Time evolution of localized states in Lieb lattices

TL;DR

This paper investigates how localized states in Lieb lattices evolve over time under a slowly increasing magnetic flux, revealing a precession-like behavior and oscillations influenced by symmetry breaking.

Contribution

It introduces a novel analogy between the time evolution of localized states in Lieb lattices and classical precession, highlighting the impact of magnetic flux on state dynamics.

Findings

Localized states precess around states with small itinerant components.

Amplitude oscillates near unity, with sharp variations at flux quanta.

Magnetic flux induces symmetry breaking and oscillatory behavior.

Abstract

We study the slow time evolution of localized states of the open-boundary Lieb lattice when a magnetic flux is applied perpendicularly to the lattice and increased linearly in time. In this system, Dirac cones periodically disappear, reappear and touch the flat band as the flux increases. We show that the slow time evolution of a localized state in this system is analogous to that of a zero-energy state in a three-level system whose energy levels intersect periodically and that this evolution can be mapped into a classical precession motion with a precession axis that rotates as times evolves. Beginning with a localized state of the Lieb lattice, as the magnetic flux is increased linearly and slowly, the evolving state precesses around a state with a small itinerant component and the amplitude of its localized component oscillates around a constant value (below but close to 1), except at multiples of the flux quantum where it may vary sharply. This behavior reflects the existence of an electric field (generated by the time-dependent magnetic field) which breaks the C4 symmetry of the constant flux Hamiltonian.