Casimir effect in critical $\mathrm{O}(N)$ models from non-equilibrium Monte Carlo simulations

TL;DR

The paper tackles universal critical Casimir forces in three-dimensional $O(N)$ vector models by developing a non-equilibrium Monte Carlo method to compute the Casimir amplitude $\Delta$ from the second derivative of the excess free energy in a slab geometry. The slab-exchange protocol, coupled with Jarzynski's equality, provides a direct, primary observable for $\Delta$ and enables systematic control of finite-size effects. High-precision results for $N=1,2,3,4,6$ reveal a non-monotonic dependence on $N$ and show tension with conformal bootstrap estimates, with $N=4$ and $N=6$ reported as new Monte Carlo results. The study emphasizes the role of accurate finite-size analyses and motivates larger-$N$ investigations and improved $1/N$ corrections to understand convergence to the large-$N$ limit.

Abstract

$\mathrm{O}(N)$ vector models in three dimensions, when defined in a geometry with a compact direction and tuned to criticality, exhibit long-range fluctuations which induce a Casimir effect. The strength of the resulting interaction is encoded in the excess free-energy density, which depends on a universal coefficient: the Casimir amplitude. We present a high-precision numerical calculation of the latter, by means of a novel non-equilibrium Monte Carlo algorithm, and compare our findings with results obtained from large-$N$ expansions and from the conformal bootstrap.

Figures (8)

• Figure 1: Protocol to compute second derivatives of free energies. Starting from two lattices of equal height $l$, panel (a), a slab is detached from the left-hand lattice, by removing a set of links (red dotted lines) with coupling $J_{\text{on}\rightarrow\text{off}}$. Simultaneously, a new set of links is introduced (blue dashed lines) with coupling $J_{\text{off}\rightarrow\text{on}}$, embedding the slab in the right-hand lattice, panels (b) and (c). In the end, panel (d), the system consists of two lattices of heights $l-1$ and $l+1$ respectively.
• Figure 2: Results of $\partial^2f_{\text{ex}}/\partial l^2$ (blue circles) for various values of the lattice size $l$ for $N=6$, appropriately normalized. The orange curve is the best fit result for the functional form of eq. \ref{['eq:fit_function_l']}.
• Figure 3: Numerical estimates of $\Delta/N$ (orange circles) compared with previous Monte Carlo results Vasilyev_2009 (cyan triangles) and large-$N$ calculations Dantchev:1998etdDiatlyk:2023msc.
• Figure 4: Comparison between different Monte Carlo determinations of $\Delta/N$ and the bootstrap result from Barrat:2024fwq.
• Figure 5: Calibration of the algorithm at the critical point $\beta=\beta_c$. The relative variance for different $\mathop{\mathrm{O}}\nolimits(N)$ models is plotted against $n_{\mathrm{dof}}/n_{\mathrm{step}}$. A data collapse is manifest for all the values of $N$ except for $N=1$.