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Effective theory of magnetization plateaus in a three-leg ladder with periodic boundary conditions

We show that the low-energy physics of the three-leg-ladder model in the presence of a critical magnetic field can be described by a broken SU(3) spin chain for a periodic rung model and a broken SU(2) spin chain for an open rung model. Using the Lieb-Schultz-Mattis theorem we characterize the possible magnetization plateaus and study the critical behaviour in the region of transition between the plateaus S z = 1/2 and S z = 3/2 by means of renormalization group calculations performed on the bosonized effective continuum field theory. We show that in certain regions of the parameter space of the effective theory the system remains gapless, and we compute the spin-spin correlation functions for these regions. We also discuss the possibility of a plateau at S z = 1, and show that although there exists in the continuum theory a term that might cause the appearance of a plateau there, such a term is unlikely to be relevant. This conjecture is proved by density-matrix renormalization group (DMRG) techniques. The modifications of the three-leg-ladder Hamiltonian that show plateaus at S z = 1, 5/6, 7/6 are discussed. We show the S z = 1 plateau in the XXZ -type modification by means of the DMRG technique. We find non-magnetic gapless excitation on this plateau in the periodic rung case.