Fractional calculus helps 1D model mimic correlations seen in 2D phase transitions

Phase transitions often depend heavily on spatial dimension. In traditional one-dimensional systems with short-range interactions, thermal fluctuations hinder continuous degrees of freedom from maintaining order over long distances. However, an exception exists in the form of the Berezinskii–Kosterlitz–Thouless (BKT) transition.

Notably, this transition does not produce long-range order but instead exhibits correlations that decay slowly with distance. Researchers at Kochi University of Technology have discovered that a one-dimensional mathematical model, when incorporating nonlocal interactions through fractional calculus—a generalization of ordinary derivatives to non-integer orders—can mimic the correlation scaling associated with the two-dimensional BKT transition.

The key to this phenomenon lies in fractional calculus. By substituting the standard spatial derivative with a half-order derivative, the researchers created a one-dimensional model wherein distant points interact directly. This approach modifies how information is transmitted through space, enabling a one-dimensional field to retain long-range spatial connections, which are absent in conventional local models.

To study this model, the team employed stochastic Gross–Pitaevskii dynamics and performed extensive simulations across various system sizes. They analyzed spatial correlations, Binder ratios, correlation ratios, effective exponents, and the free-energy response to phase twists.

At low temperatures, the researchers observed that correlations decayed algebraically, weakening as a power of distance rather than disappearing exponentially. This decay exponent varied with temperature, mirroring the quasi-long-range order observed below a conventional BKT transition. Notably, despite the similarities in correlation scaling, the one-dimensional model does not replicate every aspect of the conventional two-dimensional BKT transition.

For instance, in the standard two-dimensional XY model, the helicity modulus—indicative of superfluid stiffness—remains finite in the low-temperature phase and experiences a universal jump at the transition. Conversely, in the fractional one-dimensional model, the conventional linear-response quantity increased with system size rather than converging to a finite stiffness.

Additionally, the response to a global phase twist adapted to the nonanalytic fractional dispersion remained finite within a finite system but decreased with increasing system size in the low-temperature algebraic regime. This behavior scaled similarly to the squared zero-mode order parameter and did not manifest as an independent superfluid stiffness.

These findings highlight both an analogy and a fundamental difference between the two-dimensional BKT physics and the one-dimensional model. The one-dimensional model successfully reproduces critical correlation features of two-dimensional BKT physics but exhibits distinct response characteristics to a global phase twist. This distinction is crucial for understanding the extent to which fractional nonlocality can emulate two-dimensional phenomena in one dimension.

Furthermore, the high-temperature regime of the fractional model differs from that of conventional short-range systems due to the presence of a weak long-distance algebraic tail, consistent with an inverse-square decay. This tail is not indicative of quasi-long-range order but is instead a direct consequence of the long-range interaction itself.

In contrast, the low-temperature regime features a more gradually decaying correlation branch superimposed on this background, creating a subtler transition region than the traditional contrast between exponential decay above a transition and algebraic decay below it.

Led by Hitomi Endo, a second-year master's student at Kochi University of Technology who has previously explored fractional derivatives, the study began with a simple inquiry: could the principles of fractional calculus, already familiar to Endo, be employed to reveal new physical insights? The numerical results confirmed robust BKT-like behavior within the finite system sizes investigated.

However, the researchers stress that an important theoretical question remains unanswered: whether the model displays a genuine finite-temperature phase transition in the infinite-size limit or an exceptionally long crossover associated with zero-temperature criticality. Resolving this uncertainty necessitates larger-scale calculations and a detailed renormalization-group analysis of the compact nonlocal phase model.

Specifically, the study aims to ascertain the existence of a well-defined defect-based description in one dimension and to explore how such defects interact with long-wavelength phase fluctuations. Additionally, the researchers plan to examine the effects of varying the order of the fractional derivative to investigate the connections between true long-range order, BKT-like algebraic behavior, and disorder across different degrees of nonlocality.

In summary, fractional calculus offers a mathematical framework to encode long-range interactions without increasing spatial dimension, as demonstrated by this study, which shows how altering the order of a spatial derivative can enable a one-dimensional model to replicate part of the correlation physics typically associated with a two-dimensional topological transition.