Eliashberg theory of quantum criticality - when it is valid and when it fails.

We consider a set of models of itinerant 2D fermionis interacting with a soft boson that becomes gapless at a quantum-critical point (QCP) . Our goal is to understand when Eliadhberg theory is valid all the way to a QCP and when it breaks down before a QCP is reached. We argue that this depends on how flat bosonic dispersion is near a relevant momentum. We argue that for strong enough dispersion, Eliashberg theory is valid, while for weaker dispersion it breaks down. For the latter case, we go beyond Eliashber theory and employ eikonal computational procedure -- a summation of infite diagrammatic series for the fermionic self-energy and bosonic polarization, assuming that a fermionic frequency is much larger than typical frequency of a boson. We argue that this procedure is complimentary to Eliashberg theory, i.e., it become valid when Eliashberg theory breaks down. We show the results of eikonal summation and argue that they are very different for the cases when a boson is of density origin and when it is of spin origin. In the latter case, we show that the system develops precursors to the ordered state already at T=0.

In collaboration with Shangshun Zhang, Zachary Raines (Minnesota), Mengxing Ye (Utah)