Generalized Brillouin zone and non-Hermitian band theory

2021 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Keyword(s):  
Brillouin Zone ◽  
Band Theory

scholarly journals Observation of non-Bloch parity-time symmetry and exceptional points

2020 ◽  
Author(s):  
Peng Xue ◽  
Lei Xiao ◽  
Tianshu Deng ◽  
Kunkun Wang ◽  
Zhong Wang ◽  
...  
Keyword(s):  
Brillouin Zone ◽  
Skin Effect ◽  
Quantum Walks ◽  
Pt Symmetry ◽  
Great Promise ◽  
Single Photons ◽  
Band Theory ◽  
Exceptional Points ◽  
Time Symmetry ◽  
Transition Points

Abstract Parity-time (PT)-symmetric Hamiltonians have widespread significance in non-Hermitian physics. A PT-symmetric Hamiltonian can exhibit distinct phases with either real or complex eigen spectrum, while the transition points in between, the so-called exceptional points, give rise to a host of critical behaviors that holds great promise for applications. For spatially periodic non-Hermitian systems, PT symmetries are commonly characterized and observed in line with the Bloch band theory, with exceptional points dwelling in the Brillouin zone. Here, in non-unitary quantum walks of single photons, we uncover a novel family of exceptional points beyond this common wisdom. These "non-Bloch exceptional points" originate from the accumulation of bulk eigenstates near boundaries, known as the non-Hermitian skin effect, and inhabit a generalized Brillouin zone. Our finding opens the avenue toward a generalized PT-symmetry framework, and reveals the intriguing interplay among PT symmetry, non-Hermitian skin effect, and non-Hermitian topology.

scholarly journals Non-Hermitian Bulk-Boundary Correspondence and Singular Behaviors of Generalized Brillouin Zone

2021 ◽  
Author(s):  
Gang-Feng Guo ◽  
Xi-Xi Bao ◽  
Lei Tan
Keyword(s):  
Brillouin Zone ◽  
Closed Loop ◽  
Boundary Energy ◽  
Topological Invariant ◽  
Open Boundary ◽  
Band Theory ◽  
Boundary Correspondence ◽  
Bloch Band ◽  
Definition Of ◽  
The Continuum

Abstract The bulk boundary correspondence, which connects the topological invariant, the continuum band and energies under different boundary conditions, is the core concept in the non-Bloch band theory, in which the generalized Brillouin zone (GBZ), appearing as a closed loop generally, is a fundamental tool to rebuild it. In this work, it can be shown that the recovery of the open boundary energy spectrum by the continuum band remains unchanged even if the GBZ itself shrinks into a point. Contrastively, if the bizarreness of the GBZ occurs, the winding number will become illness. Namely, we find that the bulk boundary correspondence can still be established whereas the GBZ has singularities from the perspective of the energy, but not from the topological invariant. Meanwhile, regardless of the fact that the GBZ comes out with the closed loop, the bulk boundary correspondence can not be well characterized yet because of the ill-definition of the topological number. Here, the results obtained may be useful for improving the existing non-Bloch band theory.

scholarly journals Non-Bloch band theory and bulk–edge correspondence in non-Hermitian systems

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Kazuki Yokomizo ◽  
Shuichi Murakami
Keyword(s):  
Brillouin Zone ◽  
Tight Binding ◽  
Topological Invariant ◽  
Edge States ◽  
Energy Bands ◽  
Open Chain ◽  
Band Theory ◽  
Continuum Energy ◽  
Bloch Band ◽  
Simple Models

Abstract In this paper, we review our non-Bloch band theory in 1D non-Hermitian tight-binding systems. In our theory, it is shown that in non-Hermitian systems, the Brillouin zone is determined so as to reproduce continuum energy bands in a large open chain. By using simple models, we explain the concept of the non-Bloch band theory and the method to calculate the Brillouin zone. In particular, for the non-Hermitian Su–Schrieffer–Heeger model, the bulk–edge correspondence can be established between the topological invariant defined from our theory and existence of the topological edge states.

Ab initio band theory approach to electron energy loss near edge structures

1988 ◽  
Vol 46 ◽  
pp. 506-507
Author(s):  
Xudong Weng ◽  
O.F. Sankey ◽  
Peter Rez
Keyword(s):  
Ab Initio ◽  
Local Density ◽  
Interpolation Method ◽  
Atomic Orbital ◽  
Plane Waves ◽  
Large Set ◽  
Theory Approach ◽  
Band Theory ◽  
Atomic Orbitals ◽  
Pseudopotential Calculations

Single electron band structure techniques have been applied successfully to the interpretation of the near edge structures of metals and other materials. Among various band theories, the linear combination of atomic orbital (LCAO) method is especially simple and interpretable. The commonly used empirical LCAO method is mainly an interpolation method, where the energies and wave functions of atomic orbitals are adjusted in order to fit experimental or more accurately determined electron states. To achieve better accuracy, the size of calculation has to be expanded, for example, to include excited states and more-distant-neighboring atoms. This tends to sacrifice the simplicity and interpretability of the method.In this paper. we adopt an ab initio scheme which incorporates the conceptual advantage of the LCAO method with the accuracy of ab initio pseudopotential calculations. The so called pscudo-atomic-orbitals (PAO's), computed from a free atom within the local-density approximation and the pseudopotential approximation, are used as the basis of expansion, replacing the usually very large set of plane waves in the conventional pseudopotential method. These PAO's however, do not consist of a rigorously complete set of orthonormal states.

A New Condition of Formation and Stablity of All Crystalline Systems in a Good Agreement with Experimental Data

2015 ◽  
Vol 11 (3) ◽  
pp. 3224-3228
Author(s):  
Tarek El-Ashram
Keyword(s):  
Experimental Data ◽  
Brillouin Zone ◽  
Electron Concentration ◽  
Free Electron ◽  
Lattice Point ◽  
Valence Electron ◽  
Fermi Gas ◽  
Valence Electron Concentration ◽  
Fermi Sphere ◽  
Good Agreement

In this paper we derived a new condition of formation and stability of all crystalline systems and we checked its validity andit is found to be in a good agreement with experimental data. This condition is derived directly from the quantum conditionson the free electron Fermi gas inside the crystal. The new condition relates both the volume of Fermi sphere VF andvolume of Brillouin zone VB by the valence electron concentration VEC as ;𝑽𝑭𝑽𝑩= 𝒏𝑽𝑬𝑪𝟐for all crystalline systems (wheren is the number of atoms per lattice point).

Unemployment as an indicator of economic security in the employment sector: A regional dimension

2020 ◽  
Vol 16 (4) ◽  
pp. 715-729
Author(s):  
T.N. Savina
Keyword(s):  
Comparative Analysis ◽  
Age Groups ◽  
National Development ◽  
Federal District ◽  
Economic Security ◽  
Employment Policy ◽  
Multivariate Statistical ◽  
Band Theory ◽  
The Republic ◽  
The Impact

Subject. To achieve a high level of economic security is a key priority of national development. Employment reveals one of the most important aspects of social development of the individual that is associated with his or her needs satisfaction in the sphere of employment and is boon to economic security. Objectives. The purpose of the study is to show the impact of unemployment on economic security in employment. Methods. I apply such scientific methods as dialectical, historical and logical unity, structural and functional analysis, traditional techniques of economic analysis and synthesis. The methods of multivariate statistical and comparative analysis serve as a methodological basis of the study. To determine the indicator of unemployment, I use the band theory. Results. I underpin the growing role of employment in ensuring economic security. The paper presents a comprehensive assessment of the unemployment status and a comparative analysis of the indicator in the Republic of Mordovia, the Volga Federal District, and the Russian Federation as a whole. I identify trends in the average duration of unemployment, show the distribution of unemployed by level of education and age groups. Conclusions. The average annual unemployment rate in the Republic of Mordovia is lower than in Russia and the Volga Federal District. The findings may be useful for public authorities to substantiate their employment policy at both macro- and meso-levels, for designing programs and strategies for socio-economic development of regions and the social security doctrine, as well as in practical activities of employment services.

The Chemical Bond Across the Periodic Table: Part 1 – First Row and Simple Metals

2020 ◽  
Author(s):  
Gabriel Freire Sanzovo Fernandes ◽  
Leonardo dos Anjos Cunha ◽  
Francisco Bolivar Correto Machado ◽  
Luiz Ferrão
Keyword(s):  
Physical Chemistry ◽  
Chemical Bond ◽  
Materials Science ◽  
Chemical Elements ◽  
Orbital Theory ◽  
Solid State Physics ◽  
Band Theory ◽  
Van Der Waals Clusters ◽  
Earth Systems ◽  
Chemical Content

<p>Chemical bond plays a central role in the description of the physicochemical properties of molecules and solids and it is essential to several fields in science and engineering, governing the material’s mechanical, electrical, catalytic and optoelectronic properties, among others. Due to this indisputable importance, a proper description of chemical bond is needed, commonly obtained through solving the Schrödinger equation of the system with either molecular orbital theory (molecules) or band theory (solids). However, connecting these seemingly different concepts is not a straightforward task for students and there is a gap in the available textbooks concerning this subject. This work presents a chemical content to be added in the physical chemistry undergraduate courses, in which the framework of molecular orbitals was used to qualitatively explain the standard state of the chemical elements and some properties of the resulting material, such as gas or crystalline solids. Here in Part 1, we were able to show the transition from Van der Waals clusters to metal in alkali and alkaline earth systems. In Part 2 and 3 of this three-part work, the present framework is applied to main group elements and transition metals. The original content discussed here can be adapted and incorporated in undergraduate and graduate physical chemistry and/or materials science textbooks and also serves as a conceptual guide to subsequent disciplines such as quantum chemistry, quantum mechanics and solid-state physics.</p>

scholarly journals Computational Complexity Reduction of Neural Networks of Brain Tumor Image Segmentation by Introducing Fermi–Dirac Correction Functions

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 223
Author(s):  
Yen-Ling Tai ◽  
Shin-Jhe Huang ◽  
Chien-Chang Chen ◽  
Henry Horng-Shing Lu
Keyword(s):  
Neural Networks ◽  
Deep Learning ◽  
Computational Complexity ◽  
High Performance ◽  
Low Cost ◽  
Structural Complexity ◽  
Correction Function ◽  
Computational Time ◽  
Learning Methods ◽  
Band Theory

Nowadays, deep learning methods with high structural complexity and flexibility inevitably lean on the computational capability of the hardware. A platform with high-performance GPUs and large amounts of memory could support neural networks having large numbers of layers and kernels. However, naively pursuing high-cost hardware would probably drag the technical development of deep learning methods. In the article, we thus establish a new preprocessing method to reduce the computational complexity of the neural networks. Inspired by the band theory of solids in physics, we map the image space into a noninteraction physical system isomorphically and then treat image voxels as particle-like clusters. Then, we reconstruct the Fermi–Dirac distribution to be a correction function for the normalization of the voxel intensity and as a filter of insignificant cluster components. The filtered clusters at the circumstance can delineate the morphological heterogeneity of the image voxels. We used the BraTS 2019 datasets and the dimensional fusion U-net for the algorithmic validation, and the proposed Fermi–Dirac correction function exhibited comparable performance to other employed preprocessing methods. By comparing to the conventional z-score normalization function and the Gamma correction function, the proposed algorithm can save at least 38% of computational time cost under a low-cost hardware architecture. Even though the correction function of global histogram equalization has the lowest computational time among the employed correction functions, the proposed Fermi–Dirac correction function exhibits better capabilities of image augmentation and segmentation.

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