Geometric mass acquisition via a quantum metric: An effective-band-mass theorem for the helicity bands

By fully taking into account the virtual interband processes along with the intraband ones, here we propose an effective-band-mass theorem that is suitable for a wide class of single-particle Hamiltonians exhibiting multiple energy bands. Then, for the special case of two-band systems, we show that the interband contribution to the effective band mass of a particle at a given quantum state is directly controlled by the quantum metric of the corresponding state. As an illustration, we consider a spin-orbit-coupled spin-1/2 particle and calculate its effective band mass at the band minimum of the lower helicity band. Independent of the coupling strength, we find that the bare mass m(0) of the particle jumps to 2m(0) for the Rashba coupling and to 3m(0) for the Weyl coupling. This geometric mass enhancement is a nonperturbative effect, uncovering the mystery behind the effective mass of the two-body bound states in the noninteracting limit. As a further illustration, we show that a massless Dirac quasiparticle acquires a linearly dispersing band mass (equivalent to the effective cyclotron mass up to a prefactor) with its momentum through the same mechanism.