A generalization of the Artin-Tate formula for fourfolds

We give a new formula for the special value at s=2 of the Hasse-Weil zeta function for smooth projective fourfolds under some assumptions (the Tate and Beilinson conjecture, finiteness of some cohomology groups, etc.). Our formula may be considered as a generalization of the Artin-Tate(-Milne) formula for smooth surfaces, and expresses the special zeta value almost exclusively in terms of inner geometric invariants such as higher Chow groups (motivic cohomology groups). Moreover we compare our formula with Geisser's formula for the same zeta value in terms of Weil-\'etale motivic cohomology groups, and as a consequence (under additional assumptions) we obtain some presentations of weight two Weil-\'etale motivic cohomology groups in terms of higher Chow groups and unramified cohomology groups.Comment: 18 page

The band-gap structures and recovery rules of generalized n-component Fibonacci piezoelectric superlattices

The spectral evolution from periodic structure to random structure has always been an interesting topic in solid state physics, the generalized n-component Fibonacci sequences (n- CF) provide a convenient tool to investigate such process since its randomness can be controlled via the parameter n. In this letter, the band-gap structures of n-CF piezoelectric superlattices have been calculated using the transfer-matrix-method, the self-similarity behavior and recovery rule have been systematically analyzed. Consistent with the rigorous mathematical proof by Hu et al.[A. Hu et al. Phys. Rev. B. 48, 829 (1993)], we find that the n-CF sequences with 2 \leq n \leq 4 are identified as quasiperiodic. The imaginary wave numbers are characterized by the self-similar spectrum, their major peaks can all be properly indexed. In addition, we find that the n = 5 sequence belongs to a critical case which lies at the border between quasiperiodic to aperiodic structures. The frequency range of self-similarity pattern approaches to zero and a unique indexing of imaginary wave numbers becomes impossible. Our study offers the information on the critical 5-CF superlattice which was not available before. The classification of band-gap structures and the scaling laws around fixed points are also given

Zero modes and the edge states of the honeycomb lattice

The honeycomb lattice in the cylinder geometry with zigzag edges, bearded edges, zigzag and bearded edges (zigzag-bearded), and armchair edges are studied. The tight-binding model with nearest-neighbor hoppings is used. Edge states are obtained analytically for these edges except the armchair edges. It is shown, however, that edge states for the armchair edges exist when the the system is anisotropic. These states have not been known previously. We also find strictly localized states, uniformly extended states and states with macroscopic degeneracy.Comment: 6 pages 8 figure

Correlations in one-dimensional disordered electronic systems with interaction

We investigate the effects of randomness in a strongly correlated electron model in one-dimension at half-filling. The ground state correlation functions are exactly written by products of 3$\times$3 transfer matrices and are evaluated numerically. The correlation lengths depend on randomness when the interaction is effectively weak. On the contrary, they are completely insensitive to randomness when the interaction is effectively strong.Comment: 7 pages, Revte

Localization problem of the quasiperiodic system with the spin orbit interaction

We study one dimensional quasiperiodic system obtained from the tight-binding model on the square lattice in a uniform magnetic field with the spin orbit interaction. The phase diagram with respect to the Harper coupling and the Rashba coupling are proposed from a number of numerical studies including a multifractal analysis. There are four phases, I, II, III, and IV in this order from weak to strong Harper coupling. In the weak coupling phase I all the wave functions are extended, in the intermediate coupling phases II and III mobility edges exist, and accordingly both localized and extended wave functions exist, and in the strong Harper coupling phase IV all the wave functions are localized. Phase I and Phase IV are related by the duality, and phases II and III are related by the duality, as well. A localized wave function is related to an extended wave function by the duality, and vice versa. The boundary between phases II and III is the self-dual line on which all the wave functions are critical. In the present model the duality does not lead to pure spectra in contrast to the case of Harper equation.Comment: 10 pages, 11 figure