Odd relaxation in three-dimensional Fermi liquids

This paper demonstrates that isotropic three-dimensional Fermi liquids exhibit a hierarchy of long-lived, non-hydrodynamic modes where odd-parity relaxation is significantly slower than even-parity relaxation due to Pauli blocking and interaction effects, a phenomenon previously thought unique to two dimensions.

The Gist

Imagine a crowded dance floor where everyone is trying to move without bumping into each other. In the world of physics, this dance floor is a Fermi liquid—a sea of electrons moving through a metal. Usually, when these electrons bump into each other, they scatter chaotically, and the system quickly settles down into a calm, balanced state. This is what we call "hydrodynamic" behavior, similar to how water flows.

However, physicists recently discovered a weird, "super-organized" way these electrons can behave, but they thought it only happened in two-dimensional (2D) worlds (like a flat sheet of graphene). They called this the "Tomographic Effect."

Here is the simple breakdown of what this paper says, using some everyday analogies:

1. The Old Belief: "It's Only a 2D Thing"

In a flat, 2D world, there is a special rule called the Pauli Exclusion Principle. Think of it as a strict bouncer at a club who says, "You can't sit in a seat that's already taken."

Because of this bouncer, in a 2D world, electrons can only relax (calm down) if they hit each other head-on (like two cars crashing front-to-front).

• The Twist: If the electrons are moving in a pattern that is "symmetrical" (like a mirror image), they crash head-on easily and calm down fast.
• The Oddity: If the electrons are moving in an "asymmetrical" pattern (like a lopsided wobble), head-on collisions are impossible due to the bouncer's rules. These "odd" patterns get stuck in a loop, refusing to calm down. They become long-lived ghosts that linger much longer than the symmetrical ones.

Scientists thought this "ghostly lingering" was a unique quirk of flat, 2D worlds.

2. The New Discovery: "It Happens in 3D Too!"

This paper asks: "What if we go into a 3D room?" (Like a real block of copper or gold).

In 3D, the geometry is different. Electrons have more room to dodge. They don't have to hit head-on; they can graze each other from the side. The old theory said, "Okay, since they can dodge, the 'ghosts' will disappear, and everything will calm down at the same speed."

The authors say: "Not so fast!"

They found that even in 3D, the "ghosts" (the odd patterns) still linger longer than the symmetrical ones, though the effect is a bit subtler than in 2D.

• The Analogy: Imagine a 3D room full of people. Even though they can dodge each other, if they try to move in a specific "wobbly" way, the crowd naturally resists it more than a "smooth" way. The "wobbly" movement takes about 40% longer to die out than the smooth movement, even without the strict 2D rules.

3. Why Does This Matter? (The "Tomographic" Regime)

The authors call this a "Tomographic" regime. Think of a medical CT scan (Tomography). A CT scan takes a 3D object and slices it into layers to see the inside structure.

In these electron fluids, the "odd" patterns act like those slices. Because they relax so slowly, they allow us to "see" the internal structure of how electrons interact in a way we couldn't before. It's like the electrons are holding up a mirror to their own collisions.

4. How Do We See This? (The Experiment)

How do you prove this is happening in a chunk of metal? You can't just look at the electrons. You have to measure how electricity flows.

• The Test: The authors suggest measuring transverse conductivity. Imagine pushing a river of electrons with a magnetic field or a specific type of electric pulse.
• The Signature: If you look at how the electricity flows at different speeds and distances, you will see a "bump" or a specific change in the flow pattern. This bump is the fingerprint of those "ghostly" odd patterns that are taking their time to relax.
• The Bonus: If you use materials where electrons like to bounce off each other at sharp angles (like billiard balls hitting the cushions), this effect gets even stronger, making it easier to spot.

Summary

• The Myth: Only flat (2D) electron worlds have these weird, long-lasting "odd" patterns.
• The Reality: Even in 3D metals, these patterns exist and relax much slower than normal patterns.
• The Impact: This changes how we understand how electricity moves in metals. It suggests that 3D metals have a hidden, complex layer of behavior (a "tomographic" layer) that we can now detect and potentially use in future electronics.

In a nutshell: The authors found that even in a 3D crowd, the "awkward dancers" (odd patterns) take much longer to stop dancing than the "smooth dancers" (even patterns), and we can finally see this happening in real-world metals.

Technical Summary

1. Problem Statement

Recent theoretical developments have identified a "tomographic" transport regime in two-dimensional (2D) Fermi liquids. In this regime, quasiparticle relaxation rates exhibit a stark separation based on parity:

• Even-parity modes (symmetric under inversion) relax rapidly via standard Fermi liquid scattering ($\sim T^2$).
• Odd-parity modes (antisymmetric) relax anomalously slowly because Pauli blocking severely restricts the phase space for "head-on" scattering, which is the dominant relaxation channel in 2D.

The central question addressed in this work is whether this even-odd staggering of relaxation rates is an anomaly unique to the kinematic constraints of 2D, or if it is a more general feature of Fermi liquids. Specifically, the authors investigate three-dimensional (3D) isotropic Fermi liquids, where conventional wisdom suggests head-on scattering is not enforced by Pauli exclusion (due to an additional free azimuthal angle in the scattering geometry), implying that even and odd modes should relax at similar rates.

2. Methodology

The authors employ a rigorous microscopic approach based on the linearized Boltzmann equation and the collision integral for electron-electron scattering.

• Angular Decomposition: They expand deviations from the equilibrium Fermi-Dirac distribution ($\delta f$) in a basis of spherical harmonics $Y_{lm}(\theta, \phi)$.
  • In 3D, parity is determined by the angular momentum index $l$ (even $l$ = even parity, odd $l$ = odd parity).
  • The radial dependence is treated in the leading-order approximation (rigid Fermi surface shift), focusing on the decay rates $\gamma_l$ of the angular modes.
• Collision Integral Diagonalization: The authors diagonalize the linearized collision operator $L$. They derive an exact expression for the decay rates $\gamma_l$ at low temperatures ($T \ll T_F$) for generic interaction potentials $V(q)$.
• Interaction Models: They analyze two specific interaction scenarios:
  1. Constant Interaction: A momentum-independent repulsion (representing strong screening).
  2. Angle-Dependent Interactions:
     • Large-angle scattering: A potential favoring large momentum transfers.
     • Small-angle scattering: A screened Coulomb interaction (Thomas-Fermi model).
• Transport Signatures: To connect microscopic rates to observables, they solve the full Boltzmann equation to calculate the static transverse conductivity $\sigma_\perp(q)$ and analyze the collective modes of the transverse current.

3. Key Contributions and Results

A. Existence of the Even-Odd Effect in 3D

Contrary to the assumption that 3D Fermi liquids lack this effect, the authors prove that odd-parity modes relax significantly slower than even-parity modes, even though the leading-order temperature scaling remains $\gamma \sim T^2$ for both.

• Magnitude of the Effect: For a constant interaction potential, the lowest-order odd-parity decay rate ($\gamma_3$) is only ~60% of the even-parity rate ($\gamma_2$).
• Relative Difference: The relative difference between odd and even rates, $|\gamma_l - \gamma_{l-1}|/\gamma_l$, peaks at 40% for $l=3$ and remains substantial for higher $l$.
• Mechanism: While 3D does not enforce strict head-on scattering (unlike 2D), the phase space for scattering that relaxes odd-parity deformations is still suppressed relative to even-parity deformations due to the geometry of the momentum transfer vectors and Pauli blocking.

B. Dependence on Scattering Potential

The magnitude of the even-odd staggering is tunable and depends strongly on the interaction potential:

• Large-Angle Scattering: Interactions that favor large momentum transfers (and thus favor the specific geometric configuration $\phi_2 \approx \pi$) enhance the even-odd separation. In this regime, the odd-mode relaxation can be suppressed by a factor of 40 relative to even modes.
• Small-Angle Scattering: Screened Coulomb interactions (favoring small angles) suppress the effect, but a ~10% difference persists even for weak interactions ($r_s \ll 1$).
• Physical Implication: This suggests that in real 3D materials with specific band structures or lattice effects that enhance large-angle scattering, the tomographic regime could be even more pronounced than in 2D.

C. Experimental Signatures

The authors identify clear experimental probes for this effect:

1. Static Transverse Conductivity ($\sigma_\perp$):
   • In the hydrodynamic limit ($v_F q \ll \gamma$), conductivity follows standard scaling.
   • In the collisionless limit ($v_F q \gg \gamma$), it scales linearly with $q$.
   • Tomographic Regime: In the intermediate regime where $v_F q$ is larger than odd-mode rates but smaller than even-mode rates, $\sigma_\perp$ exhibits a distinct crossover behavior (neither constant nor linear). This intermediate scaling regime is a direct signature of the suppressed odd-mode scattering.
2. Collective Modes:
   • The transverse conductivity supports collective modes whose damping is determined by the slowest relaxing mode.
   • In the limit $\gamma_{odd} \ll v_F q \ll \gamma_{even}$, a collective mode emerges with a damping rate $\omega = -i\gamma_{odd}$. Observing this mode allows for the direct measurement of the odd-parity scattering rate.
3. Fermi Surface Deformations:
   • In the tomographic regime, the electric field induces Fermi surface deformations where odd-parity modes are anomalously enhanced compared to even-parity modes, a stark contrast to the hydrodynamic limit where only the conserved current ($l=1$) is excited.

4. Significance

• Paradigm Shift: The work overturns the conventional wisdom that the "tomographic" even-odd effect is exclusive to 2D systems. It establishes that 3D Fermi liquids possess a similar hierarchy of long-lived modes.
• Experimental Feasibility: The authors argue that the effect may be easier to observe in 3D than in 2D. Since 3D metals have much higher Fermi temperatures ($T_F \sim 10^4$ K) compared to 2D systems ($T_F \sim 1-100$ K), the absolute energy scale of the separation between even and odd rates is larger, potentially making the effect accessible at experimentally reachable temperatures.
• New Frontier: This discovery opens a new avenue for studying non-hydrodynamic transport in 3D materials (e.g., high-purity metals, Weyl semimetals) and suggests that transport measurements (specifically transverse conductivity and collective mode spectroscopy) can reveal the microscopic details of electron-electron interactions and parity-dependent relaxation.

Conclusion

Musser et al. demonstrate that 3D Fermi liquids exhibit a robust even-odd staggering in quasiparticle relaxation rates, driven by Pauli blocking and interaction geometry. This "tomographic" regime, previously thought to be a 2D anomaly, manifests in 3D with significant magnitude, offering new experimental signatures in transverse transport and collective modes to probe the fundamental dynamics of interacting electrons.