⚛️ quantum physics

Universal dynamics of quantum phase transitions with fractional long-range interactions

This study characterizes the full defect counting statistics in a nonintegrable fractional Ising model, revealing that defect cumulants follow fractional-order-dependent power laws that deviate from or align with Kibble-Zurek predictions, while demonstrating that long-range interactions induce enhanced kink correlations that violate exact Gaussian bounds yet still yield a Gaussian distribution for large defect numbers.

Original author: Andraás Grabarits, Lincoln D. Carr, Adolfo del Campo

Published 2026-11-04

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CC BY 4.0

Original author: Andraás Grabarits, Lincoln D. Carr, Adolfo del Campo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, matter does not always settle smoothly into its new state when conditions change. Imagine a magnet cooling down; as it passes a critical temperature, its atoms suddenly align in a specific direction. But because the change happens everywhere at once, different regions of the material cannot communicate fast enough to agree on which way to point. This lack of coordination leaves behind permanent scars: tiny defects where the alignment flips. This phenomenon, known as the Kibble-Zurek mechanism, predicts that the faster you force a system through this transition, the more defects you will create. For decades, physicists have tested this rule in simple, idealized systems where the math is clean and the particles do not interact in complicated ways. However, nature is rarely so simple. In many real materials and advanced quantum simulators, particles talk to each other across vast distances, creating a web of interactions that defies the standard rules. Understanding how these long-range connections change the formation of defects is crucial for building reliable quantum computers and understanding exotic states of matter, yet it has remained a mystery because the equations describing such systems are too complex to solve exactly.

A team of researchers has now tackled this problem by simulating a quantum chain where the interactions between particles are governed by a "fractional" rule, a mathematical concept that allows the strength of the connection to decay in a continuous, tunable way rather than dropping off sharply. Instead of just counting the average number of defects, the team mapped out the entire statistical landscape of how these defects appear, looking at not just the mean, but the fluctuations and correlations that reveal how the defects behave as a group. Using powerful computer simulations based on matrix-product states—a method that efficiently handles the complexity of many interacting particles—they drove the system through a phase transition at different speeds and recorded the resulting patterns. They found that when the interactions are very long-range, the standard rules of the Kibble-Zurek mechanism break down. The number of defects does not decrease as quickly as the theory predicts when the transition is slowed down. Instead, the defects bind together into pairs, behaving more like a single unit than independent particles. This binding creates a stronger correlation between them, causing the number of defects to fluctuate much more wildly than in simpler systems.

The researchers compared their complex simulations against a simplified model where the particles are treated as non-interacting free fermions, a common approximation that works well for short-range interactions. In the long-range regime, this simplified model failed to capture the true behavior of the system. The simulations showed that the variance in the number of defects—the measure of how much the count jumps around—exceeded the strict limit that applies to any system of free particles. This violation is a direct signature of the defects being confined by their mutual attraction, a phenomenon that the free-particle model cannot describe. However, as the researchers tuned the interactions to become shorter-range, the complex simulations began to align perfectly with the simpler free-particle predictions. The transition from the exotic, bound state to the standard, independent state was smooth and continuous, controlled entirely by the fractional order of the interaction.

What makes this discovery particularly significant is that the researchers did not just observe the average number of defects; they reconstructed the full probability distribution of how many defects appear. They found that even in the complex, long-range regime where the defects are strongly correlated, the distribution of defect numbers still eventually settles into a bell curve, or Gaussian shape, as the system size grows. This suggests a deep underlying order that persists even when the standard scaling laws fail. The team also derived exact mathematical bounds on how these defects can correlate with one another, proving that in a system of free particles, the total correlation between any two points cannot exceed a specific limit. Their simulations showed that in the long-range regime, the system violates these bounds, confirming that the defects are indeed interacting in a way that traps them together.

The study highlights a continuous spectrum of behavior in quantum phase transitions. By adjusting a single parameter that controls the range of the interaction, the system moves from a regime where defects are tightly bound and the standard theory fails, to a regime where they are free and the standard theory holds. This provides a new testbed for understanding how causality and information spread in quantum systems. In the long-range regime, the "light cone" of influence—the speed at which information travels—widens, allowing distant parts of the chain to coordinate in ways that prevent the formation of independent defects. The researchers suggest that this breakdown of the standard mechanism is not just a numerical curiosity but a fundamental consequence of how fractional interactions alter the causal structure of the quantum world. Their work offers a precise roadmap for future experiments on quantum simulators, such as those using trapped ions or Rydberg atoms, where these fractional interactions can be engineered and measured directly. By counting the defects and analyzing their statistics, experimentalists can now probe the transition between bound and free states, testing the limits of our understanding of quantum criticality in a way that was previously impossible.

Technical Summary: Universal Dynamics of Quantum Phase Transitions with Fractional Long-Range Interactions

Problem Statement
The statistical characterization of topological defects generated during quantum phase transitions (QPTs) has been extensively studied in exactly solvable models, particularly free-fermion systems. However, the full counting statistics of defects in non-integrable systems, specifically those with long-range interactions, remains largely unexplored. A central open question is whether the Kibble-Zurek (KZ) mechanism, which relates defect density to the quench rate via critical exponents, holds universally in these regimes. Furthermore, it is unclear if strong long-range interactions induce domain-wall confinement (binding defects into pairs) sufficiently to break the assumption of independent defect formation inherent in free-fermion descriptions, thereby altering the defect counting statistics and violating standard bounds on cumulants.

Methodology
The authors investigate the fractional transverse-field Ising chain, a non-integrable model where interactions decay as a power law $r^{-(1+q)}$ , governed by a continuously tunable fractional order $q$ . The interactions are derived from the Riesz fractional derivative.

Numerical Simulations: The primary results are obtained using Time-Dependent Variational Principle (TDVP) simulations within the Matrix Product State (MPS) and Matrix Product Operator (MPO) formalism. The system size is set to $L=150$ spins with open boundary conditions. The Hamiltonian is represented exactly using binomial coefficients for the fractional couplings, avoiding exponential fits.
Critical Exponent Extraction: To test the KZ mechanism, the authors independently extract the dynamical critical exponent $z$ $z$ and the correlation length exponent $\nu$ $ν$ .
- $z$ and $\nu$ are determined via finite-size scaling of the energy gap minimum using Density Matrix Renormalization Group (DMRG).
- $z$ is also extracted from the "entanglement light cone" generated by a local perturbation at criticality, measuring the spreading of bond entanglement entropy.
Free-Fermion Benchmark: The authors construct an exactly solvable free-fermion approximation (a fractional $p$ -wave superconductor) by truncating the Jordan-Wigner string operators. This allows for analytical derivation of defect statistics, Landau-Zener excitation probabilities, and exact bounds on cumulants and correlations.
Statistical Analysis: The full counting statistics of kinks (defects) are reconstructed via the characteristic function (Fourier transform of the probability distribution). The authors analyze the first three cumulants ( $\kappa_1, \kappa_2, \kappa_3$ ) and their ratios to determine the distribution shape and correlations.

Key Results

KZ Scaling and Critical Exponents:
- For intermediate and short-range orders ( $q \gtrsim 1$ ), the decay of the mean kink number with driving time $\tau_Q$ follows the KZ prediction $\kappa_1 \sim \tau_Q^{-\nu/(z\nu+1)}$ , with exponents extracted from the gap and light-cone analyses matching the numerical data within systematic uncertainties (approx. 10–20%).
- For low fractional orders ( $q \lesssim 1$ ), deep in the long-range regime, the mean defect number decays significantly slower than the KZ prediction based on the extracted exponents.
- The critical exponents $z$ and $\nu$ vary continuously with $q$ . For $q < 1$ , the system exhibits behavior distinct from the short-range universality class.
Full Counting Statistics and Cumulants:
- Gaussian Limit: In regimes where the mean kink number is large, the defect distribution approaches a Gaussian.
- Cumulant Ratios: For $q \gtrsim 1.25$ , the ratios of higher cumulants to the mean ( $\kappa_2/\kappa_1$ , $\kappa_3/\kappa_1$ ) align with free-fermion predictions.
- Violation of Free-Fermion Bounds: Deep in the long-range regime ( $q \lesssim 0.3$ ), the variance of the kink number ( $\kappa_2$ ) exceeds twice the mean ( $\kappa_1$ ). This violates the exact bound $\kappa_2 \leq 2\kappa_1$ derived for any free-fermion state. The ratio $\kappa_2/\kappa_1$ reaches values between 2.2 and 5.1 depending on $q$ and $\tau_Q$ .
- Confinement: This violation is attributed to the confinement of domain walls into bound pairs, a phenomenon captured by the full many-body interactions but absent in the truncated free-fermion model.
Correlations:
- Kink-kink correlation functions show that for $q \leq 0.5$ , kinks are positively correlated (created close together), consistent with confinement. For $q \geq 1.5$ , they are anticorrelated.
- In the long-range regime ( $q=0.2$ ), the sum of connected correlations for the central bond exceeds the local variance, violating the bound $|\sum_{r \neq 0} C(r)| \leq C(0)$ that holds for free fermions. This confirms that the breakdown of independent defect formation occurs in the bulk of the chain, not just at boundaries.
Free-Fermion Approximation Validity:
- The free-fermion model provides a quantitatively accurate description of the mean and variance only in the short-range limit ( $q \gtrsim 2$ ).
- It improves progressively as $q$ increases but fails to capture the super-Poissonian statistics and confinement effects at low $q$ .
- Analytical results for the free-fermion model reveal a crossover at $q=1/4$ : for $q < 1/4$ and slow ramps, high-momentum modes dominate, leading to a $\tau_Q^{-2}$ scaling for all cumulants, distinct from the low-momentum Landau-Zener scaling.

Significance and Claims
The paper establishes that fractional long-range interactions provide a continuous parameter to tune the universality class of quantum critical dynamics. The authors demonstrate that while the KZ mechanism holds for intermediate and short-range interactions, it breaks down in the deep long-range regime ( $q \lesssim 1$ ).

Crucially, the work shows that the full counting statistics of defects in non-integrable long-range systems deviates fundamentally from free-fermion predictions. The violation of the variance bound ( $\kappa_2 > 2\kappa_1$ ) and the enhanced correlations indicate that domain-wall confinement prevents the independent formation of defects assumed in standard free-fermion descriptions. The authors suggest that this breakdown may be linked to the altered causal structure of the system, where correlations spread along "Lévy light cones" rather than linear light cones, weakening the premise of independent ordering in distant regions. The fractional Ising chain thus serves as a testbed for understanding defect formation in systems with algebraic causal horizons and confinement physics, relevant to current quantum simulation platforms like trapped ions and Rydberg arrays.

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