Pinning of Fermionic Occupation Numbers Christian Schilling ETH Zürich in collaboration with M.Christandl, D.Ebler, D.Gross Phys. Rev. Lett. 110, (2013)

Outline 1)Motivation 2)Generalized Pauli Constraints 3)Application to Physics 4)Pinning Analysis 5)Physical Relevance of Pinning

1) Motivation Pauli’s exclusion principle (1925): `no two identical fermions in the same quantum state’ mathematically: relevant when Aufbau principle for atoms (quasi-) pinned by

`quantum states of identical fermions are antisymmetric’ strengthened by Dirac & Heisenberg in (1926): implications for occupation numbers ? further constraints beyond but only relevant if (quasi-) pinned (?)

mathematical objects ? N-fermion states 1-particle reduced density operator natural occupation numbers partial trace translate antisymmetry of to 1-particle picture

Q: Which 1-RDO are possible? 2) Generalized Pauli Constraints (Fermionic Quantum Marginal Problem) describe this set unitary equivalence: only natural occupation numbers relevant A:A:

0 1 1 Pauli exclusion principle [A.Klyachko., CMP 282, p , 2008] [A.Klyachko, J.Phys 36, p72-86, 2006] Polytope

polytope intersection of finitely many half spaces = facet: half space:

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

Position of relevant states (e.g. ground state) ? or here ? (pinning) here ? point on boundary : kinematical constraints generalization of: decay impossible ) Application to Physics

N non-interacting fermions: effectively 1-particle problem with solution with N-particle picture: 1-particle picture: ( )

Pauli exclusion principle constraints exactly pinned! Slater determinants

requirements for non-trivial model? N identical fermions with coupling parameter analytical solvable: depending on

Hamiltonian: diagonalization of length scales:

Now: Fermions restrict to ground state: [Z.Wang et al., arXiv , 2011] if non-interacting

properties of : depends only on i.e. on non-trivial duality weak-interacting from now on :

`Boltzmann distribution law’: hierarchy: Thanks to Jürg Fröhlich

too difficult/ not known yet instead: check w.r.t 4) Pinning Analysis

relevant as long as lower bound on pinning order

relevant as long as quasi-pinning

moreover : larger ? - quasi-pinning poster by Daniel Ebler excitations ? first few still quasi-pinned weaker with increasing excitation quasi-pinning a ground state effect !? quasi-pinnig only for weak interaction ? No!:

saturated by : Implication for corresponding ? 5) Physical Relevance of Pinning Physical Relevance of Pinning ?

generalization of: stable:

Selection Rule:

Example: Pinning of dimension

Application: Improvement of Hartree-Fock approximate unknown ground state Hartree-Fock much better:

Conclusions antisymmetry of translated to 1-particle picture Generalized Pauli constraints study of fermion – model with coupling Pauli constraints pinned up to corrections Generalized Pauli constraints pinned up to corrections improve Hartree-Fock e.g. Pinning is physically relevant Fermionic Ground States simpler than appreciated (?)

Outlook Hubbard model Quantum Chemistry: Atoms Physical & mathematical Intuition for Pinning HOMO- LUMO- gap Strongly correlated Fermions Antisymmetry Energy Minimization generic for:

Thank you!