Pinning of Fermionic Occupation Numbers Christian Schilling ETH Zürich in collaboration with...
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Transcript of Pinning of Fermionic Occupation Numbers Christian Schilling ETH Zürich in collaboration with...
Pinning of Fermionic Occupation Numbers
Christian SchillingETH Zürich
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:
relevant when
Aufbau principle for atoms
(quasi-) pinned by(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 occupationnumbers
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:
0
1
1
Pauli exclusion principle[A.Klyachko., CMP 282, p287-322, 2008][A.Klyachko, J.Phys 36, p72-86, 2006]
Polytope
polytope
intersection offinitely 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:
decayimpossible
0
1
1
3) 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!
0
1
1
Slaterdeterminants
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 :
depends only on i.e. on
non-trivial duality
weak-interacting
from now on :
`Boltzmann distribution law’:
hierarchy:
Thanks toJürg 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-pinningposter 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-Focke.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!