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Pinning of Fermionic Occupation Numbers

Pinning of Fermionic Occupation Numbers. Christian Schilling ETH Zürich. in collaboration with M.Christandl , D.Ebler , D.Gross. Phys. Rev . Lett . 110 , 040404 (2013). Outline. Motivation Generalized Pauli Constraints Application to Physics Pinning Analysis

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Pinning of Fermionic Occupation Numbers

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  1. PinningofFermionicOccupation Numbers Christian Schilling ETH Zürich in collaborationwith • M.Christandl, D.Ebler,D.Gross Phys. Rev. Lett. 110, 040404 (2013)

  2. Outline • Motivation • Generalized Pauli Constraints • ApplicationtoPhysics • Pinning Analysis • PhysicalRelevanceofPinning

  3. 1) Motivation Pauli’sexclusionprinciple (1925): `notwoidenticalfermions in the same quantumstate’ mathematically: (quasi-) pinnedby (quasi-) pinnedby relevant when Aufbau principleforatoms

  4. strengthenedby Dirac & Heisenberg in (1926): `quantumstatesofidentical fermionsareantisymmetric’ implicationsforoccupationnumbers ? furtherconstraintsbeyond butonly relevant if (quasi-) pinned (?)

  5. mathematicalobjects ? N-fermion states partial trace 1-particle reduceddensityoperator naturaloccupation numbers • translateantisymmetryof • to 1-particle picture

  6. 2) Generalized Pauli Constraints describethisset Q:Which1-RDO arepossible? (Fermionic Quantum Marginal Problem) A: unitaryequivalence: onlynaturaloccupationnumbersrelevant

  7. Polytope 1 1 0 [A.Klyachko., CMP 282, p287-322, 2008] [A.Klyachko, J.Phys 36, p72-86, 2006] Pauli exclusionprinciple

  8. polytope = intersectionof finitelymany half spaces facet: half space:

  9. Example: N = 3 & d= 6 [Borland&Dennis, J.Phys. B, 5,1, 1972] [Ruskai, Phys. Rev. A, 40,45, 2007]

  10. 3) ApplicationtoPhysics orhere? (pinning) Position of relevant states (e.g. groundstate) ? 1 here? point on boundary : 1 0 kinematicalconstraints decay impossible generalizationof:

  11. N non-interactingfermions: with 1-particle picture: N-particlepicture: ( ) ( ) effectively 1-particle problem withsolution

  12. Slater determinants Pauli exclusionprincipleconstraints 1 exactlypinned! 1 0

  13. requirementsfor non-trivial model? N identicalfermionswithcouplingparameter analyticalsolvable: depending on

  14. Hamiltonian: diagonalizationof lengthscales:

  15. Now: Fermions restrictto groundstate: [Z.Wang et al., arXiv 1108.1607, 2011] if non-interacting

  16. propertiesof : i.e. on dependsonly on fromnow on : non-trivial duality weak-interacting

  17. `Boltzmann distributionlaw’: Thanksto Jürg Fröhlich hierarchy:

  18. 4) Pinning Analysis toodifficult/ not knownyet instead: check w.r.t

  19. relevant aslong as lowerbound on pinningorder

  20. relevant aslong as quasi-pinning

  21. moreover : quasi-pinnigonlyforweakinteraction ? No!: larger ? - quasi-pinning posterby Daniel Ebler excitations ? firstfewstill quasi-pinned weakerwithincreasingexcitation quasi-pinning a groundstateeffect !?

  22. 5) PhysicalRelevanceofPinning saturatedby : Implicationforcorresponding ? PhysicalRelevanceofPinning ?

  23. generalizationof: stable:

  24. SelectionRule:

  25. Example: dimension Pinningof

  26. Application: ImprovementofHartree-Fock approximateunknowngroundstate Hartree-Fock muchbetter:

  27. Conclusions • antisymmetryof • translatedto1-particle picture Generalized Pauli constraints studyoffermion – modelwithcoupling Pauli constraintspinneduptocorrections Generalized Pauli constraintspinneduptocorrections Pinningisphysically relevant e.g. improveHartree-Fock FermionicGround States simpler thanappreciated (?)

  28. Outlook genericfor: Hubbard model Quantum Chemistry: Atoms HOMO- LUMO- gap Physical & mathematical Intuition forPinning StronglycorrelatedFermions AntisymmetryEnergyMinimization

  29. Thankyou!

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