On a ring, a single Jordan-Wigner transformation between the Kitaev model and the spin model suffers redundant degrees of freedom. However, we can establish a complete quaternary Jordan-Wigner mapping involving two Kitaev rings and two spin rings with periodic or antiperiodic boundary conditions in the transverse direction. This mapping facilitates us to demonstrate exactly how a topological extended-kink (TEK) phase develops in the interacting Kitaev ring with geometrical ring frustration. Unlike the usual topological phases protected by energy gap in noninteracting systems, this topological phase is gapless. And because the spectra of low energy excitations are quadratic, the specific heat per site approaches a half of Boltzmann constant near absolute zero temperature. More important, the ground state is unique, immune to spontaneous symmetry breaking, and exhibits long-range correlation but without local order parameter. We also demonstrate that concomitant localized kink zero modes (KZM's) take place by introducing a type of bond defect.

We study the quantum phase transitions driven by a point impurity in a chain seamed with ring frustration. Rich phases and quantum phase transitions are uncovered and characterized by both bulk and impurity correlation functions. Nonlocality of the correlation functions are emphasized in manifesting the novel features in the system. We demonstrate that the long-range correlation function can be factorized into local and nonlocal factors in the thermodynamic limit. The gapless topological extended-kink (TEK) phase is disclosed to exhibit long-range correlation but without long-range order, because its ground state is nondegenerate and thus immune to spontaneous symmetry breaking. This conclusion is also true in the classical impurity limit, which is significantly different from that for the open boundary chain without ring frustration. However, spontaneous symmetry breaking does occur in the gapped kink zero mode (KZM) phase and leads to the antiferromagnetic zero mode (AFZM), in which antiferromagnetic order develops in the bulk while entangled states persists locally around the impurity. And as a new feature of quantum phase transition induced by impurity, the transition from the TEK phase to the KZM-AFZM phase is reflected by a steplike nonlocal factor of the correlation function.

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