Some Physics and Mathematics

Nontrivial Riemann Zeros as Spectrum

Define $\Upsilon(s) := \Gamma(s+1)\, (1-2^{1-s}) \, \zeta(s)$ , and let $\mathcal{Z} := \left\{\gamma \in \mathbb{C} \;\middle|\; \Upsilon (\gamma)=0 \right\}$ denote its zero set, consisting of the nontrivial zeta zeros $\rho$ , together with the zeros of $(1-2^{1-s})$ , excluding $s = 1$ . We introduce a non-symmetric operator

$\displaystyle \hat{\mathcal{R}} \colon \mathcal{D}(\hat{\mathcal{R}}) \subset L^2([0,\infty)) \to L^2([0,\infty)) \, ,$

with spectrum

$\displaystyle \sigma(\hat{\mathcal{R}}) = \left\{ i\left(1/2- \gamma \right) \;\middle|\; \gamma \in \mathcal{Z} \right\} \, .$

Assuming simplicity of all nontrivial zeta zeros, we construct a positive semidefinite operator $\hat{W}$ that intertwines $\hat{\mathcal{R}}$ and its adjoint on the corresponding spectral subspace, $\hat{\mathcal{R}}^\dagger \hat{W} = \hat{W} \hat{\mathcal{R}}$ . The positivity of $\hat{W}$ , which represents an operator-theoretic form of (Bombieri’s refinement of) Weil’s positivity criterion, enforces $\Re(\rho)=1/2$ for all $\rho$ , in accordance with the Riemann Hypothesis. The same positivity condition naturally yields a self-adjoint operator whose spectrum coincides with the imaginary parts of the nontrivial zeta zeros. We further extend the framework to accommodate higher-order zeta zeros, should they exist, and observe that it generalizes to any Mellin-transformable $L$ -function satisfying a functional equation.

https://arxiv.org/abs/2408.15135v14

The most updated version (PDF)

Topologically Charged Vortices at Superconductor-Quantum Hall Interfaces

We explore interface states between a type-II s-wave superconductor (SC) and a $\nu =1$ quantum Hall (QH) state. We show that the effective interaction gives rise to two emergent Abelian Higgs fields, representing paired electrons localized at the SC-QH interface. These fields interact with each other in the presence of a Chern-Simons term originating from the QH sector. The Chern-Simons term leads to a topological contribution to the photon mass and imparts a fractional topological charge of $e/2$ to the interface vortices. The topological mass modifies the vortex lattice at the interface, while the topological charge leads to formation of vortex quadruplets. We predict that these effects lead to a topological Abrikosov lattice at the interface.

https://arxiv.org/abs/2501.12908v2

Molecular Impurities as a Realization of Anyons on the Two-Sphere

Studies on experimental realization of two-dimensional anyons in terms of quasiparticles have been restricted, so far, to only anyons on the plane. It is known, however, that the geometry and topology of space can have significant effects on quantum statistics for particles moving on it. Here, we have undertaken the first step towards realizing the emerging fractional statistics for particles restricted to move on the sphere, instead of on the plane. We show that such a model arises naturally in the context of quantum impurity problems. In particular, we demonstrate a setup in which the lowest-energy spectrum of two linear bosonic/fermionic molecules immersed in a quantum many-particle environment can coincide with the anyonic spectrum on the sphere. This paves the way towards experimental realization of anyons on the sphere using molecular impurities. Furthermore, since a change in the alignment of the molecules corresponds to the exchange of the particles on the sphere, such a realization reveals a novel type of exclusion principle for molecular impurities, which could also be of use as a powerful technique to measure the statistics parameter. Finally, our approach opens up a new numerical route to investigate the spectra of many anyons on the sphere. Accordingly, we present the spectrum of two anyons on the sphere in the presence of a Dirac monopole field.

https://arxiv.org/abs/2009.05948v2

Quantum Groups as Hidden Symmetries of Quantum Impurities

We present an approach to interacting quantum many-body systems based on the notion of quantum groups, also known as $q$ -deformed Lie algebras. In particular, we show that if the symmetry of a free quantum particle corresponds to a Lie group $G$ , in the presence of a many-body environment this particle can be described by a deformed group, $G_q$ . Crucially, the single deformation parameter, $q$ , contains all the information about the many-particle interactions in the system. We exemplify our approach by considering a quantum rotor interacting with a bath of bosons, and demonstrate that extracting the value of $q$ from closed-form solutions in the perturbative regime allows one to predict the behavior of the system for arbitrary values of the impurity-bath coupling strength, in good agreement with non-perturbative calculations. Furthermore, the value of the deformation parameter allows to predict at which coupling strengths rotor-bath interactions result in a formation of a stable quasiparticle. The approach based on quantum groups does not only allow for a drastic simplification of impurity problems, but also provides valuable insights into hidden symmetries of interacting many-particle systems.

https://arxiv.org/abs/1809.00222

Experimental Evidence for Quantum Tunneling Time

The first hundred attoseconds of the electron dynamics during strong field tunneling ionization are investigated. We quantify theoretically how the electron’s classical trajectories in the continuum emerge from the tunneling process and test the results with those achieved in parallel from attoclock measurements. An especially high sensitivity on the tunneling barrier is accomplished here by comparing the momentum distributions of two atomic species of slightly deviating atomic potentials (argon and krypton) being ionized under absolutely identical conditions with near-infrared laser pulses (1300 nm). The agreement between experiment and theory provides clear evidence for a nonzero tunneling time delay and a nonvanishing longitudinal momentum of the electron at the “tunnel exit.”

https://arxiv.org/abs/1611.03701

recent posts

Some Physics and Mathematics