Tomographic collective modes in a magnetic field

This paper investigates the transition from tomographic to conventional transport in two-dimensional Fermi liquids under a magnetic field by using a numerically exact solution of the linearized Boltzmann equation to demonstrate that a critical magnetic field causes one of two diffusive tomographic collective modes to disappear, leaving a remaining mode that becomes increasingly hydrodynamic at higher fields.

The Gist

Imagine a crowded dance floor where thousands of people (electrons) are moving around. Usually, in a messy crowd, people bump into each other constantly, getting stuck or changing direction randomly. This is like normal electricity flowing through a wire with impurities.

But in ultra-clean materials, something magical happens. The dancers start moving together like a fluid, like water flowing down a river. They push and pull each other, creating waves and swirls. This is called electron hydrodynamics.

This paper explores a very specific, weird behavior in this "electron fluid" when you add a magnetic field. Here is the story in simple terms:

1. The "Odd-Even" Dance Rule

In this electron fluid, there's a strange rule about how the crowd deforms (changes shape) when pushed.

• Even Deformations: Imagine the crowd squeezing in and out symmetrically (like a breathing chest). These shapes are very "social." The dancers bump into each other constantly to fix these shapes, so they disappear (relax) very quickly.
• Odd Deformations: Imagine the crowd tilting to the left or right, or forming a wave that goes one way and then the other. These shapes are "antisocial." Because of the rules of physics in 2D, these dancers rarely bump into each other in a way that fixes the tilt. They keep their shape for a very long time.

This difference creates a "Tomographic" regime. Think of "Tomographic" like an X-ray that sees different layers. In this regime, the "fast" even shapes act like a thick, sticky fluid (hydrodynamics), while the "slow" odd shapes act like a ghostly, frictionless stream (collisionless). The material behaves like a mix of two different worlds at the same time.

2. The Magnetic Field: The Spin Cycle

Now, imagine turning on a giant magnet. This forces every dancer to spin in circles (cyclotron motion) instead of just wandering.

The paper asks: What happens to our special "Odd-Even" dance when everyone is forced to spin?

The authors found that the magnetic field acts like a "reset button" for the odd shapes.

• Weak Magnet: The dancers spin slowly. The "odd" shapes still manage to avoid collisions. The special "Tomographic" behavior survives.
• Critical Magnet: As the magnet gets stronger, the dancers spin faster and tighter. Eventually, they spin so fast that they can't avoid bumping into each other anymore, even for the "antisocial" odd shapes.
• The Result: At a specific "Critical Magnetic Field," one of the two special "Tomographic" waves vanishes completely. It's like one of the two distinct sounds in a duet suddenly going silent.

3. The Two Waves and the "Landau" Personality

The paper discovered that there are two of these special waves (let's call them the Green Wave and the Purple Wave).

• Which one disappears first depends on the "personality" of the electrons, defined by something called Landau parameters.
• If the electrons are "shy" (low Landau parameter), the Purple Wave disappears first.
• If the electrons are "bold" (high Landau parameter), the Green Wave disappears first.
• There is a specific "personality" where both waves merge into one and then vanish together.

4. The Visual Metaphor: The Spinning Pizza

Imagine the surface of the electron fluid as a pizza dough.

• No Magnet: You can stretch the dough in two very specific, weird ways (the Tomographic modes). One stretches fast and smooths out; the other stretches slowly and stays weird for a long time.
• Add Magnet: Now, spin the pizza on a plate.
  • At first, the weird stretches still happen.
  • But as you spin faster, the dough gets flung outward. The "slow" weird stretch gets crushed by the spinning motion.
  • Eventually, the dough can only stretch in the way that matches the spin (the standard hydrodynamic way). The unique "Tomographic" stretching is gone.

Why Does This Matter?

This isn't just about math; it's about finding new ways to move electricity.

• The Problem: It's hard to prove this "Tomographic" stuff exists because impurities in real materials hide it.
• The Solution: The authors suggest that by using a magnetic field, we can actually see these modes. By watching how the electricity damps out (slows down) as we turn up the magnet, we can spot the moment one of the special waves dies. This confirms that the "Odd-Even" effect is real.

In a nutshell:
The paper shows that in a super-clean electron fluid, there are two special ways the electrons can wiggle. A magnetic field acts like a strong wind that blows one of these wiggles away. By measuring exactly when that wiggle disappears, scientists can prove that electrons are behaving like a strange, layered fluid, opening the door to new types of electronic devices.

Technical Summary

Here is a detailed technical summary of the paper "Tomographic collective modes in a magnetic field" by Jeff Maki and Johannes Hofmann.

1. Problem Statement

The paper investigates the behavior of two-dimensional (2D) Fermi liquids at low temperatures, specifically focusing on the transition between tomographic transport and conventional transport regimes under the influence of a magnetic field.

• Background: In ultra-clean 2D materials, electron-electron interactions dominate, leading to hydrodynamic flow. A key feature of these systems is an odd-even effect in quasiparticle relaxation rates: even-parity deformations of the Fermi surface relax rapidly ($\gamma_e \sim T^2$), while odd-parity deformations relax much more slowly ($\gamma_o \sim T^4$).
• Tomographic Regime: This hierarchy creates a "tomographic" transport limit where odd-parity modes are effectively collisionless while even-parity modes remain hydrodynamic. This regime is predicted to exhibit unique signatures, such as anomalous scaling of transverse conductivity and additional diffusive collective modes.
• The Gap: While the odd-even effect is established at zero field, it is theoretically predicted that a magnetic field should suppress these signatures once the cyclotron radius becomes smaller than the odd-parity mean free path. However, the specific mechanism of this transition, particularly regarding the collective mode spectrum and the structure of Fermi surface deformations, had not been fully explored.
• Objective: The authors aim to characterize the crossover from the tomographic limit (small fields) to conventional hydrodynamic/collisionless regimes (large fields) by analyzing the collective modes of the transverse conductivity tensor.

2. Methodology

The authors employ a rigorous theoretical framework combining numerical exact solutions with variational approaches:

• Linearized Boltzmann Equation: They solve the kinetic theory for a 2D Fermi liquid in a magnetic field ($B \hat{z}$) and an oscillating electric field. The distribution function is expanded in angular harmonics ($e^{im\theta_p}$).
• Relaxation Time Approximation: The collision integral is modeled using a generalized relaxation time approximation that distinguishes between:
  • Even-parity modes ($|m| \ge 2$): Relax at rate $\gamma_e$.
  • Odd-parity modes ($|m| \ge 3$): Relax at rate $\gamma_o \ll \gamma_e$.
  • Conservation Laws: $m=0$ (particle number) and $m=\pm 1$ (momentum) have zero relaxation rates in the absence of impurities.
  • Two models are tested: a constant odd-parity rate ($\gamma_o(m) = \gamma_o$) and an $m$-dependent rate ($\gamma_o(m) \propto m^{-4}$).
• Numerical Solution: The infinite set of coupled linear equations for the angular harmonics is truncated and solved numerically using Lentz's algorithm to evaluate continued fractions. This allows for the calculation of the conductivity tensor $\sigma(\omega, k)$ and the identification of collective mode poles (where $\sigma^{-1} = 0$).
• Variational Ansatz: To gain physical insight into the Fermi surface deformation $h(\theta_p)$, the authors construct a variational ansatz. They propose Gaussian-like trial functions for the odd-parity deformation (peaked at $\theta = \pm \pi/2$ for the upper branch and $\theta = 0, \pi$ for the lower branch) to minimize the transverse conductivity bound. This allows them to estimate the critical magnetic field analytically.

3. Key Contributions

• Mapping the Crossover: The paper provides a comprehensive map of how the collective mode spectrum evolves from the zero-field tomographic regime to the high-field hydrodynamic regime.
• Critical Magnetic Field Identification: The authors identify a critical magnetic field strength ($\omega_c \sim \gamma_o$) at which one of the two diffusive tomographic collective modes vanishes.
• Role of Landau Parameters: They demonstrate that the specific mode that vanishes (upper vs. lower branch) depends critically on the Landau parameter $F_1$ (which governs p-wave interactions). They define a critical value $F_1^*$ where the two branches coalesce at the critical field.
• Microscopic Structure Analysis: By analyzing the angular momentum decomposition of the Fermi surface deformation, they show how the "odd-parity" nature of the modes is progressively destroyed by the magnetic field, eventually becoming dominated by hydrodynamic current modes ($m=\pm 1$).
• Variational Validation: The variational approach successfully captures the angular structure of the deformations and accurately predicts the critical magnetic field strength derived from the variance of the angular momentum.

4. Key Results

A. Collective Mode Spectrum at Zero Field

• In the absence of a magnetic field, the transverse conductivity exhibits two diffusive collective modes (upper and lower branches) separated by a branch cut in the complex frequency plane.
• These modes exist in the "tomographic regime" ($k\xi \sim 1$, where $\xi$ is the characteristic length scale).
• The upper branch exists at all momenta, while the lower branch only emerges above a critical Landau parameter $F_c$ or a critical momentum.

B. Evolution with Magnetic Field

• Suppression of Modes: As the magnetic field increases, the system undergoes a two-step suppression:
  1. First Step (Critical Field $\omega_c \sim \gamma_o$): One of the two diffusive tomographic modes disappears.
     • If $F_1 < F_1^*$, the lower branch vanishes.
     • If $F_1 > F_1^*$, the upper branch vanishes.
     • At $F_1 = F_1^*$, the two branches merge and vanish simultaneously.
  2. Second Step (High Field $\omega_c \sim \gamma_e$): The surviving mode transitions from a tomographic state to a conventional hydrodynamic diffusive mode (shear sound) and eventually becomes collisionless as the field increases further.
• Mode Mixing: In finite magnetic fields, longitudinal and transverse modes mix. However, the residue of the tomographic modes in the longitudinal response is found to be negligible.

C. Fermi Surface Deformation Structure

• Zero Field: The deformations are highly anisotropic. The upper branch is peaked at $\theta = \pm \pi/2$, while the lower branch is peaked at $\theta = 0, \pi$. Both involve a superposition of many odd-parity angular momentum modes.
• Finite Field: As $B$ increases, the deformations rotate and broaden. The weight shifts toward the hydrodynamic current modes ($m = \pm 1$).
• Critical Condition: The disappearance of a mode occurs when the cyclotron frequency associated with the dominant angular momentum of the deformation ($\sqrt{\langle m^2 \rangle}\omega_c$) exceeds the odd-parity relaxation rate ($\gamma_o$). At this point, the mode becomes collisionless, and the tomographic signature is lost.

D. Variational Results

• The variational ansatz confirms that the root-mean-square angular momentum $\sqrt{\langle m^2 \rangle}$ determines the stability of the mode against the magnetic field.
• The analytical estimate for the critical field derived from the variational width matches the numerical results from the full Boltzmann equation.

5. Significance

• Fundamental Physics: The work clarifies the robustness of the odd-even effect and tomographic transport in the presence of magnetic fields, a crucial consideration for experimental verification.
• Experimental Guidance: It suggests that detecting tomographic modes via transverse conductivity is challenging in high magnetic fields because one of the two characteristic modes vanishes at relatively low fields ($\omega_c \sim \gamma_o$).
• Observability: The authors propose that while direct shear sound measurement is difficult, the mixing of longitudinal and transverse modes in finite fields could allow for the observation of these effects in standard longitudinal probes, although the signal is expected to be weak.
• Broader Applicability: The theoretical framework applies not only to condensed matter systems (like graphene or PdCoO$_2$) but also to ultracold atomic gases with synthetic gauge fields, where the odd-even effect is also predicted to exist.

In summary, the paper establishes that the "tomographic" transport regime is fragile against magnetic fields, characterized by a specific critical field where the unique diffusive collective modes collapse, driven by the interplay between cyclotron motion and the angular momentum structure of the Fermi surface deformation.