We study heat conduction in one dimensional (1D) anharmonic lattices analytically and numerically by using an effective phonon theory. It is found that every effective phonon mode oscillates quasi-periodically. By weighting the power spectrum of the total heat flux in the Debye formula, we obtain a unified formalism that can explain anomalous heat conduction in momentum conserved lattices without on-site potential and normal heat conduction in lattices with on-site potential. Our results agree very well with numerical ones for existing models such as the Fermi-Pasta-Ulam model, the Frenkel-Kontorova model and the φ 4 model etc.

The spontaneous parity-time (PT ) symmetry breaking is discussed in non-Hermitian PT -symmetric Kitaev and extended Kitaev models whose Hermiticity is broken by the presence of two conjugated imaginary potentials ±iγ at two end sites. In the case of the non-Hermitian Kitaev model, a spontaneous PT -symmetry breaking transition (SPT BT ) occurs at a certain γ c in the topologically trivial phase (TTP) region, similar to that of the Su-Schrieffer-Heeger (SSH) model. However, unlike the SSH model, the system also undergoes such a transition in the topologically nontrivial phase (TNP) region. We study an extended Kitaev model by combining the superconducting pairing in the Kitaev model and the staggered hopping in the SSH model. This model contains three different topological phases: the TTP, the Kitaev-like TNP, and the SSH-like TNP. For the non-Hermitian extended Kitaev model, a SPT BT occurs in the Kitaev-like TNP region, as well as in part of the TTP and SSH-like TNP regions, whereas the PT symmetry is broken for an arbitrary nonzero γ in the rest of the TTP and SSH-like TNP regions. Therefore, we can conclude that there is no universal correlation between topological properties and the SPT BT .

In an extended Harper model, the fidelity for the two lowest band-edge states corresponding to different model parameters, the fidelity susceptibility and the von Neumann entropy of the lowest band-edge states, and the spectrum-averaged von Neumann entropy are studied numerically. The fidelity is near one when parameters are in the same phase or same phase boundary; otherwise it is close to zero. There are drastic changes in fidelity when one parameter is at phase boundary. For the fidelity susceptibility the finite scaling analysis is performed. The critical exponents ␣, ␤, and depend on system sizes for the metal-metal phase transition, while this is not so for the metal-insulator phase transition. At both phase transitions / ␣ Ϸ 2. The von Neumann entropy is near one for the metallic phase, while it is small for the insulating phase. There are sharp changes in the von Neumann entropy at phase boundaries. According to the variations of the fidelity, fidelity susceptibility, and the von Neumann entropy with model parameters, the phase diagram, which includes two metallic phases and one insulating phase separated by three critical lines with one bicritical point, can be completely characterized. These numerical results indicate that the three quantities are suited for revealing all the critical phenomena in the model.