arXiv subjects

Alessandro Galvani

Publications and source records attributed to Alessandro Galvani.

3 recordsLinked to original sources

arXiv2021

High-precision anomalous dimension of $3d$ percolation from giant cluster slicing

We apply the critical geometry approach for bounded critical phenomena [1] to $3d$ percolation. The functional shape of the order parameter profile $ϕ$ is related via the fractional Yamabe equation to its scaling dimension $Δ_ϕ$. We obtain $Δ_ϕ= 0.4785(7)$ from which the anomalous dimension $η$ is found to be $η=-0.0431(14)$, a value compatible with, and more precise than, its previous direct measurements. A test of hyperscaling is also performed.

cond-mat.stat-mech↗

arXiv2021

Critical $1$- and $2$-point spin correlations for the $O(2)$ model in $3d$ bounded domains

We study the critical properties of the $3d$ $O(2)$ universality class in bounded domains through Monte Carlo simulations of the clock model. We use an improved version of the latter, chosen to minimize finite-size corrections at criticality, with 8 orientations of the spins and in the presence of vacancies. The domain chosen for the simulations is the slab configuration with fixed spins at the boundaries. We obtain the universal critical magnetization profile and two-point correlations, which favorably compare with the predictions of the critical geometry approach based on the Yamabe equation. The main result is that the correlations, once the dimensionful contributions are factored out with the critical magnetization profile, are shown to only depend on the distance between the points computed using a metric found solving the fractional Yamabe equation. The quantitative comparison with the corresponding results for the Ising model at criticality is shown and discussed. Moreover, from the magnetization profiles the critical exponent $η$ is extracted and found to be in reasonable agreement with up-to-date results.

hep-lat↗

arXiv2021

Magnetization profiles at the upper critical dimension as solutions of the integer Yamabe problem

We study the connection between the magnetization profiles of models described by a scalar field with marginal interaction term in a bounded domain and the solutions of the so-called Yamabe problem in the same domain, which amounts to finding a metric having constant curvature. Taking the slab as a reference domain, we first study the magnetization profiles at the upper critical dimensions $d=3$, $4$, $6$ for different (scale invariant) boundary conditions. By studying the saddle-point equations for the magnetization, we find general formulas in terms of Weierstrass elliptic functions, extending exact results known in literature and finding new ones for the case of percolation. The zeros and poles of the Weierstrass elliptic solutions can be put in direct connection with the boundary conditions. We then show that, for any dimension $d$, the magnetization profiles are solution of the corresponding integer Yamabe equation at the same $d$ and with the same boundary conditions. The magnetization profiles in the specific case of the $4$-dimensional Ising model with fixed boundary conditions are compared with Monte Carlo simulations, finding good agreement. These results explicitly confirm at the upper critical dimension recent results presented in [1].

cond-mat.stat-mech↗