Logarithmic Quasi-Long-Range Order and Novel Continuous Phase Transition in the Two-Dimensional XY Model with 1/r4 Interaction
Abstract
We investigate the phase diagram and critical properties of the two-dimensional classical XY model with interactions decaying as 1/r2+σ. At the marginal case σ=2, we show that the low-temperature phase is characterized by logarithmic quasi-long-range order (log-QLRO), where spin correlations decay as a power of the logarithm of distance, C(r)∼(ln r)−ηℓ with ηℓ∝ T, and that the transition into this phase is a continuous transition beyond both the Ginzburg-Landau-Wilson paradigm and the Berezinskii-Kosterlitz-Thouless mechanism. Our analysis combines a near-exact Gaussian spin-wave theory, an adiabatic renormalization-group analysis, and large-scale Monte Carlo simulations. We find that the 1/r4 interaction logarithmically modifies the spin-wave stiffness kernel as γ(k)≃κk2ln(1/k), which suppresses vortex proliferation and gives rise to a nonuniversal correlation-length exponent ν∝1/√(2πκ). Crucially, our simulations reveal that the short-range algebraic QLRO phase is unstable against a weak long-range perturbation for σ≤2. These results establish σ=2 as the crossover boundary between the long-range and short-range universality classes and provide a critical evaluation of recent theoretical proposals.
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BibTeX
@misc{pan2026logarithmic,
title = {{Logarithmic Quasi-Long-Range Order and Novel Continuous Phase Transition in the Two-Dimensional $XY$ Model with $1/r^4$ Interaction}},
author = {Xianzhi Pan and Zhijie Fan and Youjin Deng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39267},
eprint = {2609.39267},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39267},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Logarithmic Quasi-Long-Range Order and Novel Continuous Phase Transition in the Two-Dimensional XY Model with 1/r^4 Interaction
AU - Xianzhi Pan
AU - Zhijie Fan
AU - Youjin Deng
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39267
UR - https://arxiv.org/abs/2609.39267
AB - We investigate the phase diagram and critical properties of the two-dimensional classical XY model with interactions decaying as 1/r^2+σ. At the marginal case σ=2, we show that the low-temperature phase is characterized by logarithmic quasi-long-range order (log-QLRO), where spin correlations decay as a power of the logarithm of distance, C(r)∼(ln r)^−η_ℓ with η_ℓ∝ T, and that the transition into this phase is a continuous transition beyond both the Ginzburg-Landau-Wilson paradigm and the Berezinskii-Kosterlitz-Thouless mechanism. Our analysis combines a near-exact Gaussian spin-wave theory, an adiabatic renormalization-group analysis, and large-scale Monte Carlo simulations. We find that the 1/r^4 interaction logarithmically modifies the spin-wave stiffness kernel as γ(k)≃κk^2ln(1/k), which suppresses vortex proliferation and gives rise to a nonuniversal correlation-length exponent ν∝1/√(2πκ). Crucially, our simulations reveal that the short-range algebraic QLRO phase is unstable against a weak long-range perturbation for σ≤2. These results establish σ=2 as the crossover boundary between the long-range and short-range universality classes and provide a critical evaluation of recent theoretical proposals.
ER -