Where is the quantum critical point in the cuprate superconductors?

Abstract

Transport measurements in the hole-doped cuprates show a "strange metal" normal state with an electrical resistance which varies linearly with temperature. This strange metal phase is often identified with the quantum critical region of a zero temperature quantum critical point (QCP) at hole density x=x_m, near optimal doping. A long-standing problem with this picture is that low temperature experiments within the superconducting phase have not shown convincing signatures of such a optimal doping QCP (except in some cuprates with small superconducting critical temperatures). I review theoretical work which proposes a simple resolution of this enigma. The crossovers in the normal state are argued to be controlled by a QCP at x_m linked to the onset of spin density wave (SDW) order in a "large" Fermi surface metal, leading to small Fermi pockets for x<x_m. A key effect is that the onset of superconductivity at low temperatures disrupts the simplest canonical quantum critical crossover phase diagram. In particular, the competition between superconductivity and SDW order_shifts_ the actual QCP to a lower doping x_s < x_m in the underdoped regime, so that SDW order is only present for x<x_s. I review the phase transitions and crossovers associated with the QCPs at x_m and x_s: the resulting phase diagram as a function of x, temperature, and applied magnetic field consistently explains a number of recent experiments.