Quantum buckling

We study the mechanical buckling of a freestanding superfluid layer. A topological defect in the phase of the quantum order parameter distorts the underlying metric into a surface of negative Gaussian curvature, irrespective of the sign of the defect charge. The resulting instability is in striking contrast with classical buckling, where the in-plane strain induced by positive (negative) disclinations is screened by positive (negative) curvature. We derive the conditions under which the quantum buckling instability occurs in terms of the dimensionless ratio between superfluid stiffness and bending modulus. An ansatz for the resulting shape of the buckled surface is analytically and numerically confirmed.

Figure 1

Illustrative plots for surface buckling corresponding to (a) positive disclination, (b) negative disclination, (c) vortex, and (d) antivortex. Positive (negative) topological defects are shown with red (yellow) dots, where a positive (negative) disclination is screened by a surface with positive (negative) Gaussian curvature in the top row, while a vortex (antivortex) buckles the underlying metric into a negative Gaussian curvature surface in the bottom row.

Figure 2

Normalized energy $E\left(m\right)=E/[\left(\pi K\ln \left(\frac{R}{a_0}\right)\right]$ as a function of parameter $m^2$ for $r=\frac{4}{3}{r}_c$ (small dots black), $r=r_c$ (solid line black), $r=\frac{3}{4}{r}_c$ (solid red), $r=\frac{1}{2}{r}_c$ (dash-dot green), and $r=\frac{2}{5}{r}_c$ (dot blue), where $r_c\sim 0.17$ is the critical ratio of bending rigidity $\kappa$ to superfluid stiffness $K$, below which the substrate can buckle. For $r<r_c$, the normalized energy curves show a minima below 1 (top). Bottom plot shows $E\left(m^{*}\right)$ as a function of $r$ also for the saddle surface, where $m^{*}$ is the value of $m$ for which $E\left(m\right)$ has a minimum. The plateau for $r>r_c=0.17$ shows that the substrate cannot buckle for a ratio higher than the critical value.