Phase transitions are strongly constrained by spatial dimension. In conventional one-dimensional systems with short-range interactions, thermal fluctuations generally prevent continuous degrees of freedom from maintaining order over long distances.

A notable exception to conventional ordering is the Berezinskii–Kosterlitz–Thouless, or BKT, transition. Best known in two-dimensional superfluids, superconducting films, and magnetic systems, the BKT transition does not produce ordinary long-range order. Instead, the low-temperature phase exhibits correlations that decay slowly with distance.

Researchers at Kochi University of Technology have now shown that a one-dimensional mathematical model with nonlocal interactions can display correlation scaling similar to that associated with the two-dimensional BKT transition.

The key ingredient is fractional calculus—a generalization of ordinary derivatives to non-integer orders. By replacing the usual spatial derivative with a half-order derivative, the researchers constructed a one-dimensional model in which distant points interact directly.

“Rather than geometrically compressing a two-dimensional system into one dimension, fractional calculus changes how information is transmitted through space,” explains Professor Michikazu Kobayashi of Kochi University of Technology. “It allows a one-dimensional field to retain long-range spatial connections that are absent in an ordinary local model.”

Numerical evidence for BKT-like correlation scaling

The team studied a one-dimensional fractional U(1) Ginzburg–Landau model, a mathematical field theory used to describe ordering and phase transitions.

To generate equilibrium states at different temperatures, they used stochastic Gross–Pitaevskii dynamics and performed large-scale simulations for several system sizes. They then analyzed spatial correlations, Binder ratios, correlation ratios, effective exponents, and the free-energy response to an imposed phase twist.

At low temperatures, correlations decayed algebraically: they weakened as a power of distance rather than disappearing exponentially. The decay exponent varied with temperature, resembling the quasi-long-range order found below a conventional BKT transition.

The work was published in the open-access journal Mathematics on October 9, 2026. MDPI

Similar correlations, but a different thermodynamic response

Despite the similarities in correlation scaling, the one-dimensional model does not reproduce every defining feature of the conventional two-dimensional BKT transition.

In the standard two-dimensional XY model, the helicity modulus—or superfluid stiffness—remains finite in the low-temperature phase and undergoes a universal jump at the transition.

The fractional one-dimensional model behaved differently.

The conventional linear-response quantity increased with system size rather than approaching a finite stiffness. A twist response adapted to the nonanalytic fractional dispersion remained finite in a finite system, but decreased with increasing system size in the low-temperature algebraic regime. It scaled in the same way as the squared zero-mode order parameter and therefore did not behave as an independent superfluid stiffness.

The model thus shows an unusual combination:

• correlations resembling BKT physics;
• logarithmic finite-size scaling;
• but no conventional universal stiffness jump.

This results identify both an analogy and a fundamental difference. The one-dimensional model reproduces important correlation features of two-dimensional BKT physics, but its response to a global phase twist is not the same. This distinction is central to understanding what fractional nonlocality can—and cannot—carry from two dimensions into one.

A weak algebraic tail also survives at high temperature

The high-temperature regime also differs from that of a conventional short-range system.

Because of the fractional nonlocal kernel, correlations retain a weak long-distance algebraic tail even at high temperature. In the model studied, this background tail is consistent with an inverse-square decay.

This high-temperature tail is not quasi-long-range order. Instead, it is imposed by the long-range interaction itself. The low-temperature regime is distinguished by the emergence of a much more slowly decaying correlation branch on top of that background.

This makes the transition region subtler than the familiar contrast between exponential decay above a transition and algebraic decay below it.

A student-led project

The study was led by Hitomi Endo, a second-year master’s student at Kochi University of Technology, who had worked with fractional derivatives since her undergraduate research.

This project began with a simple question: could the fractional calculus that Ms. Endo had already studied be used to uncover or understand new physics?

Is this a true thermodynamic transition?

The numerical results show robust BKT-like behavior over the finite system sizes studied. However, the researchers emphasize that an important theoretical question remains unresolved.

The current evidence does not yet determine whether the model exhibits a genuine finite-temperature phase transition in the infinite-size limit, or an exceptionally long crossover associated with zero-temperature criticality.

Resolving this question will require larger-scale calculations and a controlled renormalization-group analysis of the compact nonlocal phase model. In particular, it remains necessary to establish whether a well-defined defect-based description exists in one dimension and, if so, how such defects interact with long-wavelength phase fluctuations.

Future studies will also vary the order of the fractional derivative to investigate how true long-range order, BKT-like algebraic behavior, and disorder connect across different degrees of nonlocality.

Why this matters

Fractional calculus provides a mathematical way to encode long-range interactions without increasing spatial dimension.

This study shows that changing the order of a spatial derivative can allow a one-dimensional model to reproduce part of the correlation physics normally associated with a two-dimensional topological transition.

At the same time, the absence of the conventional stiffness jump demonstrates that similar correlation scaling does not necessarily imply identical thermodynamic behavior.

The work offers a framework for separating the roles of dimensionality, interaction range, and phase rigidity in nonlocal statistical systems. It may also help guide future studies of long-range superfluids, atomic condensates, optical systems, and other platforms in which effective nonlocal interactions can arise.