Nature of the spin liquid in underdoped cuprate superconductors

In the present paper, we address a long-standing problem of the magnetic ground state and magnetic excitations in underdoped cuprates. Modeling cuprates by the extended $t\ensuremath{-}J$ model, we show that there is a dimensionless parameter $\ensuremath{\lambda}$ which drives quantum magnetic criticality at low doping $x$. Hence we derive the zero temperature $\ensuremath{\lambda}\ensuremath{-}x$ phase diagram of the model. It is argued that all underdoped cuprates are close to the quantum tricritical point $x=0,\ensuremath{\lambda}=1$. The three phases ``meet'' at the tricritical point: (i) N\'eel antiferromagnet, (ii) spin spiral with antinodal direction of the spiral wave vector, (iii) algebraic spin liquid. We argue that underdoped cuprates belong either to the spin-liquid phase or they are on the borderline between the spin liquid and the spin spiral. We calculate the energy position ${E}_{\text{cross}}$ of the inelastic neutron scattering response maximum at $\mathbit{q}=(\ensuremath{\pi},\ensuremath{\pi})$ and compare our results with experiments. We also explain softening of magnons in the intermediate regime observed in inelastic neutron scattering.