On the construction of Wannier functions in topological insulators: the 3D case

We investigate the possibility of constructing exponentially localized composite Wannier bases, or equivalently smooth periodic Bloch frames, for 3-dimensional time-reversal symmetric topological insulators, both of bosonic and of fermionic type, so that the bases in question are also compatible with time-reversal symmetry. This problem is translated in the study, of independent interest, of homotopy classes of continuous, periodic, and time-reversal symmetric families of unitary matrices. We identify three $\mathbb{Z}_2$-valued complete invariants for these homotopy classes. When these invariants vanish, we provide an algorithm which constructs a "multi-step" logarithm that is employed to continuously deform the given family into a constant one, identically equal to the identity matrix. This algorithm leads to a constructive procedure to produce the composite Wannier bases mentioned above.Comment: 29 pages. Version 2: minor corrections of misprints, corrected proofs of Theorems 2.4 and 2.9, added references. Accepted for publication in Annales Henri Poicar\'

Beyond Diophantine Wannier diagrams: Gap labelling for Bloch-Landau Hamiltonians

It is well known that, given a $2d$ purely magnetic Landau Hamiltonian with a constant magnetic field $b$ which generates a magnetic flux $\varphi$ per unit area, then any spectral island $\sigma_b$ consisting of $M$ infinitely degenerate Landau levels carries an integrated density of states $\mathcal{I}_b=M \varphi$. Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any $2d$ Bloch-Landau operator $H_b$ which also has a bounded $\mathbb{Z}^2$-periodic electric potential. Assume that $H_b$ has a spectral island $\sigma_b$ which remains isolated from the rest of the spectrum as long as $\varphi$ lies in a compact interval $[\varphi_1,\varphi_2]$. Then $\mathcal{I}_b=c_0+c_1\varphi$ on such intervals, where the constant $c_0\in \mathbb{Q}$ while $c_1\in \mathbb{Z}$. The integer $c_1$ is the Chern marker of the spectral projection onto the spectral island $\sigma_b$. This result also implies that the Fermi projection on $\sigma_b$, albeit continuous in $b$ in the strong topology, is nowhere continuous in the norm topology if either $c_1\ne0$ or $c_1=0$ and $\varphi$ is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields.Comment: 20 pages, no figures. Appendix C added. Final version accepted for publication in Journal of the European Mathematical Societ

Discrete port-controlled Hamiltonian dynamics and average passivation

The paper discusses the modeling and control of port-controlled Hamiltonian dynamics in a pure discrete-time domain. The main result stands in a novel differential-difference representation of discrete port-controlled Hamiltonian systems using the discrete gradient. In these terms, a passive output map is exhibited as well as a passivity based damping controller underlying the natural involvement of discrete-time average passivity

Parseval frames of exponentially localized magnetic Wannier functions

Motivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension $d \le 3$, we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states. When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model $m$ occupied energy bands by a real-analytic and $\mathbb Z^{d}$-periodic family $\{P({\bf k})\}_{{\bf k} \in \mathbb R^{d}}$ of orthogonal projections of rank $m$. A moving orthonormal basis of $\mathrm{Ran} P({\bf k})$ consisting of real-analytic and $\mathbb Z^d$-periodic Bloch vectors can be constructed if and only if the first Chern number(s) of $P$ vanish(es). Here we are mainly interested in the topologically obstructed case. First, by dropping the generating condition, we show how to algorithmically construct a collection of $m-1$ orthonormal, real-analytic, and periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of $m+1$ real-analytic and periodic Bloch vectors which generate $\mathrm{Ran} P({\bf k})$. Both algorithms are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. A moving Parseval frame of analytic, periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of magnetic Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting. Our results are illustrated in crystalline insulators modelled by $2d$ discrete Hofstadter-like Hamiltonians, but apply to certain continuous models of magnetic Schr\"{o}dinger operators as well.Comment: 40 pages. Improved exposition and minor corrections. Final version matches published paper on Commun. Math. Phy

Long-term radial-velocity variations of the Sun as a star: The HARPS view

Stellar radial velocities play a fundamental role in the discovery of extrasolar planets and the measurement of their physical parameters as well as in the study of stellar physical properties. We investigate the impact of the solar activity on the radial velocity of the Sun using the HARPS spectrograph to obtain measurements that can be directly compared with those acquired in the extrasolar planet search programs. We use the Moon, the Galilean satellites, and several asteroids as reflectors to measure the radial velocity of the Sun as a star and correlate it with disc-integrated chromospheric and magnetic indexes of solar activity that are similar to stellar activity indexes. We discuss in detail the systematic effects that affect our measurements and the methods to account for them. We find that the radial velocity of the Sun as a star is positively correlated with the level of its chromospheric activity at about 95 percent significance level. The amplitude of the long-term variation measured in the 2006-2014 period is 4.98 \pm 1.44 m/s, in good agreement with model predictions. The standard deviation of the residuals obtained by subtracting a linear best fit is 2.82 m/s and is due to the rotation of the reflecting bodies and the intrinsic variability of the Sun on timescales shorter than the activity cycle. A correlation with a lower significance is detected between the radial velocity and the mean absolute value of the line-of-sight photospheric magnetic field flux density. Our results confirm similar correlations found in other late-type main-sequence stars and provide support to the predictions of radial velocity variations induced by stellar activity based on current models.Comment: 11 pages, 7 figures, 2 tables, 1 Appendix; accepted by Astronomy and Astrophysic

High frequency acoustic modes in vitreous Beryllium Fluoride probed by inelastic X-ray scattering

Inelastic X-ray Scattering measurements of the dynamics structure factor have bene performed on vitreous Beryllium Fluoride ({\it v-}BeF$_2$) at $T$=297 K in the momentum transfer, $Q$, range $Q$=1.5$\div$10 nm$^{-1}$. We find evidence of well defined high frequency acoustic modes. The energy position and linewidth of the excitations disperse with $Q$ as $\propto Q$ and $\propto Q^2$, respectively up to about one half of the first maximum of the static structure factor. Their magnitude compare favorably with low-frequency sound velocity and absorption data. The results indicate worth mentioning similarities of the high frequency collective dynamics of different network forming glasses such as {\it v-}B$_2$O$_3$ and {\it v-}SiO$_2$.Comment: 17 pages, 5 .ps fig

Biogeographical patterns of endolithic infestation in an invasive and an indigenous intertidal marine ecosystem engineer

By altering the phenotypic properties of their hosts, endolithic parasites can modulate the engineering processes of marine ecosystem engineers. Here, we assessed the biogeographical patterns of species assemblages, prevalence and impact of endolithic parasitism in two mussel species that act as important ecosystem engineers in the southern African intertidal habitat, Perna perna and Mytilus galloprovincialis. We conducted large-scale surveys across three biogeographic regions along the South African coast: the subtropical east coast, dominated by the indigenous mussel, P. perna, the warm temperate south coast, where this species coexists with the invasive Mediterranean mussel, M. galloprovincialis, and the cool temperate west coast dominated by M. galloprovincialis. Infestation increased with mussel size, and in the case of M. galloprovincialis we found a significantly higher infestation in the cool temperate bioregion than the warm temperate region. For P. perna, the prevalence of infestation was higher on the warm temperate than the subtropical region, though the difference was marginally non-significant. On the south coast, there was no significant difference in infestation prevalence between species. Endolith-induced mortality rates through shell collapse mirrored the patterns for prevalence. For P. perna, endolith species assemblages revealed clear grouping by bioregions. Our findings indicate that biogeography affects cyanobacteria species composition, but differences between biogeographic regions in their effects are driven by environmental conditions.Agência financiadora Número do subsídio Fundacao para a Ciencia e Tecnologia (FCT-MEC, Portugal) UID/Multi/04326/2019 IF/01413/2014/CP1217/CT0004 South African Research Chairs Initiative (SARChI) of the Department of Science and Technology National Research Foundationinfo:eu-repo/semantics/publishedVersio

Intra-molecular origin of the fast relaxations observed in the Brillouin light scattering spectra of molecular glass-formers

The Brillouin light scattering spectra of the o-terphenyl single crystal are compared with those of the liquid and the glass phases. This shows: i) the direct evidence of a fast relaxation at 5 GHz in both the single crystal and the glass; ii) a similar temperature dependence for the attenuation of the longitudinal sound waves in the single crystal and the glass; and iii) the absence of coupling between the fast relaxation and the transverse acoustic waves. These results allow us to assign such a relaxation to the coupling between the longitudinal acoustic waves and intra-molecular vibrations, and therefore to exclude any relationship between it and the glass transition.Comment: 4 Pages, 4 Figure

Constraining Cosmological Models by the Cluster Mass Function

We present a comparison between two observational and three theoretical mass functions for eight cosmological models suggested by the data from the recently completed BOOMERANG-98 and MAXIMA-1 cosmic microwave background (CMB) anisotropy experiments as well as peculiar velocities (PVs) and type Ia supernovae (SN) observations. The cosmological models have been proposed as the best fit models by several groups. We show that no model is in agreement with the abundances of X-ray clusters at $\sim 10^{14.7} h^{-1}M_{\odot}$.On the other hand, we find that the BOOM+MAX+{\sl COBE}:I, Refined Concordance and $\Lambda$MDM are in a good agreement with the abundances of optical clusters. The P11 and especially Concordance models predict a slightly lower abundances than observed at $\sim 10^{14.6} h^{-1}M_{\odot}$. The BOOM+MAX+{\sl COBE}:II and PV+CMB+SN models predict a slightly higher abundances than observed at $\sim 10^{14.9} h^{-1}M_{\odot}$. The nonflat MAXIMA-1 is in a fatal conflict with the observational cluster abundances and can be safely ruled out.Comment: 17 pages, 2 figures, reference added, figures changes, substantial revision mad