Berry phase effect on metal-insulator transition in chiral symmetry class

The interplay between disorder and a weak topological index in chiral symmetry classes leads to an intermediate phase between metal and Anderson localized phase, dubbed as the "quasi-localized" phase, where wave function along a spatial direction with the topological index is delocalized but exponentially localized along the other directions. In this report, we present a theory of the Anderson transition in 2D chiral symmetry classes with the weak topological index, and clarify that the quasi-localized phase universally emerges from a Berry phase effect associated with the topological index. Field variable in the non-linear sigma model for the chiral symmetry classes has U(1) phase degree of freedom, and Anderson transition therein is induced by a proliferation of vortex-antivortex pair of the U(1) phase. The weak topological index endows the vortex-antivortex pair with a Berry phase that depends on a spatial polarization of the pair. Based on this observation, we argue that the Berry phase leads to a spatial polarization of the pair along the topological direction through a quantum interference effect. In the theory, we first obtain a dual theory of the non-linear sigma model by a duality mapping, that takes a form of matrix-version of sine-Gordon model with a magnetic potential. We analyze the dual theory by the renormalization group (RG) study. The RG analyses as well as a transfer matrix study of lattice models demonstrate a universal emergence of the quasi-localized phase.