Momentum-gapped quasiparticles in disordered metals

Abstract

Nature contains massless particles with linear dispersions, and massive particles whose energies depend quadratically on their momenta with finite mass gaps. Both have equivalents in condensed matter physics in the form of collective modes and quasiparticles, measurable excitations with well-defined energy-momentum relations. A hypothesised third particle type - the super-luminal tachyon - would have an undefined energy at low momentum. A similar collective mode - long hypothesised within the hydrodynamic theory of matter - would have a purely imaginary energy at low momentum, corresponding to a finite lifetime. This third possibility has never been directly observed in a quantum system. Through a careful comparison of hydrodynamics with microscopic models of metals, we establish that this previously unseen third dispersion occurs in correlated quantum matter whenever the electronic fluid undergoes momentum relaxation due to explicit breaking of translation by impurities. As a specific example of these momentum-relaxed modes we consider the recent discovery of an acoustic plasmon - dubbed Pines' demon - in Sr$_2$RuO$_4$. The observed dispersion of this neutral mode differed significantly from the massless linear behaviour predicted by the random phase approximation. We demonstrate that the observed dispersion corresponds, in fact, to a momentum-gapped quasiparticle.

Figures (8)

• Figure 1: Possible dispersion relations in Nature. Particles, quasiparticles and collective modes with finite energy at zero momentum $q$ feature a dispersion (dark blue) of the form $\omega^2\sim q^2+\Delta^2$ with an energy gap $\Delta$ corresponding to their mass or energy at zero momentum, while the dispersion of massless particles, quasiparticles and collective modes $\omega \sim q$ is linear and intercepts the origin (light blue), having zero energy at zero momentum. Massless phonons can also have an imaginary part at finite $q$. The yellow dispersion $\omega^2\sim q^2-\Delta^2$, describing hypothetical superluminal elementary particles (tachyons), is purely imaginary below a critical momentum $q_c$, with an imaginary positive part ranging from $\Delta$ (the imaginary mass term) to zero, and becomes purely real for $q>q_c$. The momentum--gapped dispersion is purely imaginary for $q<q_c$, with negative values ranging from $0$ and $-\Gamma$ at $q=0$ depending on the branch of the mode, to $-\Gamma/2$. Here $\Gamma$ is the momentum relaxation rate. The momentum gapped dispersion maintains a finite imaginary part for $q>q_c$, lying in a different plane (red) than the rest of the dispersions (blue). Solid (dotted) lines indicate purely real (imaginary) dispersions; dot-dashed indicate complex dispersions.
• Figure 2: Momentum-gapped dispersion of Pines' demon in Sr$_2$RuO$_4$. The theoretical predictions based on the hydrodynamic and microscopic theories for the acoustic plasmon dispersion (solid red) are indistinguishable by eye, and are compared to the experimental m-EELS measurements (black and white diamonds) presented in Ref. husain_pines_2023. Note that the lowest-$q$ data are only error bars, matching our prediction that the demon is purely dissipative at low momentum. We show the loss function $S(q,\omega)$ as a heatmap to aid the visualisation of the acoustic mode as a brighter region in the $(q,\omega)$ plane. We constrain the one free parameter in our model, the momentum relaxation rate $\Gamma$, fitting it to optical conductivity measurements stricker_optical_2014. The lower panels in both figures present the imaginary component of the dispersion with $q$, showing the evolution of the positive branch in Eq. \ref{['eqn:hydro_dispersion']} from $\mathrm{Im}~\omega=0$ for $q=0$ to a finite value $\mathrm{Im}~\omega=-\Gamma/2$ (dashed gray lines) for $q=q_c$ in each case.
• Figure 3: Fit of the parabolic bands to the relevant bands of the tight-binding model of Sr$_2$RuO$_4$ in Ref. zabolotnyy_renormalized_2013. The Fermi surface of the tight-binding model is indicated in solid colors for the $\gamma$ (red) and $\beta$ (blue) bands. The Fermi surface of the parabolic bands is indicated by dashed lines. The high-symmetry directions $\Gamma X$ and $\Gamma M$ of the tight-binding model are indicated, where $\Gamma=(0,0)$, $M=(\pi,\pi)$, and $X=(\pi,0)$.
• Figure 4: Fermi velocities of the $\beta$ (red) and $\gamma$ (blue) bands of Sr$_2$RuO$_4$ as a function of temperature. The circles correspond to the data reported in Ref. hunter_fate_2023, and the solid lines to the corresponding least squares linear fit.
• Figure 5: Estimation of $\Gamma$ from the Drude peak of optical conductivity measurements in Ref. stricker_optical_2014 (light red circles) with $\omega_{max}=0.15$ eV using Eq. \ref{['eq:Drude_fit']} with their respective standard errors. The linear fit (blue) is used to estimate the value of $\Gamma$ at $T=300$ K (dark red circle) $\Gamma(300{\rm~K})=51\pm 2$ meV.