Magnetic excitation spectrum of dimerized antiferromagnetic chains

Motivated by recent measurements on CuGe${\mathrm{O}}_3$, the spectrum of magnetic excitations of an antiferromagnetic $S=\frac{1}{2}$ chain with alternating coupling strength is investigated. Wave-vector-dependent magnons and a continuum with square-root behavior at the band edges are found. The spectral density of the continua is calculated. Spin-rotation symmetry fixes the gap of the continuum to be twice the elementary magnon gap. This is in excellent agreement with experimental results. The existence of (anti)bound states of two magnons is predicted: below the continuum a singlet and a triplet, above the continuum a quintuplet.