[Paper Review] Dynamics of magnetization at infinite temperature in a Heisenberg spin chain

Eliott Rosenberg, Trond Andersen+179 more|arXiv (Cornell University)|Jun 15, 2023

Theoretical and Computational Physics5 references4 citations

TL;DR

This study investigates quantum magnetization dynamics in a 46-qubit superconducting processor simulating a Heisenberg spin chain at infinite temperature. By measuring the full probability distribution of transferred magnetization, including higher moments, the experiment finds superdiffusive behavior consistent with KPZ universality in the first two moments, but rules out the KPZ conjecture based on third and fourth moments, highlighting the critical role of higher-order statistics in identifying universal dynamic behavior in quantum systems.

ABSTRACT

Understanding universal aspects of quantum dynamics is an unresolved problem in statistical mechanics. In particular, the spin dynamics of the 1D Heisenberg model were conjectured to belong to the Kardar-Parisi-Zhang (KPZ) universality class based on the scaling of the infinite-temperature spin-spin correlation function. In a chain of 46 superconducting qubits, we study the probability distribution, $P(\mathcal{M})$, of the magnetization transferred across the chain's center. The first two moments of $P(\mathcal{M})$ show superdiffusive behavior, a hallmark of KPZ universality. However, the third and fourth moments rule out the KPZ conjecture and allow for evaluating other theories. Our results highlight the importance of studying higher moments in determining dynamic universality classes and provide key insights into universal behavior in quantum systems.

Motivation & Objective

• To test the conjecture that spin dynamics in the 1D Heisenberg model belong to the Kardar-Parisi-Zhang (KPZ) universality class at infinite temperature.
• To investigate whether higher moments of the magnetization transfer distribution can falsify or confirm proposed universality classes in quantum many-body systems.
• To explore the role of full counting statistics—beyond mean values—in characterizing universal quantum dynamics in non-equilibrium systems.
• To determine whether superdiffusive scaling in the first two moments is sufficient to confirm KPZ universality or if higher-order moments are necessary for definitive classification.
• To leverage high-fidelity, high-sampling-rate superconducting quantum processors to probe experimentally inaccessible regimes of quantum dynamics.

Proposed method

• The Heisenberg spin chain is simulated using a 46-qubit superconducting quantum processor with periodic Floquet application of high-fidelity fSim gates.
• The system is initialized in a domain-wall state with infinite-temperature initial condition (μ → ∞), corresponding to maximal magnetization imbalance.
• The probability distribution P(ℳ) of magnetization transferred across the chain’s center is measured via full quantum state tomography and repeated sampling.
• The first four moments of P(ℳ) are extracted to analyze scaling behavior and test universality conjectures.
• The fSim gate parameters (θ, φ) are tuned to approximate the XXZ Hamiltonian with Δ = sin(φ/2)/sin(θ), enabling simulation of the Heisenberg model at Δ = 1.
• The experiment is designed to avoid finite-size effects by restricting measurements to times t ≤ NQ/2, ensuring the system remains in the transient, non-equilibrium regime.

Figure 1: Domain wall relaxation in the Heisenberg XXZ spin chain . ( A ) Schematic of the unitary gate sequence used in this work, where fSim gates are applied in a Floquet scheme on a 1D chain of $N_{Q}=46$ qubits. ( B ) Relaxation dynamics as a function of site and cycle number for $\mu=\infty,0.

Experimental results

Research questions

• RQ1 Does the superdiffusive scaling of the first two moments of magnetization transfer in the Heisenberg spin chain at infinite temperature confirm KPZ universality?
• RQ2 Can higher moments (skewness and kurtosis) of the magnetization distribution falsify the KPZ conjecture, even when lower moments appear consistent with it?
• RQ3 To what extent do full counting statistics—beyond expectation values—improve the classification of universal dynamic behavior in quantum systems?
• RQ4 How robust are the results to variations in the initial state’s symmetry-breaking parameter μ, particularly in the limit μ → 0?
• RQ5 What is the role of integrability and conserved quantities in determining the universality class of non-equilibrium quantum dynamics?

Key findings

• The first two moments of the magnetization transfer distribution exhibit superdiffusive scaling with a dynamical exponent z ≈ 3/2, consistent with KPZ universality.
• The third moment (skewness) of the distribution is non-zero and significantly deviates from the symmetric prediction of the KPZ class, contradicting the KPZ conjecture.
• The fourth moment (kurtosis) also disagrees with KPZ predictions, further ruling out the KPZ universality class for the observed dynamics.
• The results demonstrate that higher moments are essential for distinguishing between competing universality classes and cannot be inferred from low-order statistics alone.
• The experiment confirms that superconducting quantum processors enable high-precision measurement of full probability distributions, allowing falsification of theoretical conjectures in quantum dynamics.
• The findings show that even in integrable systems with global SU(2) symmetry, non-equilibrium dynamics do not conform to the KPZ class, challenging previous conjectures based on lower-order moment analysis.

Figure 2: Mean and variance in various transport regimes . ( A ) Mean of transferred magnetization $\langle\mathcal{M}(t)\rangle$ as a function of cycle number for initial states with $\mu=0.5$ and for $\Delta=0.16$ (purple triangles), 1 (orange squares), 1.6 (green circles). Light and dark curves s