Beyond Equilibrium: Universal Signatures of Quantum Phase Transitions

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Author: Denis Avetisyan

New research reveals a unifying principle governing how quantum systems transition between phases, even when driven far from equilibrium.

The study of the dissipative Dicke model reveals that near the critical point, the thermodynamic metric $ζ$ scales according to $ζ \sim |g - g_{c}|^{x}$, with a numerically determined exponent of $x = -2.9953 \pm 0.0009$ - a value remarkably consistent with the theoretical prediction of $x = -(\alpha + \gamma)$, given parameters $\alpha = 2$ and $\gamma = 1$.
The study of the dissipative Dicke model reveals that near the critical point, the thermodynamic metric $ζ$ scales according to $ζ \sim |g – g_{c}|^{x}$, with a numerically determined exponent of $x = -2.9953 \pm 0.0009$ – a value remarkably consistent with the theoretical prediction of $x = -(\alpha + \gamma)$, given parameters $\alpha = 2$ and $\gamma = 1$.

A universal scaling law connecting entropy production, driving speed, and the Liouvillian gap characterizes dissipative quantum phase transitions.

While conventional analyses of critical phenomena largely focus on closed quantum systems, understanding thermodynamic irreversibility in open, dissipative environments remains a significant challenge. This work, ‘Thermodynamic universality across dissipative quantum phase transitions’, establishes a universal framework for characterizing these transitions, demonstrating predictable scaling of nonadiabatic entropy production with driving duration and the Liouvillian gap. These findings reveal a fundamental connection between thermodynamic irreversibility and critical slowing down in driven-dissipative systems, echoing the Kibble-Zurek mechanism. Could this framework provide a pathway to controlling and optimizing the response of complex quantum devices operating far from equilibrium?

Beyond Equilibrium: The Inevitable Dance of Dissipation

The conventional understanding of phase transitions hinges on systems reaching equilibrium – a stable, unchanging state isolated from external influences. However, this framework struggles to capture the behavior of the vast majority of physical systems, which are inherently open and constantly interacting with their surroundings. These systems are not merely passively responding to stimuli, but are actively driven by energy or matter flow, establishing a nonequilibrium steady state. Consider a laser, a biological cell, or even a stirred fluid – each maintains a specific state not through isolation, but through a continuous exchange with its environment. This constant interplay fundamentally alters the conditions for phase transitions, necessitating a new theoretical approach that moves beyond the limitations of equilibrium physics and acknowledges the pervasive role of dissipation – the irreversible loss of energy – in shaping the behavior of matter.

Dissipative Quantum Phase Transitions, or DQPTs, represent a significant broadening of the conventional understanding of how matter changes state. Unlike traditional phase transitions which occur in isolated, equilibrium systems, DQPTs describe shifts in systems constantly exchanging energy with their surroundings – a far more realistic depiction of most physical processes. These transitions aren’t defined by the minimization of free energy, but rather by changes in the steady-state properties of a system driven out of equilibrium. This means a system can undergo a dramatic change in behavior – for example, transitioning from a localized to a delocalized state – not because it’s seeking a lower energy configuration, but because of the interplay between internal dynamics and external driving forces. The study of DQPTs therefore necessitates a departure from established theoretical tools and offers a pathway to understanding complex phenomena in areas like open quantum systems, driven-dissipative lattices, and even certain biological systems.

Investigating Dissipative Quantum Phase Transitions (DQPTs) demands a departure from the conventional toolkit of equilibrium statistical mechanics. Traditional methods, built upon the foundations of closed systems and thermodynamic equilibrium, falter when applied to the open, driven systems where DQPTs arise. These transitions are characterized by steady states maintained by constant energy exchange with the environment, necessitating new theoretical approaches. Researchers are now developing frameworks centered around concepts like non-equilibrium Green’s functions, quantum trajectories, and effective Hamiltonians that incorporate dissipation. This shift involves reframing the order parameter not as a property of the system’s ground state, but as a characteristic of the steady state, and defining new criteria for identifying transitions based on the system’s response to perturbations. Ultimately, a broadened theoretical landscape is crucial to fully characterize and predict the behavior of DQPTs, unlocking insights into a vast range of physical phenomena from driven optical lattices to biological systems.

The Liouvillian spectrum reveals a finite gap separating stable and unstable states until the critical coupling is reached, at which point the system's evolving state diverges from its adiabatic trajectory, defining a freeze-out point where the system transitions from quasiadiabatic to impulsive behavior.
The Liouvillian spectrum reveals a finite gap separating stable and unstable states until the critical coupling is reached, at which point the system’s evolving state diverges from its adiabatic trajectory, defining a freeze-out point where the system transitions from quasiadiabatic to impulsive behavior.

Modeling Open Systems: The Master Equation as a Necessary Fiction

The Master Equation is a Lindblad equation, a first-order differential equation that governs the time evolution of the density matrix, $ \rho $, for an open quantum system. Unlike the Schrödinger equation which describes isolated systems, the Master Equation explicitly accounts for the system’s interaction with an external environment. This interaction leads to dissipation and decoherence, phenomena not captured by unitary time evolution. The equation’s general form is $ \frac{d\rho}{dt} = -i\hbar^{-1}[H, \rho] + \mathcal{L}[\rho] $, where $ H $ is the system Hamiltonian and $ \mathcal{L}[\rho] $ is the Lindblad superoperator describing the effects of the environment. Solving the Master Equation yields the time-dependent density matrix, allowing for the prediction of observable quantities and a comprehensive understanding of the system’s dynamics in the presence of environmental influences.

Dissipation in open quantum systems, arising from interactions with the environment, is mathematically represented within the Master Equation formalism through Lindblad operators. These operators, denoted as $L_i$, describe the rates and mechanisms by which the system loses energy or coherence to the environment. The Master Equation incorporates these operators to modify the system’s time evolution, ensuring that the resulting density matrix remains completely positive and physically realistic. Specifically, the Lindblad term adds a contribution proportional to $L_i \rho L_i^\dagger$ and $L_i^\dagger \rho L_i$ to the time derivative of the density operator $\rho$, effectively modeling decoherence and decay processes without violating the constraints of quantum mechanics. The precise form of the $L_i$ depends on the specific environmental interaction and the nature of the dissipation.

The Kerr Parametric Oscillator (KPO) and the Open Dicke Model are frequently utilized as benchmark systems for evaluating the Master Equation approach to open quantum systems. The KPO, exhibiting nonlinear interactions between cavity modes, provides a well-defined Hamiltonian allowing for analytical and numerical comparisons with Master Equation solutions, particularly regarding squeezed state generation and photon loss. Similarly, the Open Dicke Model, describing the interaction between an ensemble of two-level atoms and a single mode of the electromagnetic field subject to dissipation, facilitates validation of the Master Equation’s ability to accurately model collective atomic behavior and spontaneous emission. These systems offer quantifiable metrics – such as photon statistics, coherence properties, and population dynamics – against which the accuracy and efficiency of different Master Equation implementations can be rigorously assessed, enabling refinement of numerical methods and theoretical approximations.

In the dissipative Dicke model, the soft mode symplectic eigenvalue scales with the square root of the inverse Liouvillian gap, indicating a sensitivity to dissipation and coupling strength parameters (ωc=1, ωz=1, κ=0.2, g0=gc−5×10−3, gf=gc−10−8, τq=102…105).
In the dissipative Dicke model, the soft mode symplectic eigenvalue scales with the square root of the inverse Liouvillian gap, indicating a sensitivity to dissipation and coupling strength parameters (ωc=1, ωz=1, κ=0.2, g0=gc−5×10−3, gf=gc−10−8, τq=102…105).

Unveiling Criticality: The Language of Scaling and the Thermodynamic Metric

Dynamical Quantum Phase Transitions (DQPTs), while occurring in nonequilibrium systems, demonstrate critical behavior analogous to equilibrium phase transitions. This similarity manifests in the presence of scaling exponents that characterize the divergence or vanishing of certain physical quantities as the control parameter approaches the transition point. Specifically, these exponents dictate how observables change in the vicinity of the DQPT, with universal values often observed across different physical systems exhibiting DQPTs. The identification and analysis of these scaling exponents, such as the critical exponent $v$ relating the correlation length to the control parameter, provides a framework for classifying and understanding the universality classes of DQPTs, mirroring the approach used in the study of static critical phenomena.

The Thermodynamic Metric ($g_{ij}$) provides a quantifiable measure of the separation between distinct nonequilibrium steady states in a system’s parameter space. Mathematically, it is defined through the inverse of the Fisher Information Matrix, calculated from the probability distribution of fluctuations around each steady state. This metric is not merely a descriptive tool; it directly influences the determination of critical scaling exponents. Specifically, the eigenvalues of the Thermodynamic Metric near a critical point dictate the characteristic divergence or vanishing of physical quantities, thereby establishing the values of the exponents that define the universality class of the dynamical phase transition. Variations in the metric’s behavior-such as its scaling with system size or proximity to the critical point-directly correlate with observed critical phenomena.

The Quantum Fisher Information (QFI) provides a quantifiable metric for system sensitivity, proving useful in characterizing critical behavior near dynamical phase transitions. Analysis of the KMB QFI-a specific formulation-reveals a consistent scaling exponent of $\alpha = 2$ in both the Dicke model, a system modeling light-matter interactions, and the Kerr model, a nonlinear optical system. This consistent value of $\alpha$ across distinct physical models suggests a universal characteristic of these dynamical phase transitions, allowing for precise determination of the transition point based on QFI measurements and facilitating a deeper understanding of the system’s response to external perturbations.

Finite-size scaling analysis reveals the critical behavior of the quantum Fisher information in the dissipative Kerr model, demonstrating its divergence as the system approaches the thermodynamic limit.
Finite-size scaling analysis reveals the critical behavior of the quantum Fisher information in the dissipative Kerr model, demonstrating its divergence as the system approaches the thermodynamic limit.

Universal Behavior and the Kibble-Zurek Mechanism: Echoes of Inevitable Failure

Recent investigations into dynamical quantum phase transitions (DQPTs) reveal a surprisingly consistent thermodynamic behavior across diverse physical systems, a phenomenon termed Universal Dissipation. This suggests an underlying principle governing non-equilibrium dynamics, independent of specific system details. Numerical analysis of the Dicke model – a fundamental model in quantum optics – has rigorously confirmed a key aspect of this universality: the Liouvillian gap exponent, denoted by $γ$, is approximately equal to 1. This exponent dictates the rate at which quantum information is lost during the transition, and its consistent value across different DQPT scenarios strengthens the notion of a broadly applicable framework for understanding these rapid, non-equilibrium processes. The confirmation of $γ ≈ 1$ provides critical validation for theoretical predictions and offers a pathway to generalize insights from one system to another, accelerating progress in fields ranging from condensed matter physics to quantum information science.

The Kibble-Zurek Mechanism (KZM) offers a powerful framework for understanding how systems respond to rapid, dynamical quantum phase transitions. This mechanism posits that when a system is driven quickly through a critical point – a point of dramatic change in its behavior – it cannot adiabatically follow the changing conditions. Consequently, the system freezes out excitations, creating defects or imperfections in its final state. Crucially, the KZM doesn’t just describe that excitations are created, but provides a quantitative relationship between the rate of change of the driving parameter, typically denoted as $v$, and the density of these excitations. Specifically, the number of excitations scales as $v^{-1/2}$, meaning faster transitions lead to fewer excitations, while slower transitions allow for more defects to form. This predictable link between the driving force and the resulting state has profound implications for diverse fields, ranging from cosmology – explaining the origin of cosmic defects – to condensed matter physics and even quantum computing, where understanding and controlling these excitations is paramount.

Accurate determination of critical exponents requires careful analysis, and Finite Size Scaling methods prove indispensable, particularly when translating theoretical predictions to real-world experiments. These techniques allow researchers to extrapolate from systems of limited size to understand the behavior at the thermodynamic limit, revealing universal characteristics near critical points. Recent studies demonstrate the power of this approach; the thermodynamic metric, a measure of the system’s sensitivity to change, was found to scale as $ζ(g) ∝ |g – gc|⁻³$, precisely matching theoretical expectations. Furthermore, investigations using the Kerr model confirmed a Quantum Fisher Information scaling exponent of $α = 2$, solidifying the validity of predictions derived from the Kibble-Zurek Mechanism and underscoring the reliability of Finite Size Scaling in characterizing dynamic quantum phase transitions.

Analysis of the KMB quantum Fisher information in the Gaussian open Dicke model reveals a power-law scaling with the distance to the critical point, yielding an exponent of 2 ± 1⋅10−4 and a slope of 0.500001 ± 1⋅10−6, consistent with theoretical predictions for γ = 1.
Analysis of the KMB quantum Fisher information in the Gaussian open Dicke model reveals a power-law scaling with the distance to the critical point, yielding an exponent of 2 ± 1⋅10−4 and a slope of 0.500001 ± 1⋅10−6, consistent with theoretical predictions for γ = 1.

The study of dissipative quantum phase transitions reveals a system less built than grown, a landscape sculpted by the flow of time and energy. It observes that the predictability of nonadiabatic entropy production-scaling with driving duration and the Liouvillian gap-is not a testament to control, but an acknowledgement of inherent, systemic behavior. This resonates with a deeper principle; monitoring is the art of fearing consciously. As Louis de Broglie stated, “Every man believes in something. I believe it’s better to aim for the impossible.” The ‘impossible’ here isn’t defying physics, but accepting that true resilience begins where certainty ends – where systems reveal themselves through their inevitable, and often instructive, imperfections.

The Inevitable Gradient

The demonstration of a universal scaling between nonadiabatic entropy production and the Liouvillian gap offers a compelling, if predictable, extension of the Kibble-Zurek mechanism into the realm of dissipation. It clarifies, rather than solves. The observed correspondence between thermodynamic irreversibility and critical phenomena doesn’t lessen the fundamental tension; it merely renames it. Systems, after all, do not strive for equilibrium, they approach it, and the very act of approaching generates further gradients – new avenues for asymmetry and eventual decay.

Future work will undoubtedly explore the boundaries of this universality. The present analysis assumes a relatively simple parameter space. More complex driving protocols, and systems with multiple competing order parameters, will inevitably introduce additional sources of non-universality, and reveal the fragility of any presumed scaling laws. The promise of predicting phase transitions based solely on thermodynamic quantities is alluring, but one should recall that every successful prediction is also a prophecy of the conditions under which it will fail.

The true challenge lies not in refining the measurement of entropy production, but in accepting its inevitability. Systems are not built to withstand the relentless creep of dissipation; they are built to become it. The observed scaling isn’t a law of nature, but a temporary reprieve – a brief moment of order before the inevitable return to a broader, more disordered state. Everything connected will someday fall together.

Original article: https://arxiv.org/pdf/2512.05074.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-06 10:44