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Davide E. Galli Dipartimento di Fisica Università degli Studi di Milano D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 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 Università degli Studi di Milano

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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, ItalyRPMBT15 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 i.e. it is not possible to find a single theory whose prediction fits the data limited set of observations + noise ill-posed QMC: imaginary-time correlation functions Spectral functions: dynamical properties T=0 Laplace transform: Università degli Studi di Milano

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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 QMC D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Density fluctuation single excitation peaks multi excitation component Maximum Entropy method (MEM): qualitative agreement with exp. (Boninsegni & Ceperley JLTP ‘96) sharp features cannot be recovered uncontrolled approximation: entropic prior Università degli Studi di Milano Superfluid 4 He (Moroni & Baroni PRL ‘99)

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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 them 2.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 Fine!But how can we implement this? Can we obtain more information? D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Università degli Studi di Milano

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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 information 2. We need a falsification procedure relying on the (discrete and noisy QMC) “observations” of the imaginary time correlation function, f l. s( ) is real-valued and non negative thus we chose as models Genetic Inversion (via) Falsification (of) Theories: the GIFT method characteristic function D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Università degli Studi di Milano s( ) differs from the physical spectral function by a factor c 0, the zero-momentum M maximum number of quanta of spectral weight width of the partition Extract s( ) from Fredholm integral equation of the first kind

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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 {f l, c 0 } –But any set {f l *, c 0 * } compatible with {f l, c 0 } provides equivalent information –we can use any set {f l *, c 0 * } obtained by sampling independent Gaussian distributions centered on the original observations with variances corresponding to their statistical uncertainties to define The GIFT Method II adjustable parameters to make the two contributions of the same order of magnitude D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Università degli Studi di Milano Eventual exactly known momenta of s( ) We have used only c 1

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Università degli Studi di Milano The 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 between two intervals s( ) D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA s( )

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Università degli Studi di Milano 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 {f l *, c 0 * }. Each one provides a non-falsified model s i ( ) 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, ItalyRPMBT15 Ohio State University, Columbus, OH, USA

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Path Integral Projector MC methods: Path Integral Ground State (PIGS) 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. 2001 PIGS provides “exact” ground state expectation values via a discrete imaginary time evolved quantum state (such that T ) which gives rise to a discrete path {R 1,…,R 2P+1 } sampled with a Metropolis algorithm Università degli Studi di Milano Imaginary time Quantum particles Equation of state: liquid and metastable overpressurized liquid PIGS is robust! It converges without importance sampling. See arXiv:0907.4430

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D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Shadow-PIGS (SPIGS) = 0.0315 Å -3 = 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 SWF Galli, Reatto, Mol. Phys. 101, ‘03 Università degli Studi di Milano Solid phase: spontaneously broken translational symmetry Above the melting point, nucleation process becomes more and more efficient After few thousand of MC step Starting from a liquid-like configuration the system remains disordered for some MC time

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Università degli Studi di Milano D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA “internal” imaginary- time evolution Whole imaginary-time evolution Imaginary time correlation functions Accurate f( )’s has been computed via SPIGS; relative error ≈ 0.4% q (Å -1 ) (K -1 ) f( ) (K -1 ) Details: pair-product propagator = 1/160 K -1 Example: Superfluid 4 He =0.0218 Å -3 TOT ≈ 1.0 - 0.62 K -1 INT ≈ 0.4 K -1

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S GIFT (q, ) Example: superfluid 4 He, at equilibrium density =0.0218 Å -3 GA details: =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 4 He D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Università degli Studi di Milano

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The multi-phonon contribution very 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, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Università degli Studi di Milano = 0.0262 Å -3 = 0.0218 Å -3 S GIFT (q, ) q=1.755 Å -1 = 0.0218 Å -3

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Università degli Studi di Milano 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? Exp.: Gibbs et al. J. Phys.: Condens Matter 11, ‘99 Yes, we can … obtain more! D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Exp.: Cowley and Woods Can. J. Phys 49, ‘71

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D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Over-pressurized 4 He Università 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 4 He 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

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Superfluid 4 He with one 3 He atom Impurity branch experimentally measured (Fåk et al. PRB 1990) f( ) computed with a SPIGS simulations with N=225 4 He atoms and one 3 He atom at =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 D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA = 0.0218 Å -3 Università degli Studi di Milano

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Vacancy-wave in solid 4 He single vacancy in hcp solid 4 He at =0.0293 Å -3 f( ) computed with SPIGS by considering r vac 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 m He connected to the motion of the vacancy between different basal planes KK MM AA D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Università degli Studi di Milano m*= 0.46±0.03 m He m*= 0.55±0.1 m He m*= 0.46 m He

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Conclusions 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 4 He 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 see arXiv: cond-mat 0905.4406 D.E. Galli.Università degli Studi di Milano, ItalyRPMBT15 Ohio State University, Columbus, OH, USA Università degli Studi di Milano

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