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.