🔬 condensed matter

Kibble-Zurek Dynamics in Two-dimensional Frustrated Systems with a Neural Foundation-state Subspace Method

This paper introduces a Neural Foundation-state Subspace (NFS) method that efficiently simulates near-adiabatic dynamics in two-dimensional frustrated quantum systems, successfully validating Kibble-Zurek scaling in the transverse-field Ising model and providing strong numerical evidence for continuous critical behavior across the Néel-to-spin-liquid transition in the $J_1$ - $J_2$ Heisenberg model.

Original authors: Linda Mauron, Luciano Loris Viteritti, Zakari Denis, Riccardo Rende, Giuseppe Carleo

Published 2026-09-09

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CC BY 4.0

Original authors: Linda Mauron, Luciano Loris Viteritti, Zakari Denis, Riccardo Rende, Giuseppe Carleo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of physics, there is a special class of moments when matter fundamentally changes its nature. These are phase transitions, the same kind of shifts that turn water into ice or iron into a magnet. In the quantum world, where particles behave more like waves than solid objects, these transitions can be incredibly subtle. When a system is pushed slowly through such a change, it usually follows the path of least resistance, staying in its lowest energy state. But if the push is too fast, the system gets left behind, unable to keep up with the changing conditions. This lag creates a universal pattern of behavior, a kind of cosmic fingerprint that depends only on the type of transition and not on the specific materials involved. Scientists call this the Kibble-Zurek mechanism. It is a powerful idea because it suggests that by watching how a system fails to adapt, we can learn deep truths about how it behaves when it is perfectly still. However, watching these quantum systems in real time is notoriously difficult, especially when they are large and the particles are frustrated, meaning they are pulled in conflicting directions by their neighbors.

A team of researchers at the École Polytechnique Fédérale de Lausanne and the Flatiron Institute has found a new way to watch this drama unfold. They developed a computational method that acts like a high-speed camera for quantum systems, allowing them to simulate the exact moment a material crosses a critical threshold. Instead of trying to calculate the impossible complexity of every single particle moving at once, they built a "foundation" model. Imagine a master map that accurately describes the calm, low-energy states of a system across a wide range of conditions. The researchers trained a sophisticated computer program, based on neural networks, to create this map. Once the map was ready, they selected a small, carefully chosen group of states from it to serve as a fixed stage. They then watched how the system moved across this stage as it was driven through a phase transition. Because the stage was already built, they could run the simulation thousands of times with different speeds and system sizes without having to recalculate the underlying physics each time. This approach bypassed the usual bottlenecks that have kept scientists from studying these dynamics in two-dimensional systems.

The team first tested their method on a well-known model called the transverse-field Ising model, which describes a grid of magnetic spins. They pushed the system through its critical point and measured how much energy was left over and how the spins were correlated. The results matched the expected universal patterns perfectly, confirming that their method could capture the Kibble-Zurek mechanism with high precision. They then applied this same technique to a much more difficult problem: the frustrated square-lattice J1-J2 Heisenberg model. In this system, magnetic interactions compete with one another, creating a tug-of-war that can melt the usual magnetic order and potentially give rise to a mysterious state known as a quantum spin liquid. For years, scientists have debated whether the transition from the ordered magnetic state to this liquid state is a smooth, continuous change or a sudden jump.

By simulating this frustrated system on grids up to 16 by 16 sites, the researchers observed a clear signal of the Kibble-Zurek mechanism. The way the system responded to the driving speed revealed that the transition is indeed continuous. They extracted specific numbers that describe the critical behavior, finding values that align closely with previous static calculations but now confirmed through a dynamic lens. This provides strong, independent evidence that the transition is smooth, settling a long-standing question about the nature of this quantum phase. The study demonstrates that universal scaling laws, which were once thought to be accessible only in equilibrium, can be reliably measured in the chaotic rush of a real-time transition. By turning a complex many-body problem into the movement of a few simple numbers on a pre-built stage, the researchers have opened a practical path to exploring the dynamics of quantum criticality in systems that were previously out of reach.

Technical Summary: Kibble-Zurek Dynamics in Two-dimensional Frustrated Systems with a Neural Foundation-state Subspace Method

Problem Statement
Accessing real-time quantum dynamics in strongly interacting, two-dimensional (2D) many-body systems remains a significant numerical challenge. While one-dimensional systems are well-handled by tensor-network methods (e.g., DMRG), 2D systems suffer from rapid entanglement growth, dynamical sign problems in Quantum Monte Carlo, and stochastic noise/conditioning issues in time-dependent variational Monte Carlo (tVMC). This limitation is particularly acute for frustrated quantum magnets, such as the $J_1$ - $J_2$ Heisenberg model, where the nature of the transition between Néel antiferromagnetic order and a potential quantum spin liquid (QSL) phase is debated. Specifically, while static ground-state studies suggest a continuous transition with universal critical exponents, the non-equilibrium dynamics across this transition—governed by the Kibble-Zurek (KZ) mechanism—have remained largely unexplored due to the difficulty of simulating near-adiabatic quenches in 2D.

Methodology: Neural Foundation-state Subspace (NFS)
The authors introduce the Neural Foundation-state Subspace (NFS) method to simulate near-adiabatic dynamics. The approach constructs a low-energy variational subspace to project the many-body Schrödinger equation, reducing the problem to the evolution of a small set of linear coefficients.

Foundation Neural-Network Quantum State (FNQS): A single neural network (based on a transformer architecture) is trained once to represent the ground-state manifold across the entire range of couplings spanned by the quench (from initial $\gamma_0$ to final $\gamma_f$ ). This FNQS, denoted $|\Phi_\theta(\gamma)\rangle$ , serves as a "foundation" state.
Subspace Construction: A small, fidelity-selected subset of $M$ states ( $|\phi_k\rangle = |\Phi_\theta(\gamma_k)\rangle$ ) is extracted from the FNQS at discrete coupling points along the driving path. These states form a fixed basis for the time evolution.
Linear Variational Ansatz: The time-evolved state is expressed as a linear superposition: $|\Psi(t)\rangle = \sum_{k=1}^M c_k(t) |\phi_k\rangle$ .
Projected Dynamics: The time-dependent Schrödinger equation is projected onto this fixed subspace, yielding a system of coupled linear ordinary differential equations for the coefficients $c(t)$ $c (t)$ : $S \dot{c}(t) = -i H(\gamma(t)) c(t)$ $S \overset{c}{˙} (t) = - i H (γ (t)) c (t)$ .
- $S$ is the overlap matrix (time-independent).
- $H(\gamma(t))$ is the Hamiltonian matrix, constructed as a linear combination of precomputed projected operators $\langle \phi_i | \hat{h}_\mu | \phi_j \rangle$ .
Efficiency: Because the basis is fixed and operators are precomputed, the method avoids neural-network optimization and sampling during real-time evolution. Different ramp times ( $\tau_Q$ ) and protocols can be simulated with negligible additional cost once the subspace is established.

Key Results
The method was validated on two models:

2D Transverse-Field Ising Model:
- Benchmarked against exact diagonalization on $6 \times 6$ lattices, showing agreement up to the critical point with a maximal relative energy error of 0.05%.
- Finite-size scaling analysis ( $6 \le L \le 12$ ) recovered the expected KZ scaling.
- Extracted critical exponents: $\alpha = 0.362(7)$ , $\nu = 0.567(18)$ , and $\eta = 0.043(29)$ (with fixed dynamical exponent $z=1$ ). These values align with previous dynamical studies and static estimates.
Frustrated Square-Lattice $J_1$ - $J_2$ Heisenberg Model:
- Applied to clusters up to $16 \times 16$ ( $M=28$ basis states) to study the transition at $J_2/J_1 = 0.49$ .
- The method provided strong numerical evidence of KZ scaling across the Néel-to-spin-liquid transition.
- Extracted exponents: $\nu = 1.23(15)$ and $\eta = 0.409(19)$ (at fixed $z=1$ ).
- These dynamical estimates are in quantitative agreement with static finite-size scaling and level spectroscopy results ( $\nu \approx 1.21$ , $\eta \approx 0.410$ ), supporting the hypothesis of a continuous critical transition.

Significance and Claims
The paper claims to provide the first evidence of the Kibble-Zurek mechanism in the frustrated $J_1$ - $J_2$ Heisenberg model. By successfully extracting universal critical exponents from non-equilibrium dynamics, the NFS method offers an independent dynamical probe of the transition's nature, corroborating static ground-state findings.

The authors emphasize that the NFS method shifts the computational burden to a single ground-state optimization, avoiding the instabilities of tVMC during real-time evolution. They note that while the method is naturally tailored to slow ramps and near-critical dynamics (where the system remains close to the ground-state manifold), its accuracy diminishes for fast quenches or long post-critical evolution where the state explores directions not captured by the instantaneous ground states. Nevertheless, the work establishes a practical route for dynamical tests of quantum criticality in 2D frustrated systems where direct real-time methods are currently limited.

Technical Summary: Kibble-Zurek Dynamics in Two-dimensional Frustrated Systems with a Neural Foundation-state Subspace Method

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