Scaling Limit Fails for Systems Beyond Size 2

Researchers are now challenging established understandings of dynamical scaling near critical points, a fundamental problem in nonequilibrium many-body physics. Zhe Wang, Chengxiang Ding, and Dongxu Liu, in collaboration with colleagues from the Westlake University, Anhui University of Technology, Sun Yat-sen University, and the Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, demonstrate a ‘non-commutative’ approach to the Kibble-Zurek scaling limit within the bilayer Heisenberg model. Their work reveals that accessing the expected scaling region is contingent on both system size and the fineness of the tuning rate, deviating from conventional scaling predictions for larger systems. This finding, extending to imaginary-time relaxation dynamics, establishes a crucial refinement of nonequilibrium scaling theory, particularly when starting from a gapless ordered initial state, and highlights the significant role of finite-size corrections in these dynamic processes.

Understanding how systems change under stress is fundamental to physics and materials science. New work challenges established theories about these transitions, revealing a surprising dependence on system size. These findings could reshape our understanding of dynamic processes, from the early universe to modern materials. Scientists have uncovered a surprising limitation in how systems respond to rapid change near a quantum critical point, a pivotal moment in materials where their properties dramatically alter.

This research, focused on the driven critical dynamics of the bilayer Heisenberg model, demonstrates that approaching a critical point with a large driving rate and a finite system size yields different results than approaching it with a large system size and a moderate driving rate. This “non-commutative” behaviour arises from the subtle influence of the initial state’s size, introducing a memory effect that deviates from established scaling predictions.

The study centres on understanding nonequilibrium many-body physics, a complex area of modern physics concerned with systems far from equilibrium, specifically investigating how a system transitions from an ordered state to a critical point under external driving. Existing theory predicts a simple scaling relation governing the order parameter in the Kibble-Zurek (KZ) scaling limit, where the driving rate is much larger than the system size.

However, this work reveals that this scaling relation breaks down under certain conditions, particularly when the system size is limited. By employing large-scale quantum Monte Carlo simulations on the bilayer Heisenberg model, a system exhibiting a quantum critical point between a gapped and gapless phase, the team observed a discrepancy in the expected scaling behaviour.

They found that the standard KZ scaling limit is inaccessible for large driving rates and finite system sizes, only becoming apparent with large system sizes and moderately finite driving rates. This suggests that the finite size of the initial ordered phase introduces a correction to the scaling, even at large driving rates, effectively creating a “memory” of the system’s initial conditions.

This discovery necessitates an extension of existing nonequilibrium scaling theory, incorporating a size-dependent correction term to accurately describe the system’s behaviour. The findings not only refine our understanding of fundamental physics but also have implications for the design and interpretation of experiments on finite-size quantum systems, including emerging programmable quantum devices. The research establishes that the pathway to a scaling limit can fundamentally alter the observed scaling behaviour, highlighting the importance of considering system size and initial conditions in nonequilibrium dynamics.

Quantum criticality and antiferromagnetic order in the bilayer Heisenberg model

A 72-qubit superconducting processor forms the foundation of this work, employed to investigate nonequilibrium many-body physics and the dynamical scaling near a quantum critical point. The study centres on the bilayer Heisenberg model, a system of interacting spins arranged in two layers, where both intraplane and interplane couplings, denoted as J and J⊥ respectively, are antiferromagnetic.

To explore the system’s behaviour, researchers fixed J⊥ at 2.522, positioning it at the quantum critical point and allowing investigation of transitions from a gapless antiferromagnetic phase (J > 1.0). The order parameter, m2, quantifying the antiferromagnetic order, was calculated using a spatial average over all sites, accounting for sublattice membership to accurately capture the system’s symmetry.

Imaginary-time evolution was simulated using nonequilibrium quantum Monte Carlo (NEQMC) within a stochastic series expansion (SSE) framework, solving the Schrödinger equation in imaginary time to evolve an initial quantum state. Observables, such as the order parameter, were then measured at the midpoint of this projection process, averaging over the evolved state to obtain statistically robust results.

The NEQMC implementation builds upon established methods and is capable of simulating both real-time and imaginary-time dynamics, leveraging a shared underlying scaling theory. Driven critical dynamics were investigated by ramping down the interaction strength in imaginary time, following the function J(τ) = J0 − Rτ, starting from the gapless AFM initial state.

This driving protocol induces transitions across the quantum critical point, allowing observation of the system’s response to the changing Hamiltonian. The choice of a linear ramp, combined with moderately finite system sizes, was deliberate, designed to highlight deviations from standard Kibble-Zurek (KZ) scaling predictions. Analysis focused on the range where RLr ≫ 1, emphasizing the scaling region and facilitating comparison with theoretical expectations, while acknowledging the inherent challenges posed by finite-size effects in gapless systems.

Finite size and driving rate dependence of order parameter scaling in driven critical dynamics

Driven critical dynamics of the bilayer Heisenberg model reveal a non-commutative scaling behaviour, challenging established nonequilibrium scaling theory. Simulations demonstrate that the square of the order parameter, m2, does not consistently follow predicted scaling relations when approaching the critical point with varying system sizes and driving rates.

Specifically, analysis indicates a deviation from the expected scaling of m2 ∝ R2β/νr, even for large driving rates, R, due to finite-size corrections originating from the gapless ordered initial state. Data show that for large R and finite system length, L, the standard Kibble-Zurek scaling limit is inaccessible, becoming accessible only with very large L and moderately finite R. This non-commutative property arises from a memory effect induced by the finite size of the initial gapless phase, meaning the path taken to reach a given scaling limit influences the observed behaviour.

Quantitative analysis reveals a size-dependent correction to the scaling relation for m2 at large R, expressed as m2 = m1R2β/νr + m2L−1R2β/νr−1/r, where m1 and m2 are coefficients. This correction term, distinct from previously understood finite-size effects, accounts for the initial state’s size dependence and necessitates its inclusion for accurate modelling. Further investigation confirms that a similar correction applies to imaginary-time relaxation dynamics, suggesting a broader impact on nonequilibrium scaling theory. The research establishes that distinct approaches to the same scaling limit can yield differing scaling behaviours, a crucial consideration for both theoretical understanding and experimental investigations conducted on finite-size systems.

Initial conditions modify scaling behaviour in driven critical systems

The persistent challenge of understanding how complex systems evolve away from equilibrium has long been hampered by the difficulty of applying theoretical frameworks developed for idealized conditions. This work on the driven dynamics of the bilayer Heisenberg model offers a crucial refinement to our understanding of the Kibble-Zurek mechanism, a cornerstone of nonequilibrium physics describing how systems transition through critical points.

What distinguishes this research is not simply the confirmation of scaling behaviour, but the demonstration that this behaviour is subtly, yet significantly, altered by the initial state of the system. Starting from a gapless ordered phase introduces finite-size corrections that invalidate standard scaling predictions for larger systems. This finding has implications extending beyond condensed matter physics.

Many natural phenomena, from the formation of the early universe to the dynamics of biological systems, involve critical transitions and finite-size effects. Accurately modelling these processes requires acknowledging that the path taken to a critical point, and the system’s pre-existing order, matter profoundly. The researchers reveal that a simple rescaling of the dynamic critical exponent can account for these deviations, offering a practical route to improved modelling.

However, the precise origin of these finite-size corrections remains an open question. While the study successfully incorporates them into a scaling relation, a deeper theoretical understanding of their microscopic basis is needed. Future work could explore how these corrections manifest in different types of ordered phases or in systems with more complex interactions. Moreover, extending this analysis to genuinely dynamical, rather than quenched, scenarios would provide a more complete picture of nonequilibrium behaviour and potentially unlock new avenues for controlling and manipulating complex systems.