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) MotivationPauli’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!