Beyond Universality: Entanglement Reveals Quantum Criticality’s Hidden Depths

Author: Denis Avetisyan

New research leverages the Schmidt decomposition theorem and non-additive entropy to dissect the interplay between universal and non-universal behaviors in quantum phase transitions.

The quantum Grüneisen parameter, $ \Gamma^{\textmd{0K}}\_{q=q\_{special}}$, exhibits a strong dependence on the tuning parameter $ \lambda = J/B$ for a one-dimensional Mott transition, revealing a shift from a superposition of spin states at infinite $ \lambda$ to a quantum paramagnetic phase aligned with the external field at zero $ \lambda$, with the critical value $ \lambda_c$ being system-size dependent and characterized by a specific value of $ \Gamma^{\textmd{0K}}\_{q=q\_{special}}$ indicated for a system of eight spins and further corroborated in the thermodynamic limit for $S=1/2$ through analysis of nearest-neighbor interactions.

This study analyzes the 1D Ising model with a transverse field, demonstrating a finite-size dependent quantum Grüneisen parameter and proposing a novel photostrictive experiment to probe finite size effects.

Quantum criticality, while theoretically well-defined, often presents challenges in discerning universal behaviors from system-specific details. This is addressed in ‘Universal and non-universal facets of quantum critical phenomena unveiled along the Schmidt decomposition theorem’, where we investigate the influence of spin magnitude on critical phenomena in the one-dimensional Ising model using non-additive entropy and the Schmidt decomposition. Our analysis reveals a finite, size-dependent quantum Grüneisen parameter alongside both universal and non-universal characteristics dictated by system symmetry, offering insights into Hilbert space occupation at critical points. Could these findings pave the way for novel experimental probes of quantum criticality, such as the proposed photostrictive setup, and a deeper understanding of entanglement’s role in emergent behavior?

The Delicate Balance at Criticality

The behavior of physical systems as they approach a $Critical Point$ represents a cornerstone of modern physics, governing transitions like boiling water or the emergence of magnetism. However, conventional analytical techniques frequently falter when confronted with the intricacies arising from interacting components. These interactions, whether magnetic spins, atomic bonds, or social connections, introduce correlations that render simple approximations inadequate. As a system nears criticality, these interactions amplify, giving rise to long-range correlations and collective phenomena that defy description by methods designed for non-interacting parts. This limitation necessitates the development of more sophisticated tools capable of capturing the emergent behavior and subtle changes in system properties as the $Critical Point$ is approached, pushing the boundaries of theoretical and computational physics.

The seemingly simple $1D$ Ising Model, when subjected to a transverse magnetic field, becomes a surprisingly nuanced system for exploring quantum phase transitions. While retaining mathematical tractability, this model introduces quantum fluctuations that challenge traditional analytical techniques. Its power lies in its ability to mimic the behavior of more complex systems, offering a controlled environment to study phenomena like entanglement and critical slowing down. However, successfully characterizing this model demands analytical tools-such as quantum Monte Carlo simulations or sophisticated renormalization group methods-capable of detecting subtle changes in system properties as the transverse field is varied. These tools are essential because the model’s critical behavior isn’t captured by purely classical approaches, requiring sensitivity to quantum correlations and the emergence of non-local order.

The predictive power of many analytical techniques diminishes when applied to systems exhibiting critical behavior, especially those constrained by realistic, limited dimensions. These “finite size effects” introduce distortions in observed phenomena; quantities that would diverge infinitely in an unbounded system are instead capped, and transitions appear smoothed or shifted in temperature. Consequently, calculations based on infinite system assumptions yield inaccurate results for tangible, real-world scenarios. This discrepancy isn’t merely a matter of scale; the very nature of the critical transition can be obscured, masking true universality classes and hindering the reliable extraction of critical exponents. Researchers find that increasingly sophisticated computational methods, and a careful consideration of boundary conditions, are essential to mitigate these distortions and accurately model the behavior of systems facing these inherent dimensional limitations, particularly as they approach a critical point where even minor perturbations can dramatically alter outcomes.

At the critical point (λ=J/B=1), the q-entropy was used to analyze a spin chain composed of N spins divided into subsystems of size L and N-L, with L=2 for simplicity.

Beyond Conventional Limits: Embracing Non-Extensive Frameworks

Extensivity, a foundational principle in thermodynamics, dictates that properties like energy and entropy scale proportionally with the size of the system under consideration. This linear relationship is essential for maintaining consistency in thermodynamic calculations and interpretations. However, this principle often fails as a system approaches a critical point – the point at which a phase transition occurs. Near these critical points, fluctuations become large and long-ranged, leading to divergences in traditionally calculated quantities. The breakdown of extensivity manifests as non-linear scaling, where a small change in system size can result in a disproportionately large change in the observed property, thereby invalidating standard thermodynamic predictions and necessitating alternative frameworks to accurately describe the system’s behavior. This is because the correlations between system components become increasingly significant, rendering the assumption of independent contributions – the basis of extensivity – inaccurate.

Traditional thermodynamic entropy, based on the Boltzmann-Gibbs formalism, assumes additivity – the total entropy of a composite system is the sum of the entropies of its individual parts. However, $qq$-Entropy offers a generalized framework where entropy is not necessarily additive. This non-additivity is mathematically expressed through a $q$-parameter that deforms the logarithmic function used in the standard entropy definition. Consequently, $qq$-Entropy can effectively regularize critical phenomena, which are characterized by diverging fluctuations and non-extensive behavior, and allows for the investigation of systems where interactions or correlations lead to deviations from the assumption of extensivity. The framework provides a means to analyze systems exhibiting long-range interactions, memory effects, or other complexities not adequately described by traditional extensive thermodynamics.

The application of a specific entropic index, $q_s = 0.0828$, within the $qq$-Entropy framework addresses the issue of divergences frequently observed at critical points in traditional thermodynamic analyses. This non-extensive approach effectively regularizes critical phenomena by modifying the additive nature of entropy, preventing the unbounded growth of certain quantities as the system approaches criticality. By introducing a degree of non-additivity, $qq$-Entropy facilitates the finite characterization of systems exhibiting strong interactions and pronounced finite size effects, offering a more accurate description than methods reliant on the assumption of extensivity. This parameterization allows for the consistent calculation of thermodynamic properties even in regimes where conventional approaches fail due to singularities.

The q-generalized entropy exhibits a length-dependent scaling, transitioning from non-linear behavior for small block sizes to a linear regime (shown in panel b) for block sizes less than N/2, and differing based on the value of q.

Entanglement’s Influence: The Fabric of Collective Behavior

Quantum entanglement, a non-classical correlation between quantum particles, fundamentally impacts the collective behavior of many-body systems. This influence extends to how these systems respond to external stimuli, as entangled states create long-range correlations that modify the system’s susceptibility and dynamics. Specifically, entanglement introduces correlations beyond those described by classical statistical mechanics, leading to qualitatively different responses to perturbations. The degree of entanglement, quantified through measures like entanglement entropy, directly correlates with the strength of these non-classical effects and dictates the system’s ability to process information or exhibit novel phases of matter. In systems exhibiting collective phenomena, such as magnetism or superconductivity, entanglement serves as a crucial resource for mediating interactions and establishing long-range order, altering the system’s response functions and critical behavior compared to non-entangled scenarios.

The Schmidt Decomposition Theorem is a mathematical tool used to analyze entanglement in bipartite quantum systems. It states that any pure state of a composite system can be uniquely decomposed into a sum of product states, weighted by Schmidt coefficients. These coefficients, denoted as $ \lambda_i $, represent the contribution of each product state to the overall state and are directly related to the entanglement entropy. Specifically, the entanglement entropy, $S = – \sum_i \lambda_i^2 \log_2 \lambda_i^2 $, quantifies the degree of entanglement and provides information about the number of entangled pairs necessary to describe the system. The theorem allows for the determination of the maximum entanglement achievable between two subsystems and establishes a direct connection between entanglement, system size, and the underlying quantum correlations.

Analysis of the 1D Ising Model with a transverse magnetic field, utilizing the Schmidt Decomposition Theorem, demonstrates a clear correlation between entanglement and the model’s critical behavior. Investigations across different spin projections-specifically, $S = 1/2$, $3/2$, $5/2$, and $7/2$-reveal that the scaling of the entanglement measure $\Gamma_q$ is not universal. Each spin projection exhibits a distinct scaling behavior, indicating that the degree of entanglement, as quantified by $\Gamma_q$, responds differently to changes in system parameters near the critical point and is thus spin-dependent. This suggests that the entanglement structure itself is modified by the higher spin degrees of freedom, influencing the system’s critical properties.

Analysis of the inverse spectral function reveals that increasing spin enhances extrapolation towards infinite system sizes, with the function scaling as Γq(S) ∝ [S(S+1)]², and differing behaviors observed for nearest-neighbor versus next-nearest-neighbor interactions.

Experimental Pathways: Probing the Limits of Finite Systems

Investigating finite size effects – the alterations in a material’s properties due to its limited dimensions – presents a considerable experimental hurdle. Theoretical models often predict behaviors drastically different from those observed in macroscopic samples, but directly witnessing these size-dependent changes in real materials is exceptionally difficult. The influence of surfaces and boundaries becomes disproportionately large as dimensions shrink, masking the intrinsic properties of the bulk material and introducing artifacts. Consequently, researchers face the challenge of isolating and quantifying these effects, requiring precise control over sample geometry and surface quality, as well as highly sensitive measurement techniques capable of detecting subtle shifts in material behavior. Overcoming these obstacles is crucial for validating theoretical predictions and ultimately harnessing the unique properties that emerge at the nanoscale, potentially leading to advancements in diverse fields like catalysis and energy storage.

Photostriction, the phenomenon where materials deform under the influence of light, presents a compelling experimental method for investigating finite size effects within ferroelectric systems. Unlike traditional mechanical strain methods, photostriction allows for highly localized and precise control of deformation, bypassing many of the challenges associated with macroscopic measurements in small samples. By illuminating specific regions of a ferroelectric material, researchers can induce strain gradients and observe the resulting changes in polarization and domain structure. This approach is particularly valuable when studying the behavior of nanoscale ferroelectrics, where surface and interface effects dominate. The technique allows for the precise mapping of strain fields and the correlation of these fields with material properties, ultimately providing insights into how the size of a ferroelectric material influences its critical behavior and the interplay between $nearest neighbor$ and $next-nearest neighbor$ interactions.

The Grüneisen Ratio, a measure of the sensitivity of a material’s phase transition temperature to external pressure or strain, provides a crucial experimental handle on validating theoretical models of ferroelectric behavior. Precise analysis of this ratio allows researchers to discern the influence of competing interactions within the material; specifically, how $Nearest Neighbor Interaction$ and $Next-Nearest Neighbor Interaction$ contribute to the overall critical behavior near a phase transition. Deviations from predicted Grüneisen values can reveal the importance of previously unconsidered factors or refine existing theoretical frameworks, ultimately enhancing the understanding of collective phenomena in condensed matter systems. By meticulously mapping the relationship between external stimuli and the resulting material response, researchers can confirm the accuracy of theoretical predictions and gain insights into the fundamental forces governing phase transitions.

Increasing the illuminated area of a critical sample enhances the photostriction effect by enclosing more electric dipoles, allowing for the investigation of finite size effects.

Universal Principles and the Horizon of Criticality

Investigations into the seemingly simple $1D$ Ising Model with a transverse magnetic field reveal a surprising connection to a much wider phenomenon: universal criticality. This model, despite its basic structure, demonstrates behaviors – like diverging fluctuations and power-law correlations – that aren’t unique to it. These critical behaviors are observed in a remarkably diverse range of physical systems, from magnetic materials undergoing phase transitions to liquid-gas transformations and even certain economic models. The astonishing implication is that the specific details of a system become irrelevant near a critical point; instead, the behavior is governed by underlying, universal principles dictated by the system’s dimensionality and symmetry. This universality isn’t merely a mathematical curiosity, but a powerful tool allowing physicists to understand and predict the behavior of complex systems by focusing on these shared, fundamental characteristics.

The remarkable consistency of critical behavior across seemingly disparate systems points to a deeper, underlying principle: universality. This isn’t merely a mathematical coincidence within the $1D$ Ising model; rather, it suggests that the specific details of a material or model become irrelevant near a critical point. Instead of needing to understand every nuance of a complex substance, researchers can focus on a few key characteristics – like the dimensionality and symmetry – to predict its behavior as it transitions between phases. This powerful concept implies that the same fundamental laws govern phase transitions in magnets, fluids, alloys, and even certain biological systems, offering a unifying framework for understanding complex phenomena and allowing for cross-disciplinary insights into the behavior of matter.

The exploration of universal criticality, as demonstrated by the $1D$ Ising Model, extends far beyond the confines of condensed matter physics. Investigations into these fundamental principles promise to illuminate the behavior of diverse complex systems, offering potential breakthroughs in statistical mechanics, materials science, and even fields like biology and economics. By identifying the underlying mathematical symmetries that govern transitions between different states, researchers hope to develop predictive models applicable to a broad range of phenomena – from the collective behavior of spins in a magnet to the spread of epidemics or the fluctuations in financial markets. This interdisciplinary approach fosters a deeper understanding of how order emerges from chaos and how systems respond to external perturbations, potentially leading to the design of novel materials and more effective strategies for managing complex challenges.

The pursuit of understanding quantum criticality, as detailed in this study of the 1D Ising model, demands a careful distillation of complex interactions. It reveals how seemingly disordered systems can exhibit underlying order when examined through the lens of non-additive entropy and Schmidt decomposition. This echoes the sentiment of Richard Feynman, who once stated, “The first principle is that you must not fool yourself – and you are the easiest person to fool.” Rigorous application of mathematical tools, like those employed here to analyze finite size effects and the quantum Grüneisen parameter, is essential to avoid self-deception and reveal the true elegance of the system. The work highlights that beauty scales, clutter does not, as the underlying principles governing critical phenomena become clearer with refined analytical approaches.

Beyond the Horizon

The pursuit of quantum criticality, even within the ostensibly simple confines of the one-dimensional Ising model, reveals a landscape far more nuanced than anticipated. This work, by embracing the language of non-additive entropy and the elegant decomposition offered by Schmidt’s theorem, highlights a persistent tension: the desire for universal descriptions clashes with the inescapable reality of finite size effects. The finite, size-dependent Grüneisen parameter, far from being a mere technicality, suggests that the very notion of a universally scaling critical point requires careful reconsideration. It whispers that true universality might be an asymptotic ideal, never fully realized in any physical system.

The proposed photostrictive experiment, while conceptually direct, presents a challenge. It demands a precision in measurement that belies the inherent messiness of real materials. Yet, it is in grappling with such difficulties that deeper understanding emerges. The study doesn’t merely seek to observe critical phenomena; it strives to sculpt them, to expose their vulnerabilities, and to test the limits of current theoretical frameworks.

The path forward lies not in seeking ever more complex models, but in embracing simplicity. The goal is to find the minimal set of principles that can capture the essential physics, acknowledging that every interface element-every theoretical assumption, every experimental parameter-is part of a symphony. Future work might explore the extension of these techniques to higher-dimensional systems, or the investigation of analogous phenomena in other quantum materials, always guided by the belief that beauty in code-and in physical understanding-emerges through clarity and restraint.

Original article: https://arxiv.org/pdf/2512.11093.pdf

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