Berry phases and pairing symmetry in Holstein-Hubbard polaron systems

We study the tunneling dynamics of dopant-induced hole polarons that are self-localized by electron-phonon coupling in a two-dimensional antiferromagnet. Our treatment is based on a path-integral formulation of the adiabatic (Born-Oppenheimer) approximation, combined with many-body tight-binding, instanton, constrained lattice dynamics, and many-body exact diagonalization techniques. The applicability and limitations of the adiabatic approximation in polaron tunneling problems are discussed in detail and adiabatic results are compared to exact numerical results for a two-site polaron problem. Our results are mainly based on the Holstein-$\mathrm{tJ}$ and, for comparison, on the Holstein-Hubbard model. We also study the effects of second-neighbor hopping and long-range electron-electron Coulomb repulsion. The polaron tunneling dynamics is mapped onto an effective low-energy Hamiltonian that takes the form of a fermion tight-binding model with occupancy-dependent, predominantly second- and third-neighbor tunneling matrix elements, excluded double occupancy, and effective intersite charge interactions. Antiferromagnetic spin correlations in the original many-electron Hamiltonian are reflected by an attractive contribution to the first-neighbor charge interaction and by Berry phase factors that determine the signs of effective polaron tunneling matrix elements. In the two-polaron case, these phase factors lead to polaron-pair wave functions of either $d_{x^2-y^2}$-wave symmetry or p-wave symmetry with zero and nonzero total pair momentum, respectively. Implications for the doping-dependent isotope effect, pseudogap, and $T_c$ of a superconducting polaron-pair condensate are discussed and compared to observed properties of the cuprate high-$T_c$ materials.