Xianzhi Pan | Searcharxiv

arXiv2026

Persistence of BKT phase transition in the 2D nonanalytic XY model

We study the two-dimensional XY model with the nonanalytic pair potential $2[(1-\cosδ)/2]^{p}$, whose small-angle law $\propto|δ|^{2p}$ carries a cusp for $p<1$ and a flat bottom for $p>1$, invalidating the harmonic spin-wave expansion. Two questions arise: the nature of the low-temperature ($T$) phase and of the phase transition. A naive energetic argument would predict genuine long-range order and an enhanced transition temperature for $p<1$, and no transition at all for $p>1$. Large-scale Monte Carlo simulations contradict both: for every $p>0$ the \dengrevxxi{low-$T$ phase} is quasi-long-range ordered, with anomalous dimension $η(T)\propto T^{1/p}$, and terminates at a Berezinskii--Kosterlitz--Thouless transition. Using a vortex-free noncompact lattice-field description and utilizing a duality transformation, we show that coarse-graining drives the height-difference distribution onto a single Gaussian fixed point, renormalizing the cusp and flatness into a finite harmonic stiffness that restores the spin-wave description and the BKT scenario for all $p>0$.

cond-mat.stat-mech↗

arXiv2026

Scaling of Long-Range Loop-Erased Random Walks

We study the scaling properties of long-range loop-erased random walks (LR-LERW), where the underlying random walker performs Lévy-flight-like jumps with a power-law step-length distribution $P(\mathbf{r})\sim |\mathbf{r}|^{-(d+σ)}$. Using extensive Monte Carlo simulations, we measure the scaling relation $N \sim R^{d_N}$ between the loop-erased step number $N$ and the spatial extent $R$, and determine the geometric exponent $d_N$ for various values of $σ$ in spatial dimensions $d = 1, 2,$ and $3$, as well as at the marginal point $σ= 2$ in $d=4$ and $5$. We observe a continuous crossover from long-range (LR) to short-range (SR) behavior as $σ$ increases. Below the upper critical dimension $d<d_c=4$, for $σ< d/2$, loop erasure is asymptotically irrelevant and $d_N=σ$, consistent with Lévy-flight scaling. For $d/2 < σ< 2$, loop erasure becomes relevant and $d_N$ varies continuously toward the SR-LERW value. At the marginal points with $σ=d/2$ or $σ=2$, clear logarithmic corrections are observed. At and above the upper critical dimension, $d \geq 4$, the scaling at $σ=2$ is found to be $N \sim R^2/\ln R$, consistent with that of the corresponding Lévy flight. Our results provide a systematic numerical determination of $d_N(σ)$ for the LR-LERW across dimensions, and are consistent with $σ_* = 2$ as the boundary between LR and SR critical behaviors recently established in a broad variety of statistical models.

cond-mat.stat-mech↗