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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2410.09587 |
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| _version_ | 1866915990072197120 |
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| author | Damnjanović, Ivan Damnjanović, Milan Milošević, Ivanka Stevanović, Dragan |
| author_facet | Damnjanović, Ivan Damnjanović, Milan Milošević, Ivanka Stevanović, Dragan |
| contents | We formulate a polynomial GCD certificate for exact flat bands in finite-range periodic tight-binding Hamiltonians. Writing the characteristic polynomial of the Bloch Hamiltonian as a Laurent polynomial \( P_L(\mathbf{z},λ)=\det(λI-H_B(\mathbf{z}))=\sum_{\mathbf{t}}c_{\mathbf{t}}(λ)\mathbf{z}^{\mathbf{t}}, \) we show that the monic greatest common divisor \(G_L(λ)=\gcd_{\mathbf{t}}c_{\mathbf{t}}(λ)\) is precisely the maximum factor of \(P_L\) that depends only on the energy variable. Its roots are exactly the exact flat-band energies, and their multiplicities give common algebraic multiplicities of these flat bands throughout the Brillouin zone. The coefficient-vanishing criterion underlying this statement is known in the flat-band and periodic-graph literature; the contribution emphasized here is the compact GCD formulation, its unit cell and Bloch-gauge invariance, and its use as a symbolic computation tool for hopping parameter engineering. The method is illustrated on kagome, dice and octahedron-chain examples, including weighted kagome and dice lattices. The certificate detects exact dispersionless eigenvalues; compact localized states, band touching and topological character must be analyzed in a subsequent eigenvector or projector calculation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09587 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A polynomial GCD certificate for exact flat bands in finite-range Bloch Hamiltonians Damnjanović, Ivan Damnjanović, Milan Milošević, Ivanka Stevanović, Dragan Materials Science Mathematical Physics 82-XX We formulate a polynomial GCD certificate for exact flat bands in finite-range periodic tight-binding Hamiltonians. Writing the characteristic polynomial of the Bloch Hamiltonian as a Laurent polynomial \( P_L(\mathbf{z},λ)=\det(λI-H_B(\mathbf{z}))=\sum_{\mathbf{t}}c_{\mathbf{t}}(λ)\mathbf{z}^{\mathbf{t}}, \) we show that the monic greatest common divisor \(G_L(λ)=\gcd_{\mathbf{t}}c_{\mathbf{t}}(λ)\) is precisely the maximum factor of \(P_L\) that depends only on the energy variable. Its roots are exactly the exact flat-band energies, and their multiplicities give common algebraic multiplicities of these flat bands throughout the Brillouin zone. The coefficient-vanishing criterion underlying this statement is known in the flat-band and periodic-graph literature; the contribution emphasized here is the compact GCD formulation, its unit cell and Bloch-gauge invariance, and its use as a symbolic computation tool for hopping parameter engineering. The method is illustrated on kagome, dice and octahedron-chain examples, including weighted kagome and dice lattices. The certificate detects exact dispersionless eigenvalues; compact localized states, band touching and topological character must be analyzed in a subsequent eigenvector or projector calculation. |
| title | A polynomial GCD certificate for exact flat bands in finite-range Bloch Hamiltonians |
| topic | Materials Science Mathematical Physics 82-XX |
| url | https://arxiv.org/abs/2410.09587 |