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The description and understanding of the consequences of competinginteractions in various systems, both classical and quantum, are notoriouslydifficult due to insufficient information involved in conventional concepts,for example, order parameters and/or correlation functions. Here we go beyondthese conventional language and present a pattern picture to describe andunderstand the frustration physics by taking the one-dimensional (1D) axialnext-nearest-neighbor Ising (ANNNI) model in a transverse field as an example.The system is dissected by the patterns, obtained by diagnonalizing the modelHamiltonian in an operator space with a finite lattice size $4n$ ($n$: naturalnumber) and periodic boundary condition. With increasing the frustrationparameter, the system experiences successively various phases/metastates,identified respectively as those with zero, two, four, $\cdots$, $2n$domains/kinks, where the first is the ferromagnetic phase and the last theantiphase. Except for the ferromagnetic phase and antiphase, the others shouldbe metastates, whose transitions are crossing over in nature. The resultsclarify the controversial issues about the phases in the 1D ANNNI model andprovide a starting point to study more complicated situations, for example, thefrustration systems in high dimensions.

Pattern description of the ground state properties of the one-dimensional axial next-nearest-neighbor Ising model in a transverse field