Nonlocal effective-average-action approach to crystalline phantom membranes

We investigate the properties of crystalline phantom membranes, at the crumpling transition and in the flat phase, using a nonperturbative renormalization group approach. We avoid a derivative expansion of the effective average action and instead analyze the full momentum dependence of the elastic coupling functions. This leads to a more accurate determination of the critical exponents and further yields the full momentum dependence of the correlation functions of the in-plane and out-of-plane fluctuation. The flow equations are solved numerically for $D=2$ dimensional membranes embedded in a $d=3$ dimensional space. Within our approach we find a crumpling transition of second order which is characterized by an anomalous exponent ${\eta }_c\approx 0.63(8)$ and the thermal exponent $\nu \approx 0.69$. Near the crumpling transition the order parameter of the flat phase vanishes with a critical exponent $\beta \approx 0.22$. The flat phase anomalous dimension is ${\eta }_f\approx 0.85$ and the Poisson’s ratio inside the flat phase is found to be ${\sigma }_f\approx -1/3$. At the crumpling transition we find a much larger negative value of the Poisson’s ratio ${\sigma }_c\approx -0.71(5)$. We discuss further in detail the different regimes of the momentum dependent fluctuations, both in the flat phase and in the vicinity of the crumpling transition, and extract the crossover momentum scales which separate them.

Figure 1

Diagrammatic representation of the flow equation which determines the flow of the order parameter and which arises from the vanishing flow of the one-point vertex [28]. Wavy lines correspond to $G_{\mathit{ab}}$ propagators and solid lines to $G_{\mathit{hh}}$. Lines with a dash correspond to single-scale propagators. The small open circle with a dot denotes a derivative of the order parameter with respect to $\Lambda$.

Figure 2

Diagrams which enter the flow of the self-energy ${\Sigma }_{\mathit{hh}}$ of the out-of-plane fluctuations. The solid dot above the self-energy on the left-hand side denotes a derivative with respect to $\Lambda$; the other symbols are defined in the caption of Fig. 1.

Figure 3

Diagrams which determine the flow of the self-energy ${\Sigma }_{\mathit{ab}}$ of the in-plane fluctuations. The symbols are defined as in Figs. 1 and 2.

Figure 4

Flow of the critical exponent $\eta$ for different initial values of $J_{{\Lambda }_0}^2$ (solid dots). For finite $J^2=\underset{\Lambda \to 0}{lim}{J}_{\Lambda }^2$, ${\eta }_{\Lambda }$ saturates at the flat phase fixed point value ${\eta }_f\approx 0.85$; this value is indicated by the upper vertical line. At the crumpling transition (with $J_{{\Lambda }_0}=J_c$) ${\eta }_{\Lambda }$ saturates at the critical fixed point value ${\eta }_c\approx 0.638$ (see curve 1); this value is indicated by the lower vertical line. The values for $(J_{{\Lambda }_0}^2-J_c^2)/J_c^2$ are $8.0\times {10}^{-7}$ (curve 2), $3.0\times {10}^{-5}$ (curve 3), $5.0\times {10}^{-4}$ (curve 4), $9.1\times {10}^{-3}$ (curve 5), and $1.9\times {10}^{-1}$ (curve 6).

Figure 5

The correlation function $G_{\mathit{hh}}(k)$ (solid lines) for different values of $J_{{\Lambda }_0}^2-J_c^2$ (curves are multiplied with $k^3$ for easier visibility of the different scaling regimes). At the critical point of the crumbling transition (upmost curve) a single crossover from the perturbative scaling to a critical scaling with ${\eta }_c\approx 0.63(8)$ is observed. Slightly away from criticality (middle curve), $G_{\mathit{hh}}$ enters at small momenta the flat phase fixed point scaling with ${\eta }_f\approx 0.85$. Even further away (lower curve), the intermediate regime, where $G_{\mathit{hh}}$ obeys critical scaling, disappears and a direct crossover from perturbative to flat phase fixed point scaling is observed. Dashed lines indicate the different scaling behaviors.

Figure 6

The correlation function $G_{\perp }(k)$ (solid lines) for different values of $J_{{\Lambda }_0}^2-J_c^2$ (curves are multiplied with $k^2$ for easier visibility of the different scaling regimes). At the critical point of the crumbling transition (upmost curve) a single crossover from the perturbative scaling to a critical scaling with ${\eta }_c\approx 0.63(8)$ is observed. At criticality, we have $G_{\mathit{hh}}(k)=G_{\perp }(k)=G_{\parallel }(k)$ since the order parameter $J$ vanishes. Slightly away from criticality (middle curve), $G_{\mathit{hh}}$ enters at small momenta the flat phase fixed point scaling with ${\eta }^u=4-D-2{\eta }_f\approx 0.30$. Even further away (lower curve), the intermediate regime, where $G_{\mathit{hh}}$ obeys critical scaling, disappears and a direct crossover from perturbative to flat phase fixed point scaling is observed. Dashed lines indicate the different scaling behaviors.

Figure 7

Flow of the Poisson’s ratio ${\sigma }_{\Lambda }$. In the flat phase, we find $\sigma =-1/3$ whereas at the crumpling transition we find a much larger magnitude, $\sigma \approx -0.71(5)$. Both fixed point values for ${\sigma }_{\Lambda }$ are indicated by dashed lines. The flow in the middle shows a flow of ${\sigma }_{\Lambda }$ for a membrane close to the critical point, with the IR properties however dominated by the flat phase fixed point.

Figure 8

Dependence of the fully renormalized order parameter $J$ as a function of $\delta r={\mathrm{r̃}}_c-{\mathrm{r̃}}_0$. The critical exponent is found to be $\beta \approx 0.22$.