Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin Clusters
Spin Hamiltonians, like the Heisenberg model, are used to describe the magnetic properties of exchange-coupled molecules and solids. For finite clusters, physical quantities, such as heat capacities, magnetic susceptibilities or neutron-scattering spectra, can be calculated based on energies and eig...
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2024-10-01
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| author | Shadan Ghassemi Tabrizi Thomas D. Kühne |
| author_facet | Shadan Ghassemi Tabrizi Thomas D. Kühne |
| author_sort | Shadan Ghassemi Tabrizi |
| collection | DOAJ |
| description | Spin Hamiltonians, like the Heisenberg model, are used to describe the magnetic properties of exchange-coupled molecules and solids. For finite clusters, physical quantities, such as heat capacities, magnetic susceptibilities or neutron-scattering spectra, can be calculated based on energies and eigenstates obtained by exact diagonalization (ED). Utilizing spin-rotational symmetry SU(2) to factor the Hamiltonian with respect to total spin <i>S</i> facilitates ED, but the conventional approach to spin-adapting the basis is more intricate than selecting states with a given magnetic quantum number <i>M</i> (the spin <i>z</i>-component), as it relies on irreducible tensor-operator techniques and spin-coupling coefficients. Here, we present a simpler technique based on applying a spin projector to uncoupled basis states. As an alternative to Löwdin’s projection operator, we consider a group-theoretical formulation of the projector, which can be evaluated either exactly or approximately using an integration grid. An important aspect is the choice of uncoupled basis states. We present an extension of Löwdin’s theorem for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mstyle scriptlevel="+1"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></semantics></math></inline-formula> to arbitrary local spin quantum numbers <i>s</i>, which allows for the direct selection of configurations that span a complete, linearly independent basis in an <i>S</i> sector upon the spin projection. We illustrate the procedure with a few examples. |
| format | Article |
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| institution | Kabale University |
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| language | English |
| publishDate | 2024-10-01 |
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| series | Magnetism |
| spelling | doaj-art-eef6464eae164f1a8a4fc85e90a623b52024-12-27T14:37:19ZengMDPI AGMagnetism2673-87242024-10-014433234710.3390/magnetism4040022Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin ClustersShadan Ghassemi Tabrizi0Thomas D. Kühne1Center for Advanced Systems Understanding (CASUS), Am Untermarkt 20, 02826 Görlitz, GermanyCenter for Advanced Systems Understanding (CASUS), Am Untermarkt 20, 02826 Görlitz, GermanySpin Hamiltonians, like the Heisenberg model, are used to describe the magnetic properties of exchange-coupled molecules and solids. For finite clusters, physical quantities, such as heat capacities, magnetic susceptibilities or neutron-scattering spectra, can be calculated based on energies and eigenstates obtained by exact diagonalization (ED). Utilizing spin-rotational symmetry SU(2) to factor the Hamiltonian with respect to total spin <i>S</i> facilitates ED, but the conventional approach to spin-adapting the basis is more intricate than selecting states with a given magnetic quantum number <i>M</i> (the spin <i>z</i>-component), as it relies on irreducible tensor-operator techniques and spin-coupling coefficients. Here, we present a simpler technique based on applying a spin projector to uncoupled basis states. As an alternative to Löwdin’s projection operator, we consider a group-theoretical formulation of the projector, which can be evaluated either exactly or approximately using an integration grid. An important aspect is the choice of uncoupled basis states. We present an extension of Löwdin’s theorem for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mstyle scriptlevel="+1"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></semantics></math></inline-formula> to arbitrary local spin quantum numbers <i>s</i>, which allows for the direct selection of configurations that span a complete, linearly independent basis in an <i>S</i> sector upon the spin projection. We illustrate the procedure with a few examples.https://www.mdpi.com/2673-8724/4/4/22spin Hamiltoniansexact diagonalizationsymmetry |
| spellingShingle | Shadan Ghassemi Tabrizi Thomas D. Kühne Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin Clusters Magnetism spin Hamiltonians exact diagonalization symmetry |
| title | Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin Clusters |
| title_full | Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin Clusters |
| title_fullStr | Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin Clusters |
| title_full_unstemmed | Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin Clusters |
| title_short | Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin Clusters |
| title_sort | projective spin adaptation for the exact diagonalization of isotropic spin clusters |
| topic | spin Hamiltonians exact diagonalization symmetry |
| url | https://www.mdpi.com/2673-8724/4/4/22 |
| work_keys_str_mv | AT shadanghassemitabrizi projectivespinadaptationfortheexactdiagonalizationofisotropicspinclusters AT thomasdkuhne projectivespinadaptationfortheexactdiagonalizationofisotropicspinclusters |