Electron-Phonon Coupling in the t-J Model

We present numeric results for ground state and angle resolved photoemission spectra (ARPES) for single hole in $tt't''-J$ model coupled to optical phonons. The systematic-error free diagrammatic Monte Carlo [1,2] is employed where the Feynman graphs for the Matsubara Green function in imaginary time are summed up completely with respect to phonons variables. It is found that interplay of electron-phonon interaction with intrinsic to the holes in Mott insulator interaction with magnons can explain the main puzzles encountered in ARPES experimental studies: anomalous damping in underdoped and moderately doped regime, ``kink,'' anomalous ``universal velocity,'' and giant anomalous isotope effect. Crucial discussions with N.\ Nagaosa and Z.-X.\ Shen are acknowledged. ([1] A.\ S.\ Mishchenko {et.\ al}.\ Phys.\ Rev.\ B {\bf 62} 6317 (2000); [2] A.\ S.\ Mishchenko and N.\ Nagaosa\ Phys.\ Rev.\ Lett.\ {\bf 93} 036402 (2004).)