Extrapolating Quantum Observables with Machine Learning: Inferring Multiple Phase Transitions from Properties of a Single Phase.

We present a machine-learning method for predicting sharp transitions in a Hamiltonian phase diagram by extrapolating the properties of quantum systems. The method is based on Gaussian process regression with a combination of kernels chosen through an iterative procedure maximizing the predicting power of the kernels. The method is capable of extrapolating across the transition lines. The calculations within a given phase can be used to predict not only the closest sharp transition but also a transition removed from the available data by a separate phase. This makes the present method particularly valuable for searching phase transitions in the parts of the parameter space that cannot be probed experimentally or theoretically.

[1]  Kirk W Madison,et al.  Investigating polaron transitions with polar molecules. , 2012, Physical review letters.

[2]  Ye-Hua Liu,et al.  Discriminative Cooperative Networks for Detecting Phase Transitions. , 2017, Physical review letters.

[3]  Juan Carrasquilla,et al.  Machine learning quantum phases of matter beyond the fermion sign problem , 2016, Scientific Reports.

[4]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[5]  D. Deng,et al.  Quantum Entanglement in Neural Network States , 2017, 1701.04844.

[6]  H. Löwen,et al.  Analytical properties of polaron systems or: Do polaronic phase transitions exist or not? , 1991 .

[7]  S. Sachdev.  Quantum Phase Transitions , 1999 .

[8]  W. Marsden.  I and J , 2012 .

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

[10]  O. A. von Lilienfeld,et al.  Machine learning for many-body physics: The case of the Anderson impurity model , 2014, 1408.1143.

[11]  Richard S. Sutton,et al.  Reinforcement Learning: An Introduction , 1998, IEEE Trans. Neural Networks.

[12]  Nobuyuki Yoshioka,et al.  Learning disordered topological phases by statistical recovery of symmetry , 2017, 1709.05790.

[13]  Lei Wang,et al.  Discovering phase transitions with unsupervised learning , 2016, 1606.00318.

[14]  Matthias Troyer,et al.  Neural-network quantum state tomography , 2018 .

[15]  Roger G. Melko,et al.  Machine learning vortices at the Kosterlitz-Thouless transition , 2017, 1710.09842.

[16]  Vittorio Ferrari,et al.  Advances in Neural Information Processing Systems 24 , 2011 .

[17]  Felix A Faber,et al.  Machine Learning Energies of 2 Million Elpasolite (ABC_{2}D_{6}) Crystals. , 2015, Physical review letters.

[18]  Titus Neupert,et al.  Probing many-body localization with neural networks , 2017, 1704.01578.

[19]  Tomi Ohtsuki,et al.  Deep Learning the Quantum Phase Transitions in Random Two-Dimensional Electron Systems , 2016, 1610.00462.

[20]  M. Berciu,et al.  Sharp transition for single polarons in the one-dimensional Su-Schrieffer-Heeger model. , 2010, Physical review letters.

[21]  Antonio-José Almeida,et al.  NAT , 2019, Springer Reference Medizin.

[22]  M. Grunwald.  Principles Of Condensed Matter Physics , 2016 .

[23]  Jinguo Liu,et al.  Approximating quantum many-body wave functions using artificial neural networks , 2017, 1704.05148.

[24]  Vedika Khemani,et al.  Machine Learning Out-of-Equilibrium Phases of Matter. , 2017, Physical review letters.

[25]  M. Berciu,et al.  Systematic improvement of the momentum average approximation for the Green's function of a Holstein polaron , 2007, 0705.4154.

[26]  Roger G. Melko,et al.  Machine learning phases of matter , 2016, Nature Physics.

[27]  Mona Berciu.  Green's function of a dressed particle. , 2006, Physical review letters.

[28]  Manuel Scherzer,et al.  Machine Learning of Explicit Order Parameters: From the Ising Model to SU(2) Lattice Gauge Theory , 2017, 1705.05582.

[29]  Andrew S. Darmawan,et al.  Restricted Boltzmann machine learning for solving strongly correlated quantum systems , 2017, 1709.06475.

[30]  Dong-Ling Deng,et al.  Machine Learning Topological States , 2016, 1609.09060.

[31]  Lu-Ming Duan,et al.  Efficient representation of quantum many-body states with deep neural networks , 2017, Nature Communications.

[32]  Richard S. Sutton,et al.  Reinforcement Learning , 1992, Handbook of Machine Learning.

[33]  S. Huber,et al.  Learning phase transitions by confusion , 2016, Nature Physics.

[34]  Kristof T. Schütt,et al.  How to represent crystal structures for machine learning: Towards fast prediction of electronic properties , 2013, 1307.1266.

[35]  M. Berciu,et al.  Momentum average approximation for models with electron-phonon coupling dependent on the phonon momentum , 2008, 0810.1498.

[36]  R. Krems,et al.  Phonon-mediated repulsion, sharp transitions and (quasi)self-trapping in the extended Peierls-Hubbard model , 2017, Scientific Reports.

[37]  Sebastian Johann Wetzel,et al.  Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders , 2017, Physical review. E.

[38]  M. Heyl,et al.  Quantum dynamics in transverse-field Ising models from classical networks , 2017, 1707.06656.

[39]  J. Vybíral,et al.  Big data of materials science: critical role of the descriptor. , 2014, Physical review letters.

[40]  Felix A Faber,et al.  Crystal structure representations for machine learning models of formation energies , 2015, 1503.07406.

[41]  R. Melko,et al.  Machine Learning Phases of Strongly Correlated Fermions , 2016, Physical Review X.

[42]  M. Berciu,et al.  Single-polaron properties of the one-dimensional breathing-mode Hamiltonian , 2007, 0708.2920.

[43]  H. Fehske,et al.  Momentum average approximation for models with boson-modulated hopping: Role of closed loops in the dynamical generation of a finite quasiparticle mass , 2010, 1010.4250.

[44]  Eyal Bairey,et al.  Learning phase transitions from dynamics , 2017, Physical Review B.

[45]  Matthias Troyer,et al.  Solving the quantum many-body problem with artificial neural networks , 2016, Science.