Approximate scaling relation for the anharmonic electron-phonon problem

The interaction of electrons with anharmonic lattice vibrations is a long-standing problem that is not yet fully understood. What is surprising about this problem is that nearly all real materials are anharmonic ~as can be seen by the fact that they expand or contract upon heating!, but quasiharmonic models ~which replace the anharmonic phonons by harmonic phonons with temperature-dependent phonon frequencies! work remarkably well at describing properties of most materials. Superconductivity is described most accurately, where the theory of electrons interacting with harmonic phonons, introduced by Migdal and Eliashberg, can routinely reproduce experimental tunneling conductances to better than one part in a thousand. The explanation for this result is actually quite simple—the thermal effects that arise due to a nonuniform spacing of the anharmonic energy levels are unimportant when the temperature is much less than the effective phonon frequency ~defined by the difference in energy between the ground and the first-excited state of the anharmonic phonon!. Furthermore, quantum Monte Carlo ~QMC! studies, have shown that anharmonicity does not appear to produce any exotic behavior, such as enhancements of transition temperatures, or novel superconducting behavior. Instead, the results indicate that an effect of anharmonicity is to generically break particle-hole symmetry. The discovery we present here is that the anharmonic systems can be mapped onto harmonic ones, with results from widely different parameter regimes collapsing onto the same scaling curve. We believe that this result sheds light onto the question of why harmonic models work so well for describing properties of real materials. Our strategy is to solve anharmonic models in the limit of large spatial dimension where the lattice many-body problem can be mapped onto a self-consistently embedded impurity problem that is solved via a QMC simulation for quantum phonons, or via an iterative transcendental equation for classical phonons. The simplest electron-phonon model that includes anharmonic effects is the anharmonic Holstein model in which the conduction electrons interact with local phonon modes: