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'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
Irene D'Amico, UoY 2
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
Irene D'Amico, UoY 3
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
Irene D'Amico, UoY 4
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
Irene D'Amico, UoY 5
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
Irene D'Amico, UoY 6
{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
Irene D'Amico, UoY 7
(4) (6)
'DFT meets QI',
December 2014
Irene D'Amico, UoY 8
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
Irene D'Amico, UoY 9
• 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
Irene D'Amico, UoY 10
Natural distance between any two
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
Irene D'Amico, UoY 11
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
Irene D'Amico, UoY 12
Natural distance between any two
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
Irene D'Amico, UoY 13
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
Irene D'Amico, UoY 14
C31/p
C21/p
C11/p
DFT:
{w-ftc’s ↔ densities}
in metric spaces
'DFT meets QI',
December 2014
Irene D'Amico, UoY 15
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
Irene D'Amico, UoY 16
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
Irene D'Amico, UoY 17
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
Irene D'Amico, UoY 18
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
Irene D'Amico, UoY 19
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
Irene D'Amico, UoY 20
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
Irene D'Amico, UoY 21
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
Irene D'Amico, UoY 22
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
Irene D'Amico, UoY 23
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
Irene D'Amico, UoY 24
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
Irene D'Amico, UoY 25
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
Irene D'Amico, UoY 27
‘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
Irene D'Amico, UoY 28
'DFT meets QI',
December 2014
Irene D'Amico, UoY 29
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
Irene D'Amico, UoY 30
‘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: irene.damico@york.ac.uk
'DFT meets QI',
December 2014
Irene D'Amico, UoY 31
'DFT meets QI',
December 2014
Irene D'Amico, UoY 32
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
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