Abstract
We describe the interplay between $d$-wave superconductivity and spin density wave (SDW) order in a theory of the hole-doped cuprates at hole densities below optimal doping. The theory assumes local SDW order, and associated electron and hole pocket Fermi surfaces of charge carriers in the normal state. We describe quantum and thermal fluctuations in the orientation of the local SDW order, which lead to $d$-wave superconductivity: we compute the superconducting critical temperature and magnetic field in a ``minimal'' universal theory. We also describe the back action of the superconductivity on the SDW order, showing that SDW order is more stable in the metal. Our results capture key aspects of the phase diagram of Demler et al. [Phys. Rev. Lett. 87, 067202 (2001)] obtained in a phenomenological quantum theory of competing orders. Finally, we propose a finite temperature crossover phase diagram for the cuprates. In the metallic state, these are controlled by a ``hidden'' quantum critical point near optimal doping involving the onset of SDW order in a metal. However, the onset of superconductivity results in a decrease in stability of the SDW order, and consequently the actual SDW quantum critical point appears at a significantly lower doping. All our analysis is placed in the context of recent experimental results.
Highlights
A number of recent experimental observations have the potential to dramatically advance our understanding of the enigmatic underdoped regime of the cuprates
Chang et al.[10,11] have provided detailed studies of the spin dynamics in the vicinity of Hsdw, including observation of a gapped spin collective mode for H < Hsdw whose gap vanishes as H ր Hsdw. Most recently, such observations have been extended to YBa2Cu3O6.45 by Haug et al.[12], who obtained evidence for the onset of spin density wave (SDW) order at H ≈ 15 T. These observations were all on systems which do not have SDW order at H = 0; they build on the earlier work of Lake et al.[13] who observed enhancement of prexisting SDW order at H = 0 by an applied field in La2−x SrxCuO4 with x = 0.10
This paper has described the phase diagram of a simple ’minimal model’ of the underdoped, hole-doped cuprates contained in Eqs. (1.4), (1.5), and (1.6)
Summary
A number of recent experimental observations have the potential to dramatically advance our understanding of the enigmatic underdoped regime of the cuprates. We will use our theory to propose a finite temperature phase diagram for the hole-doped cuprates; in particular, we will argue that it helps resolve a central puzzle on the location of the quantum critical point important for the finite temperature crossovers into the ‘strange metal’ phase
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