Geometric Delocalization in Two Dimensions

Abstract

We demonstrate the existence of transient two-dimensional surfaces where a random-walking particle escapes to infinity in contrast to localization in standard flat 2D space. We first prove that any rotationally symmetric 2D membrane embedded in flat 3D space cannot be transient. Then we formulate a criterion for the transience of a general asymmetric 2D membrane. We use it to explicitly construct a class of transient 2D manifolds with a non-trivial metric and height function but ``zero average curvature,'' which we dub tablecloth manifolds. The absence of the logarithmic infrared divergence of the Laplace-Beltrami operator in turn implies the absence of weak localization, non-existence of bound states in shallow potentials, and breakdown of the Mermin-Wagner theorem and Kosterlitz-Thouless transition on the tablecloth manifolds, which may be realizable in both quantum simulators and corrugated two-dimensional materials.

Figures (5)

• Figure 1: Two examples of tablecloth manifolds. On the right, a generic tablecloth manifold, and on the left the simplified example constructed in Eq. \ref{['eqn:h']}. The colors demonstrate the ratio of the volume element at any point on the manifold, $\sqrt{g(r,\theta)}drd\theta$, to the regular flat volume element at the same point, $r dr d\theta$. Therefore, the colors encode the value of $\sqrt{g(r,\theta)}/r$. This allows us to compare the volume growth to the regular flat one as we go away from the origin with increasing $r$. Purple corresponds to the regular $\pi r^2$ volume growth of a flat disk, while other colors designate faster growths.
• Figure 2: Calculating the resistance along a thin strip with angle $\delta\theta$. The volume (which in two dimensions is the surface area) of the shaded region,$\sqrt{g(r,\theta)}\,\delta\theta\,\delta r$, is given by the product of its physical length $A(r,\theta)\delta r$ and its physical cross-section which therefore is given by $\sqrt{g(r,\theta)}\delta\theta \delta r/A(r,\theta)\delta r$. The resistance of the shaded region is proportional to its physical length divided by its cross-section. Consequently, the resistance of the whole strip is proportional to $\int_{r_0}^\infty dr A^2(r,\theta)/\sqrt{g(r,\theta)}\,\delta\theta$.
• Figure 3: A disk shaped tablecloth has no wrinkles when lying flat on a flat two dimensional surface. When the tablecloth is set on a circular table and drapes from it, it needs to fit into a new geometry, transitioning from disk $ds_D^2 = dr_{\text{ph}}^2 + r_{\text{ph}}^2 d\theta^2$ to cylinder $ds_C^2 = dr_{\text{ph}}^2 + R^2 d\theta$ with $R$ being the radius of the circular table. Because of the volume mismatch, $\sqrt{g_D}/\sqrt{g_C}=r_{\text{ph}}/R$, the tablecloth needs to wrinkle up. Also note how the radial coordinate, $r$, differs from $r_{\text{ph}}$. $r_{\text{ph}}$ is the physical distance a traveler takes on the tablecloth, while $r$ is identified with the radial component of the cylindrical coordinates.
• Figure 4: Drawing of example bump functions $\chi_n(r)$, $n=1,2,3$. Each function $\chi_n$ is supported in $[n(n-1)/2,n(n+1)/2]$, and has a flat plateau within that interval where it is exactly equal to one, along with a small interval near the endpoints where it is exactly equal to zero. Precise requirements for the bump functions are given in the Supplemental Material sm.
• Figure 5: Evenly spaced plateaus, leading to a specific example that fails to satisfy Eq. \ref{['eqn:transience-cond']}.

Theorems & Definitions (1)

• Claim 1