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

Technical Summary: Kibble–Zurek Scaling in the Dicke Model at Mesoscopic Scales

Problem Statement
The Dicke model serves as a paradigmatic framework for collective light–matter physics and superradiant phase transitions. While the underlying Hamiltonian is simple, quantitatively extracting critical exponents in experimentally accessible mesoscopic regimes (finite particle number $N$) remains a significant challenge. Two primary obstacles obscure the asymptotic scaling behavior:

1. Slow Divergence of Correlation Time: In all-to-all coupled systems, the critical correlation time grows slowly with $N$, causing finite-size corrections to dominate and bias estimates of the dynamical exponent.
2. Dissipative Crossover: Unavoidable photon loss acts as a relevant perturbation, driving the system toward a distinct dissipative universality class. At mesoscopic sizes, the crossover between the coherent (closed) and dissipative (open) limits is broad, meaning apparent exponents often fail to coincide with either asymptotic fixed point.
   Furthermore, existing dynamical scaling laws for all-to-all models are formulated for the asymptotic large-$N$ limit, leaving a gap in controlled extraction methods for finite systems where strong finite-size drift and intertwined crossover effects (size, dissipation, ramp speed) complicate the analysis.

Methodology
The authors develop a unified mesoscopic scaling framework that treats static, spectral, and driven (ramping) dynamics on equal footing for both closed and open Dicke models.

1. Large-$N$ Field Theory: The authors utilize a unified Keldysh field theory to analyze the critical exponents. By keeping leading $1/N$ corrections explicit, they identify distinct fixed points:

   • Closed Case ($\kappa=0$): The system belongs to the all-to-all Ising universality class with exponents $z=1/3$ and $\nu=3/2$.
   • Open Case ($\kappa \neq 0$): The system flows to the Model-A universality class with exponents $z=1/2$ and $\nu=2$.
   • The theory establishes the Kibble–Zurek (KZ) scaling relation $1/\mu = 1/\nu + z$, where $\mu$ characterizes the ramping speed.

2. Unified Scaling Protocol: To extract exponents from mesoscopic data, the authors propose a two-stage protocol that explicitly accounts for the leading irrelevant correction (exponent $\omega$):

   • Stage I (Fixing $\omega$): A dimensionless observable (e.g., the Binder ratio $U$) is analyzed at the critical point. The finite-$N$ drift is fitted to a truncated series in powers of $N^{-\omega}$ to determine the irrelevant exponent.
   • Stage II (Extracting $z, \nu, \mu$): Using the fixed $\omega$, the authors independently extract the remaining exponents from:
     • $z$: The scaling of the critical gap (Liouvillian gap for open systems).
     • $\nu$: The size dependence of the derivative of the dimensionless observable with respect to the tuning parameter.
     • $\mu$: The size dependence of the leading speed correction in ramped dynamics (e.g., $\partial_s U$ or $\partial^2_s U$).

Key Results
Numerical simulations and large-$N$ expansions were performed to validate the framework:

• Closed Dicke Model:

  • The irrelevant exponent was determined as $\omega \approx 0.372$.
  • Extracted exponents recovered the theoretical large-$N$ values: $z \approx 0.33$ (theoretical $1/3$), $\nu \approx 1.52$ (theoretical $3/2$), and $\mu \approx 0.99$ (theoretical $1$).
  • The KZ relation $1/\mu = 1/\nu + z$ was verified.
  • Ramped dynamics showed a quadratic speed dependence (absence of linear correction) consistent with time-reversal symmetry.

• Open Dicke Model:

  • A joint fit across dissipation rates $\kappa \in [0.1, 3]$ yielded $\omega \approx 0.363$.
  • For $\kappa=1$, exact diagonalization gave $z \approx 0.48$. However, for generic $\kappa$, small sizes accessible to exact diagonalization ($N \le 24$) exhibited mode crossings and rearrangements that obscured the asymptotic scaling.
  • To resolve this, a numerical large-$N$ expansion was performed for $N \in [10^5, 10^7]$, confirming that the extracted exponent approaches the theoretical value $z=1/2$ for different $\kappa$.
  • The exponent $\nu$ was extracted as $\nu \approx 2.02$ (at $\kappa=1$) and fluctuated around the expected value for different $\kappa$.
  • The exponent $\mu$ was found to be $\approx 0.96$ (at $\kappa=1$) and $\approx 1.02(5)$ across different $\kappa$.
  • The KZ relation held for the open case as well.
  • Ramped dynamics exhibited a linear speed correction due to broken time-reversal symmetry.

• Crossover Dynamics:

  • The study clarified the competition between finite size ($N$), dissipation ($\kappa$), and ramp speed ($s$).
  • A crossover scaling variable $\kappa N^{1/3}$ was identified, distinguishing "closed-like" behavior in small/weakly dissipative systems from "open-like" behavior.
  • The interplay between the KZ time scale ($t_{KZ} \sim s^{-1/3}$) and the dissipative time scale ($t_\kappa \sim \kappa^{-1}$) explains why raw ramp data can appear complex, necessitating the unified scaling treatment to disentangle these effects.

Significance and Claims
The paper claims to establish a unified framework for resolving both static and dynamical critical scaling in closed and open quantum systems with long-range interactions. Its primary contributions are:

1. Resolution of Mesoscopic Ambiguity: By incorporating leading irrelevant corrections, the protocol recovers the correct thermodynamic exponents from mesoscopic data where standard finite-size scaling would fail or yield biased estimates.
2. Verification of KZ Scaling: It provides the first controlled verification of Kibble–Zurek scaling in a fully connected model without spatial domain formation, confirming the relation $1/\mu = 1/\nu + z$ in both coherent and dissipative regimes.
3. General Applicability: The authors state that this framework is not limited to cavity-QED but applies broadly to other mesoscopic systems with long-range interactions, such as trapped ions and Sachdev–Ye–Kitaev-type models, particularly where controlled large-$N$ theories are unavailable and direct data extraction is essential.

The work emphasizes that while the thermodynamic limit is clear in principle, experimental accessibility is often restricted to mesoscopic scales; this protocol provides a practical route to probe universality classes in that regime.