Pinning of Fermionic - PowerPoint Presentation

Slide1

Pinning of Fermionic Occupation Numbers

Christian SchillingETH Zürich

in collaboration with M.Christandl, D.Ebler, D.Gross

Phys. Rev. Lett. 110, 040404 (2013)

Slide2

OutlineMotivation

Generalized Pauli ConstraintsApplication to Physics

Pinning AnalysisPhysical Relevance of Pinning

Slide3

1) MotivationPauli’s

exclusion principle (1925):`no two identical fermions

inthe same quantum state’mathematically:relevant when

Aufbau

principle for atoms

(quasi-)

pinned

by

(quasi-)

pinned

by

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`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 (?)

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mathematical objects ?N-fermion states1-particle reduced density operator

natural occupationnumbers

partial trace

translate antisymmetry of to

1-particle picture

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Q: Which 1-RDO

are possible?

2) Generalized Pauli Constraints

(Fermionic

Quantum Marginal Problem)

d

escribe

this

set

unitary

equivalence

:

only

natural

occupation

numbers

relevant

A

:

Slide7

0

1

1Pauli exclusion principle

[

A.Klyachko

., CMP 282, p287-322, 2008]

[

A.Klyachko

,

J.Phys

36, p72-86, 2006]

Polytope

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polytope

i

ntersection

of

f

initely

many

half

spaces

=

f

acet

:

h

alf

space

:

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Example: N = 3 & d= 6[Borland&Dennis, J.Phys. B, 5,1, 1972][

Ruskai, Phys. Rev. A, 40,45, 2007]

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Position of relevant

states(e.g. ground

state) ?or here ? (pinning

)

here

?

point

on

boundary

:

kinematical

constraints

generalization

of

:

decay

impossible

0

1

1

3

)

Application

to

Physics

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N non-interacting fermions:

effectively

1-particle problem with solution

with

N-

particle

picture

:

1

-particle

picture

:

( )

( )

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Pauli

exclusion

principle

constraints

exactly

pinned

!

0

1

1

Slater

determinants

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requirements for non-trivial model?N identical fermions with coupling parameter

analytical solvable:

d

epending

on

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Hamiltonian:

diagonalization

of

l

ength

scales

:

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Now: Fermions

restrict to

ground state: [Z.Wang et al., arXiv 1108.1607, 2011]

i

f non-interacting

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properties of : depends

only on i.e. on

non-trivial duality

weak-interacting

f

rom

now

on :

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`Boltzmann distribution law’:

h

ierarchy

:

Thanks

to

Jürg Fröhlich

Slide18

too

difficult/ not known yetinstead: check w.r.t

4) Pinning Analysis

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r

elevant

as

long

as

l

ower

bound

on

p

inning

order

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relevant as long as

q

uasi-pinning

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moreover :larger ?

- quasi-

pinningposter by Daniel Ebler

excitations ?first few

still quasi-pinnedweaker with

increasing

excitation

q

uasi-

pinning

a

ground

state

effect

!?

quasi-

pinnig

only

for weak interaction ?No!:

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saturated by :Implication for corresponding ?

5

) Physical Relevance of

Pinning

Physical Relevance of

Pinning

?

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generalization of:

stable:

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Selection

Rule:

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Example:

Pinning of

dimension

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Application: Improvement of Hartree-Fockapproximate unknown ground state

Hartree

-Fock

much

better:

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Conclusionsantisymmetry of translated to 1-particle picture

Generalized

Pauli constraintsstudy of fermion – model with

coupling

Pauli

constraints

pinned

up

to

corrections

Generalized

Pauli

constraints

pinned

up

to

corrections

i

mprove Hartree-Focke.g.

Pinning is physically relevantFermionic Ground States simpler than appreciated (?)

Slide29

OutlookHubbard modelQuantum Chemistry: Atoms

Physical & mathematical Intuition for Pinning

HOMO-

LUMO-

gap

Strongly

correlated

Fermions

Antisymmetry

Energy

Minimization

g

eneric

for

:

Slide30

Thank you!