Download - Quantum Mechanics in Metric Space: Results for DFT and CDFT · Irene D'Amico, UoY 17 Metric space for ground state wave functions and densities • ground state (GS) particle densities,

'DFT meets QI',

December 2014

Irene D'Amico, UoY 1

Quantum Mechanics in Metric

Space: Results for DFT and CDFT

Irene D'Amico

Department of Physics

'DFT meets QI',

December 2014


Jeremy Coe Department of Chemistry, University of Herriot-Watt, Edinburg, UK

Klaus Capelle Centro de Ciencias Naturais e Humanas, UFABC, Sao Paulo, Brazil

Vivian França Department of Physical Chemistry, UNESP, Araraquara Brazil

Paul Sharp Department of Physics, University of York, York, UK

'DFT meets QI',

December 2014


General Motivation: Conservation Laws →

Metrics → New tool for understanding physics Conservation laws are a central tenet of our understanding

of the physical world and a fundamental tool for

developing theoretical physics

If we can deduce ‘natural metrics’ for the quantities related

to conservation laws, we have a new tool for

understanding these quantities

(and the related physical systems)

This may be useful when considering many-body systems,

often too complex when considered within the usual

coordinate space analysis.

Would many-body systems become ‘simpler’ when looked

at using metric spaces?

P. Sharp and I. D’Amico, PRB 89, 115137 (2014)

'DFT meets QI',

December 2014


Outline

• Metric spaces

• From conservation laws to metric spaces

• Geometry of these metric spaces:

‘Onion shell’ geometry

• Wave functions, Particle densities, and Paramagnetic

currents as metric spaces

• Gauge invariance of metrics

• Hohenberg-Kohn theorem in metric space:

DFT and CDFT

• [Looking at approximations for v_xc in metric spaces:

some results for LDA & QI-related possibilities]

[if time allows…]

'DFT meets QI',

December 2014


Metric space

• Metric space: it is possible to assign a

distance between any two elements of the

space

given any A, B, C in metric space M, we can

assign D(A,B) such that:

D(A,B)≥0, D(A,B)=0 if and only if A=B

D(A,B)=D(B,A) and

D(A,B)≤D(A,C)+D(C,B) (triangular inequality)

Conservation Laws

↓

p-Norm

↓ Canonical Metric

↓ ‘Natural’ Metric

'DFT meets QI',

December 2014


{f} physical

functions

{f} physical

functions

{f} vector

space

{f} vector

space

P. Sharp and I. D’Amico, PRB 89, 115137 (2014)

Conservation Laws → ‘Natural’ Metrics

'DFT meets QI',

December 2014


(4) (6)

'DFT meets QI',

December 2014


Natural distance between any two

N-particle densities • it is derived from particle conservation

with the density corresponding to a N-particle

wave function given by

'DFT meets QI',

December 2014


• with this distance the densities form a

metric space

• contrary to wave functions, the densities

are though NOT a vector space NOR a

Hilbert space.

• metric spaces give a structure to the

densities’ ensemble

'DFT meets QI',

December 2014



N-particle wave functions • it is derived from wave-function norm

• however discriminates

between wave functions which differ by a

global phase only.

• ≠ 0 for most

• This is unphysical,

it does not satisfy gauge invariance

'DFT meets QI',

December 2014


We consider then the physically meaningful

classes {ΨeiΦ} and define the related distance

where the phase Φ is defined by

This restores the physically expected property

0 for any

It can be shown that is indeed

a distance. This metrics is gauge invariant.

'DFT meets QI',

December 2014



N-particle paramagnetic currents • the conservation of z-component of angular momentum

generates a metric for the paramagnetic current,

suggesting this to be the fundamental variable in the

presence of a magnetic field [CDFT?]

Gauge invariance for

paramagnetic current metric

'DFT meets QI',

December 2014


Reference gauge such that [Lz,H] = 0, then in any gauge

Then in any gauge there is

the constant of motion:

where {m} are the eigenstates of Lz in the reference gauge.

and

The gauge-invariant paramagnetic current metric is then

~ ~ ~ ~

which reduces to the previous one when [Lz,H] = 0

Geometry of these metric

spaces: ‘Onion Shell’ geometry • Conservation laws naturally build within the related

metric spaces a hierarchy of concentric spheres, or

‘onion shell’ geometry.

• center: zero function f(0)(x)≡0

• then Df (f, f(0))=|| f (x) ||p= p-norm

• but:

• and (conservation law)

• so for each c we get a sphere of radius

Df (f, f(0))=|| f (x) ||p= c1/p

'DFT meets QI',

December 2014


C31/p

C21/p

C11/p

DFT:

{w-ftc’s ↔ densities}

in metric spaces

'DFT meets QI',

December 2014


D’Amico, Coe, França, Capelle PRL 106, 050401 (2011)

D’Amico, Coe, França, Capelle PRL 107, 188902 (2011)

N-particle densities spheres:

centre ρ(0)(x)≡0 and radius

Dρ(ρ, ρ(0))=N

'DFT meets QI',

December 2014


Geometry of particle density and

wave functions metric space • Both spaces display a onion-shell geometry

Fock space stratifies in an

onion-shell geometry

Gauge invariance implies

only half sphere is occupied

'DFT meets QI',

December 2014


Metric space for ground state

wave functions and densities • ground state (GS) particle densities, and

GS N-particle wave functions do NOT form

a vector space NOR a Hilbert space

• GS N-particle wave functions are a

metric space, (it follows from the same

definitions just discussed).

• GS N-particle densities are a metric

space (it follows from the same definitions

just discussed)

'DFT meets QI',

December 2014


Hohenberg-Kohn theorem

• the Hohenberg-Kohn theorem establishes

a one-to-one mapping between GS

wavefunctions and their densities.

• It is at the core of Density Functional

Theory which allows to effectively

calculate the properties of realistic large

many-body systems

'DFT meets QI',

December 2014


Metric spaces and Hohenberg-

Kohn (H-K) theorem

• We see then that H-K mapping is indeed

a mapping between metric spaces

• Since

the H-K theorem implies that GS wave

functions with nonzero distance are

mapped onto densities with nonzero

distance.

'DFT meets QI',

December 2014


Metric spaces and Hohenberg-

Kohn (H-K) theorem • the H-K theorem implies that, the plot of

Dρ versus Dψ has a positive slope at the

origin

• It also guarantees that the origin is the

only point with Dρ =0, so distances

between densities are good to discriminate

between different quantum systems

• We will derive other properties using

numerical calculations

'DFT meets QI',

December 2014


Family of GS’s

•Each family of GS’s

ψ1… ψM and related

densities are defined

by varying a single

system parameter.

•Distances are then

calculated between

the north pole

(reference state)

and the other states.

'DFT meets QI',

December 2014


Hooke’s atom

all curves are almost linear

in a large range: when looking

at distances, the HK theorem

is a very simple mapping

similar results for

Helium series and

1D Hubbard model

'DFT meets QI',

December 2014


C-DFT:

w-ftc’s ↔ {particle and

paramagnetic current densities}

in metric spaces

more at Paul Sharp’s poster

tomorrow afternoon

P. Sharp and I. D’Amico, PRB 89, 115137 (2014),

P. Sharp and I. D’Amico, preprint (2014)

'DFT meets QI',

December 2014


Geometry of paramagnetic

current density metric space • all paramagnetic current densities with a z-

component of the angular momentum

equal to ± m lie on spheres of radius |m|.

• onion-shell geometry

• in general as |m| changes we

‘jump’ from one sphere to the

next even for the same N. |m3|

|m2|

|m1|

'DFT meets QI',

December 2014


Systems

• magnetic Hooke's Atom and inverse

square interaction (ISI) systems

Ground

State

Results:

Djp vs

Dρ and DΨ

'DFT meets QI',

December 2014

26

positive slope

piece-wise

linearity NO universality for

same number

of particles

GAPS!

‘Band structure’

• We find a ‘band-gap’ structure’, i.e.

regions of allowed (`bands') and forbidden

(`gaps') distances, whose widths depend

on the value of |m|.

• It is induced by the application of a

magnetic field: now GS may correspond

to different, finite |m| and have finite jp

'DFT meets QI',

December 2014


‘Band structure’ and C-DFT H-K Theorem • In contrast with DFT analysis, we find that GS currents

populate a well-defined, limited region of each sphere,

whose size and position display monotonic behaviour

with respect to the quantum number m.

• This regular behaviour is not at all expected, by the

CDFT-HK theorem

'DFT meets QI',

December 2014


'DFT meets QI',

December 2014


Current work • Role of jp and ρ in

CDFT HK-like mapping Ψ↔{ρ,jp}

• Characterisation of approximations within Density

functional theory. We are currently characterising

LDA (one of the most used approximations within

density functional theory) in terms of metric spaces

and distances.

• Lattice-DFT: demonstration of: vext ↔ n

J. Coe, I. D’Amico, V. França, submitted (2014)

P. Sharp and I. D’Amico, preprint (2014)

J. Coe, I. D’Amico, V. França (alphabetical order)

in progress (2014)

'DFT meets QI',

December 2014


‘QI’-related possibilities

Can metric space analysis provide new tools to

track the system dynamics?

e.g. :

fidelity is not a proper metric but still tries to track

‘distance’ between the wished and actual state.

Could using a proper metric provide a better

tools?

• Postdoctoral position available:

Crossover between Quantum Information

and Density Functional Theory

• USP-S.Carlos

• For inquiry: [email protected]

'DFT meets QI',

December 2014


'DFT meets QI',

December 2014


Summary • Proposed a metric space formulation of

conservation laws (‘natural’ metrics)

• Demonstrated the onion shell geometry of these

metric spaces

• Shown that GS, their particle densities and current

densities form metric spaces

• use this to characterised the Hohenberg-Kohn

theorem in DFT and CDFT.

• Shown that in both DFT and CDFT the HK theorem

is strikingly simpler in metric spaces than in

coordinate spaces.

• Discussed some work in progress

Download - Quantum Mechanics in Metric Space: Results for DFT and CDFT · Irene D'Amico, UoY 17 Metric space for ground state wave functions and densities • ground state (GS) particle densities,

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