Fractional calculus brings BKT-like scaling into one dimension

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, which is why textbook statistical mechanics treats one dimension as hostile territory for ordering. A notable exception to conventional ordering is the Berezinskii–Kosterlitz–Thouless, or BKT, transition. Best known from 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, a regime known as quasi-long-range order in which correlations weaken as a power of distance rather than vanishing exponentially. 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, opening an unexpected route to this celebrated physics without changing dimension at all.

The key ingredient is fractional calculus, a generalization of ordinary derivatives to non-integer orders. By replacing the usual first-order 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, allowing a one-dimensional field to retain long-range spatial connections that are absent in an ordinary local model. This nonlocality is what gives the half-order derivative its power: it encodes an effective interaction kernel that couples every point in the one-dimensional chain to every other point, mimicking some of the long-wavelength phase physics that in two dimensions emerges from the interplay of spin-wave fluctuations and vortex defects. The approach provides a mathematical way to encode long-range interactions without increasing spatial dimension, and it does so within a compact field-theoretic framework that can be simulated at scale.

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, a method that drives the field toward thermal equilibrium while preserving the phase structure of the complex order parameter. They performed large-scale simulations for several system sizes and then analyzed spatial correlations, Binder ratios, correlation ratios, effective exponents, and the free-energy response to an imposed phase twist. This battery of finite-size diagnostics is standard practice for identifying quasi-long-range order and distinguishing it from both true long-range order and ordinary short-range disorder, and applying it to a fractional nonlocal model required careful adaptation of each observable to the model’s unusual dispersion relation.

The central numerical finding concerns the low-temperature regime. Below a certain temperature range, correlations decayed algebraically: they weakened as a power of distance rather than disappearing exponentially. Crucially, the decay exponent varied with temperature, resembling the quasi-long-range order found below a conventional BKT transition, where the exponent continuously changes rather than jumping between fixed values. The finite-size scaling of the correlations also showed logarithmic features characteristic of BKT physics. Taken together, these results identify a strong analogy between the one-dimensional fractional model and the two-dimensional BKT universality class, at least as far as correlation functions are concerned. The work was published in the open-access journal Mathematics on October 9, 2026, and 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. The project began with a simple question: could the fractional calculus she had already studied be used to uncover or understand new physics?

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, a hallmark that connects the BKT transition to the unbinding of vortex-antivortex pairs. The fractional one-dimensional model behaved differently. The conventional linear-response quantity increased with system size rather than approaching a finite stiffness, which immediately signals that the usual interpretation of phase rigidity cannot be carried over unchanged from two dimensions to this nonlocal one-dimensional setting.

The researchers therefore examined a twist response adapted to the nonanalytic fractional dispersion. This adapted quantity 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 distinction is central to understanding what fractional nonlocality can, and cannot, carry from two dimensions into one. Similar correlation scaling does not necessarily imply identical thermodynamic behavior, and the absence of the stiffness jump demonstrates that the correspondence between the two models is real but partial, resting on correlation functions rather than on the full response structure of the BKT transition.

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, a kinematic consequence of the nonlocal coupling rather than a signature of an ordered phase. 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, and it means that identifying the transition in this model requires disentangling an interaction-imposed background from the genuinely BKT-like branch that emerges at low temperature.

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. In two dimensions, the BKT transition is understood through the unbinding of vortex-antivortex pairs, but a corresponding microscopic defect-unbinding mechanism has not been established for the one-dimensional fractional model, leaving the physical picture behind the observed scaling incomplete.

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. Mapping this landscape is one of the main theoretical motivations for the approach, since the derivative order acts as a tunable dial for interaction range and dispersion nonanalyticity. 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, from ultracold atoms coupled to light to photonic lattices with engineered long-range coupling. For such systems, knowing which features of two-dimensional topological transition physics survive a change of dimension, and which depend essentially on local interactions and defect physics, is directly relevant to interpreting experiments and designing simulations.

For a field in which one-dimensional models are usually dismissed as too disordered to sustain interesting critical behavior, the demonstration that a single change of derivative order can summon BKT-like correlation scaling is a striking result. It shows that the mathematical structure of nonlocality can substitute, at least partially, for the dimensional scaffolding that two-dimensional systems provide naturally. At the same time, the careful dissection of twist responses and stiffness-like quantities shows the limits of the analogy: correlation functions alone do not settle the thermodynamic identity of a phase transition. The open questions, from the existence of a true finite-temperature transition to the search for a defect-based description, now define a concrete research program. Whether the one-dimensional fractional model ultimately proves to host a genuine BKT transition or an extraordinarily long crossover, the study establishes fractional calculus as a practical tool for engineering long-range correlation physics in low dimension, and it hands the statistical mechanics community a well-posed problem with clear diagnostics for the next generation of simulations and analyses.

Subject of Research: Computational simulation of a one-dimensional fractional U(1) Ginzburg–Landau model exhibiting BKT-like correlation scaling

Article Title: Fractional calculus reveals BKT-like scaling in one dimension

Article References: Fractional calculus reveals BKT-like scaling in one dimension. Original publication (Bibliographic details unavailable.)

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Keywords: fractional calculus, BKT transition, quasi-long-range order, Ginzburg-Landau model, nonlocal interactions, one-dimensional systems, phase transitions, superfluid stiffness, statistical mechanics, numerical simulation, Fractional, calculus