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We propose a new generalized Su–Schrieffer–Heeger model with hierarchical long-range hopping based on a one-dimensional tetratomic chain. The properties of the topological states and phase transition, which depend on the cointeraction of the intracell and intercell hoppings, are investigated using the phase diagram of the winding number. It is shown that topological states with large positive/negative winding numbers can readily be generated in this system. The properties of the topological states can be verified by the ring-type structures in the trajectory diagram of the complex plane. The topological phase transition is strongly related to the opening (closure) of an energy bandgap at the center (boundaries) of the Brillouin zone. Finally, the non-zero-energy edge states at the ends of the finite system are revealed and matched with the bulk–boundary correspondence. [ABSTRACT FROM AUTHOR]

Considering that under the framework of the massive gravity theory, the interaction between the mass gravitons and Schwarzschild black hole (BH) could make it carry a scalar charge, the phase transition process caused by this scalar charge is investigated in this analysis. The phase transition grade and microstructure of those BHs are investigated from both macroscopic and microscopic points of view. From the macroscopic point of view, it is found that Ehrenfest equations are satisfied at the phase transition critical point, which implies that the phase transition grade of those BHs is second-order. Based on the BH molecules model and Landau continuous phase transition theory, the phase transition of those BHs from the microcosmic point of view is analyzed. The critical exponents obtained from the two perspectives are consistent. By investigating the Ruppeiner geometry, the microstructure feature of those BHs is revealed. These results suggest that the phase transition of BH in massive gravity is a standard second-order phase transition at the critical point, and the microscopic details of those BHs are different from the Reissnerâ€"Nordström AdS BH in standard Einstein gravity. [ABSTRACT FROM AUTHOR]

The R/Ba-ordered and R-site mixed compound Y0.5La0.5BaMn2O6 is synthesized, in which (Y, La) and Ba are regularly arranged, while Y and La randomly occupy the R-site. Y0.5La0.5BaMn2O6 has a tetragonal unit cell with a space group of P4/mmm. A structural transition between tetragonal and orthorhombic is observed at about 325 K by X-ray powder diffraction (XRD). Thermal magnetic measurement shows the occurrence of an antiferromagnetic transition at the temperature TN ~ 190 K. Anomalies in magnetization, resistivity and lattice parameters observed around 340 K indicate a charge/orbital order transition accompanying the structural phase transition. The R-site randomness effect is discussed to interpret the different properties of Y0.5La0.5BaMn2O6 between NdBaMn2O6 and SmBaMn2O6. [ABSTRACT FROM AUTHOR]