Mozib Bin Awal and Prabwal Phukon of Dibrugarh University investigated critical relaxation dynamics and their link to quantum information scrambling using a holographic model. Their work demonstrates a universal power-law divergence in autocorrelation time, τ~|ε|^(-Δ), with Δ≃ 2/3, aligning with both analytical predictions and previous gravitational results. By varying the CFT central charge alongside temperature, the researchers found the leading relaxation dynamics consistently maintained this same critical exponent. The study further explores how thermodynamic relaxation transitions into quantum scrambling, offering new insight into black hole information paradoxes.
Holographic RN-AdS Black Holes and CFT Correspondence
The extended thermodynamic description of black holes treats the cosmological constant as a thermodynamic pressure, a concept first formalized with the equation P = -Λ/(8π G_N), where Λ represents the cosmological constant and is Newton’s gravitational constant. This approach allows for a holographic matching between the bulk gravitational system and its boundary conformal field theory (CFT), preserving the first laws of thermodynamics in a generalized setting.
Visser proposed that a complete holographic thermodynamic description requires accounting for a chemical potential associated with color charge. Previous work established a dynamical critical exponent, Δ, consistently near 0.66, a value aligning with the analytical prediction of Δ = 2/3. The study builds on earlier investigations of black hole phase transitions initiated by P.C.W. Davies and P. Hut, and significantly informed by Maldacena’s 1997 formulation of the AdS/CFT correspondence.
Li and Wang previously examined black hole phase transitions by analyzing the free energy landscape, specifically the Hawking-Page transition in both Einstein and massive gravity. Their approach was then extended to the small-large black hole phase transition in Reissner-Nordström Anti-de Sitter (RN-AdS) black holes, revealing a phase structure closely mirroring the liquid-gas transition observed in the van der Waals system. This connection between gravitational systems and fluid dynamics provides a powerful analogy for understanding complex phenomena.
The current work explores the transition between the relaxation time and the scrambling time, investigating whether these timescales exhibit a predictable relationship within the holographic framework. Addressing these questions is important for resolving long-standing questions about black hole information paradoxes and the fundamental nature of quantum gravity.
Free Energy Landscapes Define Black Hole Transition Kinetics
Recent work demonstrates that the structure of these landscapes, specifically the height of free energy barriers, governs the kinetics of phase transitions in black holes, offering new insights into their dynamic behavior. Going beyond the conventional single-parameter analysis, variations of the CFT central charge were incorporated and a combined scaling variable involving temperature and central charge was identified, demonstrating that the leading relaxation dynamics retain the same critical exponent along different trajectories in the two-dimensional parameter space. The study also verified this numerically using the full nonlinear free-energy landscape.
Critical Slowing Down in Dual CFT Dynamics
Researchers at Dibrugarh University, specifically Mozib Bin Awal and Prabwal Phukon, formulated the thermodynamics using boundary conformal field theory variables and employed Langevin and Fokker-Planck formalisms to model the stochastic evolution of the horizon radius. This approach allowed for a detailed examination of the relaxation dynamics within the dual conformal field theory, revealing a pronounced slowing down as the system approached criticality.
Examining the smallest nonzero eigenvalue of the Fokker-Planck operator further substantiated these findings, confirming the suppression of rapid relaxation modes as the system approached the critical regime. The work explores the connection between thermodynamic relaxation and quantum information scrambling by comparing the relaxation timescale with the fast-scrambling timescale.
Autocorrelation Time Divergence Near Critical Points
The diminishing curvature near critical points directly enhances autocorrelation time, a measurable indicator of how quickly a system returns to equilibrium after a disturbance. A similar pattern emerges near spinodal points, where the free-energy landscape shares characteristics with those found at criticality, suggesting a shared underlying mechanism.
To quantify this slowing, researchers investigated the scaling of autocorrelation time as the system nears the critical point, extracting the dynamical critical exponent that governs the divergence of relaxation time. Their analysis revealed a power-law divergence described by the equation τ~|ε|^(-Δ), where ε represents the reduced temperature or charge and Δ is the critical exponent.
Combined Scaling of Temperature and Central Charge
Researchers discovered a combined scaling variable linking temperature and central charge, demonstrating consistent relaxation dynamics despite parameter shifts within the conformal field theory. This scaling variable, defined as u≡ + , allows for analysis of critical behavior when both temperature and central charge are varied simultaneously, rather than independently.
The study reveals that the leading relaxation dynamics maintain a critical exponent of 2/3 along different trajectories in this two-dimensional parameter space, suggesting a robustness in the system’s response. The generalized free-energy landscape governing the system, expressed as G(x,T,Q,C) = C( + 3x^4) + , exhibits nonzero coefficients indicating both temperature and central charge influence the equilibrium configuration.
Specifically, the relaxation time is inversely proportional to this curvature, with the dissipation coefficient ζ playing a role, resulting in the equation *τ = ζ/G^(′′)(x_e)*. The resulting relaxation time is governed by a dynamical exponent of Δ = 2/3, but is now characterized by the combined scaling field u. Numerical analysis confirms this behavior, yielding a relaxation time described by *τ(ε,C) = a/C|ε|^(-2/3)*, where ε represents the reduced temperature and a = (24^(2/3)ζ)/(72(||T_c)^(2/3))≃ 0.0369 ζ.
Crossover from Thermodynamic Relaxation to Quantum Scrambling
The relationship between how quickly a system relaxes to equilibrium and the speed at which quantum information spreads within it is now more clearly defined, according to new analysis of holographic conformal field theories. Researchers identified a crossover point where thermodynamic relaxation, the system returning to its lowest energy state, becomes slower than quantum scrambling, the rapid dispersal of information. This transition’s location is increasingly confined as the system’s central charge, a measure of its complexity, increases, behaving as ε~(C ln C)^(-3/2) in the large-central-charge limit.
This crossover scale was determined by comparing the relaxation timescale, derived from a generalized free-energy landscape, with the timescale for fast scrambling, a benchmark for information dispersal. The analysis reveals that as the central charge grows, the regime where thermodynamic relaxation lags behind scrambling narrows, suggesting a fundamental limit to how slowly a system can return to equilibrium before quantum effects dominate.
Crossover Scale Behavior in Large-Central-Charge Limit
Specifically, calculations reveal this scale behaves as ε~(C ln C)^(-3/2), where C represents the CFT central charge, indicating the influence of logarithmic contributions from the scrambling time. The study demonstrates that the ratio of relaxation to scrambling timescales increases with the central charge, eventually reaching unity at a specific value of ε. This suggests a hierarchy of timescales emerges within the dual CFT, with relaxation typically occurring faster than scrambling except within a narrow, diminishing region near the critical point. The analysis yields a large-C behavior of ε~(C ln C)^(-3/2) along the critical locus, further supporting this trend.
Numerical Verification of Nonlinear Free-Energy Landscape
Numerical verification of the free-energy landscape confirms a predictable collapse of relaxation times across varying parameters, bolstering the holographic framework used to model black hole dynamics. By directly comparing relaxation and scrambling timescales, researchers identified a crossover scale where these two processes become comparable. The consistency between analytical predictions and full nonlinear calculations reinforces the robustness of the model and its potential for understanding the interplay between critical phenomena and quantum information processing in strongly coupled systems.
👉 More information
🗞 Critical Slowing Down and the Crossover to Quantum Scrambling in Holographic CFTs
✍️ Mozib Bin Awal (Department of Physics, Dibrugarh University – Assam, India); Prabwal Phukon
🧠 ArXiv: https://arxiv.org/html/2611.11889v1
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