Numerical tests of the large charge expansion

TL;DR

We study the large charge expansion in the 3D critical $O(2)$ CFT by computing scaling dimensions and OPE data of highly charged operators. Using an improved Worm Monte Carlo algorithm with a continuous-time update and a ratio-based measurement scheme, we determine $D(Q)$ up to $Q=19$ and OPE coefficients in Regimes I and II. The extracted scaling data satisfy $D(Q)= c_{3/2} Q^{3/2}+ c_{1/2} Q^{1/2}+ c_0+ O(Q^{-1/2})$ with $c_{3/2}=0.339(1)$ and $c_{1/2}=0.25(1)$, and the OPE results agree with the corresponding EFT predictions within uncertainties, providing evidence for the superfluid EFT in the large-charge sector. The study also carefully analyzes lattice and finite-size corrections, outlining paths toward higher-precision tests and future extensions, such as higher-spin operators and alternative numerical approaches.

Abstract

We perform Monte-Carlo measurements of two and three point functions of charged operators in the critical O(2) model in 3 dimensions. Our results are compatible with the predictions of the large charge superfluid effective field theory. To obtain reliable measurements for large values of the charge, we improved the Worm algorithm and devised a measurement scheme which mitigates the uncertainties due to lattice and finite size effects.

Figures (13)

• Figure 1: $\Delta(1)$, extracted with eq.\ref{['eq:ratios']}, for $\alpha =2$ and $L\in[16,32]$. Notice that the region where there are significant deviations from a constant value, on the left of the vertical line, gets smaller as $L$ increases. This is expected from lattice effects. We include the previous Monte-Carlo result with error bars at 1$\sigma$, taken from Tab.I in Banerjee:2017fcx, as well as the bootstrap result Kos:2015mba.
• Figure 2: Best-fit values of $c_{3/2}$ (left) and $c_{1/2}$ (right) as a function of the minimum charge included in the fit. For larger values of $Q_\textrm{min}$ the error bars are larger than the plotted range. The coloured regions represent the 1$\sigma$ interval quoted on \ref{['eq:results conformal dimension']}.
• Figure 3: Difference between the data and the best-fit curves with the averaged parameters in eq. \ref{['eq:results conformal dimension']} (red) and with $Q_\text{min}=8$ (blue). For large values of $Q$ the uncertainties exceed the range displayed in the plot.
• Figure 4: Results for $\mathcal{A}(Q)$, defined in eq. \ref{['eq:A(Q)']}. Notice the large uncertainties despite the precision of the measurements of $\Delta(Q)$.
• Figure 5: Difference between the value of $\gamma_Q$ measured from Monte-Carlo and the theoretical prediction $2D(Q) + D(2Q)$ (see eq. \ref{['eq_ratio_OPE']}). The result is obtained from the ratio of correlation functions sampled at $L=32$ and $L=64$, at the same relative position. We used the values $D(Q)$ and $D(2Q)$ measured in the previous section for the theoretical prediction.