Magnetic phases in the S=1 Shastry-Sutherland model with uniaxial anisotropy

We explore the field induced magnetic phases of an $S=1$ $XXZ$ model with single-ion anisotropy and large Ising-like anisotropy on a Shastry Sutherland lattice over a wide range of Hamiltonian parameters and applied magnetic field. The multitude of ground state phases are characterized in detail in terms of their thermodynamic properties and the underlying classical (Ising limit) spin arrangements for the plateau phases are identified by calculating the static structure factors. The enlarged local Hilbert space of the $S=1$ spins results in several new ground state phases that are not realized for $S=1/2$ spins. These include the quantum paramagnetic state that is ubiquitous to $S=1$ spins with single ion anisotropy, two different spin supersolid phases (with distinct longitudinal ordering) and a magnetization plateau that arises as a direct descendant of the 1/3 plateau due to quantum fluctuations that are not possible for $S=1/2$ spins. We predict the same mechanism will lead to plateaus at smaller fractions of 1/3 for higher spins. The full momentum dependence of the longitudinal and transverse components of the static structure factor is calculated in the spin supersolid phase to demonstrate the simultaneous existence of diagonal and off-diagonal long-range order as well as the different longitudinal orderings.Comment: 7 pages, 7 figure

Evolutionary multiplayer games on graphs with edge diversity

Evolutionary game dynamics in structured populations has been extensively explored in past decades. However, most previous studies assume that payoffs of individuals are fully determined by the strategic behaviors of interacting parties and social ties between them only serve as the indicator of the existence of interactions. This assumption neglects important information carried by inter-personal social ties such as genetic similarity, geographic proximity, and social closeness, which may crucially affect the outcome of interactions. To model these situations, we present a framework of evolutionary multiplayer games on graphs with edge diversity, where different types of edges describe diverse social ties. Strategic behaviors together with social ties determine the resulting payoffs of interactants. Under weak selection, we provide a general formula to predict the success of one behavior over the other. We apply this formula to various examples which cannot be dealt with using previous models, including the division of labor and relationship- or edge-dependent games. We find that labor division facilitates collective cooperation by decomposing a many-player game into several games of smaller sizes. The evolutionary process based on relationship-dependent games can be approximated by interactions under a transformed and unified game. Our work stresses the importance of social ties and provides effective methods to reduce the calculating complexity in analyzing the evolution of realistic systems.Comment: 50 pages, 7 figure