General theory of emergent elasticity for second-order topological phase transitions

Abstract

The raise of the symmetry breaking mechanism by Landau[1] is a landmark in the studies of phase transitions. The Kosterlitz-Thouless phase transition[2-3] and the fractional quantum Hall effect[4], however, are believed to be induced by another mechanism: topology change. Despite rapid development of the theory of topological orders[5-7], a unified theoretical framework describing this new paradigm of phase transition and its relation to the Landau paradigm, is not seen. Here, we establish such a framework based on variational principle, and show that the critical condition for any second-order topological phase transitions is loss of positive-definiteness of a topologically protected second-order variation of the free energy. A topologically protected variation of a field solution is performed with respect to its emergent displacements, which is introduced by constructing an emergent elasticity problem for the solution. The Landau paradigm of phase transitions studies global property changes induced instability, while topological phase transitions study local property changes induced topological instability. The general effectiveness of this criterion is shown through analyzing the topological stability of several prototype solutions in both real space (two-kink solutions of the sine-Gordon model; an isolated skyrmion in chiral magnets) and reciprocal space (phonon spectrum of a monoatomic chain and a diatomic chain; band structure of a monoatomic chain within the Kronig-Penny model). Every field is emergent elastic with spatially modulated stiffness, and changes of topological property occur at its softened points. We anticipate our work to be a starting point for a general study of topological phase transitions in all field theories.