Quantized nonlinear transport and its breakdown in Fermi gases with Berry curvature

This paper demonstrates that while Berry curvature does not affect quantized nonlinear transport in translationally invariant two-dimensional Fermi gases, the introduction of spatial inhomogeneity causes this quantization to break down due to the interplay between Berry curvature and potential gradients, a phenomenon observable in trapped ultracold atoms.

The Gist

The Big Idea: A Perfectly Ordered Dance Floor

Imagine a massive, perfectly smooth dance floor where thousands of people (electrons) are moving around. In physics, we call this a metal. Usually, when you push these people, they move in predictable ways.

But sometimes, the dance floor has a hidden "twist" or "spin" to it, even if the people don't feel it directly. In physics, this hidden twist is called Berry Curvature. It's like the floor is slightly tilted or has a magnetic swirl that makes people drift sideways when they try to walk straight.

For a long time, scientists thought that if you tried to measure a very specific, super-precise movement of these people (called Quantized Nonlinear Transport), this hidden twist would ruin the measurement. They thought the "twist" would make the numbers messy and unpredictable.

This paper says: "Not so fast!"

The authors, Fan Yang and Xingyu Li, discovered that the hidden twist doesn't ruin the measurement... as long as the dance floor is perfectly flat and uniform. However, if the floor is slightly bumpy or uneven (which happens in real experiments with trapped atoms), the measurement breaks down.

---

The Experiment: The "Double-Clap" Test

To understand what they measured, imagine this thought experiment:

1. The Setup: You have a giant grid. You give two sharp "claps" (voltage pulses) to the crowd.
   • Clap 1: Happens on the left side.
   • Clap 2: Happens on the bottom side.
2. The Goal: You want to see how many people end up in the top-right corner (the first quadrant) because of both claps working together.
3. The Magic Number: In a perfect, flat world, the number of people who end up in that corner isn't random. It's a whole number (an integer) determined by the shape of the crowd's "dance pattern" (the topology of the Fermi sea). It's like saying, "No matter how fast you clap, exactly 3 people will always end up in that corner."

This is the Quantized Nonlinear Transport. It's a super-precise count that reveals the hidden shape of the universe.

Part 1: The Flat Dance Floor (Uniform Systems)

The authors first looked at a perfectly flat, uniform dance floor.

• The Twist: Even though the floor has that hidden "Berry Curvature" (the sideways drift), the authors found that when you do the "Double-Clap" test, the sideways drift cancels itself out perfectly.
• The Result: The count remains perfect. The number of people in the corner is still exactly determined by the shape of the crowd.
• The Analogy: Imagine walking on a moving walkway at the airport that is slightly tilted. If you walk straight, you drift sideways. But if you walk in a perfect square loop, you end up exactly where you started. The "drift" didn't ruin your final position because the path was closed and uniform.

Takeaway: If the system is uniform, the hidden twist doesn't matter. The "magic number" stays magic.

Part 2: The Bumpy Dance Floor (Spatial Inhomogeneity)

Now, imagine the dance floor isn't flat. Maybe it's inside a bowl (a trap), so the floor slopes upward at the edges. This is what happens in real experiments with ultracold atoms (atoms cooled to near absolute zero and trapped by lasers).

• The Problem: When the floor is bumpy, the "sideways drift" caused by the Berry Curvature changes depending on where you are.
• The Breakdown: When you do the "Double-Clap" test on this bumpy floor, the hidden twist and the slope of the floor interact. They create a new, messy current that doesn't cancel out.
• The Result: The perfect integer count is ruined. You might get 3.1 people or 2.8 people instead of exactly 3. The "magic number" breaks down.

The Analogy: Imagine trying to walk a perfect square on a trampoline that is sagging in the middle. The "drift" caused by the sagging floor changes as you move. Your final position won't be a clean, predictable spot anymore. The hidden twist and the bumpiness are fighting each other, creating a mess.

Why Does This Matter?

1. It's a New Rulebook: Before this, scientists thought Berry Curvature might always mess up these precise measurements. This paper draws a clear line: "It's fine if you are flat; it breaks if you are bumpy."
2. Real-World Application: This is crucial for experiments with ultracold atoms. Scientists use lasers to trap atoms in "bowl" shapes to study quantum physics. This paper tells them: "If you want to see this perfect quantum number, you have to be very careful about where you apply your laser pulses. If they hit a slope, the number will be wrong."
3. Future Tech: Understanding how these "magic numbers" break down helps us design better quantum computers and sensors. It teaches us how to distinguish between the "true shape" of the quantum world and the "noise" caused by uneven traps.

Summary in One Sentence

The paper reveals that while a hidden quantum "twist" (Berry Curvature) doesn't ruin precise measurements on a perfectly flat surface, it causes those measurements to fail when the surface is uneven, a fact that is critical for experiments using trapped atoms.

Technical Summary

1. Problem Statement

Quantized transport is a hallmark of topological phases. While quantized Hall conductance is well-known in gapped insulators, recent work by Kane (2022) demonstrated that quantized nonlinear conductance also exists in metallic states (specifically ballistic 2D electron gases). In the absence of Berry curvature, this nonlinear conductance is determined solely by the Euler characteristic ($\chi_F$) of the Fermi sea.

The central problem addressed in this paper is the interplay between Berry curvature (which induces anomalous Hall effects and anomalous velocities) and this quantized nonlinear transport. Specifically, the authors investigate:

1. Does non-vanishing Berry curvature at the Fermi surface destroy the quantization of nonlinear transport in spatially uniform (translationally invariant) metals?
2. How does spatial inhomogeneity (e.g., an external trap potential) combined with Berry curvature affect this quantization?

2. Methodology

The authors employ a semi-classical Boltzmann transport approach to analyze non-interacting fermionic systems in two dimensions.

• Theoretical Framework:

  • They utilize the semi-classical equations of motion for electron wavepackets, incorporating the Berry curvature ($\Omega_k$) and external electric fields ($E$) and potentials ($U_r$).
  • The distribution function $f(r, k, t)$ is solved using the collisionless Boltzmann equation via the method of characteristics.
  • They consider a specific thought experiment (based on Refs. [12, 16]) involving two voltage pulses ($V_1, V_2$) applied at different times and locations, measuring the excess charge transported to a specific region.

• Experimental Protocols Modeled:

  1. Homogeneous Case: A 2D metal with translation symmetry. The system is analyzed to see if the "pump effect" (instantaneous anomalous velocity at boundaries) alters the net charge transport.
  2. Inhomogeneous Case: A Fermi gas in a slowly varying external trap potential $U_r$ (simulating ultracold atoms in an optical lattice). The equations of motion are modified to include the force from the trap gradient and the resulting anomalous velocity term ($\dot{r} \propto \nabla_r U_r \times \Omega_k$).

• Calculation Strategy:

  • The excess charge $Q$ (or particle number $N$) is calculated by integrating the change in the distribution function $\delta f_{12}$ caused by the two pulses.
  • The calculation is split into terms dependent on the Euler characteristic and terms dependent on the Berry curvature and spatial gradients.

3. Key Contributions and Results

A. Homogeneous Systems (Translationally Invariant)

• Result: The quantized nonlinear transport remains robust in the presence of non-vanishing Berry curvature.
• Mechanism:
  • While Berry curvature induces an anomalous velocity ($\dot{r}_{anom} \propto E \times \Omega_k$), this velocity vanishes for electrons at equilibrium in a uniform system when considering the net transport over a bounded area.
  • The "pump effect" (instantaneous current at boundaries) cancels out when measuring the charge difference in a bounded region $\Sigma$.
  • The nonlinear conductance is determined solely by the energy dispersion $\epsilon_k$ at equilibrium, which defines the topology of the Fermi sea.
• Conclusion: For uniform metals, $G(\omega_1, \omega_2) \propto \chi_F$, regardless of the Berry curvature.

B. Inhomogeneous Systems (Spatially Varying Potential)

• Result: The quantization breaks down when spatial inhomogeneity is introduced in the presence of Berry curvature.
• Mechanism:
  • In a trap potential $U_r$, the equilibrium velocity acquires an anomalous component: $v_0 = \frac{1}{\hbar}\nabla_k \epsilon_k - \frac{1}{\hbar}\nabla_r U_r \times \Omega_k$.
  • This anomalous velocity exists even at equilibrium. Consequently, the velocity field $v_0$ is no longer a pure gradient field of the energy dispersion.
  • The standard link between the particle number and the Euler characteristic (via Morse theory) relies on $v_0$ being a gradient field. This link is severed.
• Mathematical Outcome:
  • The total excess particle number $N$ splits into two parts:
    $$N = N_{quantized} + N_{non-quantized}$$
  • $N_{quantized}$: Related to the indices of the zeros of the velocity field (still resembles $\chi_F$ under specific conditions).
  • $N_{non-quantized}$: A term proportional to $\Omega_k \cdot (\nabla_r U_r)$. This term depends on the local gradient of the potential and the Berry curvature at the Fermi surface.
  • The result is not solely determined by the Fermi sea topology; it depends on the specific geometry of the trap and the magnitude of the Berry curvature.
  • Time-Reversal Symmetry Breaking: The result depends on the order of the pulses ($t_1$ vs $t_2$). The average of both orders yields an antisymmetric term with respect to velocity components.

C. Experimental Observability

• The breakdown of quantization is proposed to be observable in ultracold atoms trapped in optical lattices with topological bands.
• Protocol:
  1. Create a topological band with non-trivial Berry curvature.
  2. Apply two far-detuned optical pulses to simulate voltage pulses.
  3. Use a quantum gas microscope to measure the excess particle number in the intersection region with single-atom precision.
  4. By varying the intersection location relative to the trap's local extrema (where $\nabla_r U_r = 0$), one can observe the transition from quantized (at extrema) to non-quantized transport (away from extrema).

4. Significance

• Theoretical Insight: The paper clarifies the limits of topological protection in metallic states. It demonstrates that while the Euler characteristic is a robust invariant for uniform systems, the interplay between Berry curvature and spatial gradients introduces a new channel for non-quantized transport.
• Distinction from Anomalous Hall Effect: It distinguishes between the standard Anomalous Hall Effect (linear response) and nonlinear transport, showing that while Berry curvature affects linear response, it only disrupts nonlinear quantization when spatial inhomogeneity is present.
• Experimental Guidance: It provides a concrete proposal for observing topological breakdown in ultracold atomic systems, offering a new diagnostic tool for characterizing topological bands and Berry curvature distributions in quantum simulators.
• Future Directions: The authors note that the effect of real-space magnetic fields (Landau levels) on this nonlinear transport remains an open question, suggesting distinct physics due to the asymmetry between real and momentum space.

In summary, the paper establishes that quantized nonlinear transport in metals is topologically protected only in the absence of spatial inhomogeneity. The introduction of a trap potential combined with Berry curvature leads to a measurable breakdown of this quantization, linking the transport properties to both the Fermi sea topology and the local potential landscape.