Nature of the spin liquid in underdoped cuprate superconductors

In the present work we address a long standing problem of the magnetic ground state and magnetic excitations in underdoped cuprates. Modelling cuprates by the extended $t-J$ model we show that there is a hidden dimensionless parameter $\lambda$ which drives magnetic criticality at low doping $x$. Hence we derive the zero temperature $\lambda-x$ phase diagram of the model. It is argued that all underdoped cuprates are close to the quantum tricritical point $x=0$, $\lambda=1$. 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 $E_{cross}$ of the inelastic neutron scattering response maximum at ${\bm q}=(\pi,\pi)$ and compare our results with experiments. We also explain softening of magnons in the intermediate regime observed in inelastic neutron scattering.