Robustness of Kardar-Parisi-Zhang-like transport in long-range interacting quantum spin chains

Using state-of-the-art tensor network methods, this study demonstrates that long-range interacting quantum spin chains exhibit robust, long-lived Kardar-Parisi-Zhang (KPZ) superdiffusive spin transport despite lacking integrability, a phenomenon attributed to their proximity to the integrable Inozemtsev family and observable in various experimental platforms like Rydberg atom arrays.

The Gist

The Big Picture: A Traffic Jam in a Quantum City

Imagine a long line of people (quantum spins) standing in a row. Each person is holding a tiny magnet that can point either Up or Down.

In a normal, everyday world, if you push a "Up" magnet into the middle of a line of "Down" magnets, the information about that push spreads out slowly and randomly, like a drop of ink diffusing in a glass of water. This is called diffusion. It's predictable and boring.

However, physicists have discovered that in certain special, perfectly ordered quantum systems (like the "Heisenberg model"), this doesn't happen. Instead of spreading out like ink, the "Up" magnet moves in a wild, chaotic, yet strangely organized way. It spreads faster than diffusion but slower than a bullet. This is called superdiffusion.

The paper asks a very important question: Does this wild, super-fast spreading happen only in perfectly ordered systems, or does it survive even when the system is messy and the people can talk to each other from far away?

The Cast of Characters

To answer this, the authors looked at three different "cities" (models) of quantum magnets:

1. The Neighbors (NN Model): People only talk to the person standing immediately next to them. This is the "classic" system where we know the wild spreading happens.
2. The Holographic Twins (Haldane-Shastry Model): People can talk to everyone in the line, but the strength of the conversation drops off very sharply (like $1/r^2$). This system is perfectly ordered (integrable) and behaves like a bullet—everything moves instantly (ballistic).
3. The Long-Range Talkers (Power-Law Models): This is the new, messy system. People can talk to anyone, but the strength drops off at a rate of $1/r^\alpha$ (where $\alpha$ is between 2 and infinity). These systems are not perfectly ordered; they are chaotic and "non-integrable."

The Experiment: The "Domain Wall"

The researchers set up a test. Imagine the left half of the line is full of "Up" magnets and the right half is full of "Down" magnets. This is a sharp boundary, or a Domain Wall.

They then turned on the clock and watched what happened.

• If it's Diffusion: The wall melts slowly and spreads out like a blurry cloud.
• If it's Ballistic: The wall stays sharp and shoots forward like a laser beam.
• If it's KPZ (Superdiffusion): The wall melts, but it does so in a specific, jagged, "rough" way that follows a famous mathematical pattern called the Kardar-Parisi-Zhang (KPZ) class. Think of it like a pile of sand growing unevenly; it's not smooth, but it has a specific, predictable roughness.

The Surprise Discovery

The team used powerful supercomputers (using a method called "Tensor Networks") to simulate these systems with up to 2,000 people in the line.

The Result: Even though the "Long-Range Talkers" (the Power-Law models) are messy and chaotic, they still showed the wild, KPZ superdiffusion!

Even though physics textbooks say that messy systems should eventually turn into boring diffusion, these systems held onto their wild behavior for a very long time (up to 1,000 time steps). It's as if you threw a rock into a pond, and instead of the ripples dying out, they kept bouncing around in a complex, beautiful pattern for hours.

The "Why": The Secret Neighbor

Why did the messy systems behave so well? The authors found a clever explanation.

They realized that these messy, long-range systems are actually just slightly tweaked versions of a special, perfectly ordered family of systems called the Inozemtsev models.

• The Analogy: Imagine a perfectly tuned piano (the Inozemtsev model). If you slightly loosen a few strings (add disorder), the piano might sound a little off, but for a long time, it still plays a beautiful melody. It takes a very long time for the music to turn into noise.
• The authors showed that the messy Power-Law models are so close to these perfect Inozemtsev models that they inherit their "superdiffusive" DNA. The "messiness" is too weak to kill the special behavior quickly.

Why Should You Care? (The Real-World Connection)

This isn't just math for math's sake. This matters for Quantum Simulators—the super-advanced computers scientists are building right now using:

• Rydberg Atoms: Giant atoms that can "talk" to each other from far away.
• Polar Molecules: Molecules that interact strongly over long distances.

These real-world machines naturally have "long-range interactions" (like the Power-Law models in the paper).

The Takeaway: If you build a quantum computer with these atoms, don't worry that the chaos will ruin the transport of information. The paper suggests that these machines will naturally exhibit this robust, wild, KPZ-like transport. This means we can observe these exotic quantum behaviors in the lab, even if our machines aren't perfectly tuned.

Summary in One Sentence

Even when quantum magnets are allowed to talk to each other from far away and the system is messy, they still manage to move information in a wild, organized, and surprisingly robust way, because they are secretly "cousins" to a perfectly ordered family of systems.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in non-equilibrium statistical physics: Does the Kardar-Parisi-Zhang (KPZ) universality class, which governs superdiffusive spin transport in integrable, nearest-neighbor (NN) Heisenberg chains, persist in systems with long-range interactions?

• Context: In generic, non-integrable 1D quantum systems at infinite temperature, conserved charges (like spin) are expected to diffuse normally ($z=2$). However, the integrable NN Heisenberg chain exhibits anomalous superdiffusion ($z=3/2$) belonging to the KPZ universality class.
• The Conflict: The Haldane-Shastry (HS) model, a prototypical integrable long-range interacting spin chain with $1/r^2$ interactions, exhibits ballistic transport ($z=1$), not KPZ superdiffusion.
• The Gap: It is unclear whether the transition from NN (KPZ) to HS (Ballistic) is abrupt or if there is a regime where KPZ-like dynamics survive in non-integrable, long-range interacting models. This is crucial for experimental platforms (Rydberg atoms, polar molecules, trapped ions) that naturally realize long-range interactions but are often non-integrable.

2. Methodology

The authors employ state-of-the-art tensor network methods to simulate large systems ($L \sim 2000$) and late times ($t \sim 10^3/J$), overcoming previous limitations of exact diagonalization (limited to $N \sim 20$).

• Models Investigated:
  1. Power-law Heisenberg Models: Non-integrable models with interactions $f(r) \propto 1/r^\alpha$ for $2 < \alpha < \infty$.
  2. Inozemtsev Models: A family of integrable models interpolating between HS ($\kappa=0$) and NN ($\kappa \to \infty$) via interactions $f(r) \propto \sinh^2(\kappa)/\sinh^2(\kappa r)$.
  3. Anisotropic XXZ Models: Models relevant to Rydberg arrays and polar molecules, including mixed power-law interactions.
• Numerical Techniques:
  • Doubled Hilbert Space: The mixed state density matrix $\rho$ is vectorized into a pure state in a doubled Hilbert space ($2N$ spins) to allow for Matrix Product State (MPS) evolution.
  • Time-Dependent Variational Principle (TDVP): Used to evolve the state under long-range Hamiltonians. The Hamiltonian is approximated as a sum of exponentials to represent it as a Matrix Product Operator (MPO).
  • Symmetry Exploitation: $U(1)$ charge conservation (spin $S^z$) is enforced in the doubled space to significantly reduce computational cost.
  • Initial States: Weak domain wall states ($\rho \propto e^{\lambda \hat{R}}$) are evolved to probe infinite-temperature transport.
• Observables:
  • Charge Transfer ($P_Q(t)$): Scales as $t^{1/z_Q}$ to extract the dynamical critical exponent $z$.
  • Two-Point Structure Factors: $C(r, t) = \langle \hat{q}_r(t)\hat{q}_0(0) \rangle$, rescaled by $t^{1/z}$ to test for data collapse against KPZ and Gaussian (diffusive) scaling functions.
  • Current Density: Computed via commutators of the Hamiltonian with local energy/spin densities.

3. Key Contributions

1. Discovery of Robust KPZ Transport in Non-Integrable Long-Range Systems: The authors demonstrate that despite the lack of integrability and the asymptotic expectation of diffusion, power-law interacting Heisenberg models ($2 < \alpha < \infty$) exhibit long-lived KPZ-like superdiffusion ($z \approx 3/2$) up to the longest simulable timescales.
2. Microscopic Explanation via Inozemtsev Proximity: They propose and verify that the origin of this robustness is the proximity of power-law models to the integrable Inozemtsev family. By treating the power-law interaction as a weak, symmetry-preserving perturbation to an optimal Inozemtsev model, they show the perturbation strength is small ($\epsilon \lesssim 0.1$), leading to anomalously long crossover times to diffusion.
3. Characterization of Energy Transport: Unlike spin transport, energy transport in the Inozemtsev family does not fall into a single universality class but smoothly interpolates between ballistic (HS) and superdiffusive/ballistic (NN) behaviors depending on the interaction range.
4. Experimental Relevance: The study extends to anisotropic XXZ models (relevant to Rydberg atoms and polar molecules), showing that non-diffusive transport (ballistic to subdiffusive) persists in these experimentally realizable, non-integrable settings.

4. Key Results

A. Power-Law Heisenberg Models (Non-Integrable)

• Spin Transport: For exponents $2 < \alpha \lesssim 6$, the dynamical exponent $z_S$ remains close to $3/2$ (KPZ) up to $t \sim 400/J$.
  • For $\alpha \lesssim 3$, $z_S$ shows a slow finite-time flow toward diffusion ($z=2$), but the timescale for this crossover is extremely long.
  • For $\alpha \gtrsim 4$, the flow is negligible, and the system behaves indistinguishably from the KPZ class within simulation windows.
• Scaling Functions: The spin structure factors and current densities collapse onto the KPZ scaling function (specifically the Airy$_2$ process profile) rather than the Gaussian (diffusive) profile.
• Energy Transport: Surprisingly, energy transport remains ballistic ($z_E = 1$) across all $\alpha$, independent of the spin transport behavior.

B. Inozemtsev Models (Integrable)

• Universality: For all $\kappa > 0$ (intermediate to NN), spin transport is KPZ-like ($z_S = 3/2$).
• Crossover: For small $\kappa$ (close to HS), there is a long crossover from ballistic ($z=1$) to KPZ ($z=3/2$) behavior.
• Energy Transport: Energy transport is ballistic for all $\kappa$, but the shape of the energy profile and the current density depend on $\kappa$, indicating no single universality class for energy transport in this family.

C. Perturbation Analysis

• The authors quantify the "distance" between a power-law model and the nearest Inozemtsev model. They find that power-law models can be viewed as weak perturbations ($\epsilon \sim 0.01 - 0.1$) to an Inozemtsev model.
• Citing recent theories on integrability-breaking perturbations, they argue that the crossover time to diffusion scales as $t^* \sim \epsilon^{-\nu}$ (with $\nu \approx 6$), explaining why KPZ dynamics persist for such long times.

D. Anisotropic Models (XXZ)

• In dipolar ($1/r^3$) and mixed-power-law models:
  • Spin: Exhibits a rich phase diagram ranging from ballistic ($\Delta \approx 0.3-0.5$) to subdiffusive ($\Delta \ge 2$).
  • Energy: Remains highly anomalous (superdiffusive) even in regimes where spin transport becomes diffusive.

5. Significance

• Theoretical Impact: The work challenges the assumption that long-range interactions immediately destroy KPZ universality. It suggests that KPZ dynamics are a robust feature of a broad class of 1D quantum systems, provided they are sufficiently close to an integrable point with continuous non-Abelian symmetry.
• Experimental Guidance: The results provide a strong theoretical basis for experimentalists working with Rydberg atom arrays, trapped ions, and polar molecules. These platforms naturally host long-range interactions and are often non-integrable; the paper predicts that KPZ-like superdiffusion should be observable in these systems for experimentally relevant timescales, despite the lack of fine-tuned integrability.
• Methodological Advance: The successful application of MPS/TDVP to long-range, non-integrable systems with $L \sim 2000$ sets a new benchmark for studying quantum hydrodynamics in regimes previously inaccessible to numerical simulation.

In summary, the paper establishes that KPZ-like superdiffusion is robust against long-range interactions and integrability breaking, mediated by the proximity to the Inozemtsev integrable family, and predicts that these exotic transport phenomena are readily observable in current quantum simulation experiments.