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arXiv:2304.04720 · Annals of Physics

Generic deformation channels for critical Fermi surfaces in the collisionless regime

Kazi Ranjibul Islam, Ipsita Mandal

Published Jul 6, 2023• 5 citations
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Abstract

Using a quantum Boltzmann equation framework, we analyze the nature of generic low-energy deformations of a critical surface, which exists at the non-Fermi liquid fixed point of a system consisting of fermions interacting with massless bosons. The non-Fermi liquid behaviour arises due to the itinerant quasiparticles of the Fermi surface interacting strongly with the massless bosons, which on the other hand undergo Landau damping as result of the mutual interactions. Focussing on the collisionless regime, where we neglect the collision integral, we chalk out the possible excitations spanning the entire spectrum of angular momentum (`) channels (i.e., including both small and large values of `). The excitations are of two types: particle-hole like localized excitations forming an energy band (or continuum) and delocalized collective modes with discrete energies. Although we find a collective mode analogous to the zero sound of a Fermi liquid, its dispersion shows a crossover from a Ω ∼ |q|6 / 5 behaviour to the usual Ω ∼ |q| dependence, where Ω and q represent the frequency and momentum, respectively. We estimate the frequency scale at which this crossover takes place. We also determine the boundary for the particle-hole continuum in the Ω–q plane, and observe a crossover from Ω ∼ |q|3 / 2 to Ω ∼ |q| behaviour, determined by another crossover frequency scale.

DOI
10.1016/j.aop.2023.169409

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Using a quantum Boltzmann equation framework, we analyze the nature of generic low-energy deformations of a critical surface, which exists at the non-Fermi liquid fixed point of a system consisting of fermions interacting with massless bosons. The non-Fermi liquid behaviour arises due to the itinerant quasiparticles of the Fermi surface interacting strongly with the massless bosons, which on the other hand undergo Landau damping as result of the mutual interactions. Focussing on the collisionless regime, where we neglect the collision integral, we chalk out the possible excitations spanning the entire spectrum of angular momentum () channels (i.e., including both small and large values of ). The excitations are of two types: particle-hole like localized excitations forming an energy band (or continuum) and delocalized collective modes with discrete energies. Although we find a collective mode analogous to the zero sound of a Fermi liquid, its dispersion shows a crossover from a Ω ∼ |q|6 / 5 behaviour to the usual Ω ∼ |q| dependence, where Ω and q represent the frequency and momentum, respectively. We estimate the frequency scale at which this crossover takes place. We also determine the boundary for the particle-hole continuum in the Ω–q plane, and observe a crossover from Ω ∼ |q|3 / 2 to Ω ∼ |q| behaviour, determined by another crossover frequency scale.
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 (ℓ\ellℓ) 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 ℓ=0\ell = 0ℓ=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.
We derive the quantum Boltzmann equation (QBE) by using generalized Landau-interaction parameters, obtained through the nonequilibrium Green's function technique. This is a generalization of the usual QBE formalism to non-Fermi liquid (NFL) systems, which do not have well-defined quasiparticles. We apply this framework to a controlled low-energy effective field theory for the Ising-nematic quantum critical point, in order to find the collective excitations of the critical Fermi surface in the collisionless regime. We also compute the nature of the dispersion after the addition of weak Coulomb interactions. The zero angular momentum longitudinal vibrations of the Fermi surface show a linear-in-wavenumber dispersion, which corresponds to the zero sound of Landau's Fermi liquid theory. The Coulomb interaction modifies it to a plasmon mode in the long-wavelength limit, which disperses as the square-root of the wavenumber. Remarkably, our results show that the zero sound and plasmon modes exhibit the same behaviour as in a Fermi liquid, although an NFL is fundamentally different from the former.
Semiholographic models of non-Fermi liquids have been shown to have generically stable generalized quasiparticles on the Fermi surface. Although these excitations are broad and exhibit particle-hole asymmetry, they were argued to be stable from interactions at the Fermi surface. In this work, we use this observation to compute the density response and collective behaviour in these systems. Compared to the Fermi liquid case, we find that the boundaries of the particle-hole continuum are blurred by incoherent contributions. However, there is a region inside this continuum, that we call inner core, within which salient features of the Fermi liquid case are preserved. A particularly striking prediction of our work is that these systems support a plasmonic collective excitation which is well-defined at large momenta, has an approximately linear dispersion relation and is located in the low-energy tail of the particle-hole continuum. Furthermore, the dynamic screening potential shows deep attractive regions as a function of the distance at higher frequencies which might lead to long-lived pair formation depending on the behavior of the pair susceptibility. We also find that Friedel oscillations are present in these systems but are highly suppressed.
We study the appearance of a sharp collective mode which features transverse current fluctuations within the bosonization approach to interacting two-dimensional Fermi liquids. This mode is analogous to the shear sound modes in elastic media, and, unlike the conventional zero sound mode, it is damped in weakly interacting Fermi liquids and only separates away from the particle-hole continuum when the quasiparticle mass becomes twice the transport mass ${m}^{*}\ensuremath{\gtrsim}2m$. The shear sound should be present in a large class of interacting charged and neutral Fermi liquids especially those proximate to critical points where the quasiparticle mass diverges. In metals this mode remains linearly dispersing in the presence of the long-ranged Coulomb force, unlike the conventional zero sound mode which becomes the plasma mode. We also detail a quick path between bosonization and classical Landau's Fermi liquid theory by constructing a mapping between the solutions of the classical kinetic equation and the quantized bosonic eigenmodes. By further mapping the kinetic equation into a one-dimensional tight-binding model we solve for the entire spectrum of collective and incoherent particle-hole excitations of Fermi liquids with nonzero F0{F}_{0}F0​ and F1{F}_{1}F1​ Landau parameters.
We study the fermion-boson coupled system in d=3−ϵ space dimensions near the quantum phase transition; infinite many boson modes locating on a sphere become critical simultaneously, which is dubbed “critical boson surface” (CBS). The fermions on the Fermi surface can be scattered to nearby points locating on a boson ring in the low-energy limit. The large number of the boson scattering channels N renders the well-known Landau damping effect largely suppressed. We propose an effective theory for a single point on the Fermi surface and the associated critical boson ring induced by the boson scattering channels. Based on the effective model, one-loop renormalization group analysis is performed with asymptotic ϵ-expansion. The fermion self-energy and Yukawa interaction vertex are dressed with ϵ poles and are largely enhanced due to the presence of critical boson ring. The imaginary part of the self-energy exhibits a linear-in frequency dependence and the real part gives a vanishing quasiparticle weight in the low-energy limit, which signatures the celebrated “marginal Fermi liquid” behavior. We find a novel non-Fermi liquid fixed point, at which the critical properties show features associated with the CBS.
We study the quantum theory of a Fermi surface coupled to a gapless boson scalar in $D=4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions as a simple model for non-Fermi liquids (NFL) near a quantum phase transition. Our analysis takes into account the full backreaction from Landau damping of the boson, and obtains an RG flow that proceeds through three distinct stages. Above the scale of Landau damping, the Fermi velocity flows to zero, while the coupling evolves according to its classical dimension. Once damping becomes important, its backreaction leads to a crossover regime where dynamic and static damping effects compete and the fermion self-energy does not respect scaling. Below this crossover and having tuned the boson to criticality, the theory flows to a z=3z=3z=3 scalar interacting with an NFL. We finally analyze the IR phases of the theory with arbitrary number of flavors Nc{N}_{c}Nc​. When Nc{N}_{c}Nc​ is small, the superconducting dome covers the NFL behavior; strikingly, for moderately large Nc{N}_{c}Nc​, we find that NFL effects become important first, before the onset of superconductivity. A generic prediction of the theory is that the Fermi velocity and quasiparticle residue vanish with a power law ${\ensuremath{\omega}}^{\ensuremath{\epsilon}}$ as the fixed point is approached. These features may be useful for understanding some of the phenomenology of high-Tc{T}_{c}Tc​ materials in a systematic $\ensuremath{\epsilon}$ expansion.