Distribution function of charge carriers in electron-phonon systems with spontaneously broken translational symmetry and 'vertical dispersion' in ARPES spectra of cuprates

We show that spontaneous breaking of the translational symmetry (SBTS) of strongly interacting electron and phonon fields acknowledged in lightly doped cuprates to interpret broad bands in their ARPES and optical conductivity spectra influences the carrier distribution function as charge carrier momentum is no longer certain. We develop appropriate distribution and use it to calculate the carrier concentration in different states in essentially doped systems with SBTS. This allows studying doping and temperature evolution of ARPES spectra of such systems. We reveal that at increasing both temperature or doping polarons concede dominance to bipolarons that results in a shift of the broad band maximum to higher binding energy as it occurs in ARPES spectra of cuprates. At further doping delocalized carriers appear and coexist with bipolarons. We show that intimate property of systems with SBTS is partition of the momentum space between autolocalized and delocalized carriers caused by Pauli exclusion rule. We demonstrate that in ARPES spectra such partition manifests itself as vertical dispersion or waterfalls universally observed in all the cuprates at essential doping and in undoped parent compounds. Thus, studying the carriers distribution in systems where electron-phonon interaction breaks the translation symmetry leads simultaneously to both types of waterfalls observed in significantly doped and undoped cuprates. The article also represents the methods developed to calculate the band in ARPES spectrum caused by bipolaron photodissociation and the bipolaron binding energy and radius using coherent states for the phonon field characterization.