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arXiv:1905.06688 · Physical review. B./Physical review. B

Analysis of the low-temperature phase in the two-dimensional long-range diluted XY model

Cescatti Fabiana, Miguel Ibáñez-Berganza, A. Vezzani, Raffaella Burioni

Published Aug 22, 2019• 14 citations
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Abstract

The critical behavior of statistical models with long-range interactions exhibits distinct regimes as a function of $\ensuremath{\rho}$, the power of the interaction strength decay. For large enough $\ensuremath{\rho}, \ensuremath{\rho}>{\ensuremath{\rho}}_{\mathrm{sr}}$, the critical behavior is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold ${\ensuremath{\rho}}_{\mathrm{sr}}$ in the case of the long-range XY model in two dimensions. We study the 2D XY model on the diluted graph, a sparse graph obtained from the 2D lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as ${|r|}^{\ensuremath{-}\ensuremath{\rho}}$, which is expected to feature the same critical behavior of the long-range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetization and of the Binder cumulant, we present numerical evidence that ${\ensuremath{\rho}}_{\mathrm{sr}}=4$. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless universality class for $\ensuremath{\rho}\ensuremath{\ge}4$, and to present a second-order transition for $\ensuremath{\rho}<4$.

DOI
10.1103/physrevb.100.054203

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The critical behavior of statistical models with long-range interactions exhibits distinct regimes as a function of \ensuremathρ\ensuremath{\rho}\ensuremathρ, the power of the interaction strength decay. For large enough \ensuremathρ,\ensuremathρ>\ensuremathρsr\ensuremath{\rho}, \ensuremath{\rho}>{\ensuremath{\rho}}_{\mathrm{sr}}\ensuremathρ,\ensuremathρ>\ensuremathρsr​, the critical behavior is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold \ensuremathρsr{\ensuremath{\rho}}_{\mathrm{sr}}\ensuremathρsr​ in the case of the long-range XY model in two dimensions. We study the 2D XY model on the diluted graph, a sparse graph obtained from the 2D lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as ∣r∣\ensuremath−\ensuremathρ{|r|}^{\ensuremath{-}\ensuremath{\rho}}∣r∣\ensuremath−\ensuremathρ, which is expected to feature the same critical behavior of the long-range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetization and of the Binder cumulant, we present numerical evidence that \ensuremathρsr=4{\ensuremath{\rho}}_{\mathrm{sr}}=4\ensuremathρsr​=4. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless universality class for \ensuremathρ\ensuremath≥4\ensuremath{\rho}\ensuremath{\ge}4\ensuremathρ\ensuremath≥4, and to present a second-order transition for \ensuremathρ<4\ensuremath{\rho}<4\ensuremathρ<4.
Determining the threshold value σ∗σ_*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−ηSRσ_* = 2 - η_{\mathrm{SR}}σ∗​=2−ηSR​, 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/r2+σ1/r^{2+σ}1/r2+σ. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at σ∗=2σ_* = 2σ∗​=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=8192L = 8192L=8192. By analyzing the Fortuin-Kasteleyn critical polynomial RpR_pRp​, the Binder ratio QmQ_mQm​, and the anomalous dimension ηηη, we obtain convergent and self-consistent evidence that the universality class already changes sharply at σ=2σ= 2σ=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∗=2σ_* = 2σ∗​=2.
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Abstract The crossover between short-range and long-range (LR) universal behaviors remains a central theme in the physics of LR 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 ∼1/r 2+σ . Utilizing an advanced update strategy, we conduct LR 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 < 2, which exhibit 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 ≤ 2 agree with our theoretical analysis. Our findings request further theoretical understanding and can be of practical application in cutting-edge experiments like Rydberg atom arrays.
Logarithmic Quasi-Long-Range Order and Novel Continuous Phase Transition in the Two-Dimensional XYXYXY Model with 1/r41/r^41/r4 Interaction
We investigate the phase diagram and critical properties of the two-dimensional classical XY model with interactions decaying as 1/r2+σ1/r^{2+σ}1/r2+σ. At the marginal case σ=2σ=2σ=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)−ηℓC(r)\sim(\ln r)^{-η_\ell}C(r)∼(lnr)−ηℓ​ with ηℓ∝Tη_\ell\propto Tηℓ​∝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/r41/r^41/r4 interaction logarithmically modifies the spin-wave stiffness kernel as γ(k)≃κk2ln⁡(1/k)γ(k)\simeqκk^2\ln(1/k)γ(k)≃κk2ln(1/k), which suppresses vortex proliferation and gives rise to a nonuniversal correlation-length exponent ν∝1/2πκν\propto1/\sqrt{2πκ}ν∝1/2πκ​. Crucially, our simulations reveal that the short-range algebraic QLRO phase is unstable against a weak long-range perturbation for σ≤2σ\le2σ≤2. These results establish σ=2σ=2σ=2 as the crossover boundary between the long-range and short-range universality classes and provide a critical evaluation of recent theoretical proposals.
The critical behavior of the O(2) model on dilute Levy graphs built on a 2D square lattice is analyzed. Different qualitative cases are probed, varying the exponent rho governing the dependence on the distance of the connectivity probability distribution. The mean-field regime, as well as the long-range and short-range non-mean-field regimes are investigated by means of high-performance parallel Monte-Carlo numerical simulations running on GPUs. The relationship between the long-range rho exponent and the effective dimension of an equivalent short-range system with the same critical behavior is investigated. Evidence is provided for the effective short-range dimension to coincide with the spectral dimension of the Levy graph for the XY model.
The critical behavior of the O(2) model on dilute Levy graphs built on a 2D square lattice is analyzed. Different qualitative cases are probed, varying the exponent rho governing the dependence on the distance of the connectivity probability distribution. The mean-field regime, as well as the long-range and short-range non-mean-field regimes are investigated by means of high-performance parallel Monte-Carlo numerical simulations running on GPUs. The relationship between the long-range rho exponent and the effective dimension of an equivalent short-range system with the same critical behavior is investigated. Evidence is provided for the effective short-range dimension to coincide with the spectral dimension of the Levy graph for the XY model in the mean-field regime.