Circular-like Maps: Sensitivity to the Initial Conditions, Multifractality and Nonextensivity

We generalize herein the usual circular map by considering inflexions of arbitrary power $z$, and verify that the scaling law which has been recently proposed [Lyra and Tsallis, Phys.Rev.Lett. 80 (1998) 53] holds for a large range of $z$. Since, for this family of maps, the Hausdorff dimension $d_f$ equals unity for all $z$ values in contrast with the nonextensivity parameter $q$ which does depend on $z$, it becomes clear that $d_f$ plays no major role in the sensitivity to the initial conditions.Comment: 15 pages (revtex), 8 fig

Dissipative Symmetry-Protected Topological Order

In this work, we investigate the interplay between dissipation and symmetry-protected topological order. We considered the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki model interacting with an environment where the dissipative dynamics are described by the Lindladian master equation. The Markovian dynamics is solved by the implementation of a tensor network algorithm for mixed states in the thermodynamic limit. We observe that, for time-reversal symmetric dissipation, the resulting steady state has topological signatures even if being a mixed state. This is seen in finite string-order parameters as well as in the degeneracy pattern of singular values in the tensor network decomposition of the reduced density matrix. We also show that such features do not appear for non-symmetric dissipation. Our work opens the way toward a generalized and more practical definition of symmetry-protected topological order for mixed states induced by dissipation.Comment: 5 pages, 5 figure, revised versio

Metallic-insulator phase transitions in the extended Harper model

In this work we investigate the transport properties of non-relativistic quantum particles on incommensurate multilayered structures with the thicknesses $w_n$ of the layers following an extended Harper model given by $w_n = w_0 |\cos(\pi a n^{\nu})|$. For the normal incidence case, which means an one-dimensional system, we obtained that for a specific range of energy, it is possible to see a metallic-insulator transition with the exponent $\nu$. A metallic phase is supported for $\nu<1$. We also obtained that for the specific value $\nu=1$ there is an alternation between metallic and insulator phases as we change the disorder strength $w_0$. When we integrate out all incidence angles, which means a two-dimensional system, the metallic-insulator transition can be seen for much larger range of energy compared to the normal incidence case