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
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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
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
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
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K(τ ,ω) = θ(ω)e −ωτ
Spectral functions: dynamical properties
T=0 Laplace transform:
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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
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
€
S(r q ,ω) = dt
2πN e iωt ˆ ρ r q (t ) ˆ ρ − r q (0)
−∞
∞
∫Density fluctuation
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ˆ ρ r q (t ) = e ir q ⋅
r r j (t )
j =1
N
∑
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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
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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?
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
Università degli Studi di MilanoUniversità degli Studi di Milano
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
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s() = si
MΔχ i ,i +1[ ]
(ω)i =1
N ω∑
€
si ∈ 0,1,2,L ,M{ }
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d s() =0
+∞
∫ 1
Ms i = 1
i =1
N ω∑
characteristic function
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
Università degli Studi di MilanoUniversità degli Studi di Milano
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
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s(ω)
€
€
f (τ ) = dω K(τ ,ω)s (ω)∫Extract s() from Fredholm integral equationof the first kind
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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
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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
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
Università degli Studi di MilanoUniversità degli Studi di Milano
Eventual exactly known momenta of s()We have used only c1
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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
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
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.
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
€
SGIFT (ω) =1
Ne
s i (ω)i
∑
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
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
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€
Ri = r r1i ,L ,
r r N
i{ }
€
L
€
r r11
€
L
€
L
€
L
€
L
€
L
Imaginary time
Quantumparticles
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M
€
r r21
€
r r31
€
M
€
M
€
M
€
r r12
€
r r22
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r r32
€
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
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r r12P +1
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rr 2
2P +1
€
r r32P +1
Equation of state:liquid and metastableoverpressurizedliquid
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ρfr
€
ρm
PIGS is robust!It converges withoutimportance sampling.See arXiv:0907.4430
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
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
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ψTSWF (R) = dS G(R,S ) ψT (S )∫
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ψ0(R) = dR1L dRPdS R e − τP
ˆ H R1 ×L∫
€
L × RP−1 e − τP
ˆ H RP G(RP ,S )ψT (S )
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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
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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
€
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
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
Università degli Studi di MilanoUniversità degli Studi di Milano
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
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
Università degli Studi di MilanoUniversità degli Studi di Milano
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) ⎧ ⎨ ⎩
⎫ ⎬ ⎭
∫
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• 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
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Z(r q ) ≈ dωS (
r q ,ω)
peak
∫
Yes, we can… obtain more!
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
Exp.: Cowley and WoodsCan. J. Phys 49, ‘71
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
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
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ˆ A = ˆ B + = ˆ ρ r q = e−i
r q ⋅
r r imp
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
ρ = 0.0218 Å-3
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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
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
Università degli Studi di MilanoUniversità degli Studi di Milano
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
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f (τ ) = dω K(τ ,ω)s (ω)∫
see arXiv: cond-mat 0905.4406
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
Università degli Studi di MilanoUniversità degli Studi di Milano