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We use the coupled cluster method for infinite chains complemented by exactdiagonalization of finite periodic chains to discuss the influence of athird-neighbor exchange J3 on the ground state of the spin-1/2 Heisenberg chainwith ferromagnetic nearest-neighbor interaction J1 and frustratingantiferromagnetic next-nearest-neighbor interaction J2. A third-neighborexchange J3 might be relevant to describe the magnetic properties of thequasi-one-dimensional edge-shared cuprates, such as LiVCuO4 or LiCu2O2. Inparticular, we calculate the critical point J2^c as a function of J3, where theferromagnetic ground state gives way for a ground state with incommensuratespiral correlations. For antiferromagnetic J3 the ferro-spiral transition isalways continuous and the critical values J2^c of the classical and the quantummodel coincide. On the other hand, for ferromagnetic J3 \lesssim-(0.01...0.02)|J1| the critical value J2^c of the quantum model is smaller thanthat of the classical model. Moreover, the transition becomes discontinuous,i.e. the model exhibits a quantum tricritical point. We also calculate theheight of the jump of the spiral pitch angle at the discontinuous ferro-spiraltransition.

Spiral correlations in frustrated one-dimensional spin-1/2 Heisenberg J1-J2-J3 ferromagnets