \omega / q_0$ with $q_0 = 10^{-4}$ for $\lambda_{\rm ph}/v_s=0$ [(a) and (d)], $\lambda_{\rm ph}/v_s=0.01$ [(b) and (e)], and $\lambda_{\rm ph}/v_s=0.04$ [(c) and (f)]. As $\lambda_{\rm ph}/v_s$ becomes finite, the phonon mode splits into two different modes, which move away from each other as $\lambda_{\rm ph}/v_s$ is increased further.