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Nature Communications

Finite-time scaling beyond the Kibble-Zurek prerequisite in Dirac systems

Zhi Zeng, Yin-Kai Yu, Zhixuan Li, Zi-Xiang Li, Shuai Yin

Published Jul 4, 2025• 10 citations
Publisher

Abstract

The conventional Kibble-Zurek mechanism and the finite-time scaling provide universal descriptions of the driven critical dynamics from gapped initial states based on the adiabatic-impulse scenario. Here we investigate the driven critical dynamics in two-dimensional Dirac systems, which harbor semimetal and Mott insulator phases separated by the quantum critical point triggered by the interplay between fluctuations of gapless Dirac fermions and order parameter bosons. We find that despite the existence of the gapless initial phase, the driven dynamics can still be captured by the finite-time scaling form. This leads us to propose a criterion for the validity of Kibble-Zurek mechanism with a gapless initial state. Accordingly, our results generalize the Kibble-Zurek theory to incorporate composite fluctuations and relax its requirement for a gapped initial state to systems accommodating gapless Dirac fermionic excitations. Our work not only brings fundamental perspective into the nonequilibrium critical dynamics, but also provides an approach to fathom quantum critical properties in fermionic systems. The Kibble-Zurek mechanism and finite-time scaling provide descriptions of the driven critical dynamics from gapped initial states. Here, the authors generalize these theories to Dirac systems accommodating gapless initial states.

DOI
10.1038/s41467-025-61611-6

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The conventional Kibble-Zurek mechanism and the finite-time scaling provide universal descriptions of the driven critical dynamics from gapped initial states based on the adiabatic-impulse scenario. Here we investigate the driven critical dynamics in two-dimensional Dirac systems, which harbor semimetal and Mott insulator phases separated by the quantum critical point triggered by the interplay between fluctuations of gapless Dirac fermions and order parameter bosons. We find that despite the existence of the gapless initial phase, the driven dynamics can still be captured by the finite-time scaling form. This leads us to propose a criterion for the validity of Kibble-Zurek mechanism with a gapless initial state. Accordingly, our results generalize the Kibble-Zurek theory to incorporate composite fluctuations and relax its requirement for a gapped initial state to systems accommodating gapless Dirac fermionic excitations. Our work not only brings fundamental perspective into the nonequilibrium critical dynamics, but also provides an approach to fathom quantum critical properties in fermionic systems. The Kibble-Zurek mechanism and finite-time scaling provide descriptions of the driven critical dynamics from gapped initial states. Here, the authors generalize these theories to Dirac systems accommodating gapless initial states.
The conventional Kibble-Zurek mechanism (KZM) and finite-time scaling (FTS) provide universal descriptions of the driven critical dynamics from gapped initial states based on the adiabatic-impulse scenario. In this paper, by leveraging the large-scale quantum Monte Carlo simulation, we investigate the driven critical dynamics in two-dimensional Dirac systems, which harbor semimetal and Mott insulator phases separated by the quantum critical point (QCP) triggered by the interplay between fluctuations of gapless Dirac fermions and order parameter bosons. A remarkable discovery is that, despite the existence of the gapless initial phase, the driven dynamics can still be captured by the FTS form. This leads us to propose a generalized criterion for the validity of KZM and FTS with a gapless initial state. Accordingly, our results significantly generalize the KZM theory to incorporate composite fluctuations and relax its requirement for a gapped initial state to systems accommodating gapless Dirac fermionic excitations. Through successfully extending the KZM and FTS theory to the strongly interacting Dirac QCP, our work not only brings new fundamental perspective into the nonequilibrium critical dynamics, but also provides a novel theoretical approach to fathom quantum critical properties in fermionic systems.
The Kibble-Zurek mechanism (KZM) and finite-time scaling (FTS) provide a foundational framework for driven critical dynamics, yet their predictive power has been largely confined to local observables. Here, we establish a universal finite-time scaling theory for the nonequilibrium dynamics of quantum entanglement. Using unbiased quantum Monte Carlo simulations, we investigate the corner entanglement entropy of (2+1)-dimensional interacting Dirac fermions driven from ordered phases toward a quantum critical point. We find that the corner entanglement accurately obeys a universal driven scaling governed by the driving rate and system size, persisting whether the initial ordered state is fully gapped or hosts gapless Goldstone modes. Crucially, this dynamical entanglement exhibits a logarithmic dependence on the driving rate, from which the universal corner coefficient of the underlying conformal field theory can be robustly extracted far from equilibrium. These results generalize the KZM from local observables to the intrinsic nonlocal quantum information measures, offering a practical blueprint for characterizing quantum criticality and entanglement on programmable quantum simulators.
The Kibble-Zurek mechanism demands an initial adiabatic stage before an impulse stage to have a frozen correlation length that generates topological defects in a cooling phase transition. Here we study such a driven critical dynamics but with an initial condition that is near the critical point and that is far away from equilibrium. In this case, there is no initial adiabatic stage at all and thus adiabaticity is broken. However, we show that there again exists a finite length scale arising from the driving that divides the evolution into three stages. A relaxation--finite-time-scaling--adiabatic scenario is then proposed in place of the adiabatic-impulse-adiabatic scenario of the original Kibble-Zurek mechanism. A unified scaling theory, which combines finite-time scaling with critical initial slip, is developed to describe the universal behavior and is confirmed with numerical simulations of a two-dimensional classical Ising model.
We study the effects induced by the initial conditions in the driving critical dynamics. A protocol is proposed in which a system starts with a nonequilibrium initial state near the critical point. This driving critical dynamics is remarkably different from the dynamics described by the usual adiabatic--impulse--adiabatic scenario of the Kibble-Zurek mechanism. Instead, it is described by a universal relaxation--finite-time scaling--adiabatic scenario. The distinctive feature is the appearance of the relaxation stage in which the increment of the correlation time dominates and the external driving acts only as a perturbation. A unified scaling theory, which includes the initial scaling variables, is proposed to describe the scaling behaviors. The results are confirmed by numerical simulations of the two-dimensional Ising model.
Driven dynamics across a quantum critical point is usually described by the Kibble-Zurek scaling. Although the original Kibble-Zurek scaling requires an adiabatic initial state, it has been shown that scaling behaviors exist even when the driven dynamics is triggered from a thermal equilibrium state exactly at the critical point, in spite of the breakdown of the initial adiabaticity. In this paper, we show that the existence of the scaling behavior can be generalized to the case of the initial state being a thermal equilibrium state near the critical point. We propose a scaling theory in which the initial parameters are included as additional scaling variables due to the breakdown of the initial adiabaticity. In particular, we demonstrate that for the driven critical dynamics in a closed system, the nontrivial thermal effects are closely related to the initial distance to the critical point. We numerically confirm the scaling theory by simulating the real-time dynamics of the one-dimensional quantum Ising model at both zero and finite temperatures.
Proposed as an elegant symmetry relating bosons and fermions, spacetime supersymmetry (SUSY) has been actively pursued in both particle physics and emergent phenomena in quantum critical points (QCP) of topological quantum materials. However, how SUSY casts the light on nonequilibrium dynamics remains open. In this letter, we investigate the Kibble-Zurek dynamics across a QCP with emergent N=2\mathcal{N}=2N=2 spacetime SUSY between the Dirac semimetal and a superconductor through large-scale quantum Monte Carlo simulation. The scaling behaviors in the whole driven process are uncovered to satisfy the full finite-time scaling (FTS) forms. More crucially, we demonstrate that the emergent SUSY manifests in the intimate relation between the FTS behaviors of fermionic and bosonic observables, namely the fermions and bosons acquire the identical anomalous dimensions. Our work not only brings a fundamental new ingredient into the critical theory with SUSY, but also provide the theoretical guidance to experimental detect of QCP with emergent SUSY from the perspectives of Kibble-Zurek mechanism and FTS.