Diverging exchange force and form of the exact density matrix functional
For translationally invariant one-band lattice models, we exploit the ab initio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered: First, within each symmetry sector, the interaction functional F depends only o...
| Main Authors: | , |
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| Format: | Journal article |
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American Physical Society
2019
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| _version_ | 1826298658681782272 |
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| author | Schilling, C Schilling, R |
| author_facet | Schilling, C Schilling, R |
| author_sort | Schilling, C |
| collection | OXFORD |
| description | For translationally invariant one-band lattice models, we exploit the ab initio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered: First, within each symmetry sector, the interaction functional F depends only on the natural occupation numbers n. The respective sets P^1_N and E^1_N of pure and ensemble N-representable one-matrices coincide. Second, and most importantly, the exact functional is strongly shaped by the geometry of the polytope E^1_N = P^1_N, described by linear constraints D^{(j)}(n)⩾0. For smaller systems, it follows as F[n]=\sum_{i,i'} V_{i,i'} \sqrt{D^{(i)}(n)D^{(i')}(n)}. This generalizes to systems of arbitrary size by replacing each D^{(i)} by a linear combination of {D^{(j)}(n)} and adding a non-analytical term involving the interaction V. Third, the gradient dF/dn is shown to diverge on the boundary ∂E^1_N, suggesting that the fermionic exchange symmetry manifests itself within RDMFT in the form of an ``exchange force''. All findings hold for systems with non-fixed particle number as well and V can be any p-particle interaction. As an illustration, we derive the exact functional for the Hubbard square. |
| first_indexed | 2024-03-07T04:50:12Z |
| format | Journal article |
| id | oxford-uuid:d4afc13b-fb43-4999-82e7-7419fa31ae6c |
| institution | University of Oxford |
| last_indexed | 2024-03-07T04:50:12Z |
| publishDate | 2019 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | oxford-uuid:d4afc13b-fb43-4999-82e7-7419fa31ae6c2022-03-27T08:20:27ZDiverging exchange force and form of the exact density matrix functionalJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d4afc13b-fb43-4999-82e7-7419fa31ae6cSymplectic Elements at OxfordAmerican Physical Society2019Schilling, CSchilling, RFor translationally invariant one-band lattice models, we exploit the ab initio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered: First, within each symmetry sector, the interaction functional F depends only on the natural occupation numbers n. The respective sets P^1_N and E^1_N of pure and ensemble N-representable one-matrices coincide. Second, and most importantly, the exact functional is strongly shaped by the geometry of the polytope E^1_N = P^1_N, described by linear constraints D^{(j)}(n)⩾0. For smaller systems, it follows as F[n]=\sum_{i,i'} V_{i,i'} \sqrt{D^{(i)}(n)D^{(i')}(n)}. This generalizes to systems of arbitrary size by replacing each D^{(i)} by a linear combination of {D^{(j)}(n)} and adding a non-analytical term involving the interaction V. Third, the gradient dF/dn is shown to diverge on the boundary ∂E^1_N, suggesting that the fermionic exchange symmetry manifests itself within RDMFT in the form of an ``exchange force''. All findings hold for systems with non-fixed particle number as well and V can be any p-particle interaction. As an illustration, we derive the exact functional for the Hubbard square. |
| spellingShingle | Schilling, C Schilling, R Diverging exchange force and form of the exact density matrix functional |
| title | Diverging exchange force and form of the exact density matrix functional |
| title_full | Diverging exchange force and form of the exact density matrix functional |
| title_fullStr | Diverging exchange force and form of the exact density matrix functional |
| title_full_unstemmed | Diverging exchange force and form of the exact density matrix functional |
| title_short | Diverging exchange force and form of the exact density matrix functional |
| title_sort | diverging exchange force and form of the exact density matrix functional |
| work_keys_str_mv | AT schillingc divergingexchangeforceandformoftheexactdensitymatrixfunctional AT schillingr divergingexchangeforceandformoftheexactdensitymatrixfunctional |