arXiv subjects

Boris Zhilinskii

Publications and source records attributed to Boris Zhilinskii.

7 recordsLinked to original sources

arXiv2018

Band rearrangement through the 3D-Dirac equation with boundary conditions, and the corresponding topological change

Rearrangement of energy bands against a parameter is studied through the 3D-Dirac equation on a ball in $\mathbb{R}^3$ under the APS and the chiral bag boundary conditions on the boundary two-sphere, where APS is an abbreviation of Atiyah-Patodi-Singer. The notion of spectral flow and its extension is introduced to characterize the energy eigenvalue redistribution against the parameter, which reflects an analytical property of edge eigenstates. It is shown that though band rearrangement takes place the net spectral flow is zero ($1+(-1)=0$) for both boundary conditions. The corresponding semi-quantum Hamiltonian defined on the 3D momentum space is studied in parallel and it is shown that a change against the parameter is observed in the mapping degree defined for the semi-quantum model Hamiltonian, which is viewed as reflecting a topological property of the bulk eigenstates of the full quantum system. Specifically, there are two mappings assigned, for which changes in mapping degree take the values $\pm1$. The present correspondence is viewed as a bulk-edge correspondence.

quant-ph↗

arXiv2017

Topological Phase Transition in a Molecular Hamiltonian with Symmetry and Pseudo-Symmetry, Studied through Quantum, Semi-Quantum and Classical Models

The redistribution of energy levels between energy bands is studied for a family of simple effective Hamiltonians depending on one control parameter and possessing axial symmetry and energy-reflection symmetry. Further study is made on the topological phase transition in the corresponding semi-quantum and completely classical models, and finally the joint spectrum of the two commuting observables $(H=E,J_z)$ (also called the lattice of quantum states) is superposed on the image of the energy-momentum map for the classical model. Through these comparative analyses, mutual correspondence is demonstrated to exist among the redistribution of energy levels between energy bands for the quantum Hamiltonian, the modification of Chern numbers of eigenline bundles for the corresponding semi-quantum Hamiltonian, and the presence of Hamiltonian monodromy for the complete classical analog. In particular, as far as the band rearrangement is concerned, a fine agreement is found between the redistribution of the energy levels described in terms of joint spectrum of energy and momentum in the full quantum model and the evolution of singularities of the energy-momentum map of the complete classical model. The topological phase transition observed in the present semi-quantum and the complete classical models are analogous to topological phase transitions of matter.

quant-ph↗

arXiv2014

Local description of band rearrangements. Comparison of semi-quantum and full quantum approach

Rearrangement of rotation-vibration energy bands in isolated molecules within semi-quantum approach is characterized by delta-Chern invariants associated to a local semi-quantum Hamiltonian valid in a small neighborhood of a degeneracy point for the initial semi-quantum Hamiltonian and also valid in a small neighborhood of a critical point corresponding to the crossing of the boundary between iso-Chern domains in the control parameter space. For a full quantum model, a locally approximated Hamiltonian is assumed to take the form of a Dirac operator together with a specific boundary condition. It is demonstrated that the crossing of the boundary along a path with a delta-Chern invariant equal to $\pm1$ corresponds to the transfer of one quantum level from a subspaces of quantum states to the other subspace associated with respective positive and negative energy eigenvalues of the local Dirac Hamiltonian.

math-ph↗

arXiv2013

Qualitative features of the rearrangement of molecular energy spectra from a "wall-crossing" perspective

Qualitatively different systems of molecular energy bands are studied on example of a parametric family of effective Hamiltonians describing rotational structure of triply degenerate vibrational state of a cubic symmetry molecule. The modification of band structure under variation of control parameters is associated with a topological invariant "delta-Chern". This invariant is evaluated by using a local Hamiltonian for the control parameter values assigned at the boundary between adjacent parameter domains which correspond to qualitatively different band structures.

math-ph↗

arXiv2007

Qualitative Analysis of the Classical and Quantum Manakov Top

Qualitative features of the Manakov top are discussed for the classical and quantum versions of the problem. Energy-momentum diagram for this integrable classical problem and quantum joint spectrum of two commuting observables for associated quantum problem are analyzed. It is demonstrated that the evolution of the specially chosen quantum cell through the joint quantum spectrum can be defined for paths which cross singular strata. The corresponding quantum monodromy transformation is introduced.

math-ph↗

arXiv2002

Topologically coupled energy bands in molecules

We propose a concrete application of the Atiyah-Singer index formula in molecular physics, giving the exact number of levels in energy bands, in terms of vector bundles topology. The formation of topologically coupled bands is demonstrated. This phenomenon is expected to be present in many quantum systems.

quant-ph↗

arXiv1999

Topological Chern indices in molecular spectra

Topological Chern indices are related to the number of rotational states in each molecular vibrational band. Modification of the indices is associated to the appearance of ``band degeneracies'', and exchange of rotational states between two consecutive bands. The topological dynamical origin of these indices is proven through a semi-classical approach, and their values are computed in two examples. The relation with the integer quantum Hall effect is briefly discussed.

quant-ph↗