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arXiv:2505.12595 · Chinese Physics Letters

Driven Critical Dynamics in the Tricitical Point

Ting-Long Wang, Yi‐Fan Jiang, Shuai Yin

Published Sep 15, 2025• 4 citations
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Abstract

Abstract The conventional Kibble–Zurek mechanism, describing driven dynamics across critical points based on the adiabatic-impulse scenario (AIS), has attracted broad attention. However, the driven dynamics at the tricritical point with two independent relevant directions have not been adequately studied. Here, we employ the time-dependent variational principle to study the driven critical dynamics at a one-dimensional supersymmetric Ising tricritical point. For the relevant direction along the Ising critical line, the AIS apparently breaks down. Nevertheless, we find that the critical dynamics can still be described by finite-time scaling in which the driving rate has a dimension of r μ = z + 1/ ν μ with z and ν μ being the dynamic exponent and correlation length exponent in this direction, respectively. For driven dynamics along another direction, the driving rate has a dimension of r p = z + 1/ ν p with ν p being another correlation length exponent. Our work brings a new fundamental perspective into nonequilibrium critical dynamics near the tricritical point, which could be realized in programmable quantum processors in Rydberg atomic systems.

DOI
10.1088/0256-307x/42/11/110001

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Abstract The conventional Kibble–Zurek mechanism, describing driven dynamics across critical points based on the adiabatic-impulse scenario (AIS), has attracted broad attention. However, the driven dynamics at the tricritical point with two independent relevant directions have not been adequately studied. Here, we employ the time-dependent variational principle to study the driven critical dynamics at a one-dimensional supersymmetric Ising tricritical point. For the relevant direction along the Ising critical line, the AIS apparently breaks down. Nevertheless, we find that the critical dynamics can still be described by finite-time scaling in which the driving rate has a dimension of r μ = z + 1/ ν μ with z and ν μ being the dynamic exponent and correlation length exponent in this direction, respectively. For driven dynamics along another direction, the driving rate has a dimension of r p = z + 1/ ν p with ν p being another correlation length exponent. Our work brings a new fundamental perspective into nonequilibrium critical dynamics near the tricritical point, which could be realized in programmable quantum processors in Rydberg atomic systems.
The Kibble–Zurek (KZ) mechanism has been extensively studied in various second-order phase transitions, yet the case of tricriticality—the point where second-order phase transition lines terminate—remains experimentally elusive. Here, we theoretically propose probing KZ scaling at tricritical points using Rydberg atom arrays arranged as two- and three-leg ladders, which realize the tricritical Ising and tricritical Potts universality classes. By slowly ramping the Rabi frequency and detuning, we extract two relevant tricritical exponents, ν and ν', both via conventional paths from the disordered to the ordered phase and via “tangential” paths confined entirely within the disordered phase. At faster speeds, ramping dynamics go beyond the standard KZ paradigm: data collapse analysis using the parent critical exponents (rather than the tricritical ones) reveals renormalization group flows toward the adjacent second-order critical line, and we identify it as a dynamical analog of Zamolodchikov’s c-theorem. Our protocol is readily implementable on existing Rydberg quantum simulators. This provides a direct route to measuring distinct tricritical exponents which can reveal an emergent spacetime supersymmetry constraint 1/ν − 1/ν' = 1. Moreover, this work deepens our theoretical understanding and opens new avenues for exploring beyond-KZ quantum dynamics with rich renormalization group structure. The Kibble-Zurek mechanism describes universal dynamics near criticality. Here, the authors extend it to tricriticality using Rydberg atom ladders, proposing ramping protocols that measure two tricritical exponents and reveal emergent supersymmetry and dynamical renormalization group flow.
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.
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 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.
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.
The pursuit of a universal description for nonequilibrium critical dynamics in quantum many-body systems stands as a central frontier in modern statistical physics. For driven critical dynamics starting far from the critical point, the well-known Kibble-Zurek (KZ) scaling holds only when the driving rate lies below an upper bound. Here we study driven dynamics restricted to the critical region, and show that robust dynamic scaling behavior exists for arbitrary driving rates. We develop a generalized finite-time scaling (FTS) framework, which provides a unified understanding on the driven dynamics for the full range of quench rates, bridging the KZ scaling in the slow-driving regime and the De~Grandi-Gritsev-Polkovnikov (DGP) scaling in the sudden-quench limit. We verify this unified FTS form through numerical simulations in both quantum critical and tricritical points. The good agreement between theoretical predictions and numerical results confirms the generality of our theory. Our work establishes a universal theory for nonequilibrium critical dynamics spanning the full range of driving rates, with broad implications for quantum quench experiments and out-of-equilibrium statistical mechanics.