${\varphi }^4$ lattice model with cubic symmetry in three dimensions: Renormalization group flow and first-order phase transitions

We study the three-component ${\varphi }^4$ model on the simple cubic lattice in the presence of a cubic perturbation. To this end, we perform Monte Carlo simulations in conjunction with a finite-size scaling analysis of the data. The analysis of the renormalization group (RG) flow of a dimensionless quantity provides us with the accurate estimate $Y_4-{\omega }_2=0.00081\left(7\right)$ for the difference of the RG eigenvalue $Y_4$ at the $\text{O}\left(3\right)$-symmetric fixed point and the correction exponent ${\omega }_2$ at the cubic fixed point. We determine an effective exponent ${\nu }_{\text{eff}}$ of the correlation length that depends on the strength of the breaking of the $\text{O}\left(3\right)$ symmetry. Field theory predicts that depending on the sign of the cubic perturbation, the RG flow is attracted by the cubic fixed point, or runs to an ever increasing amplitude, indicating a fluctuation-induced first-order phase transition. We demonstrate directly the first-order nature of the phase transition for a sufficiently strong breaking of the O(3) symmetry. We obtain accurate results for the latent heat, the correlation length in the disordered phase at the transition temperature, and the interface tension for interfaces between one of the ordered phases and the disordered phase. We study how these quantities scale with the RG flow, allowing quantitative predictions for weaker breaking of the $\text{O}\left(3\right)$ symmetry.

Figure 1

We plot the line of slow flow in the $(\lambda ,\mu )$ plane as parametrized by Eq. (28) using the numerical values ${\lambda }^{*}=5.12$, $c=-0.8$, $d=0.06$, and $e=0.01$ as solid black line. The solid circles give values of $(\lambda ,\mu )$ with a small correction amplitude $w$. These are used in the analysis of the RG flow below. Additional pairs of $(\lambda ,\mu )$ that are analyzed in this section are plotted as diamonds. The cross gives the improved point for the decoupled Ising system. The asterisks give points, where, below in Sec. 6, we demonstrate directly that the transition is first order.

Figure 2

We plot ${\overline{U}}_4$ versus ${\overline{U}}_C$ obtained for six different pairs of $(\lambda ,\mu )$ and a number of linear lattice sizes $L\ge 12$. The largest lattice size is $L=192$, 128, 96, 128, 64, and 48 for $(\lambda ,\mu )=(4.0,-1.2)$, $(3.7,-1.33)$, $(3.0,-1.449)$, $(2.7,-1.552)$, $(2.333,-1.764)$, and $(2.0,-1.85)$, respectively. For a discussion see the text.

Figure 3

We plot estimates of $u$, Eq. (31), computed by using Eq. (38) and the data for $(\lambda ,\mu )=(3.4,-1.566)$ and $(3.0,-1.449)$ as a function of ${\overline{U}}_C$. The data are obtained for the pairs of lattice sizes $(L_1,L_2)=(12,16)$, $(16,24)$, ..., (64,96). Since ${\overline{U}}_C$ increases with $L$ in the given range, the leftmost points correspond to $(L_1,L_2)=(12,16)$, while the rightmost ones correspond to $(L_1,L_2)=(64,96)$. For a discussion see the text.

Figure 4

We plot estimates of $u$, Eq. (31), computed by using Eq. (38) and the data for $(\lambda ,\mu )=(2.7,-1.552)$, $(3.0,-1.449)$, $(3.7,-1.3)$, $(3.8,-1.2)$, and $(4.0,-1.2)$, as a function of ${\overline{U}}_C$. For a discussion see the text.

Figure 5

We plot $u$, Eq. (31), obtained from a large set of $(\lambda ,\mu )$ as a function of ${\overline{U}}_C$. Results obtained by using Eq. (35) are given as diamonds, while those obtained by using Eq. (38) are given as squares. The filled circle gives the result for the decoupled Ising system. The dotted line shows the behavior (34) in the neighborhood of the decoupled Ising system. Details are discussed in the text.

Figure 6

We plot $\mathrm{\Delta }{y}_t$, Eq. (42), obtained for a large set of $(\lambda ,\mu )$ as a function of ${\overline{U}}_C$. Details are discussed in the text.

Figure 7

We give the histograms of the energy density at $(\lambda ,\mu )=(1.24,-2.3)$ for the linear lattice sizes $L=12$, 16, 24, 28, 32, and 40. The data are reweighted to the Boltzmann distribution for $\beta =0.3294108$, which is our estimate of the inverse of the transition temperature.