Topological invariant of periodic many body wavefunction from charge pumping simulation

This paper introduces a robust charge pumping simulation method that enables the accurate calculation of topological invariants, such as Chern numbers, for many-body systems using neural network wavefunctions, thereby overcoming a key bottleneck in studying correlated topological matter and facilitating the identification of exotic states like anomalous composite Fermi liquids.

The Gist

Imagine you are trying to figure out the "personality" of a complex, crowded dance floor. In the world of quantum physics, this dance floor is made of electrons, and the "personality" we are looking for is something called a Topological Invariant.

Think of a topological invariant like a knot. If you have a piece of string, you can twist it, stretch it, or pull it, but as long as you don't cut it, the number of knots in it stays the same. In physics, this "knot count" tells us if a material is a normal insulator (like a rubber band) or a special, exotic quantum state (like a complex knot that can't be untangled).

For a long time, scientists had a hard time counting these "knots" in complex materials, especially when using powerful new tools called Neural Networks (AI) to simulate them. The AI was great at describing the dance moves (the wavefunction), but it was terrible at counting the knots because it couldn't see the whole picture at once.

Here is how this paper solves that problem, using a simple analogy: The Charge Pumping Machine.

The Problem: The AI Can't Count the Knots

Imagine you have a simulation of electrons dancing on a torus (a shape like a donut). You want to know if this dance floor has a "fractional knot" (a fractional topological number, like 2/3 or 1/2).

• Old Method: To count the knot, you usually need to see every single dancer's exact position and energy at the same time. But with AI, we only see a "blurry" picture of the crowd, not every individual. The old counting methods failed because they needed a perfect, crystal-clear view that the AI couldn't provide.

The Solution: The "Flux" Turnstile

The authors invented a new way to count the knots by watching what happens when you push the system.

1. The Setup: Imagine the donut-shaped dance floor. The scientists insert a "magnetic flux" (think of it as a invisible wind or a turnstile) through the hole of the donut.
2. The Action: They slowly turn up this wind from zero to a full "unit" of wind.
3. The Reaction: As the wind blows, the dancers (electrons) shift their positions. This shift creates a "charge pump"—a flow of electricity.

The Magic Trick:

• If the dance floor is normal (no knots), the dancers just shuffle a little and then return to their starting spots when the wind stops. The net movement is zero.
• If the dance floor has one knot (an integer topological state), the dancers shift exactly one full step around the donut.
• If the dance floor has a fractional knot (like a Fractional Chern Insulator), the dancers shift only a fraction of a step (e.g., 2/3 of a step).

Why This is a Big Deal

The authors used this "Charge Pumping" method on their AI simulations. Instead of trying to count the knots directly (which was impossible with the blurry AI picture), they just watched how much the crowd shifted when they turned on the wind.

• For Fractional Chern Insulators (FCI): They saw the crowd shift by exactly 2/3 or 1/3 of a step. This confirmed the AI had found these exotic, fractional states.
• For Composite Fermi Liquids (CFL): This is a weird, "liquid" state that was very hard to identify before. The AI was confused about whether it was a solid crystal or a liquid. But when they turned on the wind, the crowd shifted by exactly 1/2 a step. This was the "smoking gun" that proved the AI had found this mysterious liquid state.

The Takeaway

Think of this paper as inventing a new metal detector for quantum knots.

• Before: You had to dig up the whole beach (calculate the entire energy spectrum) to find a buried treasure.
• Now: You just walk a metal detector over the sand (simulate the charge pumping). If it beeps with a specific rhythm (a fractional shift), you know exactly what kind of treasure is buried underneath, even if you can't see it directly.

This breakthrough allows scientists to use powerful AI to discover and verify these exotic quantum states much faster and more reliably, paving the way for future technologies like quantum computers that use these "knots" to store information without errors.

Technical Summary

1. Problem Statement

The identification of topological phases in strongly correlated many-body systems, such as Fractional Chern Insulators (FCIs) and Composite Fermi Liquids (CFLs), is a major challenge in condensed matter physics. While Neural Network Variational Monte Carlo (NNVMC) has emerged as a powerful tool for approximating complex many-body wavefunctions, a critical bottleneck remains: reliably extracting topological invariants (specifically the many-body Chern number) from these wavefunctions.

Existing methods face significant limitations in the context of NNVMC:

• Exact Diagonalization (ED) limitations: Traditional methods for calculating Chern numbers often require access to the full deterministic energy spectrum or all degenerate ground states, which is computationally prohibitive for large systems and incompatible with stochastic NNVMC sampling.
• Symmetry constraints: Methods relying on high-symmetry points in the Brillouin zone are restricted by specific system geometries and have not been successfully extended to fractional states.
• Polarization-based methods: Previous approaches using Resta's polarization operator were largely confined to integer topological invariants or required specific symmetries not present in general FCI states.

The authors aim to develop a robust, generalizable method to determine fractional topological invariants directly from neural network wavefunctions, even in the presence of ground state degeneracy and without requiring the full energy spectrum.

2. Methodology

The authors propose a Charge Pumping Simulation approach inspired by topological charge pumping and Laughlin's gauge argument. The core methodology involves:

• Twisted Boundary Conditions (TBC): The system is simulated on a torus with twisted boundary conditions defined by flux angles $\theta = (\theta_x, \theta_y)$.
• Polarization Operator: They utilize the many-body polarization operator $\hat{Z}_B = \exp(i \mathbf{B} \cdot \sum_i \hat{r}_i)$, where $\mathbf{B}$ is a supercell reciprocal lattice vector.
• Adiabatic Flux Insertion: The method simulates the adiabatic insertion of a magnetic flux (varying $\theta_x$ from $0$ to $2\pi$) while monitoring the phase shift of the expectation value of the polarization operator:
  $$C(\theta_x) = \frac{1}{2\pi} \text{Im} \ln \frac{\langle \Psi_{\theta_x, 0} | \hat{Z}_{B_y} | \Psi_{\theta_x, 0} \rangle}{\langle \Psi_{0, 0} | \hat{Z}_{B_y} | \Psi_{0, 0} \rangle}$$
• Chern Number Extraction: The slope of the polarization phase shift with respect to the flux insertion yields the Chern number. For fractional states, the ground state manifold is degenerate. The authors demonstrate that even if the system does not return to the exact initial state after one unit of flux (due to spectral flow among degenerate states), the average polarization phase shift per unit flux remains fractionally quantized, encoding the topological order ($p/q$).
• Neural Network Implementation: The wavefunctions $|\Psi_\theta\rangle$ are generated using a continuous real-space NNVMC framework (DeepSolid package). The authors employ an "adiabatic continuation" strategy where the neural network parameters optimized at $\theta=0$ are used as initialization for subsequent flux steps to ensure convergence and maintain the correct ground state branch.

3. Key Contributions

• Robust Fractional Invariant Extraction: The paper introduces a method to calculate fractional Chern numbers directly from stochastic neural network wavefunctions, bypassing the need for the full energy spectrum or exact diagonalization.
• Generalization to Degenerate Manifolds: The authors theoretically prove and numerically verify that the charge pumping method works for Fractional Chern Insulators (FCIs) despite ground state degeneracy. They show that the polarization phase shift of a single ground state, when tracked over a full flux cycle (or averaged over the manifold), correctly reveals the fractional topological invariant.
• Identification of Composite Fermi Liquids (CFL): This work presents the first identification of anomalous Composite Fermi Liquid states using neural network wavefunctions. They successfully distinguish CFLs from trivial Fermi liquids and Charge Density Waves (CDWs) in gapless systems where the Chern number is not strictly defined but Hall conductance remains quantized.
• Validation on Moiré Systems: The method is applied to twisted bilayer MoTe$_2$, a leading platform for realizing fractional quantum anomalous Hall effects, validating the approach against known theoretical expectations.

4. Key Results

The authors validated their method on several systems using NNVMC:

• Abelian FCI States (Odd Denominators):

  • 2/3 Filling: Calculated for a $3 \times 4$ lattice. The extracted Chern numbers for the three degenerate ground states were 0.667, 0.664, and 0.667, matching the theoretical value of $2/3$. The results also correctly reproduced the spectral flow behavior (states flowing into each other upon flux insertion).
  • 1/3 Filling: Successfully distinguished between FCI and CDW states. In a geometry incompatible with the CDW pattern ($3 \times 4$), the method yielded a Chern number of 0.344 (close to $1/3$). In a geometry compatible with CDW ($3\sqrt{3} \times 3\sqrt{3}$), the Chern number was vanishingly small, correctly identifying the trivial CDW phase.
  • 3/5 Filling: Demonstrated robust quantization with a slope of -0.589 (close to $-3/5$) for a $5 \times 3$ geometry.

• Composite Fermi Liquid (CFL) States (Even Denominators):

  • 1/2 Filling: Applied to the $4 \times 4$ lattice in twisted MoTe$_2$. The method extracted Chern numbers of -0.495, -0.496, and -0.537 for different momentum sectors. These values are consistent with the theoretical prediction of $-1/2$ for CFL states, confirming the topological nature of the gapless CFL phase.
  • The method successfully excluded competing phases (CDW and normal Fermi liquid) based on structure factors and particle distributions.

• Robustness Checks:

  • The method was shown to be stable against the direction of flux insertion (provided it is not parallel to the polarization direction).
  • It works for non-translational symmetry states (superpositions of momentum sectors), yielding similar quantized slopes.

5. Significance

• Solving a Critical Bottleneck: This work resolves a major obstacle in applying machine learning to correlated topological matter. It provides a reliable "diagnostic tool" to verify if a neural network has successfully captured a topological phase, which was previously difficult without exact diagonalization.
• Scalability: By avoiding the need for the full energy spectrum, this method is highly scalable and applicable to larger system sizes where ED fails.
• New Physics Discovery: The ability to identify CFL states via NNVMC opens new avenues for studying gapless topological phases and anyonic statistics in moiré materials.
• Future Directions: The authors suggest that the polarization matrix (off-diagonal elements) could be used to probe anyonic statistics directly, extending the utility of charge pumping beyond just Chern numbers.

In summary, the paper establishes charge pumping simulation as a universal, robust, and efficient framework for determining topological invariants in neural network-based many-body calculations, significantly advancing the study of fractional topological phases in moiré systems.