Condensation of bosons and fermions in spin chains and ladders

The closing of the Haldane gap in spin chains in a strong magnetic field can be viewed as Bose condensation of magnons [1]. This analogy helps to obtain the universal properties of spin chains near a quantum critical point separating an {\em integer} (in the sense of Oshikawa {\em et al.} [2]) magnetization plateau from an incommensurate phase [3]. We find a different universal behavior at the quantum phase transitions out of a {\em half-integer} magnetization plateau. That transition can be described as condensation of spinons---domain walls carrying spin $s_z = \pm 1/2$ and best viewed as fermions. The difference between the two types of condensation is most pronounced in the presence of a staggered transverse magnetic field that is often induced in real materials by the staggering of the $g$-tensor or by the Dzyaloshinskii-Moriya interaction. [1] I. Affleck, Phys. Rev. B {\bf 43,} 3215 (1991). [2] M. Oshikawa, M. Yamanaka, and I. Affleck, Phys. Rev. Lett. {\bf 78,} 1984 (1997). [3] J.-B. Fouet, O. Tchernyshyov, and F. Mila, Phys. Rev. B {\bf 70,} 174427 (2004).