Exact solution of a cluster model with next-nearest-neighbor interaction

Understand

A one-dimensional cluster model with next-nearest-neighbor interactions and two additional composite interactions is solved; the free energy is obtained and a correlation function is derived exactly.

• The model is diagonalized by a transformation obtained automatically from its interactions, which is an algebraic generalization of the Jordan-Wigner transformation.
• The gapless condition is expressed as a condition on the roots of a cubic equation, and the phase diagram is obtained exactly.
• We find that the distribution of roots for this algebraic equation determines the existence of long-range order, and we again obtain the ground-state phase diagram.