Construction of Fractal Order and Phase Transition with Rydberg Atoms

We propose the construction of a many-body phase of matter with fractal structure using arrays of Rydberg atoms. The degenerate low energy excited states of this phase form a self-similar fractal structure. This phase is analogous to the so-called “type-II fracton topological states.” The main challenge in realizing fractonlike models in standard condensed matter platforms is the creation of multispin interactions, since realistic systems are typically dominated by two-body interactions. In this work, we demonstrate that the van der Waals interaction and experimental tunability of Rydberg-based platforms enable the simulation of exotic phases of matter with fractal structures, and the study of a quantum phase transition involving a fractal ordered phase.

Figure 1

Left: three-body spin interaction on each downward triangle in Eq. (1). Right: one of the low energy excitations of Eq. (1); starting with the obvious ground state with ${\sigma }^z=+1$, the spins are flipped to ${\sigma }^z=-1$ on a $\ell =3$ Sierpinski triangle (blue), this configuration only costs energy on the unit triangles at the three corners (red).

Figure 2

We propose trapping atoms on both the vertices and the center of each downward triangle of the triangular lattice, which together form a honeycomb lattice. Vertices (centers) contain “target” (“auxiliary”) atoms. A Sierpinski triangle excited state is shown where ${\widehat{n}}_t=1$ (blue) and ${\widehat{n}}_a=0$ (orange). The Hamiltonian Eq. (5) reduces to $V_{\mathcal{A}\mathcal{B}}$ between auxiliary and neighboring target atoms, and interaction $V_{\mathcal{B}\mathcal{B}}$ between neighboring target atoms.

Figure 3

With Hamiltonian Eq. (5), if three target atoms at the corners of a triangle with side length $L=8$ are excited from the ground state to the Rydberg state, it costs energy $3V+3v$ at each corner. If we apply an extra detuning $-{\overline{\delta }}_C{\widehat{n}}_t$ on the three corners of the triangle, for sufficiently large ${\overline{\delta }}_C$ the ground state of the system is given by the fractal configuration in Fig. 1.