We study a class of one-dimensional chains whose topology leads to flatbands in the electronic spectrum. Using the Hubbard model, we find that these materials should exhibit femmagnetic ordering for a half-filled band, in agreement with a theorem by Lieb. Away from half filling the system displays a very rich magnetic phase diagram. Possible experimental realizations are suggested.

Functional brain networks are often constructed by quantifying correlations among brain regions.Their topological structure includes nodes, edges, triangles and even higher-dimensional objects.Topological data analysis (TDA) is the emerging framework to process datasets under this perspective. In parallel, topology has proven essential for understanding fundamental questions in physics. Here we report the discovery of topological phase transitions in functional brain networks by merging concepts from TDA, topology, geometry, physics, and network theory. We show that topological phase transitions occur when the Euler entropy has a singularity, which remarkably coincides with the emergence of multidimensional topological holes in the brain network. Our results suggest that a major alteration in the pattern of brain correlations can modify the signature of such transitions, and may point to suboptimal brain functioning. Due to the universal character of phase transitions and noise robustness of TDA, our findings open perspectives towards establishing reliable topological and geometrical biomarkers of individual and group differences in functional brain network organization.

We study the nature of collective excitations in harmonic chains with masses exhibiting longrange correlated disorder with power spectrum proportional to 1/k α , where k is the wave-vector of the modulations on the random masses landscape. Using a transfer matrix method and exact diagonalization, we compute the localization length and participation ratio of eigenmodes within the band of allowed energies. We find extended vibrational modes in the low-energy region for α > 1. In order to study the time evolution of an initially localized energy input, we calculate the second moment M2(t) of the energy spatial distribution. We show that M2(t), besides being dependent of the specific initial excitation and exhibiting an anomalous diffusion for weakly correlated disorder, assumes a ballistic spread in the regime α > 1 due to the presence of extended vibrational modes.

Topological phases of matter have attracted much attention over the years. Motivated by analogy with photonic lattices, here we examine the edge states of a one-dimensional trimer lattice in the phases with and without inversion symmetry protection. In contrast to the Su-Schrieffer-Heeger model, we show that the edge states in the inversion-symmetry broken phase of the trimer model turn out to be chiral, i.e., instead of appearing in pairs localized at opposite edges they can appear at a single edge. Interestingly, these chiral edge states remain robust to large amounts of disorder. In addition, we use the Zak phase to characterize the emergence of degenerate edge states in the inversion-symmetric phase of the trimer model. Furthermore, we capture the essentials of the whole family of trimers through a mapping onto the commensurate off-diagonal Aubry-André-Harper model, which allow us to establish a direct connection between chiral edge modes in the two models, including the calculation of Chern numbers. We thus suggest that the chiral edge modes of the trimer lattice have a topological origin inherited from this effective mapping. Also, we find a nontrivial connection between the topological phase transition point in the trimer lattice and the one in its associated two-dimensional parent system, in agreement with results in the context of Thouless pumping in photonic lattices. arXiv:1810.05566v2 [cond-mat.mes-hall]

We discuss the rich phase diagram of doped AB2 t-J chains using data from DMRG and exact diagonalization techniques. The J vs δ (hole doping) phase diagram exhibits regions of itinerant ferrimagnetism, Incommensurate, RVB, and Nagaoka States, Phase Separation, and Luttinger Liquid (LL) Physics. Several features are highlighted, such as the modulated ferrimagnetic structure, the occurrence of Nagaoka spin polarons in the underdoped regime and small values of J = 4t 2 /U , where t is the first-neighbor hopping amplitude and U is the on-site repulsive Coulomb interaction, incommensurate structures with nonzero magnetization, and the strong-coupling LL physics in the high-doped regime. We also verify that relevant findings are in agreement with the corresponding ones in the square and n-leg ladder lattices. In particular, we mention the instability of Nagaoka ferromagnetism against J and δ.