Real-space mapping of topological invariants using artificial neural networks

This paper has been financially supported in part by FEDER funds. We acknowledge financial support by Marie-Curie-ITN Grant No. 607904-SPINOGRAPH, FCT, under Projects No. PTDC/FIS-NAN/4662/2014 and No. P2020-PTDC/FIS-NAN/3668/2014, and by MINECO-Spain (Grant No.MAT2016-78625-C2). J.L.L. acknowledges financial support from the ETH Fellowship program. D.C. acknowledges the hospitality of International Iberian Nanotechnology Laboratory through its Summer Student program.

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

[2]  Liang Fu,et al.  Topological crystalline insulators. , 2010, Physical review letters.

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

[4]  W. Marsden I and J , 2012 .

[5]  K. Müller,et al.  Fast and accurate modeling of molecular atomization energies with machine learning. , 2011, Physical review letters.

[6]  Jun Ding,et al.  Quantum anomalous Hall effect in graphene from Rashba and exchange effects , 2010, 1005.1672.

[7]  E. Bakkers,et al.  Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices , 2012, Science.

[8]  Wenjian Hu,et al.  Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination. , 2017, Physical review. E.

[9]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[10]  Matthias Troyer,et al.  WannierTools: An open-source software package for novel topological materials , 2017, Comput. Phys. Commun..

[11]  S. Das Sarma,et al.  Search for Majorana fermions in multiband semiconducting nanowires. , 2010, Physical review letters.

[12]  A. Marrazzo,et al.  Locality of the anomalous Hall conductivity , 2017, 1702.08885.

[13]  F. Guinea,et al.  Spontaneous strains and gap in graphene on boron nitride , 2014, 1404.7777.

[14]  F ROSENBLATT,et al.  The perceptron: a probabilistic model for information storage and organization in the brain. , 1958, Psychological review.

[15]  Hui Li,et al.  A machine learning correction for DFT non-covalent interactions based on the S22, S66 and X40 benchmark databases , 2016, Journal of Cheminformatics.

[16]  Jing Shi,et al.  Proximity-induced ferromagnetism in graphene revealed by the anomalous Hall effect. , 2015, Physical review letters.

[17]  S. Adam,et al.  Moiré band model and band gaps of graphene on hexagonal boron nitride , 2017, 1706.06016.

[18]  G. Refael,et al.  Non-Abelian statistics and topological quantum information processing in 1D wire networks , 2010, 1006.4395.

[19]  Geoffrey E. Hinton,et al.  Learning representations by back-propagating errors , 1986, Nature.

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

[21]  Yuanbo Zhang,et al.  Gaps induced by inversion symmetry breaking and second-generation Dirac cones in graphene/hexagonal boron nitride , 2016, Nature Physics.

[22]  S. Sarma,et al.  Majorana Fermions in Semiconductor Nanowires , 2011, 1106.3078.

[23]  R. Aguado Majorana quasiparticles in condensed matter , 2017, 1711.00011.

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

[25]  Yang Qi,et al.  Self-learning Monte Carlo method: Continuous-time algorithm , 2017, 1705.06724.

[26]  J. E. Hill,et al.  Intrinsic and Rashba spin-orbit interactions in graphene sheets , 2006, cond-mat/0606504.

[27]  Raffaele Resta,et al.  Mapping topological order in coordinate space , 2011, 1111.5697.

[28]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[29]  Michael A. McGuire,et al.  Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit , 2017, Nature.

[30]  M. Troyer,et al.  Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials , 2016, 1610.08983.

[31]  Zhongqin Yang,et al.  Strong magnetization and Chern insulators in compressed graphene / CrI 3 van der Waals heterostructures , 2017, 1710.06324.

[32]  Kenji Watanabe,et al.  Ballistic Majorana nanowire devices , 2016, Nature Nanotechnology.

[33]  G. Refael,et al.  Helical liquids and Majorana bound states in quantum wires. , 2010, Physical review letters.

[34]  S. Das Sarma,et al.  Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures. , 2010, Physical review letters.

[35]  Yoshua Bengio,et al.  A Neural Probabilistic Language Model , 2003, J. Mach. Learn. Res..

[36]  Yoshua Bengio,et al.  Gradient-based learning applied to document recognition , 1998, Proc. IEEE.

[37]  C. Kane,et al.  Topological Insulators , 2019, Electromagnetic Anisotropy and Bianisotropy.

[38]  Xiao-Liang Qi,et al.  Topological Mott insulators. , 2007, Physical review letters.

[39]  X. Qi,et al.  Topological insulators and superconductors , 2010, 1008.2026.

[40]  Yi Zhang,et al.  Quantum Loop Topography for Machine Learning. , 2016, Physical review letters.

[41]  Noam Bernstein,et al.  Machine learning unifies the modeling of materials and molecules , 2017, Science Advances.

[42]  Kelvin George Chng,et al.  Unsupervised machine learning account of magnetic transitions in the Hubbard model. , 2017, Physical review. E.

[43]  J. Lado,et al.  Quantum Hall effect in gapped graphene heterojunctions , 2013, 1304.5035.

[44]  N. Mitchell,et al.  Amorphous topological insulators constructed from random point sets , 2016, Nature Physics.

[45]  T. Loring K-Theory and Pseudospectra for Topological Insulators , 2015, 1502.03498.

[46]  D. Vanderbilt,et al.  Computing topological invariants without inversion symmetry , 2011, 1102.5600.

[47]  T. Loring,et al.  Supplemental Material to : “ Aperiodic weak topological superconductors ” , 2016 .

[48]  G. Wellein,et al.  The kernel polynomial method , 2005, cond-mat/0504627.

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

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

[51]  H. J. Mclaughlin,et al.  Learn , 2002 .

[52]  Shinhyun Choi,et al.  Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state , 2013, Nature.

[53]  Murat Hüsnü Sazli,et al.  Speech recognition with artificial neural networks , 2010, Digit. Signal Process..

[54]  Helical edge states and fractional quantum Hall effect in a graphene electron-hole bilayer. , 2016, Nature nanotechnology.

[55]  Pengfei Zhang,et al.  Machine Learning Topological Invariants with Neural Networks , 2017, Physical review letters.

[56]  J. C. Budich,et al.  Equivalent topological invariants for one-dimensional Majorana wires in symmetry class D , 2013, 1306.4459.

[57]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

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