Nontrivial Character Research Articles

A Refinement of the Burgess Bound for Character Sums

In this paper we give a refinement of the Burgess bound for multiplicative character sums modulo a prime number q. This continues a series of previous logarithmic improvements, which are mostly due to Friedlander, Iwaniec, and Kowalski. In particular, for any nontrivial multiplicative character χ modulo a prime q and any integer r⩾2, we show that ∑M<n⩽M+Nχ(n)=O(N1−1/rq(r+1)/4r2(logq)1/4r), which sharpens the previous results by a factor (logq)1/4r. Our improvement comes from averaging over numbers with no small prime factors rather than over an interval as in the previous approaches.

Improved bounds on Gauss sums in arbitrary finite fields

Let [Formula: see text] be a power of a prime and let [Formula: see text] be the finite field consisting of [Formula: see text] elements. We establish new explicit estimates on Gauss sums of the form [Formula: see text], where [Formula: see text] is a nontrivial additive character. In particular, we show that one has a nontrivial upper bound on [Formula: see text] for certain values of [Formula: see text] of order up to [Formula: see text]. Our results improve on the previous best-known bound due to Zhelezov.

Character Ratios for Exceptional Groups of Lie Type

AbstractWe prove character ratio bounds for finite exceptional groups $G(q)$ of Lie type. These take the form $\dfrac{|\chi (g)|}{\chi (1)} \le \dfrac{c}{q^k}$ for all nontrivial irreducible characters $\chi$ and nonidentity elements $g$, where $c$ is an absolute constant, and $k$ is a positive integer. Applications are given to bounding mixing times for random walks on these groups and also diameters of their McKay graphs.

An automorphic approach to Darmon points

We give archimedean and non-archimedean constructions of Darmon points on modular abelian varieties attached to automorphic forms over arbitrary number fields and possibly non-trivial central character. An effort is made to present a unifying point of view, emphasizing the automorphic nature of the construction.

Polynomial bound for partition rank in terms of analytic rank

Let $$G_1, \ldots , G_k$$ be vector spaces over a finite field $${\mathbb {F}} = {\mathbb {F}}_q$$ with a non-trivial additive character $$\chi $$ . The analytic rank of a multilinear form $$\alpha :G_1 \times \cdots \times G_k \rightarrow {\mathbb {F}}$$ is defined as $${\text {arank}}(\alpha ) = -\log _q \mathop {\mathbb {E}} _{x_1 \in G_1, \ldots , x_k\in G_k} \chi (\alpha (x_1,\ldots , x_k))$$ . The partition rank $${\text {prank}}(\alpha )$$ of $$\alpha $$ is the smallest number of maps of partition rank 1 that add up to $$\alpha $$ , where a map is of partition rank 1 if it can be written as a product of two multilinear forms, depending on different coordinates. It is easy to see that $${\text {arank}}(\alpha ) \le O({\text {prank}}(\alpha ))$$ and it has been known that $${\text {prank}}(\alpha )$$ can be bounded from above in terms of $${\text {arank}}(\alpha )$$ . In this paper, we improve the latter bound to polynomial, i.e. we show that there are quantities C, D depending on k only such that $${\text {prank}}(\alpha ) \le C ({\text {arank}}(\alpha )^D + 1)$$ . As a consequence, we prove a conjecture of Kazhdan and Ziegler. The same result was obtained independently and simultaneously by Janzer.

Nontrivial topological valence bands of common diamond and zinc-blende semiconductors

The diamond and zinc-blende semiconductors are well-known and have been widely studied for decades. Yet, their electronic structure still surprises with unexpected topological properties of the valence bands. In this joint theoretical and experimental investigation, we demonstrate for the benchmark compounds InSb and GaAs that the electronic structure features topological surface states below the Fermi energy. Our parity analysis shows that the spin-orbit split-off band near the valence band maximum exhibits a strong topologically nontrivial behavior characterized by the ${\mathcal{Z}}_{2}$ invariants $(1;000)$. The nontrivial character is a consequence of the nonzero spin-orbit coupling and is imposed by the chosen constituents, in contrast to the conventional topological phase transition mechanism which relies on tuning parameters in the system Hamiltonian. Ab initio-based tight-binding calculations resolve topological surface states in the occupied electronic structure of InSb and GaAs, further confirmed experimentally by soft x-ray angle-resolved photoemission from both materials. Our findings are valid for all other materials whose valence bands are adiabatically linked to those of InSb, i.e., many diamond and zinc-blende semiconductors, as well as other related materials, such as half-Heusler compounds.

Magnetic resonance imaging of water absorption by highly porous ceramic materials

By using the magnetic resonance imaging method the nontrivial character of water absorption was demonstrated for the first time in highly porous ceramic materials. The effect of hygroscopic memory was found out which is that the preferable concentration of absorbed water in certain areas within the sample persists regardless the subsequent sample wetting history. Coating the oxide fibres with fluorine-containing hydrocarbons in supercritical CO2 in order to hydrophobize the material has been shown to affect substantially the water transport within the sample that can be referred to as an effective approach to protect the porous materials from humid environment. The results obtained demonstrate the advantages of the magnetic resonance imaging in studying the water absorption processes and visualization of water pathways in highly porous ceramic materials.

Topological magnon bands for magnonics

Topological excitations in magnetically ordered systems have attracted much attention lately. We report on topological magnon bands in ferromagnetic Shastry-Sutherland lattices whose edge modes can be put to use in magnonic devices. The synergy of Dzyaloshinskii-Moriya interactions and geometrical frustration is responsible for the topologically non-trivial character. Using exact spin wave theory we determine the finite Chern numbers of the magnon bands which give rise to chiral edge states. The quadratic band crossing point vanishes due the present anisotropies and the system enters a topological phase. We calculate the thermal Hall conductivity as an experimental signature of the topological phase. Different promising compounds are discussed as possible physical realizations of ferromagnetic Shastry-Sutherland lattices hosting the required antisymmetric Dzyaloshinskii-Moriya interactions. Routes to applications in magnonics are pointed out.

Dirac fermions in the layered titanium-based oxypnictide superconductor BaTi2Bi2O

First-principles calculations and effective model analysis are used to prove the existence of a single pair of three-dimensional Dirac points in the well-known layered titanium-based oxypnictide superconductor ${\mathrm{BaTi}}_{2}{\mathrm{Bi}}_{2}\mathrm{O}$. The nontrivial topological character of ${\mathrm{BaTi}}_{2}{\mathrm{Bi}}_{2}\mathrm{O}$ is demonstrated by calculating the surface states. The Dirac points, residing on the rotational axis, are found approximately 70 meV below the Fermi level which can be lifted to Fermi surface by Sb doping. The Dirac points and related physics are expected to be detected by angle-resolved photoemission spectroscopy. This compound is also found to be a superconductor with transition temperature $({T}_{c})4$.6 K; our results may provide an appropriate platform to study the interplay between topological Dirac fermions and superconductivity.

Fermions and bosons in nonsymmorphic PdSb2 with sixfold degeneracy

PdSb2 is a candidate for hosting 6-fold-degenerate exotic fermions (beyond Dirac and Weyl fermions).The nontrivial band crossing protected by the nonsymmorphic symmetry plays a crucial role in physical properties. We have grown high-quality single crystals of PdSb2 and characterized their physical properties under several stimuli (temperature, magnetic field, and pressure). While it is a diamagnetic Fermi-liquid metal under ambient pressure, PdSb2 exhibits a large magnetoresistance with continuous increase up to 14 T, which follows the Kohler's scaling law at all temperatures. This implies one-band electrical transport, although multiple bands are predicted by first principles calculations. By applying magnetic field along the [111] direction, de Haas-van Alphen oscillations are observed with frequency of 102 T. The effective mass is nearly zero (0.045m0) with the Berry phase close to {\pi}, confirming that the band close to the R point has a nontrivial character. Under quasihydrostatic pressure (p), evidence for superconductivity is observed in the resistivity below the critical temperature Tc. The dome-shaped Tc versus p is obtained with maximum Tc~2.9 K. We argue that the formation of Cooper pairs (bosons) is the consequence of the redistribution of the 6-fold-degenerate fermions under pressure.

Spin in p -adic AdS/CFT

We study the holographic dual of the simplest notion of spin in a p -adic field theory, namely Green’s functions which involve non-trivial sign characters over the p -adic numbers. In order to recover these sign characters from bulk constructions, we find that we must introduce a non-dynamical gauge field on the line graph of the Bruhat–Tits tree. Wilson lines of this gauge field on suitable paths yield the desired sign characters. We show explicitly how to start with complex scalars or fermions in the bulk, coupled to the gauge field, and compute the holographic two-point functions of their dual operators on the boundary.

Iteration of the exterior power on representation rings

In this note we investigate the idea of Michael Atiyah of using, as a possible approach to the Theorem of Feit–Thompson on the solvability of finite groups of odd order, the iterations of the transformation which replaces a representation of a finite group G on a finite dimensional complex vector space E by the difference between the associated representation of G on the sum of exterior powers of E and the trivial representation. We show that G has odd order if and only if the above operation extends to virtual representations and we then express it in terms of the exponential and the Adams operations in the complexified representation ring. We show the relevance of the idea in a concrete example by exhibiting convergence to a non-trivial character.

Magnetic Resonance Imaging of Water Absorption by Highly Porous Ceramic Materials

By using the magnetic resonance imaging method the nontrivial character of water absorption was demonstrated for the first time in highly porous ceramic materials. The effect of hygroscopic memory was found out which is that the preferable concentration of absorbed water in certain areas within the sample persists regardless the subsequent sample wetting history. Coating the oxide fibres with fluorine-containing hydrocarbons in supercritical CO 2 in order to hydrophobize the material has been shown to affect substantially the water transport within the sample that can be referred to as an effective approach to protect the porous materials from humid environment. The results obtained demonstrate the advantages of the magnetic resonance imaging in studying the water absorption processes and visualization of water pathways in highly porous ceramic materials.

Weighted Distribution of Low-lying Zeros of GL(2) -functions

AbstractWe show that if the zeros of an automorphic $L$-function are weighted by the central value of the $L$-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the $L$-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted $L$-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

Composantes isotypiques de pro-$p$-extensions de corps de nombres et $p$-rationalite

Soit p un nombre premier et soit K/k une extension galoisienne finie de corps de nombres de groupe de Galois ∆ d’ordre etranger a p. Soit S un ensemble fini de places finies de k contenant l’ensemble S p des places p-adiques et soit KS la pro-p-extension maximale de K non-ramifiee en dehors de S. Soi tG:“GS{Hun quotient de GS:“Gal KS/K) sur lequel ∆ agit trivialement. Posons X:“H{rH, Hs. Dans ce travail,nous etudions la φ-composante X φ de X pour les caracteres Qp-irreductibles φ de ∆ et nous montrons entre autres, sous la conjecture de Leopoldt et pour tout caractere non trivial φ, que X φ est ZpvGw-libre si et seulement si la φ-composante de la Zp-torsion de Gs /{Gs, Gs est triviale. Nous faisons egalement une etude numerique sur la liberte de Xφ dans des extensions cycliques K{Qde degre 3 et de degre 4 (`a partir de familles de polynomes donnees par Balady, Lecacheux et plus recemment par Balady et Washington),et ainsi que dans des extensions diedrales de degre 6 de Q. Les resultats de nos calculs renforcent une recente conjecture de Gras

Features of the Debye Mode in an Electron Plasma at Various Degrees of the Electron Gas Degeneracy

We have considered the response of an electron plasma with an arbitrary degree of degeneration to an alternating electromagnetic field directed perpendicularly to the boundary of the plasma layer. The plasma screening of the electric field is determined by the Debye mode. The analysis of the behavior of this mode has been carried out depending on the parameters of the problem. For the first time, it has been shown that, ar sufficiently high degrees of the electron gas degeneracy, the range of the Debye mode existence has a substantially nontrivial character, in which the ranges of existence and absence of this mode alternate with increasing electric field frequency.

Determinants of representations of Coxeter groups

In Ayyer et al. (J Comb Theory Ser A 150:208–232, 2017), the authors characterize the partitions of n whose corresponding representations of $$S_n$$ have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups W. Namely, given a nontrivial multiplicative character $$\omega $$ of W, we give a closed formula for the number of irreducible representations of W with determinant $$\omega $$ . For Coxeter groups of type $$B_n$$ and $$D_n$$ , this is accomplished by characterizing the bipartitions associated to such representations.

Two-dimensional type-II Dirac fermions in layered oxides

Relativistic massless Dirac fermions can be probed with high-energy physics experiments, but appear also as low-energy quasi-particle excitations in electronic band structures. In condensed matter systems, their massless nature can be protected by crystal symmetries. Classification of such symmetry-protected relativistic band degeneracies has been fruitful, although many of the predicted quasi-particles still await their experimental discovery. Here we reveal, using angle-resolved photoemission spectroscopy, the existence of two-dimensional type-II Dirac fermions in the high-temperature superconductor La1.77Sr0.23CuO4. The Dirac point, constituting the crossing of d_{x^2 - y^2} and d_{z^2} bands, is found approximately one electronvolt below the Fermi level (EF) and is protected by mirror symmetry. If spin-orbit coupling is considered, the Dirac point degeneracy is lifted and the bands acquire a topologically non-trivial character. In certain nickelate systems, band structure calculations suggest that the same type-II Dirac fermions can be realised near EF.

Alkali-metal-induced topological nodal line semimetal in layered XN2 (X = Cr, Mo, W)

Based on first principles calculations and the K·p effective model, we propose that alkali metal deposition on the surface of hexagonal XN2 (X = Cr, Mo, W) nanosheets induces topologically nontrivial phases in these systems. When spin orbit coupling (SOC) is disregarded, the electron-like conduction band from N-pz orbitals can be considered to cross the hole-like valence band from X-dz2 orbitals, thereby giving rise to a topological nodal line state in lithium-functionalized XN2 sheets (Li2MoN2 and Li2WN2). Such band crossing is protected by the existence of mirror reflection and time reversal symmetry. More interestingly, the bands cross exactly at the Fermi level, and the linear dispersion regions of such band crossings extend to as high as 0.9 eV above the crossing. For Li2CrN2, the results reveal the emergence of a Dirac cone at the Fermi level. Our calculations show that lattice compression decreases the thickness of a Li2CrN2 nanosheet, leading to phase transition to a nodal line semimetal. The evolution of the band gap of Li2XN2 at the Γ point indicates that the nontrivial topological character of Li2XN2 nanolayers is stable over a large strain range. When SOC is included, the band crossing point is gapped out giving rise to quantum spin Hall states in Li2CrN2 nanosheets, while for Li2MoN2, the SOC-induced gap at the crossing points is negligible.

Topological band crossings in hexagonal materials

Topological semimetals exhibit band crossings near the Fermi energy, which are protected by the nontrivial topological character of the wave functions. In many cases, these topological band degeneracies give rise to exotic surface states and unusual magnetotransport properties. In this paper, we present a complete classification of all possible nonsymmorphic band degeneracies in hexagonal materials with strong spin-orbit coupling. This includes (i) band crossings protected by conventional nonsymmorphic symmetries, whose partial translation is within the invariant space of the mirror/rotation symmetry; and (ii) band crossings protected by off-centered mirror/rotation symmetries, whose partial translation is orthogonal to the invariant space. Our analysis is based on (i) the algebraic relations obeyed by the symmetry operators and (ii) the compatibility relations between irreducible representations at different high-symmetry points of the Brillouin zone. We identify a number of existing materials where these nonsymmorphic nodal lines are realized. Based on these example materials, we examine the surface states that are associated with the topological band crossings. Implications for experiments and device applications are briefly discussed.