Finite-temperature spin diffusion in the two-dimensional XY model

This paper presents a combined theoretical and experimental study using a dynamical high-temperature expansion method and an optical lattice quantum simulator to quantify spin diffusion in the two-dimensional square lattice XY model, achieving excellent agreement that validates quantum simulation platforms beyond one dimension.

The Gist

The Big Picture: Tracking a Crowd of Spinning Tops

Imagine a giant, flat checkerboard made of tiny, spinning tops. In the world of quantum physics, these tops represent the "spin" of particles. Usually, these tops want to point in a specific direction, but in this experiment, they are free to wobble and swap their energy with their neighbors.

The scientists wanted to answer a simple question: If you create a crowd of tops all pointing one way on the left side of the board, and a crowd pointing the other way on the right, how fast does the "spin" spread out until everything is mixed evenly?

This spreading process is called diffusion. It's like dropping a drop of ink into a glass of water and watching it slowly spread until the whole glass is a uniform color. In this case, the "ink" is the magnetic spin, and the "water" is the grid of particles.

The Challenge: Two Different Ways to Look at the Problem

The researchers approached this problem from two angles, like two detectives trying to solve the same mystery:

1. The Theorists (The Mathematicians): They tried to calculate exactly how fast the spin should spread using complex math. The problem is that quantum systems are incredibly chaotic. It's like trying to predict the exact path of every single raindrop in a storm. For a long time, their math could only handle very hot temperatures or very small grids, and it wasn't accurate enough to match real life.
2. The Experimentalists (The Builders): They built a real-life version of this checkerboard using ultracold atoms (specifically Lithium) trapped in a grid of laser light (an "optical lattice"). They created a "wall" to separate the atoms, then knocked the wall down and watched how the atoms mixed.

The Breakthrough: A New Mathematical Tool

The biggest hurdle was that the experimentalists could measure the mixing speed, but the theorists couldn't calculate it accurately enough to compare. The old math tools were like trying to measure the ocean with a teaspoon; they worked for small cups of water but failed for the vast ocean of quantum interactions.

The team introduced a new mathematical method called Dyn-HTE (Dynamic High-Temperature Expansion).

• The Analogy: Imagine trying to understand a complex song. Old methods tried to listen to the whole song at once and got confused by the noise. The new method breaks the song down into its individual notes (frequency moments) and reconstructs the melody from those notes. This allowed the theorists to calculate the mixing speed with high precision, even at temperatures where the atoms are "warm" enough to be chaotic.

The Experiment: A Digital Micromirror and a Laser Grid

Here is how the experiment worked, step-by-step:

1. Setting the Stage: They used a laser grid to trap thousands of Lithium atoms. They used a special device (a Digital Micromirror Device, or DMD) to project a "wall" of light, creating two separate rooms for the atoms.
2. The Imbalance: They loaded more atoms into the left room than the right, creating an imbalance.
3. The Release: They quickly removed the wall.
4. The Observation: They took photos of the atoms over time. They watched the "imbalance" (the difference in density between the left and right sides) fade away as the atoms diffused across the grid.
5. The Thermometer: To make sure the math matched the experiment, they had to know the exact "temperature" of the atoms. They did this by looking at how close neighbors were to each other (like checking how tightly people are standing in a crowd). This allowed them to measure the temperature without disturbing the system.

The Result: A Perfect Match

When they compared the results:

• The Experiment: Measured a specific speed at which the spin spread.
• The New Math: Predicted that exact same speed.

This is a big deal. It is the first time scientists have achieved a perfect, quantitative match between a theory and an experiment for spin diffusion in two dimensions (a flat grid). Previously, this had only been done in one dimension (a single line), or the numbers didn't match up.

Why This Matters (According to the Paper)

• Validation: It proves that the new mathematical tool (Dyn-HTE) works. It also proves that the quantum simulator (the laser grid) is accurate enough to be trusted as a "supercomputer" for solving physics problems that normal computers can't handle.
• Temperature Matters: The paper highlights that you cannot just assume the system is "infinitely hot" (a common simplification). The experiment showed that the temperature did matter, and the new math was the only tool precise enough to account for it.
• Future Directions: The paper suggests this method can now be used to study more complex scenarios, such as what happens if the grid is stretched (making it harder for atoms to move in one direction than the other) or if the system is slightly "broken" to see how that changes the flow.

Summary Analogy

Think of this paper as the moment a car manufacturer finally built a car engine that runs exactly as the blueprints predicted.

• Before: The engineers (theorists) had blueprints that were slightly off, and the mechanics (experimentalists) built engines that ran, but no one knew exactly why or if the blueprints were right.
• Now: The engineers used a new, better drafting tool (Dyn-HTE) to fix the blueprints. The mechanics built the engine. They started the car, and the speedometer matched the blueprint perfectly. This proves both the new drafting tool and the engine design are correct.

Technical Summary

Technical Summary: Finite-temperature spin diffusion in the two-dimensional XY model

Problem Statement
The paper addresses the challenge of quantitatively describing spin diffusion in chaotic many-body quantum systems at finite temperature. While diffusion is a classical phenomenon characterized by a diffusion constant, its theoretical derivation in quantum systems requires solving the Schrödinger equation over large spatiotemporal scales, a task that exceeds the capabilities of current classical numerical methods, particularly in two dimensions (2D). Although spin diffusion has been extensively studied in one-dimensional (1D) systems, quantitative theoretical validation of 2D spin transport has remained elusive. A specific gap exists regarding the 2D Fermi-Hubbard model, where experimental data from Nichols et al. (2019) could not be reproduced theoretically due to the difficulty of handling disparate transport time-scales (spin-exchange vs. particle hopping). This work focuses on the simpler, paradigmatic spin-1/2 XY model on a square lattice to establish a benchmark for cross-validating analog quantum simulation and theoretical algorithms.

Methodology
The study employs a combined theoretical and experimental approach:

• Theoretical Framework: The authors utilize a recently developed Dynamic High-Temperature Expansion (Dyn-HTE) method. Unlike the Finite-Temperature Lanczos Method (FTLM), which is restricted by finite-size effects and slow convergence at lower temperatures, Dyn-HTE operates in the thermodynamic limit. It computes high-order high-temperature expansions (HTEs) for the frequency moments of the dynamical structure factor (DSF) and the spin conductivity. By reconstructing the spectral function from these moments and applying the Einstein relation, the authors calculate the spin diffusion constant $D(T)$. The theory is extended to the spin-1/2 XY model up to order 14 in the dimensionless inverse temperature $J/T$.
• Experimental Platform: The experiment uses an optical lattice quantum simulator composed of ultracold $^7$Li bosonic atoms. By operating in the hard-core limit ($t_{BH} \ll U$), the Bose-Hubbard model maps onto the spin-1/2 XY model.
  • State Preparation: A domain-wall state is prepared using a digital micromirror device (DMD) to create an imbalance in atom density (spin) across a 20 $\times$ 20 region of interest (ROI).
  • Dynamics: The system is quenched to initiate diffusion, and the time evolution of the density imbalance is tracked via fluorescence imaging.
  • Thermometry: To ensure quantitative comparison, the temperature is determined independently using nearest-neighbor density correlators, which are compared against Dyn-HTE predictions.

Key Results

1. Quantitative Agreement: The experimentally measured spin diffusion constant $D$ shows excellent agreement with the theoretical predictions derived from Dyn-HTE at the specific finite temperature determined by the experiment ($J/T \approx 0.47$). This agreement is significant because the experimental value differs substantially from the infinite-temperature ($T=\infty$) theoretical prediction, highlighting the necessity of precise thermometry.
2. Validation of Dyn-HTE: The study demonstrates that Dyn-HTE outperforms existing methods like FTLM in the 2D regime, providing converged results down to moderate temperatures ($T/J \approx 2/3$) where FTLM fails due to finite-size restrictions.
3. Integrability Breaking: Theoretically, the authors investigated the effect of lattice anisotropy ($J_x \neq J_y$) on spin diffusion. They found that in the limit of weakly coupled chains ($J_y/J_x \ll 1$), the diffusion constant scales as $D \propto (J_y/J_x)^{-2}$, consistent with Fermi's golden rule arguments for perturbed integrable systems. For stronger coupling ($J_y/J_x \gtrsim 1$), a crossover to $D \propto (J_y/J_x)^{-1}$ is observed.
4. Experimental Precision: The experimental determination of $D = 0.82(3)J$ (with $\hbar=1$) was achieved by measuring the linear relationship between the spin current and the spin gradient (Fick's law) derived from the time evolution of the domain wall.

Significance and Claims
The paper claims to present the first quantitative match between an experimentally measured spin diffusion constant and a theoretical prediction for a 2D quantum system. This achievement serves as a critical milestone in the validation of analog quantum simulators, demonstrating their capability to provide accurate data for quantum dynamics beyond the reach of classical numerical simulations.

The authors emphasize that this success relies on two key factors:

1. The development of the Dyn-HTE method, which provides a reliable theoretical tool for 2D quantum dynamics at finite temperatures.
2. The implementation of careful thermometry in the experiment, which is essential for meaningful comparison with theory, as the diffusion constant is highly temperature-dependent.

The work establishes a new standard for cross-validation in quantum simulation and opens the door for future quantitative studies of more complex observables, such as frequency-dependent spin conductivity, and other lattice geometries or models (e.g., the Fermi-Hubbard model) where theoretical understanding is currently lacking.