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
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-)
(quasi-) pinned
by 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
0
[A.Klyachko., CMP 282, p287-322, 2008]
[A.Klyachko, J.Phys 36, p72-86, 2006]
1
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
Position of relevant states
(e.g. ground state) ?
or here ?
(pinning)
1
here ?
point on boundary :
0
kinematical constraints
generalization of:
1
decay
impossible
N non-interacting fermions:
with
N-particle picture:
1-particle picture:
( )
effectively 1-particle problem
with solution
( )
Slater
determinants
Pauli exclusion principle constraints
1
exactly pinned!
0
1
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
from now on :
non-trivial duality
weak-interacting
i.e. on
`Boltzmann distribution law’:
Thanks to
Jürg Fröhlich
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:
Selection Rule:
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
Quantum Chemistry: Atoms
Physical & mathematical Intuition
HOMOLUMOgap
for Pinning
Strongly correlated Fermions
Antisymmetry
Energy Minimization
Thank you!