Polarization transitions in one-dimensional arrays of interacting rings

Abstract

Periodic nanostructures can display the dynamics of arrays of atoms while enabling the tuning of interactions in ways not normally possible in nature. We examine one-dimensional ͑1D͒ arrays of a "synthetic atom," a one-dimensional ring with a nearest-neighbor Coulomb interaction. We consider the classical limit first, finding that arrays of singly charged rings possess antiferroelectric order at low temperatures when the charge is discrete, but that they do not order when the charge is treated as a continuous classical fluid. In the quantum limit Monte Carlo simulation suggests that the system undergoes a quantum phase transition as the interaction strength is increased. This is supported by mapping the system to the 1D transverse field Ising model. Finally, we examine the effect of magnetic fields. We find that a magnetic field can alter the electrostatic phase transition producing a ferroelectric ground state, solely through its effect of shifting the eigenenergies of the quantum problem. D R 1 1 FIG. 1. ͑Color online͒ A schematic picture of the ground state of classical point electrons for 1D array of rings. The ring radius is R and the separation is D. The 1D ordering is antiferroelectric for the infinite size system and thus has a double degenerate ground state. PHYSICAL REVIEW B 78, 075411 ͑2008͒

Topological Effects in Tunneling-Coupled Systems of One-Dimensional Quantum Rings

“…These results also suggest a richness in the many body case. It is known that tunneling-free ring arrays display a quantum phase transition due to the Coulomb interaction, 6 and the inclusion of tunneling could further add depth to this transition, or result in entirely new system phases or system behavior, as system topology could lead to exotic sorts of topological phase transitions or collective excitations.…”

“…Note that while the sign of ā is arbitrary in Eqs. (5)(6), it has an effect on the wavefunctions in Eqs. (7)(8).…”

“…To find the spectra of these systems for arbitrary tunneling strength, we adapt the Hamiltonian in Eqs. (5)(6) to enforce periodic boundary conditions for the 2RS or FES. We then numerically solve the problem by discretizing angular displacement on each ring on a onedimensional mesh.…”

“…The 3RS Hamiltonian is similar to Eqs. (5)(6), with additional tunneling terms added to reflect system geometry. The numerically calculated spectra as a function of ā is plotted in Fig.…”

“…4 In particular, the excluded center and different boundary conditions of rings allows geometries and topologies to be studied that are unobtainable in nature, or even in similar quantum dot setups. 5 Previous theoretical work has revealed that, in the idealized case of a one dimensional chain of 1D rings, subject to the Coulomb interaction, this topological exclusion of the center results in a quantum phase transition to an antiferrolectrically polarized state at 0 K. 6 This work, however, considered only the limit of no inter-ring tunneling. Tunnel-coupled systems have also been studied, but in limited cases, such as a single electron on two laterally coupled thick rings, where topology impacts the "bond strengths" in the resulting artificial molecule.…”