davide e. galli dipartimento di fisica università degli studi di milano d.e. galli.università...

Davide E. GalliDipartimento di FisicaUniversità degli Studi di Milano

D.E. Galli. Università degli Studi di Milano, Italy RPMBT15 Ohio State University, Columbus, OH, USAD.E. Galli. Università degli Studi di Milano, Italy RPMBT15 Ohio State University, Columbus, OH, USA

In collaboration with

E. Vitali, M. Rossi and L. Reatto

INVERSE PROBLEMS AND QUANTUM DYNAMICS:the Genetic Inversion via Falsification of Theories (GIFT) method arXiv:0905.4406

INVERSE PROBLEMS AND QUANTUM DYNAMICS:the Genetic Inversion via Falsification of Theories (GIFT) method arXiv:0905.4406

Università degli Studi di MilanoUniversità degli Studi di Milano

Inverse Problems (IP)

Inverse Problems (IP)

direct problem: use a theory to predict the results of observations

inverse problem: use results of observations to infer the parameters representing a system


IP can be hard:cause-to-effect map is smoothing,different causes can produce almost the same effect loss of information

€

f (τ ) = dω K(τ ,ω)s (ω)−∞

+∞

∫

€

f (τ ) = eˆ H τ ˆ A e − ˆ H τ ˆ B =

i.e. it is not possible to find a singletheory whose prediction fits the data

limited set ofobservations

+ noise ill-posed

QMC: imaginary-timecorrelation functions

€

K(τ ,ω) = θ(ω)e −ωτ

Spectral functions: dynamical properties

T=0 Laplace transform:


Dynamical structure factor S(q,) : information about the excitation spectrum

QMC: imaginary time intermediate scattering function for a finite set of “instants” with unavoidable statistical errors

Dynamics from QMCDynamics from QMC


€

S(r q ,ω) = dt

2πN e iωt ˆ ρ r q (t ) ˆ ρ − r q (0)

−∞

∞

∫Density fluctuation

€

ˆ ρ r q (t ) = e ir q ⋅

r r j (t )

j =1

N

∑

€

F(r q ,lδτ ) =f l =

1

Nˆ ρ r q (lδτ ) ˆ ρ − r

q (0) l =1,K ,n

singleexcitatio

npeaks

multiexcitationcompone

nt

Maximum Entropy method (MEM): • qualitative agreement with exp. (Boninsegni & Ceperley JLTP ‘96)

• sharp features cannot be recovered• uncontrolled approximation: entropic prior


Superfluid 4He

(Moroni & BaroniPRL ‘99)

A. Tarantola (Commentary, Nature Physics, ‘06) :

“Observations cannot produce models, they can only falsify models”The setting, in principle, for an inverse problem should be as follows:1. use all available a priori information to sequentially create

models of the system, potentially an infinite number of them2. For each model, solve the direct problem, compare the

predictions to the actual observations and use some criterion to decide if the fit is acceptable or unacceptable, given the uncertainties in the observations

3. The unacceptable models have been falsified, and must be dropped

4. The collection of all the models that have not been falsified represent the solution of the inverse problem.No other a priori information that could “bias” the inferences should be used (MEM limitation)

The Falsification Principle

The Falsification Principle

Fine! But how can we implement this?Can we obtain more information?



We have introduced a new strategy based on the previous point of view and on Genetic Algorithms (GA) to face inverse problems like the following:

1. We need a (huge) space of models, s(), containing a wide collection of spectral functions consistent with any a priori information2. We need a falsification procedure relying on the (discrete and noisy QMC) “observations” of the imaginary time correlation function, fl .

s() is real-valued and non negative thus we chose as models

Genetic Inversion (via)

Falsification (of) Theories: the GIFT method

Genetic Inversion (via)

Falsification (of) Theories: the GIFT method

€

s() = si

MΔχ i ,i +1[ ]

(ω)i =1

N ω∑

€

si ∈ 0,1,2,L ,M{ }

€

d s() =0

+∞

∫ 1

Ms i = 1

i =1

N ω∑

characteristic function



s() differs from the physical spectral function by a factor c0 , the zero-momentum

M maximumnumber of quanta of spectral weight

Δ width of the partition

€

s(ω)

€

€

f (τ ) = dω K(τ ,ω)s (ω)∫Extract s() from Fredholm integral equationof the first kind

€

ˆ A = ˆ B +( )

How can we explore this (huge) space of models and falsify its elements? Genetic algorithms (GA) provide an extremely efficient tool to explore a sample space by a non-local stochastic dynamics, via a survival-to-fitness evolutionary process mimicking the natural selection.• the fitness of one particular s() should be based only on the observations, i.e. on the noisy extended ‘measured’ set {fl , c0 }

– But any set {fl*, c0

* } compatible with {fl , c0 } provides equivalent information

– we can use any set {fl*, c0

* } obtained by sampling independent Gaussian distributions centered on the original observations with variances corresponding to their statistical uncertainties to define

The GIFT Method IIThe GIFT Method II

€

fitness=−α f l∗−c0

∗ dωe −ωτ s (ω)0

+∞

∫ ⎡

⎣ ⎢

⎤

⎦ ⎥

2

l∑ − γn c n −c 0

∗ dωωns (ω)0

+∞

∫ ⎡

⎣ ⎢

⎤

⎦ ⎥

n∑

2

adjustable parameters to make the twocontributions of the same order of magnitude



Eventual exactly known momenta of s()We have used only c1


The Genetic Algorithm in GIFTThe Genetic Algorithm in GIFT

• Initial Population: construct a huge random collection of models s() , each s() is an individual

• Every generation: completely replace the population (but we use elitism: the best s() is cloned) with a new one using biological like processes:- selection: couples of individuals are selected for reproduction

with a probability proportional to their fitness. - crossover: a fixed amount of spectral weight, left in the original

intervals, is exchanged between the two selected s()

- mutation: shift of a fraction of spectral weight betweentwo intervals


Università degli Studi di MilanoUniversità degli Studi di MilanoThe GIFT “Solution”The GIFT “Solution”

• The GA dynamics performs the falsification: only the s() with the highest fitness in the last generation provides a model for the spectral function which has not been falsified

• Many independent evolutionary process may be generated by sampling different {fl

*, c0* }. Each one provides a non-falsified

model si() • The collection of all these models provides the “solution” of the

inverse problem. At this point an averaging procedure among these non-falsified model appears as the most natural way to extract physical information:

• From the analysis of exactly solvable analytical models discretized and “dirtied” with random noise to simulate actual data:no possibility to reconstruct the exact shape of s() ; access is granted to presence and position of a sharp peak to presence and position of a broad contribution, to some integral properties of s() and to its support.


€

SGIFT (ω) =1

Ne

s i (ω)i

∑


Path Integral Projector MC methods:

Path Integral Ground State (PIGS)

Path Integral Projector MC methods:

Path Integral Ground State (PIGS)

€

ψ0(R) ≅ψ τ (R) = dR1L dRP R e − τP

ˆ H R1 ×L × RP −1 e − τP

ˆ H RP ψT (RP )∫

Classical-Quantum mapping: ground state averages are equivalent to canonical averages of a classical system of special interacting linear polymers

Sarsa, Schmidt, Magro, J.Chem.Phys. 2001PIGS provides “exact” ground state expectation values via a discreteimaginary time evolved quantum state (such that ψψT) which givesrise to a discrete path {R1,…,R2P+1} sampled with a Metropolis algorithm


€

Ri = r r1i ,L ,

r r N

i{ }

€

L

€

r r11

€

L

€

L

€

L

€

L

€

L

Imaginary time

Quantumparticles

€

M

€

r r21

€

r r31

€

M

€

M

€

M

€

r r12

€

r r22

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r r32

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r r1P +1

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r r2P +1

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r r3P +1

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r r32P

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r r22P

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r r12P

€

r r12P +1

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rr 2

2P +1

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r r32P +1

Equation of state:liquid and metastableoverpressurizedliquid

€

ρfr

€

ρm

PIGS is robust!It converges withoutimportance sampling.See arXiv:0907.4430


Shadow-PIGS (SPIGS)Shadow-PIGS (SPIGS)

ρ = 0.0315 Å-3

ρ = 0.033 Å-3

• SWF: single (variationally optimized) projection step of a Jastrow wave function

Vitiello, Runge, Kalos, PRL ’88

– Implicit correlations (all orders)– Bose symmetry preserved

• SPIGS: PIGS which projects a SWFGalli, Reatto, Mol. Phys. 101, ‘03

€

ψTSWF (R) = dS G(R,S ) ψT (S )∫

€

ψ0(R) = dR1L dRPdS R e − τP

ˆ H R1 ×L∫

€

L × RP−1 e − τP

ˆ H RP G(RP ,S )ψT (S )


Solid phase: spontaneously brokentranslational symmetryAbove the melting point, nucleation process becomes moreand more efficient

After few thousandof MC step

Starting from a liquid-likeconfiguration thesystem remainsdisordered for some MC time



€

ˆ ρ − r q = e −i

r q ⋅

r r j (0)

j =1

N

∑

€

ˆ ρ r q (τ = lδτ ) = e −ir q ⋅

r r j (lδτ )

j =1

N

∑

€

f l =1

Nˆ ρ r q (lδτ ) ˆ ρ − r

q (0) l =1,K ,n

“internal” imaginary-time evolution

Whole imaginary-time evolution

Imaginary time correlation functionsImaginary time correlation functions

Accurate f()’s has been computed via SPIGS; relative error ≈ 0.4%

q (Å-1)

(K-1)

f()

f()

(K-1)

Details:pair-productpropagator = 1/160 K-1

Example:Superfluid 4Heρ=0.0218 Å-3

TOT ≈ 1.0 - 0.62 K-1

INT ≈0.4 K-1

SGIFT(q,) SGIFT(q,)

• Example: superfluid 4He, at equilibrium density ρ=0.0218 Å-3 • GA details: Δ=0.25 K, solution=average over ≈640 non-falsified s(), Initial

population of 25000 s(), mortality rate=5% down to a minimum of 400 s() , number of quanta of spectral weight M=5000

• Sharp peaks in S(q,) indicating energies of elementary excitations

• First evidence of a multi-excitation component in S(q,)

Results: superfluid 4HeResults: superfluid 4He



The multi-phonon contributionThe multi-phonon contributionvery good agreement with the experimental results (Cowley & Woods, Can. J. Phys. 1971)

One can add the MEM entropic term in the fitness:

with m()=cost. broadening strongly dependent on which hide the multi-phonon component



ρ = 0.0262 Å-3ρ = 0.0218 Å-3

SG

IFT(q

,)

q=1.755 Å-1

ρ = 0.0218 Å-3

€

− d s(ω)lns (ω)

m(ω)

⎡

⎣ ⎢ ⎤

⎦ ⎥− s (ω) + m(ω) ⎧ ⎨ ⎩

⎫ ⎬ ⎭

∫


• the spectral weight under the single-excitation peak, Z(q)

is very good quantitative agreement with experiments. This confirms that the multi-phonon features extracted via GIFT are indeed physical information • Also the static density response function

turns out to be in good agreement with experiments

It is physics?It is physics?

€

χ(r q ) = dω

S (r q ,ω)

ω0

+∞

∫

Exp.: Gibbs et al. J. Phys.:Condens Matter 11, ‘99

€

Z(r q ) ≈ dωS (

r q ,ω)

peak

∫

Yes, we can… obtain more!


Exp.: Cowley and WoodsCan. J. Phys 49, ‘71


Over-pressurized 4HeOver-pressurized 4HeUniversità degli Studi di MilanoUniversità degli Studi di Milano

• Experiments (Pearce et al. PRL 93, ‘04) show rotons to all pressures up to solidification, even for pressures higher than freezing• We have studied over-pressurized liquid 4He with SPIGS up to densities in the metastable region (and more) and extracted roton energies via GIFT from the imaginary time intermediate scattering function• Agreement with experiments (when available) is very good

Linear fit

Roton energy

Superfluid 4He with one 3He atomSuperfluid 4He with one 3He atom

• Impurity branch experimentally measured (Fåk et al. PRB 1990)

• f() computed with a SPIGS simulations with N=225 4He atoms and one 3He atom at ρ=0.0218 Å-3

• the calculation requires

• GIFT reproduce a sharp peak in very good agreement with the experimental results roboust check of validity of our approach

€

ˆ A = ˆ B + = ˆ ρ r q = e−i

r q ⋅

r r imp


ρ = 0.0218 Å-3


Vacancy-wave in solid 4HeVacancy-wave in solid 4He

• single vacancy in hcp solid 4He at ρ=0.0293 Å-3

• f() computed with SPIGS by considering

• rvac is a many-body variable determined in two different ways• coarse grain procedure

(CGR)• Hungarian method (HUN)

• good agreement with a tight binding model (T-B) except in the M direction• novel vacancy-roton mode

with E = 2.6±0.4 K and m*= 0.46 mHe

• connected to the motion of the vacancy between different basal planes

€

ˆ ρ r q = e ir q ⋅

r r vac

K

M

A



m*= 0.46±0.03 mHe

m*= 0.46±0.03 mHe

m*= 0.55±0.1 mHe

m*= 0.46 mHe

ConclusionsConclusions

• We have built up a new strategy to face a huge class of inverse problems of the form

• We have applied it to the extraction of information about the real time dynamics in quantum many-body systems from noisy QMC imaginary time correlations functions– very accurate results in the 4He case– major improvements with respect to previous studies

• GIFT can be extended to include different constrains or additional information like cross correlations between the statistical noise

• details of GA can be devised depending on the specific problem – basis set different from step functions– non uniform discretization– non Gaussian distribution of noise

€

f (τ ) = dω K(τ ,ω)s (ω)∫

see arXiv: cond-mat 0905.4406



davide e. galli dipartimento di fisica università degli studi di milano d.e. galli.università...

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