Simulation of the electron-phonon interaction in infinite dimensions

The electron-phonon interaction corresponding to the Holstein model (with Coulomb repulsion) is simulated in infinite dimensions using a novel quantum Monte Carlo algorithm. The thermodynamic phase diagram includes commensurate charge-density-wave phases, incommensurate chargedensity-wave phases, and superconductivity. The crossover from a weak-coupling picture (where pairs both form and condense at Tc) to a strong-coupling picture (where preformed pairs condense at Tc) is illustrated with the onset of a double-well structure in the effective phonon potential. 1. Infinite-dimensional formalism Strong electron-electron correlations are responsible for many important and exotic phenomena in condensed-matter systems including superconductivity, magnetism, heavy fermions, etc. Strongly correlated electronic systems are those in which the average electronic correlation energy is equal to or larger than the electronic kinetic energy. Exotic phenomena arise from the competition of simultaneously minimizing the kinetic and potential energy of the electrons. Models of these systems usually do not have analytic solutions. However, recently, Metzner and Vollhardt [1] discovered that these many-body problems simplify in the limit of infinite spatial dimensions. The limit must be taken in such a fashion that the electronic kinetic energy remains finite, so that the effects of the strong electron correlations remain. Consider the electronic kinetic energy determined by a tight-binding model with hopping between nearest-neighbor sites (with hopping integral t) on a hypercubic lattice in d dimensions. The band structure ǫ(k) becomes