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

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

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

Defect formation in superconducting rings: external fields and finite-size effects

Consistent with the predictions of Kibble and Zurek, scaling behaviour has been seen in the production of fluxoids during temperature quenches of superconducting rings. However, deviations from the canonical behaviour arise because of finite-size effects and stray external fields. Technical developments, including laser heating and the use of long Josephson tunnel junctions, have improved the quality of data that can be obtained. With new experiments in mind we perform large-scale 3D simulations of quenches of small, thin rings of various geometries with fully dynamical electromagnetic fields, at nonzero externally applied magnetic flux. We find that the outcomes are, in practice, indistinguishable from those of much simpler Gaussian analytical approximations in which the rings are treated as one-dimensional systems and the magnetic field fluctuation-free.Comment: 10 pages, 3 figures, presentation at QFS2012, to appear in JLT

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

Abundance ratios of red giants in low mass ultra faint dwarf spheroidal galaxies

Low mass dwarf spheroidal galaxies are key objects for our understanding of the chemical evolution of the pristine Universe and the Local Group of galaxies. Abundance ratios in stars of these objects can be used to better understand their star formation and chemical evolution. We report on the analysis of a sample of 11 stars belonging to 5 different ultra faint dwarf spheroidal galaxies (UfDSph) based on X-Shooter spectra obtained at the VLT. Medium resolution spectra have been used to determine the detailed chemical composition of their atmosphere. We performed a standard 1D LTE analysis to compute the abundances. Considering all the stars as representative of the same population of low mass galaxies, we found that the [alpha/Fe] ratios vs [Fe/H] decreases as the metallicity of the star increases in a way similar to what is found for the population of stars belonging to dwarf spheroidal galaxies. The main difference is that the solar [alpha/Fe] is reached at a much lower metallicity for the UfDSph than the dwarf spheroidal galaxies. We report for the first time the abundance of strontium in CVnI. The star we analyzed in this galaxy has a very high [Sr/Fe] and a very low upper limit of barium which makes it a star with an exceptionally high [Sr/Ba] ratio. Our results seem to indicate that the galaxies which have produced the bulk of their stars before the reionization (fossil galaxies) have lower [X/Fe] ratios at a given metallicity than the galaxies that have experienced a discontinuity in their star formation rate (quenching).Comment: 22 pages, 12 figures, submitted to A&

Gaussianity revisited: Exploring the Kibble-Zurek mechanism with superconducting rings

In this paper we use spontaneous flux production in annular superconductors to shed light on the Kibble-Zurek scenario. In particular, we examine the effects of finite size and external fields, neither of which is directly amenable to the KZ analysis. Supported by 1D and 3D simulations, the properties of a superconducting ring are seen to be well represented by analytic Gaussian approximations which encode the KZ scales indirectly. Experimental results for annuli in the presence of external fields corroborate these findings.Comment: 20 pages, 10 figures; submitted to J. Phys: Condens. Matter for the special issue 'Condensed Matter Analogues of Cosmology'; v2: considerably reduced length, incorporation of experimental details into main text, discussion improved, references added, version accepted for publicatio

Chemical abundances in the nucleus of the Sagittarius dwarf spheroidal galaxy

We present Iron, Magnesium, Calcium, and Titanium abundances for 235 stars in the central region of the Sagittarius dwarf spheroidal galaxy (within 9.0 arcmin ~70 pc from the center) from medium-resolution Keck/DEIMOS spectra. All the considered stars belong to the massive globular cluster M54 or to the central nucleus of the galaxy (Sgr,N). In particular we provide abundances for 109 stars with [Fe/H] > -1.0, more than doubling the available sample of spectroscopic metallicity and alpha-elements abundance estimates for Sgr dSph stars in this metallicity regime. Also, we find the first confirmed member of the Sagittarius dwarf spheroidal with [Fe/H]< -2.0 based on analysis of iron lines. We find for the first time a metallicity gradient in the Sgr,N population, whose peak iron abundance goes from [Fe/H]=-0.38 for R < 2.5 arcmin to [Fe/H]=-0.57 for 5.0 < R < 9.0 arcmin. On the other hand the trends of [Mg/Fe], [Ca/Fe], and [Ti/Fe] with [Fe/H] are the same over the entire region explored by our study. We reproduce the observed chemical patterns of the Sagittarius dwarf spheroidal as a whole with a chemical evolution model implying a high mass progenitor ( M_(DM)=6 X 10^{10} Msun ) and a significant event of mass-stripping occurred a few Gyr ago, presumably starting at the first peri-Galactic passage after infall.Comment: Accepted for publication to A&A, 12 pages, 14 figures, 1 tabl