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arXiv:2608.20067

Kibble--Zurek Scaling in the Dicke Model at Mesoscopic Scales

Haowei Li, Hanteng Wang

cond-mat.quant-gascond-mat.stat-mechquant-phSubmitted Aug 20, 2026• Announced Aug 21, 2026• 0 citations
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Abstract

The Dicke model is a paradigmatic setting for collective light-matter physics and the superradiant phase transition. Yet extracting the critical exponents is challenging at experimentally accessible mesoscopic sizes, due to the slow divergence of the correlation time under all-to-all coupling and a photon-loss-driven crossover to a distinct dissipative universality class. Here, we perform a large-$N$ analysis that identifies distinct coherent and dissipative fixed points for the closed and open Dicke models. We then develop a unified mesoscopic scaling framework that incorporates the leading irrelevant correction and, going beyond static and spectral probes, brings ramping dynamics under the same scaling description. It recovers the corresponding exponents, verifies Kibble-Zurek scaling, and clarifies how finite size, dissipation, and speed compete in the ramping dynamics. Our work thus establishes a unified framework for resolving static and dynamical critical scaling in closed and open quantum systems, with broader applicability to mesoscopic systems with long-range interactions.

Comments
9 pages, 5 figures

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The Dicke model is a paradigmatic setting for collective light-matter physics and the superradiant phase transition. Yet extracting the critical exponents is challenging at experimentally accessible mesoscopic sizes, due to the slow divergence of the correlation time under all-to-all coupling and a photon-loss-driven crossover to a distinct dissipative universality class. Here, we perform a large-NNN analysis that identifies distinct coherent and dissipative fixed points for the closed and open Dicke models. We then develop a unified mesoscopic scaling framework that incorporates the leading irrelevant correction and, going beyond static and spectral probes, brings ramping dynamics under the same scaling description. It recovers the corresponding exponents, verifies Kibble-Zurek scaling, and clarifies how finite size, dissipation, and speed compete in the ramping dynamics. Our work thus establishes a unified framework for resolving static and dynamical critical scaling in closed and open quantum systems, with broader applicability to mesoscopic systems with long-range interactions.
We investigate the quench dynamics of an open quantum system involving a quantum phase transition. In the isolated case, the quench dynamics involving the phase transition exhibits a number of scaling relations with the quench rate as predicted by the celebrated Kibble-Zurek mechanism. In contact with an environment however, these scaling laws break down and one may observe an anti-Kibble-Zurek behavior: slower ramps lead to less adiabatic dynamics, increasing thus nonadiabatic effects with the quench time. In contrast to previous works, we show here that such anti-Kibble-Zurek scaling can acquire a universal form in the sense that it is determined by the equilibrium critical exponents of the phase transition, provided the excited states of the system exhibit singular behavior, as observed in fully connected models. This demonstrates novel universal scaling laws granted by a system-environment interaction in a critical system. We illustrate these findings in two fully connected models, namely, the quantum Rabi and the Lipkin-Meshkov-Glick models. In addition, we discuss the impact of nonlinear ramps and finite-size systems.
We analyze quantum systems under a broad class of protocols in which the temperature and a Hamiltonian control parameter are ramped simultaneously and, in general, in a nonlinear fashion toward a quantum critical point. Using an open-system version of a Kitaev quantum wire as an example, we show that, unlike finite-temperature protocols at fixed temperature, these protocols allow us to probe, in an out-of-equilibrium situation and at finite temperature, the universality class (characterized by the critical exponents ννν and zzz) of an equilibrium quantum phase transition at zero temperature. Key to this is the identification of ramps in which both coherent and incoherent parts of the open-system dynamics affect the excitation density in a non-negligible way. We also identify the specific ramps for which subleading corrections to the asymptotic scaling laws are suppressed, which serves as a guide to dynamically probing quantum critical exponents in experimentally realistic finite-temperature situations.
When a system is swept through a quantum critical point, the Kibble-Zurek mechanism predicts that the average number of topological defects follows a universal power-law scaling with the ramp timescale. This scaling behavior is determined by the equilibrium critical exponents of the underlying phase transition. We show that the correspondence between Kibble-Zurek scaling and quantum criticality does not hold generally. In particular, the defect density can exhibit a suppression faster than the Kibble-Zurek prediction even when the quench crosses a critical point, while conventional Kibble-Zurek scaling may persist for quenches through a noncritical point. Our results, based on models representative of a broad class of quasi-one-dimensional Fermi systems, identify the dynamical conditions under which universal defect scaling emerges and clarify the relation between defect generation and equilibrium criticality. © 2026 authors. Published by the American Physical Society.
The Dicke model famously exhibits a phase transition to a superradiant phase with a macroscopic population of photons and is realized in multiple settings in open quantum systems. In this paper, we study a variant of the Dicke model where the cavity mode is lossy due to the coupling to a Markovian environment while the atomic mode is coupled to a colored bath. We analytically investigate this model by inspecting its low-frequency behavior via the Schwinger-Keldysh field theory and carefully examine the nature of the corresponding superradiant phase transition. Integrating out the fast modes, we can identify a simple effective theory allowing us to derive analytical expressions for various critical exponents including the dynamical exponent. We find excellent agreement with previous numerical results when the non-Markovian bath is at zero temperature; however, contrary to these studies, our low-frequency approach reveals that the same exponents govern the critical behavior when the colored bath is at finite temperature unless the chemical potential is zero. Furthermore, we show that the superradiant phase transition is classical in nature, while it is genuinely nonequilibrium. We derive a fractional Langevin equation and conjecture the associated fractional Fokker-Planck equation that captures the system's long-time memory as well as its nonequilibrium behavior. Finally, we consider finite-size effects at the phase transition and identify the finite-size scaling exponents, unlocking a rich behavior in both statics and dynamics of the photonic and atomic observables.
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.