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A Quantum Fidelity Study of the Anisotropic Next-Nearest-Neighbour Triangular Lattice Heisenberg Model

Ground- and excited-state quantum fidelities in combination with generalized quantum fidelity susceptibilites, obtained from exact diagonalizations, are used to explore the phase diagram of the anisotropic next-nearest-neighbour triangular Heisenberg model. Specifically, the $J'-J_2$ plane of this model, which connects the $J_1-J_2$ chain and the anisotropic triangular lattice Heisenberg model, is explored using these quantities. Through the use of a quantum fidelity associated with the first excited-state, in addition to the conventional ground-state fidelity, the BKT-type transition and Majumdar-Ghosh point of the $J_1-J_2$ chain ($J'=0$) are found to extend into the $J'-J_2$ plane and connect with points on the $J_2=0$ axis thereby forming bounded regions in the phase diagram. These bounded regions are then explored through the generalized quantum fidelity susceptibilities $χ_ρ$, $χ_{120^{\circ}}$, $χ_D$ and $χ_{CAF}$ which are associated with the spin stiffness, $120^{\circ}$ spiral order parameter, dimer order parameter and collinear antiferromagnetic order parameter respectively. These quantities are believed to be extremely sensitive to the underlying phase and are thus well suited for finite-size studies. Analysis of the fidelity susceptibilities suggests that the $J',J_2 \ll J$ phase of the anisotropic triangular model is either a collinear antiferromagnet or possibly a gapless disordered phase that is directly connected to the Luttinger phase of the $J_1-J_2$ chain. Furthermore, the outer region is dominated by incommensurate spiral physics as well as dimer order.