A preprint reports that the Dicke model follows different critical-scaling patterns depending on whether dissipation is absent or present. In the closed case, fits gave a dynamical exponent z=0.33 and a critical-window exponent ν=1.52. In the open case, the large-N analysis approached z=1/2 and gave ν=2.02 at κ=1, with ν reported as 1.97(9) across κ.
Posted as arXiv:2608.20067v1 on 20 Aug 2026, the work is a modeling study of N two-level atoms coupled to one cavity mode. It combines analytical and numerical finite-size calculations for closed conditions (κ=0) and open conditions (κ≠0).
One model, two regimes
The aim is to treat static and driven behavior, under both coherent and dissipative conditions, within one mesoscopic finite-size scaling framework. In the setup, κ=0 denotes a closed system, while κ≠0 denotes an open, dissipative one.
The fitted quantities are scaling exponents: z is obtained from critical spectral gaps, ν from how the study’s Binder-ratio diagnostic changes near the critical point, and μ from the leading response to a slow linear ramp. After estimating the leading correction for finite-size drift, the analysis uses that same correction exponent in the separate fits.
The numbers split between closed and open regimes
Without dissipation, the closed-model fit was designed to capture the way finite systems drift as N changes. Truncated at correction order j=2 and using sizes up to N=100, it estimated the leading irrelevant exponent ω=0.372 and an infinite-size Binder ratio of about 0.102.
With ω fixed, the closed critical-gap and Binder-ratio-derivative fits gave z=0.33 and ν=1.52, close to the large-N predictions z=1/3 and ν=3/2. Under a linear ramp, the leading response was quadratic in speed and gave μ=0.99.
With dissipation, small-system calculations became harder to interpret. Exact diagonalization at N≤24 showed crossings and rearrangements among low-lying Liouvillian modes, the mathematical modes used in the open-system calculation. The analysis therefore turned to a numerical large-N expansion, which approached z=1/2; ν was 2.02 at κ=1 and 1.97(9) across κ.
The open ramp response was linear in speed, with μ=0.96 at κ=1 and 1.02(5) across different κ values. That contrasts with the quadratic closed response.
An independent Hartree–Fock–Bogoliubov calculation of the open soft-mode gap at κ=1 showed asymptotic N−1/2 decay over the reported large-N range. That result supported the soft-mode scaling used in the main analysis.
Why size and ramp speed matter
The reported crossover between the two regimes is controlled jointly by dissipation and size. Near the closed fixed point, the crossover variable is κN^(1/3): when it is much smaller than 1, behavior is closed-like; at larger values, the system enters the dissipative regime.
For dynamics, the estimated crossover scale at nonzero κ was s∼κ^3. Fast ramps with s≫κ^3 could appear closed-like, while slower ramps crossed toward open dissipative scaling.
The independently extracted exponents were checked against the Kibble–Zurek relation 1/μ=1/ν+z in both closed and open models. In this work, the equation serves as a consistency check between the static and driven fits.
A model-based result still awaiting wider tests
The work remains a theoretical and numerical model calculation. It is a preprint, and the open exact-diagonalization results are limited to N≤24; the asymptotic open scaling relies on a numerical large-N expansion and an HFB soft-mode estimator.
The closed result also depends on a finite-size extrapolation from calculations up to N=100, and the preprint reports no formal uncertainty intervals for most fitted quantities. Its ramp fits focus on the leading small-speed response, so they do not provide a complete account of faster ramps.
The authors present the framework as a practical route for probing universality in other long-range or all-to-all systems. Whether the same scaling survives experimental testing or applies beyond the reported model and parameter ranges remains open.
Paper data and sources
Original title: Kibble--Zurek Scaling in the Dicke Model at Mesoscopic Scales
Authors: Haowei Li, Hanteng Wang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text