Simulating topological phases and topological phase transitions with classical strings

Abstract: The discovery of the topological insulators has fueled a surge of interests in the topological phases in periodic systems. Topological insulators have bulk energy gap and topologically protected gapless edge states. The edge states in electronic systems have been detected by observing the transport properties the Hgte quantum wells. The electromagnetic analogues of such electronic edge states have been predicted and observed in photonic crystals, coupled resonators and linear circuits. However, the edge state spectrums of the two dimensional insulators and their electromagnetic analogues haven't been directly measured and thus the gaplessness of the edge states hasn't been experimentally confirmed. Here I show the classical strings are more convenient choice to study the topological phases than the electromagnetic waves. I found that classical strings with periodic densities can simulate a variety of topological phases in condensed matter physics including two dimensional topological insulators, three dimensional topological semimetals and weak topological insulators. Because the eigenfrequencies and eigenfunctions of the strings can be easily measured, not only the gapless edge state spectrum but also the bulk topological invariant can now be directly observed. Further more I show the topological phase transitions can be simulated by the strings. My results show the richness of topological phases of mechanical waves. I anticipate my work to be a starting point to study the topological properties of mechanical waves. Even richer topological phases other than those have been found in electronics systems can be explored when we use more general mechanical waves e.g. waves in membranes with periodic densities. These phases may provide guides to hunt novel topological properties in other branches of science.