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BKT-like Correlation Scaling and Twist Responses in a One-Dimensional Fractional $U(1)$ Ginzburg--Landau Model

This paper demonstrates that a one-dimensional fractional $U(1)$ Ginzburg--Landau model with marginal dispersion ( $\sigma=1$ ) exhibits BKT-like correlation scaling and a finite-temperature transition, despite lacking the universal helicity modulus jump characteristic of the two-dimensional XY model.

Original authors: Hitomi Endo, Michikazu Kobayashi

Published 2026-09-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Hitomi Endo, Michikazu Kobayashi

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 authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of physics, materials often reveal their true nature through how they change when heated or cooled. When a substance undergoes a phase transition, like water turning to ice, its atoms rearrange into a new, more ordered pattern. For decades, scientists believed that in very thin, one-dimensional lines of atoms, such orderly patterns could never truly form at any temperature above absolute zero. This idea, known as the Mermin-Wagner theorem, suggested that thermal jitters would always be strong enough to scramble any attempt at alignment. However, nature is full of exceptions. In two-dimensional layers, a special kind of transition was discovered where order does not appear as a solid, rigid structure, but rather as a delicate, long-range connection that fades slowly over distance. This phenomenon, called the Berezinskii-Kosterlitz-Thouless transition, relies on the binding and unbinding of tiny whirlpools in the atomic field. It is a subtle dance of order and chaos that has been observed in superfluids and superconductors, but it has always been thought to require a two-dimensional space to exist.

A team of researchers at Kochi University of Technology has now explored whether this special kind of order can emerge in a one-dimensional line if the atoms interact in a very unusual way. Instead of only talking to their immediate neighbors, they imagined a system where every point feels the influence of every other point, but with a strength that fades slowly over distance. This is a "nonlocal" interaction, a concept that changes how fluctuations behave. By using powerful computer simulations to model this strange one-dimensional line, the researchers found that it does indeed exhibit a transition that looks very much like the famous two-dimensional version. They discovered that below a certain temperature, the system develops a slow, algebraic decay of order, where correlations between distant points linger much longer than expected. However, the study also revealed a crucial difference: while the pattern of order looks similar, the physical "stiffness" that usually signals this transition in two dimensions is missing here. The transition exists in the way the system connects over long distances, but it lacks the universal jump in rigidity that physicists have come to expect.

To investigate this, the researchers built a mathematical model of a one-dimensional line where the energy cost of twisting the atomic field depends on a fractional power of the distance between points. In standard physics, the energy cost usually depends on the square of the distance, which leads to short-range interactions. In this model, the researchers adjusted the rules so that the interaction strength followed a specific fractional power, creating a situation where the fluctuations of the field grow logarithmically, much like they do in the two-dimensional case. They used a method called stochastic dynamics to simulate the system at different temperatures, essentially letting the computer run a virtual experiment where the atoms jiggle and settle into equilibrium. They then measured how the atoms at one end of the line correlated with atoms at the other end, looking for signs of this special, slow-decaying order.

The results showed a clear change in behavior as the temperature was lowered. At high temperatures, the correlations between distant points dropped off quickly, but not in the simple, exponential way seen in ordinary one-dimensional systems. Instead, the nonlocal nature of the interactions created a long-distance tail where correlations faded slowly, following a specific power law. As the temperature dropped below a critical point, estimated to be around 0.35 in the units used, a new, even slower pattern of order emerged. In this low-temperature regime, the system behaved as if it had a quasi-long-range order, where the connection between distant points persisted for a very long time, decaying only as a power of the distance. This is the hallmark of the Berezinskii-Kosterlitz-Thouless transition. The researchers confirmed this by analyzing how the system's properties changed with its size; the data collapsed onto a single curve when plotted against a specific variable involving the temperature and the logarithm of the system size, a signature pattern that strongly suggests this type of transition.

However, the study also uncovered a significant departure from the standard two-dimensional story. In the classic two-dimensional transition, there is a quantity called the helicity modulus, which measures the stiffness of the system against a twist. This stiffness remains finite in the ordered phase and jumps to a specific value at the transition, acting as a definitive fingerprint of the phenomenon. When the researchers tried to measure a similar stiffness in their one-dimensional model, they found that it did not behave the same way. The ordinary measure of stiffness grew larger as the system got bigger, rather than settling on a finite value. They also tried a different, more specialized way of measuring the response to a twist, which accounted for the nonlocal nature of the interactions. While this new measure was finite for any specific system size, it still vanished as the system became infinitely large, scaling in the same way as the overall order of the system itself. This means that unlike its two-dimensional cousin, this one-dimensional transition does not possess a universal, finite stiffness that survives in the infinite limit.

The findings suggest that the Berezinskii-Kosterlitz-Thouless transition is more about the way correlations decay over long distances than about a specific type of stiffness. The transition in this one-dimensional model is driven by the same logarithmic interactions that govern the two-dimensional case, allowing for a phase where order is maintained over vast distances without ever becoming truly rigid. The researchers conclude that while the transition shares the essential scaling features of the famous two-dimensional version, it lacks the universal jump in stiffness that has long been considered a defining characteristic. This distinction highlights that the physics of phase transitions can be richer and more varied than previously thought, depending on how the parts of a system talk to each other across space. The work opens the door to understanding how nonlocal interactions can create complex ordered states in low-dimensional systems, potentially offering new insights into the behavior of exotic materials and quantum fluids where long-range forces play a dominant role.

Technical Summary: BKT-like Correlation Scaling and Twist Responses in a One-Dimensional Fractional U(1) Ginzburg–Landau Model

Problem Statement
The paper investigates whether a one-dimensional (1D) complex field model with a nonlocal fractional gradient energy can exhibit a Berezinskii–Kosterlitz–Thouless (BKT) transition. Standard BKT physics, characterized by quasi-long-range order (QLRO) and a universal jump in helicity modulus, is typically associated with two-dimensional (2D) systems with short-range interactions or specific long-range models. The authors focus on a 1D fractional U(1) Ginzburg–Landau model where the quadratic part of the Hamiltonian is governed by a Fourier multiplier $|k|^\sigma$ . The study specifically targets the marginal case $\sigma = 1$ . In this regime, the spin-wave fluctuations become logarithmic, theoretically suggesting the possibility of BKT-like behavior despite the reduced dimensionality. The central challenge is to determine if such a transition exists and, if so, how it should be characterized given that the system is 1D and the interactions are nonlocal.

Methodology
The authors employ a numerical approach based on stochastic dynamics to sample the equilibrium Gibbs measure of the model.

Model Definition: The system is defined on a periodic domain of length $L$ with a complex order parameter $q(x)$ . The energy functional includes a nonlocal fractional kinetic term (defined via the fractional Laplacian $(-\Delta)^{\sigma/2}$ with $\sigma=1$ ) and a local quartic potential. The nonlocal term corresponds to a real-space interaction kernel decaying as $1/|x-y|^2$ .
Sampling Technique: The equilibrium measure is sampled using stochastic Gross–Pitaevskii (SGP) dynamics. This equation combines a dissipative gradient-flow term, a Hamiltonian (conservative) term, and thermal noise satisfying the fluctuation–dissipation relation. The authors demonstrate that this dynamics preserves the Gibbs measure, allowing for the calculation of static equilibrium observables.
Observables: The study analyzes several key quantities:
- Correlation Function ( $C(r)$ ): To detect algebraic decay.
- Binder Ratio ( $U_L$ ) and Correlation Ratio ( $R_C$ ): Dimensionless quantities used to identify phase transitions and finite-size scaling.
- Effective Exponent ( $\eta_{\text{eff}}$ ): Derived from the correlation ratio to estimate the decay exponent.
- Twist Responses: Two types of twist responses are examined to probe stiffness:
  1. Ordinary Linear-Response: Based on a Peierls-type twist in real space, assuming an analytic $\delta^2$ response.
  2. Spectral Cusp Response: Based on shifting the Fourier multiplier ( $|k| \to |k+\delta|$ ), which yields a non-analytic, cusp-like response ( $\propto |\Delta|$ ) for $\sigma=1$ .

Key Contributions and Results

Existence of a BKT-like Transition:
The numerical results confirm that the $\sigma=1$ model exhibits a transition consistent with BKT scaling.
- Low-Temperature Phase: The correlation function $C(r)$ displays algebraic decay $C(r) \sim r^{-\eta(T)}$ with a temperature-dependent exponent $\eta(T)$ . The Binder ratio and correlation ratio become nearly size-independent in this regime, indicating QLRO.
- High-Temperature Phase: The system does not exhibit simple exponential decay. Instead, the nonlocal fractional kernel induces a long-distance algebraic tail $C(r) \sim r^{-2}$ . The transition is thus characterized by the emergence of a slower algebraic branch (with $\eta(T) < 2$ ) dominating over the background $r^{-2}$ tail.
- Finite-Size Scaling: The dimensionless ratios ( $R_C$ and $U_L$ ) collapse when plotted against the BKT-type scaling variable $(T - T_{\text{BKT}})(\log L)^2$ . This collapse supports the hypothesis that the crossover length diverges with an essential singularity, consistent with $T_{\text{BKT}} \approx 0.35$ .
Breakdown of the Helicity Modulus Analogue:
A significant finding is the failure of twist-response quantities to act as a finite stiffness analogous to the helicity modulus in the 2D XY model.
- The ordinary linear-response quantity (assuming a quadratic twist cost) grows with system size $L$ in the low-temperature phase, failing to converge to a finite value. This is attributed to the non-analytic nature of the fractional dispersion and the long-range nature of the real-space kernel.
- The spectral cusp response ( $Y_{\text{cusp}}$ ), which is the natural response for the non-analytic dispersion, remains finite for finite $L$ but scales as $L^{-\eta(T)}$ in the thermodynamic limit. It closely tracks the scaling of the squared zero-mode order parameter $\langle |m|^2 \rangle$ . Consequently, $Y_{\text{cusp}}$ vanishes in the thermodynamic limit and does not exhibit a universal jump.
Nature of the Disordered Phase:
The paper clarifies that the disordered phase in this nonlocal model is distinct from short-range 1D systems. Due to the $|k|^\sigma$ dispersion, the high-temperature phase possesses an intrinsic algebraic tail ( $r^{-2}$ for $\sigma=1$ ). The transition is therefore not a change from exponential to algebraic decay, but a crossover from a "fast" algebraic tail (induced by the kernel) to a "slow" algebraic branch (induced by phase fluctuations).

Significance and Claims
The authors claim that the one-dimensional fractional U(1) model with $\sigma=1$ realizes a BKT-like transition in terms of correlation scaling and finite-size behavior, but lacks the universal helicity-modulus jump characteristic of the standard 2D XY model.

Similarity to BKT: The transition is driven by the logarithmic growth of phase fluctuations (marginal case $\sigma=1$ ), leading to algebraic order and essential singularities in the crossover length.
Difference from BKT: The absence of a finite, size-independent stiffness (helicity modulus) distinguishes this transition. The twist response is intrinsically tied to the finite-size scaling of the order parameter rather than acting as an independent thermodynamic rigidity.
Theoretical Implication: The results suggest that while the mechanism of phase-slip interactions (logarithmic at $\sigma=1$ ) drives the transition, the nonlocal nature of the kinetic term fundamentally alters the definition and behavior of stiffness. The transition must be characterized primarily through correlation functions and dimensionless ratios rather than stiffness jumps.

The paper concludes that this model provides a useful arena for understanding how nonlocal operators modify the balance between fluctuations and ordering, offering a variant of BKT physics where the "stiffness" is not a robust thermodynamic quantity in the thermodynamic limit.

Technical Summary: BKT-like Correlation Scaling and Twist Responses in a One-Dimensional Fractional U(1) Ginzburg–Landau Model

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