Functional renormalization group approach to noncollinear magnets

A functional renormalization group approach to $d$-dimensional, $N$-component, noncollinear magnets is performed using various truncations of the effective action relevant to study their long distance behavior. With help of these truncations we study the existence of a stable fixed point for dimensions between $d=2.8$ and $d=4$ for various values of $N$ focusing on the critical value $N_c\left(d\right)$ that, for a given dimension $d$, separates a first-order region for $N<N_c\left(d\right)$ from a second-order region for $N>N_c\left(d\right)$. Our approach concludes to the absence of a stable fixed point in the physical—$N=2,3$ and $d=3$—cases, in agreement with the $\epsilon =4-d$ expansion and in contradiction with previous perturbative approaches performed at fixed dimension and with recent approaches based on the conformal bootstrap program.

Figure 1

$N=6, d=3$ case: critical exponents $\omega$ (top) and $\nu$ (bottom) as functions of the regulator parameter $\alpha$ [see Eq. (41)] calculated with $p_{\max }=2$ (dotted curve), 3 (dashed curve), and 4 (solid curve). The black dots indicate the position of the minima of the curves.

Figure 2

$N_c(d,\alpha )$ obtained with the LPA with $p_{\max }=2$ (dotted curve), $p_{\max }=3$ (dashed curve) and $p_{\max }=4$ (solid curve) for, from top to bottom: $d=3.8, 3.5$, 3, and $2.8$. For $d=3.8$ and $3.5$, the two curves obtained with $p_{\max }=3$ and 4 are superimposed at this scale. The black dots indicate the position of the maxima of the curves.

Figure 3

Critical exponents $\omega$ (top) and $\eta$ (bottom) as functions of the parameter $\alpha$ for $N=7, d=3, p_{\max }=4$. Curves of different styles correspond to various values of ${\rho }_{\mathrm{fix}}$: the dotted curve corresponds to ${\rho }_{\mathrm{fix}}=0.4$, the dashed curve to ${\rho }_{\mathrm{fix}}=0.5$, the dot-dashed curve to ${\rho }_{\mathrm{fix}}=0.6$, the long dot-dashed curve to ${\rho }_{\mathrm{fix}}=0.7$, the solid curve to ${\rho }_{\mathrm{fix}}=0.9$, and the long dashed curve to ${\rho }_{\mathrm{fix}}=1.2$. The black dots indicate the position of the minima of the curves.

Figure 4

Critical exponent $\nu$ as a function of the parameter $\alpha$ for $N=7, d=3, p_{\max }=4$. The dotted curve corresponds to ${\rho }_{\mathrm{fix}}=0.4$, the dashed curve to ${\rho }_{\mathrm{fix}}=0.5$, the dot-dashed curve to ${\rho }_{\mathrm{fix}}=0.6$, the long dot-dashed curve to ${\rho }_{\mathrm{fix}}=0.7$, the solid curve to ${\rho }_{\mathrm{fix}}=0.9$, and the long dashed curve to ${\rho }_{\mathrm{fix}}=1.2$. There are no extrema for the different curves.

Figure 5

Critical exponents $\omega$ (top) and $\eta$ (bottom) as functions of the parameter $\alpha$ for $N=6, d=3, p_{\max }=4$. Curves of different styles correspond to various values of ${\rho }_{\mathrm{fix}}$: the dotted curve corresponds to ${\rho }_{\mathrm{fix}}=0.4$, the dashed curve to ${\rho }_{\mathrm{fix}}=0.5$, the dot-dashed curve to ${\rho }_{\mathrm{fix}}=0.6$, the long dot-dashed curve to ${\rho }_{\mathrm{fix}}=0.8$, the solid curve to ${\rho }_{\mathrm{fix}}=0.9$, and the long dashed curve to ${\rho }_{\mathrm{fix}}=1.2$.The black dots indicate the position of the minima of the curves.

Figure 6

Critical exponent $\nu$ as a function of the parameter $\alpha$ for $N=6, d=3, p_{\max }=4$. The dotted curve corresponds to ${\rho }_{\mathrm{fix}}=0.4$, the dashed curve to ${\rho }_{\mathrm{fix}}=0.5$, the dot-dashed curve to ${\rho }_{\mathrm{fix}}=0.6$, the long dot-dashed curve to ${\rho }_{\mathrm{fix}}=0.8$, the solid curve to ${\rho }_{\mathrm{fix}}=0.9$, and the long dashed curve to ${\rho }_{\mathrm{fix}}=1.2$. There are no extrema for the different curves.

Figure 7

$N_c(d=3,\alpha )$ for $p_{\max }=4$. The dotted curve corresponds to ${\rho }_{\mathrm{fix}}=0.4$, the dashed curve to ${\rho }_{\mathrm{fix}}=0.5$, the dot-dashed curve to ${\rho }_{\mathrm{fix}}=0.6$, the long dot-dashed curve to ${\rho }_{\mathrm{fix}}=0.8$, the solid curve to ${\rho }_{\mathrm{fix}}=0.9$, and the long dashed curve to ${\rho }_{\mathrm{fix}}=1.2$. The black dots indicate the position of the maxima of the curves.

Figure 8

Curves $N_c\left(d\right)$. Solid curve (this work): LPA' with four functions. Diamond (this work) $N_c(d=3)$ obtained by LPA' with the full potential. Dashed curve: perturbative five loops ($\overline{MS}$ scheme) with $\epsilon$ expansion [23]. Dot-dashed curve with black points: perturbative six loops ($\overline{MS}$ scheme) without $\epsilon$ expansion [12].