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Magnetic properties of the spin $S=1/2$ Heisenberg chain with hexamer modulation of exchange

We consider the spin-1/2 Heisenberg chain with alternating spin exchange %on even and odd sites in the presence of additional modulation of exchange on odd bonds with period three. We study the ground state magnetic phase diagram of this hexamer spin chain in the limit of very strong antiferromagnetic (AF) exchange on odd bonds using the numerical Lanczos method and bosonization approach. In the limit of strong magnetic field commensurate with the dominating AF exchange, the model is mapped onto an effective $XXZ$ Heisenberg chain in the presence of uniform and spatially modulated fields, which is studied using the standard continuum-limit bosonization approach. In absence of additional hexamer modulation, the model undergoes a quantum phase transition from a gapped string order into the only one gapless Lüttinger liquid (LL) phase by increasing the magnetic field. In the presence of hexamer modulation, two new gapped phases are identified in the ground state at magnetization equal to 1/3 and 2/3 of the saturation value. These phases reveal themselves also in magnetization curve as plateaus at corresponding values of magnetization. As the result, the magnetic phase diagram of the hexamer chain shows seven different quantum phases, four gapped and three gapless and the system is characterized by six critical fields which mark quantum phase transitions between the ordered gapped and the LL gapless phases.