This article has been reviewed according to Science X's editorial process and policies. Editors have highlighted the following attributes while ensuring the content's credibility: 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." 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 journal Mathematics on October 9, 2026. 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.
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