Anisotropic scale invariance and the uniaxial Lifshitz point from the nonperturbative renormalization group

TL;DR

This work uses nonperturbative renormalization group methods with a derivative expansion to address anisotropic scale invariance at Lifshitz points, focusing on the uniaxial case in three dimensions. By employing a DE(2)+W_{\parallel}(\rho) truncation and careful regulator analysis, it demonstrates the existence of a non-classical Lifshitz fixed point with $\theta<1/2$ and extracts a set of critical exponents, including $\theta \approx 0.455$, $\eta_\perp \approx 0.125$, $\eta_{\parallel} \approx -0.12$, $\nu_\perp \approx 0.679$, $\phi \approx 0.48$, and $\nu_{\parallel} \approx 0.309$. The results are compared with $\epsilon$-expansion and $1/N$ approaches, showing qualitative agreement for some indices and notable differences for others, underscoring the value and limitations of NPRG in anisotropic critical phenomena. The work also outlines systematic error estimates and confirms the necessity of higher-order gradient terms to accurately capture Lifshitz criticality, paving the way for more precise future studies.

Abstract

We employ the derivative expansion of the nonperturbative renormalization group to address the phenomenon of anisotropic scale invariance and the associated functional fixed points, also known as Lifshitz points, in systems characterized by a scalar order parameter. We demonstrate the existence of the Lifshitz fixed point featuring a non-classical value of the anisotropy exponent $θ<1/2$ and provide estimates for values of a set of critical exponents in the physically most relevant case of the three-dimensional uniaxial Lifshitz point $(d,m)=(3,1)$, $m$ denoting the anisotropy index. We compare our predictions with existing estimates from perturbative expansions around dimensionality $d=4+\frac{1}{2}$ as well as those from the $1/N$ expansion.

Figures (6)

• Figure 1: The derivative of the fixed point effective potential $\tilde{u}'(\tilde{\rho})$ at the uniaxial ($m=1$) Lifshitz point in $d=3$. The inset shows the corresponding shape of the function $\tilde{z}_\perp(\tilde{\rho})$. The plot corresponds to the exponential cutoff with ${\alpha=\alpha^{(PMS)}_{\eta_\perp}=0.95}$.
• Figure 2: The fixed point function $\tilde{z}_\parallel(\tilde{\rho})$ at the uniaxial ($m=1$) Lifshitz point in $d=3$. The function is negative for small $\tilde{\rho}$. The inset shows the corresponding shape of the function $\tilde{w}_\parallel(\tilde{\rho})$. The plot corresponds to the exponential cutoff with ${\alpha=\alpha^{(PMS)}_{\eta_\perp}=0.95}$.
• Figure 3: Dependence of the exponent $\eta_\perp$ on the cutoff parameter $\alpha$ in the DE(2)+W$_\parallel(\rho)$ calculation, plotted for the three considered families of regulators. The PMS value of $\alpha$ as well as the shape of the curve differs substantially depending on the regulator type. However, the PMS value of the exponent differs by the relatively small amount of around 3% depending on the regulator.
• Figure 4: Dependence of the anisotropy exponent $\theta$ on the cutoff parameter $\alpha$ in the DE(2)+W$_\parallel(\rho)$ calculation, plotted for the three considered families of regulators. The PMS value of $\alpha$ as well as the shape of the curve differs substantially depending on the regulator type; the PMS value of the exponent is however almost insensitive with respect to the regulator choice.
• Figure 5: Dependence of the $\nu_\perp$ exponent on the cutoff parameter $\alpha$ in the DE(2)+W$_\parallel(\rho)$ calculation, plotted for the three considered families of regulators. The PMS value of $\alpha$ as well as the shape of the curve differs substantially depending on the regulator type; the PMS value of the exponent is however almost insensitive with respect to the regulator choice, the difference being of order $0.3\%$ of the value of $\nu_{\perp}$.