Pinning of Fermionic Occupation Numbers: Generalized Constraints and Applications
Mathematical underpinnings and physics applications of pinning Fermionic occupation numbers, delving into Pauli's exclusion principle, generalized constraints, and their relevance in quantum physics. Discover the significance of pinning in defining fermion states and explore its implications in various scenarios.
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Presentation Transcript
Pinning of Fermionic Occupation Numbers Christian Schilling ETH Z rich in collaboration with M.Christandl, D.Ebler, D.Gross Phys. Rev. Lett. 110, 040404 (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: (quasi-) pinned by (quasi-) pinned by relevant when Aufbau principle for atoms
strengthened by Dirac & Heisenberg in (1926): `quantum states of identical fermions are antisymmetric implications for occupation numbers ? further constraints beyond but only relevant if (quasi-) pinned (?)
mathematical objects ? N-fermion states partial trace 1-particle reduced density operator natural occupation numbers translate antisymmetry of to 1-particle picture
2) Generalized Pauli Constraints describe this set Q: Which 1-RDO are possible? (Fermionic Quantum Marginal Problem) A: unitary equivalence: only natural occupation numbers relevant
Polytope 1 1 0 [A.Klyachko., CMP 282, p287-322, 2008] [A.Klyachko, J.Phys 36, p72-86, 2006] Pauli exclusion principle
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]
3) Application to Physics or here ? (pinning) Position of relevant states (e.g. ground state) ? 1 here ? point on boundary : 1 0 kinematical constraints decay impossible generalization of:
N non-interacting fermions: with 1-particle picture: N-particle picture: ( ) ( ) effectively 1-particle problem with solution
Slater determinants Pauli exclusion principle constraints 1 exactly pinned! 1 0
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 1108.1607, 2011] if non-interacting
properties of : i.e. on depends only on from now on : non-trivial duality weak-interacting
`Boltzmann distribution law: Thanks to J rg Fr hlich hierarchy:
4) Pinning Analysis too difficult/ not known yet instead: check w.r.t
relevant as long as lower bound on pinning order
relevant as long as quasi-pinning
moreover : quasi-pinnig only for weak interaction ? No!: larger ? - quasi-pinning poster by Daniel Ebler excitations ? first few still quasi-pinned weaker with increasing excitation quasi-pinning a ground state effect !?
5) Physical Relevance of Pinning saturated by : Implication for corresponding ? Physical Relevance of Pinning ?
generalization of: stable:
Example: dimension Pinning of
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 Pinning is physically relevant e.g. improve Hartree-Fock Fermionic Ground States simpler than appreciated (?)
Outlook generic for: Hubbard model HOMO- LUMO- gap Quantum Chemistry: Atoms Physical & mathematical Intuition for Pinning Strongly correlated Fermions Antisymmetry Energy Minimization
