Monte Carlo study of the $O(2)$-invariant $φ^4$ theory with a cubic perturbation in three dimensions

TL;DR

The paper investigates the RG flow of the 3D two-component $\phi^4$ theory with cubic anisotropy ($N=2$) using Monte Carlo simulations combined with finite-size scaling. It identifies a line of slow RG flow connecting the decoupled Ising and $O(2)$ fixed points and extends to fluctuation-induced first-order transitions, extracting a precise RG exponent $Y_4 = -0.1118(10)$ at the $O(2)$ point. A line of improved parameters in the $(\lambda,\mu)$ plane minimizes leading corrections, enabling accurate access to the slow-flow regime on lattice sizes up to $L\sim 72$, and revealing universal ratios at first-order transitions. The results reinforce a coherent picture of how cubic perturbations influence critical behavior and provide quantitative benchmarks for related lattice models, including Ashkin–Teller-type systems.

Abstract

We study the $2$-component $φ^4$ model on the simple cubic lattice in the presence of a cubic, or equivalently, a $\mathbb{D}_4$ invariant perturbation. To this end, we perform Monte Carlo simulations in conjunction with a finite size scaling analysis of the data. We follow previous work on the $3$-component case. We study the RG flow from the decoupled Ising fixed point into the $O(2)$-invariant one and towards the fluctuation induced first order transition. To this end we study the behavior of phenomenological couplings. At the $O(2)$-invariant fixed point we obtain the estimate $Y_4=-0.1118(10)$ of the RG-exponent of the perturbation. Note that the small modulus of $Y_4$ means that the RG flow is slow. Hence, in order to interpret experiments or Monte Carlo simulations of lattice models, which are effectively described by the $φ^4$ model with a cubic term, we have to consider the RG flow beyond the neighborhood of the fixed points.

Figures (12)

• Figure 1: We have numerically integrated the 1-loop flow equations (\ref{['RGeqCardy']}) for $N=2$ and $\epsilon=1$. The fixed points are given as solid circles. The fixed points are labeled as G (Gaussian), I (decoupled Ising), and XY ($O(2)$ invariant). Selected RG-trajectories are given by dotted lines. Subsequent dots are separated by a scale factor of $b=2^{1/8}$. Hence the larger the distance between the dots, the faster the flow. The arrows indicate the direction of the flow.
• Figure 2: We plot the values of $(\lambda,\mu)$, where we simulated at. The decoupled Ising values are given by a dash-dotted black line. The black square indicates the improved point on this line. The size of the square gives the error. The blue dashed line represents our numerical estimate of the improved line in the $(\lambda,\mu)$ plane. It is given by $\lambda=\lambda^* + b_2 \mu^2 + b_4 \mu^4 + b_6 \mu^6$, using the values given in Eqs. (\ref{['lambdasval']},\ref{['b2val']},\ref{['b4val']},\ref{['b6val']}), respectively. The red dotted line is chosen such that it is quadratic in $\mu$ and goes through $\lambda^*$ at $\mu=0$ and the improved decoupled Ising point. The green crosses indicate values, where a first order phase transition occurs. The blue circles are values with a second order transition close to the improved line. The triangles give values that are further apart from the improved line. These are used to check the effect of corrections. For a detailed discussion see the text.
• Figure 3: We plot estimates of $\lambda^*$ obtained by using four different Ansätze and sets of $(\lambda,\mu)$ versus the minimal lattice size $L_{min}$ that is taken into account. In particular in the case of fit F1 we take set 3 and the Ansatz is characterized by $(m_{max},i_w,i_{sub})=(6,4,1)$. For fit F2 we take set 3 and $(m_{max},i_w,i_{sub})=(6,4,2)$. For fit F3 we take set 2m and $(m_{max},i_w,i_{sub})=(8,4,2)$. For fit F4 we take set 1m and $(m_{max},i_w,i_{sub})=(10,6,2)$. The solid line gives our final estimate of $\lambda^*$. The dashed lines indicate the preliminary estimate of the error. The values of $L_{min}$ are slightly shifted to avoid overlap of the symbols.
• Figure 4: We plot numerical estimates of $u(\overline{U}_C)$ obtained for two different ranges of lattice sizes $L_{min} \le L \le L_{max}$. The black cross gives the decoupled Ising fixed point. The dashed line gives the result for the flow in the neighborhood of the decoupled Ising fixed point, eq. (\ref{['uDI']}).
• Figure 5: We plot numerical estimates of $y_{t,eff}$ versus $\overline{U}_C$. At $\overline{U}_C=0$ we give the estimate $y_t = 1.48872(5)$ for the XY universality class my3Nclock as a black square and at $\overline{U}_C=-0.17455125(500)$ the estimate $y_t = 1.58737472(29)$ for the Ising universality class CB_Ising_2024. The dashed line is the result of a fit with the Ansatz (\ref{['yfit']}). Details are discussed in the text.