Topological Mechanics at Finite Temperature

Topological states have been studied extensively in mechanical lattices at T=0, leading to remarkable observations from topologically protected edge modes, one-way waves, to stress focusing. Little is known about how thermal fluctuations affect these phenomena. Here, we introduce a theoretical framework to analyze the behavior of topological mechanical states when the influence of temperature is significant, important for systems at small scales. We analyze effects of fluctuations on topological mechanics, employing Feynman diagram expansions, thereby estimating fluctuation corrections to the model’s band structure. This enables us to establish the condition under which thermal fluctuations alter the topology of the system Hamiltonian. Furthermore, we can design metamaterials whose mechanical properties, influenced by topology, undergo desirable changes as a function of temperature. We demonstrate our findings using lattice models, with agreement between analytic theory and molecular dynamics simulations, revealing changes in edge modes as a consequence of the altered topology under the influence of temperature.