Beyond Equilibrium: A New Universality Class in Quantum Systems

Author: Denis Avetisyan

Researchers have identified a previously unknown universality class governing the behavior of dissipative quantum systems where dipole moments are conserved, impacting our understanding of charge transport.

A one-dimensional quantum spin model conserves dipole moment through the coherent exchange of spins-denoted as $ss$-coupled with flips of dipole spins, $ \Delta $, while dissipative processes are structured to remain compatible with this symmetry, exhibiting varying strengths of dipole conservation.

This work reveals a novel non-equilibrium universality class in dissipative quantum systems with dipole moment conservation, described by Keldysh field theory and exhibiting subdiffusive charge transport.

Understanding the dynamics of quantum many-body systems far from equilibrium remains a central challenge in modern physics. This Letter, ‘Emergent Universality Class in Dissipative Quantum Systems with Dipole Moment Conservation’, investigates a novel universality class arising in such systems when dipole moment conservation is enforced. We demonstrate the existence of a strongly interacting, non-equilibrium fixed point governing phase fluctuations and revealing distinct charge transport behaviors-subdiffusion with strong dipole symmetry, and diffusion in the weakly symmetric limit. How do these kinetic constraints and dissipative effects collectively shape the emergent behavior of broader classes of driven quantum systems?

Breaking the Equilibrium: A Dance of Order and Decay

The study of quantum many-body systems frequently centers on understanding their behavior far from equilibrium, a condition ubiquitous in natural phenomena and increasingly relevant in emerging quantum technologies. However, traditional theoretical approaches – such as perturbation theory or mean-field approximations – often falter when confronted with the strong correlations and complex interactions inherent in these systems. These methods typically struggle to accurately account for the dissipation of energy, which arises from interactions with an external environment and fundamentally alters the system’s dynamics. The challenge lies in capturing how quantum coherence is lost and how energy is redistributed amongst the many interacting particles, requiring advanced techniques capable of describing the intricate interplay between quantum mechanics and classical irreversibility. Consequently, developing new theoretical frameworks that can effectively model non-equilibrium dynamics with dissipation is paramount to unlocking a deeper understanding of these complex quantum systems and predicting their behavior.

Quantum systems possessing internal dipole moments exhibit a fascinating tension between the conservation of that moment and the inevitable dissipation arising from interactions with their environment. This interplay profoundly complicates the prediction of long-term system behavior and the understanding of transport phenomena. While dipole conservation might suggest persistent oscillations or localized excitations, dissipation introduces decay mechanisms that gradually erode these effects, often leading to complex, non-equilibrium states. Accurately modeling this requires going beyond simple perturbative approaches, as the coupling between dipole dynamics and dissipation can generate strong correlations and emergent behaviors. The challenge lies in capturing how energy is redistributed within the system and lost to the environment, influencing everything from spin relaxation to energy transport, and ultimately determining the stability and functionality of quantum devices.

Describing the behavior of complex quantum systems necessitates theoretical tools that move beyond standard perturbative approaches. Researchers are increasingly turning to frameworks like the Quantum Master Equation and the Density Matrix Renormalization Group, adapted to explicitly account for dissipation and decoherence. Crucially, a precise understanding of the system’s symmetries – such as conservation of dipole moment – is paramount; these symmetries constrain the possible dynamics and significantly reduce the computational complexity of modeling. Ignoring these inherent symmetries can lead to inaccurate predictions and a failure to capture the essential physics governing the long-time behavior and transport properties of these materials, while leveraging them allows for efficient simulations and a deeper insight into the emergence of novel quantum phenomena.

Numerical simulations of the Langevin equation reveal a new non-equilibrium universality class emerging at long timescales, observable across system sizes of 60, 80, and 100 with parameters g=0.5, D~2=0, D~=1, and C=1 for exponents χ=1/2 and χ=2.

Deconstructing Complexity: Effective Fields and Bosonized Representations

An effective field theory, coupled with a large-$S$ expansion, provides a means of analyzing the quantum spin model by systematically integrating out high-energy degrees of freedom. This approach focuses on the low-energy, long-wavelength collective modes, which dominate the system’s behavior at large distances and low energies. The large-$S$ expansion, where $S$ represents the spin magnitude, is a perturbative scheme valid when the spin is much larger than other energy scales in the problem; it allows for a controlled approximation by expanding in powers of $1/S$. By employing this methodology, the complex many-body problem is reduced to an analysis of these relevant long-wavelength fluctuations, significantly simplifying the calculations and enabling analytical treatment of the system’s properties.

Bosonization is a mathematical technique used to transform the quantum spin model into an equivalent representation where spin operators are expressed in terms of bosonic fields. This transformation is achieved by introducing Schwinger bosons, effectively mapping spin-$1/2$ operators to bosonic creation and annihilation operators. The resulting bosonized Hamiltonian describes collective phase fluctuations of the spin system and facilitates analysis through perturbative methods, such as a large-S expansion, which would be considerably more difficult with the original fermionic (spin) operators. This approach allows for the treatment of long-wavelength fluctuations and provides a means to calculate correlation functions and other physical observables related to the collective behavior of the spins.

The bosonized model achieves a reduction in computational complexity by representing the original quantum spin system in terms of bosonic degrees of freedom, effectively mapping interacting spins to collective density fluctuations. This transformation simplifies calculations, allowing for the treatment of long-wavelength behavior without needing to directly address the many-body fermionic nature of the initial problem. Specifically, the bosonization procedure allows for the application of perturbative techniques and the calculation of correlation functions which are intractable in the original fermionic formulation. This ultimately enables analytical investigations of the system’s phase transitions and critical behavior, providing insights inaccessible through direct numerical simulations of the original model.

Symmetry as a Constraint: Diffusion Versus Sub-Diffusion

Analysis of charge transport dynamics demonstrates a strong correlation between dipole symmetry and the resulting transport regime. Systems exhibiting weak dipole symmetry facilitate diffusive behavior, where the mean squared displacement scales linearly with time, consistent with the Edwards-Wilkinson universality class. Conversely, strong dipole symmetry constrains particle motion, leading to sub-diffusion characterized by a slower-than-linear growth of the mean squared displacement with time. This transition between regimes is directly observable in the scaling of the density correlator, which for strong dipole symmetry follows a $t^{-d/4}$ relationship, definitively indicating sub-diffusive characteristics.

Sub-diffusion, observed under conditions of strong dipole symmetry, is characterized by anomalous transport where the mean squared displacement scales non-linearly with time. Specifically, the MSD increases at a rate slower than that predicted by standard Brownian motion, where $MSD \propto t$. This constrained dynamic behavior arises from the strong dipole symmetry limiting the available pathways for particle movement, effectively reducing the diffusion coefficient. The resulting sub-diffusive regime indicates a more localized and sluggish transport process compared to typical diffusion.

Weak dipole symmetry in the analyzed system results in diffusive behavior, characterized by a linear relationship between the mean squared displacement and time. This diffusive regime aligns with the predictions of the Edwards-Wilkinson universality class, which describes systems exhibiting fluctuations that propagate and decay over time. Specifically, the observed behavior indicates that charge carriers undergo Brownian motion with a diffusion coefficient consistent with this universality class. The density correlator for weakly symmetric systems scales according to $t^{-d/2}$, further supporting the classification within the Edwards-Wilkinson framework, where d represents the dimensionality of the system.

Experimental results demonstrate a clear correlation between dipole symmetry and charge transport behavior. Systems exhibiting weak dipole symmetry consistently display diffusive charge transport, characterized by a mean squared displacement that grows linearly with time. Conversely, systems with strong dipole symmetry exhibit sub-diffusive charge transport, where the mean squared displacement increases at a slower rate than linear with time. This distinction indicates that the degree of dipole symmetry fundamentally alters the mechanisms governing charge carrier movement within the material, shifting the transport regime from typical diffusion to a constrained, sub-diffusive process.

Analysis of systems exhibiting strong dipole symmetry demonstrates that the density correlator scales as $∝t^{-d/4}$, where $t$ represents time and $d$ is the dimensionality of the system. This specific time-dependent scaling provides quantitative confirmation of sub-diffusive behavior. In sub-diffusion, the mean squared displacement increases at a rate slower than linear with time, and this scaling relationship of the density correlator is a characteristic signature of this anomalous transport regime. The exponent of $-d/4$ indicates a fractional dynamic exponent, consistent with models predicting sub-diffusive dynamics constrained by strong dipolar interactions.

Beyond Established Orders: A New Landscape of Non-Equilibrium Physics

Recent investigations have revealed a previously unrecognized universality class in non-equilibrium systems, diverging from the well-established Kardar-Parisi-Zhang (KPZ) equation. This novel class isn’t defined by a specific microscopic model, but rather by how systems behave at large scales, demonstrating that collective behavior can emerge from fundamentally different mechanisms than previously understood. The defining characteristic of this new class is a delicate balance between symmetry and dissipation; systems within it exhibit unique scaling properties dictated by the interplay of these two forces. Specifically, the symmetries present within the system constrain the possible forms of fluctuations, while dissipation governs how these fluctuations evolve over time, leading to a distinct set of critical exponents and long-time behaviors not predicted by the KPZ framework. This discovery expands the known landscape of non-equilibrium fixed points, suggesting a richer variety of collective phenomena than previously appreciated and opening new avenues for understanding complex systems.

The identification of this new universality class significantly refines the map of non-equilibrium fixed points, those stable states systems tend toward when driven far from equilibrium. These fixed points dictate the long-time behavior of a surprisingly broad range of physical phenomena, from the growth of crystals and the formation of dunes to the dynamics of interfaces and the evolution of pattern formation. The research reveals that the interplay of symmetries and dissipation can give rise to entirely new critical behaviors, distinct from previously known models like the Kardar-Parisi-Zhang equation. Consequently, understanding these newly discovered fixed points is paramount for accurately predicting how systems evolve over extended periods and for discerning the underlying mechanisms governing their complex dynamics; it offers a pathway to anticipate and potentially control the emergent properties observed in these non-equilibrium systems, opening doors to material design and technological innovation.

The identification of universality classes represents a powerful predictive tool across a surprisingly broad spectrum of physical phenomena. These classes categorize systems exhibiting disparate microscopic details yet sharing identical macroscopic behavior near critical points or during non-equilibrium dynamics. Consequently, characterizing these classes isn’t merely an academic exercise; it allows researchers to extrapolate findings from well-understood models to entirely new systems, bypassing the need for detailed simulations or experiments on each individual material. This predictive capability is particularly valuable in materials science, where understanding the collective behavior of atoms and molecules is key to designing materials with specific, tailored functionalities – for example, predicting the surface roughness of a growing film, the flow of fluids through porous media, or the response of a material to external stresses. By recognizing the underlying universality, scientists can accelerate the discovery of novel materials and optimize their properties for advanced applications, ranging from energy storage to biomedical devices.

The research delves into the heart of dissipative quantum systems, seeking to understand how conservation laws-specifically, dipole moment conservation-shape their emergent behavior. This pursuit echoes a fundamental principle: to truly grasp a system, one must dissect its components and challenge its boundaries. As Richard Feynman once said, “The first principle is that you must not fool yourself – and you are the easiest person to fool.” This investigation doesn’t simply observe charge transport and subdiffusion; it actively seeks the underlying rules governing these phenomena, stripping away assumptions to reveal a novel universality class. It’s a testament to the idea that knowledge isn’t passively received, but actively constructed through rigorous questioning and a willingness to dismantle established paradigms.

Beyond Equilibrium’s Shadows

The identification of a novel non-equilibrium universality class, predicated on dipole moment conservation, feels less like a destination and more like the clearing of underbrush. The work suggests that prior assumptions regarding charge transport in dissipative quantum systems-often built on the scaffolding of equilibrium expectations-may be fundamentally flawed. The true utility lies not in confirming existing models, but in exposing their limitations. One naturally wonders where else such hidden conservation laws might subtly dictate behavior, quietly enforcing constraints we’ve overlooked.

Future investigations should not shy away from deliberately introducing imperfections. The current framework, while elegant, relies on certain idealized conditions. A rigorous examination of the effects of spatial inhomogeneities, or deviations from perfect dipole conservation, could reveal surprising robustness-or, more interestingly, a cascade of emergent phenomena. The exploration of higher-order interactions, currently treated as perturbations, may also be warranted – sometimes, the ‘small’ terms are where the real physics resides.

Ultimately, the value of this line of inquiry isn’t necessarily in predicting specific experimental outcomes, but in refining the questions. It’s a reminder that the universe rarely conforms to our neat categorizations, and that genuine understanding requires a willingness to dismantle, probe, and occasionally, embrace the messiness of reality.

Original article: https://arxiv.org/pdf/2512.17210.pdf

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

Breaking the Equilibrium: A Dance of Order and Decay

Deconstructing Complexity: Effective Fields and Bosonized Representations

Symmetry as a Constraint: Diffusion Versus Sub-Diffusion

Beyond Established Orders: A New Landscape of Non-Equilibrium Physics

Beyond Equilibrium’s Shadows

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