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The discovery of the topological insulators has fueled a surge of interestsin the topological phases in periodic systems. Topological insulators have bulkenergy gap and topologically protected gapless edge states. The edge states inelectronic systems have been detected by observing the transport properties theHgte quantum wells. The electromagnetic analogues of such electronic edgestates have been predicted and observed in photonic crystals, coupledresonators and linear circuits. However, the edge state spectrums of the twodimensional insulators and their electromagnetic analogues haven't beendirectly measured and thus the gaplessness of the edge states hasn't beenexperimentally confirmed. Here I show the classical strings are more convenientchoice to study the topological phases than the electromagnetic waves. I foundthat classical strings with periodic densities can simulate a variety oftopological phases in condensed matter physics including two dimensionaltopological insulators, three dimensional topological semimetals and weaktopological insulators. Because the eigenfrequencies and eigenfunctions of thestrings can be easily measured, not only the gapless edge state spectrum butalso the bulk topological invariant can now be directly observed. Further moreI show the topological phase transitions can be simulated by the strings. Myresults show the richness of topological phases of mechanical waves. Ianticipate my work to be a starting point to study the topological propertiesof mechanical waves. Even richer topological phases other than those have beenfound in electronics systems can be explored when we use more generalmechanical waves e.g. waves in membranes with periodic densities. These phasesmay provide guides to hunt novel topological properties in other branches ofscience.

Simulating topological phases and topological phase transitions with classical strings