Flat phase of quantum polymerized membranes

We investigate the flat phase of quantum polymerized phantom membranes by means of a nonperturbative renormalization group approach. We first implement this formalism for general quantum polymerized membranes and derive the flow equations that encompass both quantum and thermal fluctuations. We then deduce and analyze the flow equations relevant to study the flat phase and discuss their salient features: quantum to classical crossover and, in each of these regimes, strong to weak coupling crossover. We finally illustrate these features in the context of free-standing graphene physics.

Figure 1

The RG flow in the $(\overline{\lambda },\overline{\mu })$ plane. The shaded area corresponds to the forbidden region $\overline{\lambda }+\overline{\mu }<0$; the borderline $\overline{\lambda }=-\overline{\mu }$ is repulsive as the line $\overline{\mu }=0$. The dashed line corresponds to the attractive line $\overline{\lambda }=-\overline{\mu }/2$, along which all trajectories converge.

Figure 2

Evolution of the dimensionless quantum coupling constants ${\overline{\mu }}_k$ and ${\overline{\lambda }}_k$ with RG time $-t=ln\mathrm{\Lambda }/k$. The left vertical line corresponds to the quantum to classical crossover momentum scale $k_T$ that is located at $-t_T=1.44$; the right vertical line corresponds to the classical Ginzburg momentum scale $k_G^{\text{cl}}$ that is located at $-t_G^{\text{cl}}=2.30$. $t_T$ marks a crossover between a regime controlled by the quantum Gaussian fixed point and a regime where the vanishing behavior is enhanced by the effective temperature ${\overline{T}}_k$.

Figure 3

Evolution of the dimensionless classical coupling constants ${\overline{\mu }}_k^{\text{cl}}$ and ${\overline{\lambda }}_k^{\text{cl}}$ with RG time $-t=ln\mathrm{\Lambda }/k$. $t_G^{\text{cl}}$ marks a clear crossover between a regime controlled by a classical Gaussian fixed point $\left({\overline{\mu }}_{*}^{\text{cl}}={\overline{\lambda }}_{*}^{\text{cl}}=0\right)$ and a regime controlled by the infrared nontrivial classical fixed point $\left({\overline{\mu }}_{*}^{\text{cl}}\simeq 6.21,{\overline{\lambda }}_{*}^{\text{cl}}\simeq -3.10\right)$.

Figure 4

Evolution of the running anomalous dimension ${\eta }_k$ with the RG time $-t=ln\mathrm{\Lambda }/k$. ${\eta }_k$ evolves from a quasi-Gaussian (quantum) regime to a nontrivial (classical) regime, where it reaches the fixed point value ${\eta }^{*}=0.849$. The oscillations at the beginning of the flow are an artifact related to the use of the $\mathrm{\Theta }$ cutoff.

Figure 5

Evolution of the dimensionless classical bulk modulus ${\overline{K}}_k^{\text{cl}}$ with the RG time $-t=ln\mathrm{\Lambda }/k$ that converges to the value ${\overline{K}}_{*}^{\text{cl}}\simeq 3.11$.

Figure 6

Evolution of the dimensionless classical Young modulus ${\overline{\mathcal{Y}}}_k^{\text{cl}}$ with the RG time $-t=ln\mathrm{\Lambda }/k$ that converges to the value ${\overline{\mathcal{Y}}}_{*}^{\text{cl}}\simeq 8.30$.

Figure 7

Evolution of the Poisson ratio ${\nu }_k$ with the RG time $-t=ln\mathrm{\Lambda }/k$ that converges to the value ${\nu }_{*}=-1/3$.