Generic deformation channels for critical Fermi surfaces including the impact of collisions

TL;DR

This work analyzes how a critical Fermi surface at a two-dimensional Ising-nematic quantum critical point deforms under low-energy excitations in a non-Fermi liquid. Using the quantum Boltzmann equation framework within a Keldysh formalism and patch theory, it derives a self-consistent, angular-momentum–resolved description of Fermi-surface deformations, incorporating boson collisions and Landau damping. The authors identify two classes of excitations: a long-lived zero-sound mode in the $\ell=0$ channel and an infinite set of higher-harmonic discrete modes in higher $\ell$ channels, plus a continuous particle-hole continuum; damping is present but the zero-sound mode remains sharply defined across regimes. Including bosonic dynamics does not alter the basic deformation solutions, providing a detailed picture of NFL collective modes with potential implications for transport and thermodynamics near quantum criticality.

Abstract

This paper constitutes a sequel to our theoretical efforts to determine the nature of generic low-energy deformations of the Fermi surface of a quantum-critical metal, which arises at the stable non-Fermi liquid (NFL) fixed point of a quantum phase transition. The emergent critical Fermi surface, arising right at the Ising-nematic quantum critical point (QCP), is a paradigmatic example where an NFL behaviour is induced by the strong interactions of the fermionic degrees of freedom with those of the bosonic order parameter. It is an artifact of the bosonic modes becoming massless at the QCP, thus undergoing Landau-damping at the level of one-loop self-energy. We resort to the well-tested formalism of the quantum Boltzmann equations (QBEs) for identifying the excitations. While in our earlier works, we have focused on the collisionless regime by neglecting the collision integral and assuming the bosons to be in equilibrium, here we embark on a full analysis. In particular, we take into account the bosonic part of the QBEs as well, which, however, turn out to have no effect on the solutions. Decomposing the master equation into angular-momentum ($\ell$) channels, the emergent modes are of two types: Fermi-surface deformations with discrete spectra and particle-hole excitations forming a continuous band. The long-lived zero-sound mode, which corresponds to $\ell = 0$, is found to be robust against damping effects. Intriguingly, we have an infinite family of discrete modes corresponding to higher-order harmonics of the net deformation.

Figures (5)

• Figure 1: Frequency dependence of the generalized Landau parameter in the $\ell =0$ angular-momentum component of $F(\theta,\Omega)$ [cf. Eq. \ref{['eqF0']}], after resolving it into the real and imaginary parts.
• Figure 2: Dispersion relation of the zero-sound mode, showing the behaviour of the real and imaginary parts, $\bar{\Omega}_r$ and $\bar{\Omega}_i$. In subfigures (a) and (b), we have compared the numerical solutions of Eqs. \ref{['eq-dis-damp']} and \ref{['imaginary part']} with the analytically-estimated power-law fittings, depending on the energy scales, as discussed in the text. Subfigure (c) shows that the ratio of $|\Omega_i|$ and $|\Omega_r|$ (obtained from the numerical solutions) falls off rapidly with increasing $|\bar{\mathbf{q}}|$.
• Figure 3: Contourplots of the magnitude of the lowest eigenvalue of $M$ [cf. Eq. \ref{['eqM']}], for some values of $\left ( \Omega_0, \bar{q} \right )$, in the complex plane of $\bar{\Omega}$. While subfigure (a) corresponds to the $F_0$-model of Sec. \ref{['seczero']}, subfigures (b)-(d) capture the results for the $F_\ell$ model of Sec. \ref{['seccode']}.
• Figure 4: Comparing the dispersion relation of the zero-sound mode, capturing the behaviour of both the real and imaginary parts ($\bar{\Omega}_r$ and $\bar{\Omega}_i$), calculated from the $F_0$ and $F_\ell$ models. While the left panel shows the extreme small-$q$ regime, the right one corresponds to somewhat larger-$q$ regime.
• Figure 5: Schematic plot of the deformations of the Fermi surface arising from the collective-mode solutions of the three tiny islands visible in Fig. \ref{['figevalue']}(c), corresponding to $\bar{q} = 0.01$. The $\bar{\Omega}$-values of the centres of these islands are shown in the plot-legends. The blue circle represents the undeformed Fermi surface.