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

This paper extends the analysis of critical Fermi surface deformations at an Ising-nematic quantum critical point by employing quantum Boltzmann equations that include collision effects, revealing that while bosonic dynamics do not alter the solutions, the system supports a robust zero-sound mode and an infinite family of discrete higher-order harmonic modes alongside a continuous particle-hole band.

The Gist

Here is an explanation of the paper using simple language, analogies, and metaphors.

The Big Picture: When the Rules of the Road Break Down

Imagine a crowded dance floor. In a normal metal (like the copper wire in your phone), the electrons are like well-behaved dancers. They move in a predictable rhythm, bumping into each other occasionally but mostly following the rules of a "Fermi Liquid." If you push them, they glide smoothly.

But in certain exotic materials (like high-temperature superconductors), something strange happens near a "Quantum Critical Point." The electrons stop dancing to the beat. They become chaotic, interacting so intensely that the usual rules of physics break down. This is called a Non-Fermi Liquid (NFL).

This paper asks: If we try to push this chaotic crowd, how do they move? Do they form waves? Do they crash? And what happens if we account for the fact that they bump into each other (collisions)?

The Cast of Characters

1. The Electrons (The Dancers): These are the fermions. In this specific scenario, they are stuck in a "critical" state where they are constantly interacting with something else.
2. The Bosons (The Music/Order): These are the "order parameter" fluctuations. Think of them as the music playing on the dance floor. At the critical point, this music becomes "massless" (it has no weight) and vibrates wildly.
3. The Collision: In the past, scientists studied these dancers assuming they never bumped into each other (the "collisionless" regime). This paper says, "Wait, they do bump into each other. Let's see what happens when we include the bumps."

The Main Analogy: The "Wobbly Ferris Wheel"

Imagine the Fermi surface (the boundary of the electron crowd) as a giant, perfect Ferris wheel.

• Normal Metal: If you poke the wheel, it wobbles slightly and then settles back into a perfect circle. The wobble is a "sound wave" (Zero Sound).
• Critical Metal (NFL): Because the electrons are so strongly linked to the wild "music" (bosons), the wheel is made of jelly. When you poke it, it doesn't just wobble; it deforms in complex shapes.

The authors are trying to map out all the possible ways this jelly wheel can deform when you push it.

The Two Scenarios They Studied

The paper looks at two ways to analyze this jelly wheel:

1. The "Simple" View (The $F_0$ Model)

In earlier work, the authors looked at the wheel from a distance, only noticing if the whole thing got bigger or smaller (the "Zero Sound" or $\ell=0$ mode).

• The Discovery: They found that at low energies, the wheel doesn't wobble at a normal speed. Instead, the speed of the wobble follows a weird, fractional power law (like $|q|^{6/5}$ instead of just $|q|$).
• The New Twist: In this paper, they added "friction" (collisions). They found that the wobble does lose energy (it damps), but it loses it very slowly. The wave is still very "long-lived." It's like a jellyfish that wobbles for a long time before settling down.

2. The "Detailed" View (The $F_\ell$ Model)

This is the big breakthrough of the paper. Instead of just looking at the whole wheel, they looked at the wheel from every angle, breaking the movement down into specific shapes (harmonics).

• The Analogy: Imagine the Ferris wheel isn't just getting bigger or smaller. It's turning into a square, then a triangle, then a star, then a weird blob.
• The Surprise: When they included the "bumps" (collisions) and looked at these complex shapes, they found an infinite family of new waves.
  • In the simple view, there was just one main wave (Zero Sound) and a messy background of chaos (particle-hole continuum).
  • In the detailed view, they found many new, distinct waves appearing between the main wave and the chaos.
  • The "Infinite" Family: As the momentum gets smaller (the push gets gentler), the number of these new waves grows infinitely. They form a bridge connecting the main wave to the chaotic background.

The "Ghost" of the Bosons

One of the most interesting findings is about the "bosons" (the music).

• The authors asked: "What if the music itself gets out of sync? What if the music isn't in equilibrium?"
• The Result: They did the math and found that it doesn't matter. Even if the music is chaotic, the way the electrons (the dancers) move remains exactly the same as if the music were perfect. The "bumps" between electrons and the "bumps" between electrons and the music cancel each other out in a way that leaves the final dance steps unchanged. This is a very robust result.

Summary of Findings (The "Takeaway")

1. The "Zero Sound" is Tough: Even with all the collisions and chaos, the main wave (Zero Sound) survives. It is very stable and doesn't die out quickly.
2. A Zoo of New Waves: When you look closely (using higher angular momentum channels), you don't just see one wave. You see an infinite family of discrete waves that appear as you slow down the system.
3. Robustness: The behavior of these electrons is surprisingly stable. Even if you change the conditions of the "music" (the bosons), the "dance" (the electron deformations) stays the same.

Why Does This Matter?

Understanding how these "jelly-like" electron surfaces move helps scientists understand strange metals and high-temperature superconductors. If we can figure out how these waves travel, we might eventually figure out how to make materials that conduct electricity with zero resistance at room temperature, which would revolutionize our power grids and electronics.

In a nutshell: The authors took a chaotic, jelly-like electron system, added the realistic factor of "bumping into each other," and discovered that while the main wave is stable, there is actually a hidden, infinite orchestra of new waves playing underneath, waiting to be heard.

Technical Summary

Here is a detailed technical summary of the paper "Generic deformation channels for critical Fermi surfaces including the impact of collisions" by Islam, Savanur, and Mandal.

1. Problem Statement

The paper addresses the nature of low-energy excitations and collective modes in Non-Fermi Liquid (NFL) metals, specifically those arising at the Ising-nematic Quantum Critical Point (QCP) in two-dimensional systems.

• Context: In standard Fermi liquids, Landau quasiparticles are well-defined, and collective modes (like zero sound) are stable. However, at an Ising-nematic QCP, strong interactions between fermions and gapless bosonic order parameters destroy the quasiparticle picture (the self-energy scales as $|\omega|^{2/3}$ rather than $\omega^2$).
• The Gap: Previous theoretical work by the authors (and others) analyzed these systems in the collisionless limit, neglecting the collision integral in the kinetic equations and assuming bosons remained in equilibrium. This left open the question of how damping effects (collisions) and bosonic non-equilibrium dynamics affect the stability and dispersion of collective modes on a critical Fermi surface.
• Goal: To determine the dispersion relations and stability of Fermi surface deformations by performing a full analysis of the Quantum Boltzmann Equations (QBEs), including collision terms and bosonic kinetic equations.

2. Methodology

The authors employ a rigorous field-theoretic approach based on the Keldysh formalism for non-equilibrium systems.

• Model: They utilize the effective field theory for the Ising-nematic QCP, involving itinerant fermions coupled to a real scalar bosonic field ($\phi$). The bosons acquire Landau damping ($\Pi \sim |q_0|/|q|$) due to coupling with the fermions, leading to an overdamped propagator.
• Quantum Boltzmann Equations (QBEs):
  • Instead of relying on the quasiparticle approximation (which fails here), they derive QBEs using generalized distribution functions (Wigner distributions) obtained by integrating the Keldysh Green's functions over the energy shell near the Fermi surface.
  • The generalized Landau interaction functions (GLFs) are introduced, which are frequency-dependent due to Landau damping.
• Scenarios Analyzed:
  1. Bosons in Equilibrium: The bosonic distribution is fixed to the equilibrium form ($n_B=0$ at $T=0$), and only the fermionic QBE is solved.
  2. Bosons Out of Equilibrium: The authors derive the coupled QBEs for both fermions and bosons to check if bosonic density fluctuations ($\delta n$) modify the fermionic collective modes.
• Angular Momentum Decomposition: The master QBE is decomposed into angular momentum channels ($\ell$). The authors analyze:
  • The $\ell=0$ channel (Zero Sound) analytically.
  • Generic $\ell$ channels ($\ell > 0$) numerically, treating the system as a non-Hermitian tight-binding model in $\ell$-space.

3. Key Contributions

1. Inclusion of Collision Integrals: Unlike previous studies that ignored the imaginary part of the GLF (damping), this work explicitly includes the collision integral to quantify the decay rates of collective modes.
2. Bosonic Dynamics Check: The authors demonstrate that including the bosonic kinetic equation (allowing bosons to fluctuate out of equilibrium) does not alter the collective mode spectrum of the fermions at $T=0$. The correction to the collision integral vanishes upon integration over frequency due to symmetry.
3. Discovery of Infinite Discrete Modes: By solving the full angular-momentum coupled equations (beyond the $\ell=0$ approximation), the authors discover an infinite family of discrete collective modes that exist between the zero-sound mode and the particle-hole continuum.

4. Key Results

A. Zero Sound Mode ($\ell=0$)

• Dispersion Relation: The real part of the frequency ($\Omega_r$) exhibits a crossover behavior:
  • Low Momentum ($|q| \ll q_c$): $\Omega_r \propto |q|^{6/5}$. This fractional power law is a hallmark of the NFL fixed point.
  • High Momentum ($|q| \gg q_c$): $\Omega_r \propto |q|$, recovering linear dispersion similar to Fermi liquids.
• Stability (Damping): The inclusion of collisions introduces an imaginary part ($\Omega_i$), representing damping.
  • Crucially, $|\Omega_i| \ll \Omega_r$. The ratio $|\Omega_i/\Omega_r|$ decays rapidly as momentum increases.
  • Conclusion: The zero-sound mode remains long-lived and robust against damping, validating the existence of well-defined collective modes even in the NFL regime.

B. Generic Angular Momentum Channels ($\ell > 0$)

• Emergence of New Modes: When solving the coupled equations for higher $\ell$, the authors find that the "particle-hole continuum" is not a simple band. Instead, an infinite number of discrete modes emerge, forming "islands" in the complex frequency plane between the zero sound and the continuum.
• Limit Behavior: As momentum $q \to 0$, the number of these discrete modes proliferates ($n \to \infty$), effectively connecting the zero-sound solution to the continuum.
• Dispersion Correction: In the extreme low-momentum limit, the zero-sound dispersion in the full $F_\ell$ model becomes linear ($\Omega_r \propto |q|$), differing from the $|q|^{6/5}$ scaling found in the simplified $F_0$ model. This suggests the $F_0$ approximation underestimates the stability and alters the scaling at very low energies.
• Mode Structure: The higher-order modes correspond to higher-order harmonics of the Fermi surface deformation (e.g., elliptical, quadrupolar distortions).

5. Significance

• Validation of NFL Hydrodynamics: The paper confirms that despite the destruction of Landau quasiparticles, collective modes (hydrodynamic modes) survive in NFLs. The zero sound is robust, suggesting that transport and response properties in NFL metals can still be described by collective excitations.
• Beyond Collisionless Approximations: By proving that the collision integral does not destabilize the zero sound, the work justifies the use of collisionless approximations for qualitative features while providing the necessary corrections for quantitative decay rates.
• New Spectral Features: The discovery of an infinite tower of discrete modes challenges the standard view of NFL excitations as merely a continuum. These modes could have observable consequences in optical conductivity or neutron scattering experiments on materials near nematic QCPs (e.g., cuprates, iron-based superconductors).
• Methodological Advancement: The successful application of the full Quantum Boltzmann formalism with coupled fermion-boson dynamics sets a new standard for analyzing transport and collective phenomena in strongly correlated quantum critical systems.

In summary, the paper establishes that critical Fermi surfaces in Ising-nematic NFLs support a rich spectrum of long-lived collective modes, including a robust zero sound and an infinite family of higher-order harmonic modes, even when collisional damping and bosonic fluctuations are fully accounted for.