Statistical and computational intelligence approach to analytic continuation in Quantum Monte Carlo

Graphical Abstract Abstract The term analytic continuation emerges in many branches of Mathematics, Physics, and, more generally, applied Science. Generally speaking, in many situations, given some amount of information that could arise from experimental or numerical measurements, one is interested in extending the domain of such information, to infer the values of some variables which are central for the study of a given problem. For example, focusing on Condensed Matter Physics, state-of-the-art methodologies to study strongly correlated quantum physical systems are able to yield accurate estimations of dynamical correlations in imaginary time. Those functions have to be extended to the whole complex plane, via analytic continuation, in order to infer real-time properties of those physical systems. In this review, we will present the Genetic Inversion via Falsification of Theories method, which allowed us to compute dynamical properties of strongly interacting quantum many–body systems with very high accuracy. Even though the method arose in the realm of Condensed Matter Physics, it provides a very general framework to face analytic continuation problems that could emerge in several areas of applied Science. Here, we provide a pedagogical review that elucidates the approach we have developed.

[1]  Microscopic Study of Static and Dynamical Properties of Dilute One-Dimensional Soft Bosons , 2016, 1607.05308.

[2]  D. E. Galli,et al.  Superfluid State of 4He on Graphane and Graphene–Fluoride: Anisotropic Roton States , 2013 .

[3]  M. Motta,et al.  One-Dimensional Liquid ^{4}He: Dynamical Properties beyond Luttinger-Liquid Theory. , 2014, Physical review letters.

[4]  Computation of dynamical correlation functions for many-fermion systems with auxiliary-field quantum Monte Carlo , 2016, 1606.04785.

[5]  M. Boninsegni,et al.  Bose soft discs: a minimal model for supersolidity , 2011 .

[6]  D. E. Galli,et al.  Accurate Density Response Function of Superfluid 4He at Freezing Pressure: Is DFT Successful for Superfluid Freezing? , 2011 .

[7]  I. D. Marco,et al.  Analytic continuation by averaging Pade approximants , 2015, 1511.03496.

[8]  M. Nava,et al.  Equation of state of two-dimensional 3 He at zero temperature , 2011, 1103.0915.

[9]  S. Fahy.  Quantum Monte Carlo Methods , 1996 .

[10]  D. E. Galli,et al.  The Shadow Path Integral Ground State Method: Study of Confined Solid 4He , 2004 .

[11]  K. Fiedler,et al.  Monte Carlo Methods in Ab Initio Quantum Chemistry , 1995 .

[12]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[13]  K. H. Andersen,et al.  The collective excitations of normal and superfluid : the dependence on pressure and temperature , 1999 .

[14]  D. Ceperley,et al.  Density fluctuations in liquid4He. Path integrals and maximum entropy , 1996 .

[15]  T. Pruschke,et al.  Extracting spectral properties from Keldysh Green functions. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  R. Cowley,et al.  Inelastic Scattering of Thermal Neutrons from Liquid Helium , 1971 .

[17]  Joseph A. O'Sullivan,et al.  Deblurring subject to nonnegativity constraints , 1992, IEEE Trans. Signal Process..

[18]  Gary James Jason,et al.  The Logic of Scientific Discovery , 1988 .

[19]  A. Tarantola.  Popper, Bayes and the inverse problem , 2006 .

[20]  L. Reatto,et al.  Roton Excitations and the Fluid–Solid Phase Transition in Superfluid 2D Yukawa Bosons , 2016, 1605.01295.

[21]  A. Sandvik.  Stochastic method for analytic continuation of quantum Monte Carlo data , 1998 .

[22]  M. Motta,et al.  Linear Response of One-Dimensional Liquid $$^4\hbox {He}$$4He to External Perturbations , 2016, 1607.05301.

[23]  O. Syljuåsen.  Using the average spectrum method to extract dynamics from quantum Monte Carlo simulations , 2007, 0705.4173.

[24]  M. W. Cole,et al.  Novel behavior of monolayer quantum gases on graphene, graphane and fluorographene , 2013, Journal of physics. Condensed matter : an Institute of Physics journal.

[25]  A. Roggero,et al.  Dynamical structure functions from quantum Monte Carlo calculations of a proper integral transform , 2013 .

[26]  J. Miao,et al.  Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens , 1999, Nature.

[27]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[28]  D. E. Galli,et al.  Density Functional Theory and Bose Statistics for the Freezing of Superfluid 4He , 2013 .

[29]  Thierry Giamarchi,et al.  Quantum physics in one dimension , 2004 .

[30]  M. Motta,et al.  Dynamical structure factor of one-dimensional hard rods , 2016, 1608.07722.

[31]  Akihiko Sakamoto,et al.  Diagrammatic quantum Monte Carlo study of the Fröhlich polaron , 2000 .

[32]  D. E. Galli,et al.  Exact ground state Monte Carlo method for Bosons without importance sampling. , 2009, The Journal of chemical physics.

[33]  K. S. D. Beach.  Identifying the maximum entropy method as a special limit of stochastic analytic continuation , 2004 .

[34]  O. Gunnarsson,et al.  Analytical continuation of imaginary axis data using maximum entropy , 2010, 1001.4351.

[35]  M. W. Cole,et al.  Novel substrates for Helium adsorption: Graphane and Graphene—Fluoride , 2012, 1204.3061.

[36]  D. E. Galli,et al.  Excitation spectrum in two-dimensional superfluid 4He , 2013, 1305.3732.

[37]  Vitiello,et al.  Variational calculations for solid and liquid 4He with a "shadow" wave function. , 1988, Physical review letters.

[38]  I. Csiszár.  Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems , 1991 .

[39]  A. Sandvik.  Constrained sampling method for analytic continuation. , 2015, Physical review. E.

[40]  D. E. Galli,et al.  Real time dynamics from quantum Monte Carlo data: A genetic algorithm approach , 2009 .

[41]  R. Rota,et al.  Quantum Monte Carlo estimation of complex-time correlations for the study of the ground-state dynamic structure function. , 2015, The Journal of chemical physics.

[42]  Richard Phillips Feynman,et al.  Atomic Theory of the Two-Fluid Model of Liquid Helium , 1954 .

[43]  R. Bryan,et al.  Maximum entropy analysis of oversampled data problems , 1990, European Biophysics Journal.

[44]  R. Cowley,et al.  Structure and excitations of liquid helium , 1973 .

[45]  Faming Liang,et al.  Statistical and Computational Inverse Problems , 2006, Technometrics.

[46]  Aziz,et al.  Ab initio calculations for helium: A standard for transport property measurements. , 1995, Physical review letters.

[47]  A. Rothkopf,et al.  Bayesian approach to spectral function reconstruction for Euclidean quantum field theories. , 2013, Physical review letters.

[48]  Mark Jarrell,et al.  Analytic continuation of quantum Monte Carlo data by stochastic analytical inference. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[49]  S. Giorgini,et al.  Many-body Bose systems and the hard-sphere model: dynamic properties from the weak to the strong interaction regime , 2014 .

[50]  L. Pollet,et al.  Numerical analytic continuation: Answers to well-posed questions , 2016, 1609.01260.

[51]  D. E. Galli,et al.  Recent progress in simulation of the ground state of many Boson systems , 2003 .

[52]  J. Dawidowski,et al.  Superfluid 4He dynamics beyond quasiparticle excitations , 2016, 1605.02638.

[53]  M. Boninsegni,et al.  Excitation spectrum of a supersolid. , 2012, Physical review letters.

[54]  Marcel Abendroth,et al.  Quantum Field Theory And Critical Phenomena , 2016 .

[55]  Yew-Soon Ong,et al.  Memetic Computation—Past, Present & Future [Research Frontier] , 2010, IEEE Computational Intelligence Magazine.

[56]  D. Thouless,et al.  Ordering, metastability and phase transitions in two-dimensional systems , 1973 .

[57]  Lev Davidovich Landau,et al.  Theory of the Superfluidity of Helium II , 1941 .

[58]  Stefano Baroni,et al.  Reptation Quantum Monte Carlo: A Method for Unbiased Ground-State Averages and Imaginary-Time Correlations , 1999 .

[59]  N. Mermin,et al.  Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models , 1966 .

[60]  Edmonton,et al.  Reliable Padé analytical continuation method based on a high-accuracy symbolic computation algorithm , 2000 .

[61]  Alessandro Colombo,et al.  Facing the phase problem in Coherent Diffractive Imaging via Memetic Algorithms , 2017, Scientific Reports.

[62]  Kevin Schmidt,et al.  A path integral ground state method , 2000 .

[63]  R. A. Aziz,et al.  An accurate intermolecular potential for helium , 1979 .

[64]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[65]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[66]  Mark Jarrell,et al.  Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data , 1996 .

[67]  Ab initio low-energy dynamics of superfluid and solid 4He , 2009, 0905.4406.

[68]  Silver,et al.  Maximum-entropy method for analytic continuation of quantum Monte Carlo data. , 1990, Physical review. B, Condensed matter.

[69]  E. Rabani,et al.  Analytic continuation average spectrum method for quantum liquids. , 2009, The Journal of chemical physics.

[70]  Frederick R. W. McCourt,et al.  A new determination of the ground state interatomic potential for He2 , 1987 .

[71]  P. Leiderer,et al.  4He films on graphite studied by neutron scattering , 1992 .