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 - J model, we show that there is a dimensionless parameter lambda which drives quantum 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) Neel 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 q = (pi, pi) and compare our results with experiments. We also explain softening of magnons in the intermediate regime observed in inelastic neutron scattering.