arXiv subjects

Tianning Xiao

Publications and source records attributed to Tianning Xiao.

12 recordsLinked to original sources

arXiv2026

Two-dimensional percolation with algebraically decaying interactions II: Critical exponents in the long-range regime

We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities $p(r)\propto 1/r^{2+σ}$, establishing the universality diagram in the long-range (LR) regime for $σ\le2$. Using the event-based ensemble method, we simulate systems with linear sizes up to $L=16384$ and investigate three universality regimes: LR Wilson--Fisher (WF) A ($1<σ\le2$), LR Wilson--Fisher B ($2/3<σ\le1$), and LR mean-field (MF) ($0<σ\le2/3$). In the LR-WF-B regime, the anomalous dimension is consistent with $η=2-σ$, in agreement with mathematical results for $2/3<σ<1$, while the correlation-length exponent $ν(σ)$ exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although $η$ remains close to $2-σ$ for smaller $σ$, statistically resolvable deviations $δη(σ)=η-(2-σ)>0$ start to appear near $σ\simeq3/2$ and grow toward the short-range crossover at $σ=2$. Finally, by complementing the event-based simulations with conventional ensemble simulations, we reveal the coexistence of complete-graph asymptotics and LR Gaussian-fixed-point scaling in the LR-MF regime. These results further clarify the critical properties in long-range percolation and provide crucial benchmarks for long-range statistical systems.

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arXiv2026

Spontaneous Symmetry Breaking in Two-dimensional Long-range Heisenberg Model

Algebraically decaying interactions $\sim 1/r^{d+σ}$ can lead to nontrivial universality beyond short-range (SR) theories and spontaneous symmetry breaking in low-dimensional systems. We perform large-scale Monte Carlo simulations for the classical long-range (LR) Heisenberg model in two dimensions (2D) up to linear size $L=8192$. We show that the system enters a long-range-ordered phase through a single continuous phase transition for all $σ\leq 2$, including the marginal case $σ=2$. In contrast, for $σ> 2$ it recovers the SR asymptotically free behavior with no finite-temperature transition. This places the LR--SR crossover threshold at $σ_* = 2$. To characterize the ordered phase, we introduce an LR simple random walk with a fixed total length $\mathcal{L} \sim\mathcal{O}(L^d)$. This fixed-$\mathcal L$ walk reproduces the finite-size scaling of the Goldstone-mode fluctuations in the LR Heisenberg model in both two and three dimensions, including the logarithmic scaling at $σ= 2$. These results further motivate a general criterion for the existence of finite-temperature long-range order in LR systems with continuous symmetry in any spatial dimension.

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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.

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arXiv2026

Universality Diagram of Phase Transitions in Long-range Statistical Systems

The percolation, Ising, and O($n$) models constitute fundamental systems in statistical and condensed matter physics. For short-range-interacting cases, the nature of their phase transitions is well established by renormalization-group theory. However, the universality of the transitions in these models remains elusive when algebraically decaying long-range interactions $\sim 1/r^{d+σ}$ are introduced, where $d$ is the dimensionality and $σ$ is the decay exponent. Building upon insights from Lévy flight, i.e., long-range simple random walk, we propose three universality diagrams in the $(d,σ)$ plane for the percolation model, the O($n$) model, and the Fortuin-Kasteleyn Ising model, respectively. The conjectured universality diagrams are consistent with recent high-precision numerical studies and rigorous mathematical results, offering a unified perspective on critical phenomena in systems with long-range interactions.

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arXiv2026

On Sak's criterion for statistical models with long-range interaction

Determining the threshold value $σ_*$ that separates the short-range (SR) and long-range (LR) universality classes in phase transitions remains a controversial issue. While Sak's criterion, $σ_* = 2 - η_{\mathrm{SR}}$, has been widely accepted, recent studies of two-dimensional (2D) models with long-range interactions have challenged it. In this work, we focus on the crossover between LR and SR criticality in several classical 2D statistical models, including the XY, Heisenberg, percolation, and Ising models, whose interactions decay as $1/r^{2+σ}$. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at $σ_* = 2$. Here, we complete the picture by performing large-scale Monte Carlo simulations of the 2D LR-Ising model, reaching lattice sizes up to $L = 8192$. By analyzing the Fortuin-Kasteleyn critical polynomial $R_p$, the Binder ratio $Q_m$, and the anomalous dimension $η$, we obtain convergent and self-consistent evidence that the universality class already changes sharply at $σ= 2$. Taken together, these results establish a unified scenario for LR interacting systems: across all studied models, the crossover from LR to SR universality occurs at $σ_* = 2$.

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arXiv2025

Fate of Berezinskii-Kosterlitz-Thouless Paired Phase in Coupled $XY$ Models

Intriguing phases may emerge when two-dimensional systems are coupled in a bilayer configuration. In particular, a Berezinskii-Kosterlitz-Thouless (BKT) paired superfluid phase was predicted and claimed to be numerically observed in a coupled $XY$ model with ferromagnetic interlayer interactions, as reported in [\href{https://doi.org/10.1103/PhysRevLett.123.100601}{Phys. Rev. Lett. 123, 100601 (2019)}]. However, both our Monte Carlo simulations and analytical analysis show that this model does not exhibit a BKT paired phase. We then propose a new model incorporating paired-phase gradient interlayer interactions to realize the BKT paired phase. Moreover, we observe that the anomalous magnetic dimension varies along the phase transition line between the disordered normal phase and the BKT paired phase. This finding requires an understanding beyond the conventional phase transition theory.

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arXiv2025

Two-dimensional percolation model with long-range interaction

We perform large-scale simulations of the two-dimensional long-range bond percolation model with algebraically decaying percolation probabilities $\sim 1/r^{2+σ}$, using both conventional ensemble and event-based ensemble methods for system sizes up to $L=16384$. We accurately determine the critical points, the universal values of several dimensionless quantities, and the corresponding critical exponents. Our results provide compelling evidence that the system undergoes a crossover from short-range to long-range universality at $σ= 2$, in contradiction to Sak's criterion. Notably, we observe a pronounced jump in the universal values and critical exponents at $σ= 2$, a feature absent from previous studies.

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arXiv2025

Quantum Path-integral Method for Fictitious Particle Hubbard Model

We formulate a path-integral Monte Carlo algorithm for simulating lattice systems consisting of fictitious particles governed by a generalized exchange statistics. This method, initially proposed for continuum systems, introduces a continuous parameter $ξ$ in the partition function that interpolates between bosonic ($ξ= 1$) and fermionic ($ξ= -1$) statistics. We generalize this approach to discrete lattice models and apply it to the two-dimensional Hubbard model of fictitious particles, including the Bose- and Fermi-Hubbard models as special cases. By combining reweighting and $ξ$-extrapolation techniques, we access both half-filled and doped regimes. In particular, we demonstrate that the method remains effective even in strongly correlated, doped systems where the fermion sign problem hinders conventional quantum Monte Carlo approaches. Our results validate the applicability of the fictitious particle framework on lattice models and establish it as a promising tool for sign-problem mitigation in strongly interacting fermionic systems.

cond-mat.str-el↗

arXiv2025

The Nonclassical Regime of the Two-dimensional Long-range XY Model: a Comprehensive Monte Carlo Study

The two-dimensional (2D) XY model plays a crucial role in statistical and condensed matter physics. With the introduction of long-range interactions, the system exhibits a richer set of physical phenomena and a crossover between non-classical and short-range universality classes.In this work, we investigate the 2D XY model with algebraically decaying interactions $\sim 1/r^{2+σ}$, and provide a comprehensive numerical analysis of its thermodynamic properties. We demonstrate that for $σ\leq 2$, the system undergoes a second-order phase transition into a ferromagnetic phase characterized by the emergence of long-range order. In the low-temperature phase, due to the presence of the Goldstone mode, the correlation function saturates to a non-zero constant in the form of a power law for $σ< 2$, with decaying exponent $2-σ$, and in the form of the inverse logarithm of distance for $σ=2$. Moreover, the critical points and exponents are also determined for various $σ$. We provide compelling evidence that the crossover between non-classical and short-range regimes occurs at $σ=2$. This work presents a detailed account of the simulation methodology, extensive numerical data, and new insights into the physics of long-range interacting systems.

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arXiv2024

Logarithmic Finite-Size Scaling of the Four-Dimensional Ising Model

Field-theoretical calculations predict that, at the upper critical dimension $d_c=4$, the finite-size scaling (FSS) behaviors of the Ising model would be modified by multiplicative logarithmic corrections with thermal and magnetic correction exponents $(\hat{y}_t, \hat{y}_h)=(1/6,1/4)$. Using high-efficient cluster algorithms and the lifted worm algorithm, we present a systematic study of the FSS of the four-dimensional Ising model in the Fortuin-Kasteleyn (FK) bond and loop representations. Our numerical results reveal the FSS behaviors of various geometric and physical quantities in the three representations, offering robust evidence for the logarithmic correction form conjectured by the field theory. In particular, clear evidence is obtained for the existence of $\hat{y}_t=1/6$ in the loop representation, while it is difficult to extract in the spin representations, because of mixing with the Gaussian-fixed-point asymptotics. In the FK-bond representation, the multiplicative logarithmic correction for the second-largest cluster is also numerically observed to be governed by an exponent $\hat{y}_{h2} = -1/4$ with its exact value unknown yet.

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arXiv2024

Two-dimensional XY Ferromagnet Induced by Long-range Interaction

The crossover between short-range and long-range (LR) universal behaviors remains a central theme in the physics of long-range interacting systems. The competition between LR coupling and the Berezinskii-Kosterlitz-Thouless mechanism makes the problem more subtle and less understood in the two-dimensional (2D) XY model, a cornerstone for investigating low-dimensional phenomena and their implications in quantum computation. We study the 2D XY model with algebraically decaying interaction $\sim1/r^{2+σ}$. Utilizing an advanced update strategy, we conduct large-scale Monte Carlo simulations of the model up to a linear size of $L=8192$. Our results demonstrate continuous phase transitions into a ferromagnetic phase for $σ\leq 2$, which exhibits the simultaneous emergence of a long-ranged order and a power-law decaying correlation function due to the Goldstone mode. Furthermore, we find logarithmic scaling behaviors in the low-temperature phase at $σ= 2$. The observed scaling behaviors in the low-temperature phase for $σ\le 2$ agree with our theoretical analysis. Our findings request further theoretical understandings and can be of practical application in cutting-edge experiments like Rydberg atom arrays.

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arXiv2024

Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation

Besides its original spin representation, the Ising model is known to have the Fortuin-Kasteleyn (FK) bond and loop representations, of which the former was recently shown to exhibit two upper critical dimensions $(d_c=4,d_p=6)$. Using a lifted worm algorithm, we determine the critical coupling as $K_c = 0.077\,708\,91(4)$ for $d=7$, which significantly improves over the previous results, and then study critical geometric properties of the loop-Ising clusters on tori for spatial dimensions $d=5$ to 7. We show that, as the spin representation, the loop Ising model has only one upper critical dimension at $d_c=4$. However, sophisticated finite-size scaling (FSS) behaviors, like two length scales, two configuration sectors and two scaling windows, still exist as the interplay effect of the Gaussian fixed point and complete-graph asymptotics. Moreover, using the Loop-Cluster algorithm, we provide an intuitive understanding of the emergence of the percolation-like upper critical dimension $d_p=6$ in the FK-Ising model. As a consequence, a unified physical picture is established for the FSS behaviors in all the three representations of the Ising model above $d_c=4$.

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