Novel Quantum States in Condensed Matter 2017 2017/10/27 University of Tokyo Nobuyuki Yoshioka Collaborators: Yutaka Akagi, Hosho Katsura arXiv: 1709.05790 LEARNING DISORDERED TOPOLOGICAL PHASES BY STATISTICAL RECOVERY OF SYMMETRY
▸ Introduction Objective of Machine Learning Application to Physics ▸ Method and Hamiltonian Problem set up Classification by Artificial Neural Network ▸ Result and Discussion 2
▸ Introduction Objective of Machine Learning Application to Physics ▸ Method and Hamiltonian Problem set up Classification by Artificial Neural Network ▸ Result and Discussion 3
Machine Learning in Ordinary Life Image recognition Machine translation Google translation Well understanding on non-metamessage. UC Berkeley Computer Vision Group Triumph of Go/Shogi AI in 2017 DeepMind group, Nature 550, 354 (2017). Denou Sen Website 4 http://denou.jp/2017/
Element-wise understanding of ML Machine Learning = Computer algorithm that gives prediction/knowledge from huge amount of data beyond human resources. ‣ Machine task = Construction of highly-nonlinear function. Recent progress: discovery of convolutional NN, ResNet etc. Input Output Possibility for a label ✓ Image recognition e.g.) mountain, tree, pipe RGB values 1. Next move suggestion ✓ Go/Shogi 2. Current win probability Piece configuration ‣ Learning task = Optimization of “the Loss function” i.e. the “performance” of the machine e.g.) For data , label , and some parametrized classifier F p , take y x L ( x ) = |F p ( x ) − y | prediction error 5 and update p → p − η∂ p L in a stochastic manner.
Application to Physics 6
Three main approaches for <ML|cond-mat> Glasser etal. arXiv:1710.04045 (’17) ▸ 1. Neural network as wave func. ansatz Clark arXiv:1710.03545 (’17) Quantum many-body solver RBM <—> Tensor network GS energy error of Carleo&Troyer Science 355(’17) Chen etal arXiv:1701.04831 (’17) Nomura etal. arXiv:1709.06475 periodic 10x10 AFH Restricted Boltzmann Machine (RBM) Falicov-Kimball model, 8x8 periodic square lattice Wang arXiv:1702.05856(’17) 2. Speeding up Monte Carlo ▸ Huang&Wang PRB 95(‘17) Efficient cluster-update by learning thermal distribution 7
Three main approaches for <ML|cond-mat> Tanaka&Tomiya JPSJ 86 (’17) Broecker etal. SciRep 7 (’17) ▸ 3. Classification of phases Ch’ng etal. PRX 7 (’17) Zhang&Kim PRL 118 (‘17) Supervised learning : Teach the pattern. Then let it predict. Possibility of disordered phases in Possibility of low/high-temp. phase topological insulator in 2D square Ising w/ Total Sz = 0 Ohtsuki&Ohtsuki JPSJ 85(’16), 86(‘17) Carrasquilla&Melko NatPhys 13 (’17) L. Wang PRB 94 (’16), S. Wetzel PRE 96 (’17) Data science methods e.g. PCA, VAE, tSNE Ch’ng etal. (’17), Extract patterns of spin configurations Learning transition point without teaching the notion of “phase”. 8
Today’s talk Learn the clean, classify the disordered by Neural Network ▸ Goal Phase classification of disordered TSC (e.g. 2d, class DIII) Break down of well-known formulae in lattice system Obstacle Use of NN. Solution Trained merely at clean limit, disordered phases classified correctly. Phases not present at clean limit also correctly detected. : Probability of the corresponding phases : Extract the feature of the data : Statistical average of quasiparticle distribution 9
▸ Introduction Objective of Machine Learning Application to Physics ▸ Method and Hamiltonian Problem set up Classification by Artificial Neural Network ▸ Result and Discussion 10
2d Class DIII Topological Superconductor Schnyder etal. PRB 78 (’08) ▸ Class DIII in the “Periodic table” Kitaev AIP Conf. Proc. 1134 (’09) BdG system with TRS, w/o SU(2) symm. Dimension TRS: IQHE IQHE PHS: CHS: Today’s topic Majorana edge mode as topo. feature 3D TI e.g.) Cu x Bi 2 Se 3 , Hasan group(’10) QSHE quasi-2D, surface modes ▸ BdG Hamiltonian in k-space, square-lattice Sato&Fujimoto PRB 79(’09) Diez etal NewJPhys 16(‘14) ✓ ˆ ˆ ◆ ∆ k ✏ k Kane&Mele PRL 95(’05) H k = Z2 invariant ∆ † ˆ Fu&Kane PRB 76(‘07) − ˆ ✏ − k k TRIM Z 2 inv. ✏ k = 2 t (cos k x + cos k y ) − µ ˆ Kinetic NN hopping chemical pot. (sin k x , − sin k y , 0) “Sewing matrix” = ˆ ∆ k = i σ y ( ∆ p ( d k · σ ) + ∆ s σ 0 ) helical Pairing s-wave p-wave
2d Class DIII Topological Superconductor II: Dirty Hikami PRB 24 (’81) ▸ Phases by intermediate disorder (NLSM analysis) Evers&Mirlin RevModPhys 80 (’08) An Anderson localization Topo/Triv Weak anti-localization Weak anti-localization (clean) (disorder) aka Thermal metal aka Thermal metal No marginal perturbation Back scattering scatterer suppression In lattice model…? For example, Uniform ! " on-site # box distribution randomness " # -W/2 W/2 Break down of top.inv. formulae (e.g. Kane-Mele, Niu-Thouless-Wu)… New approach introduced. NN.
Classification by ANN 13
@X X A Single Neuron (perceptron) Rosenblatt PsycoRev 65(’58) ! " %, ' 1. weighing -> linear combination of input ( = ⋮ ! = #(% ⋅ ' ⃗ + *) 2. activation -> operate nonlinear function ������ ! $ e.g.) ( (Sigmoid) 1 / (1 + e − z ) ����� σ ( z ) = (ReLU) max (0 , z ) 14
@X X A Rumelhart etal. Nature 323(’86) Deep Neural Network Hinton etal. Science 313 (‘06) Weighing and activation sequentially/simultaneously. Hidden Layers extracts abstract feature e ffi ciently. Output: X y i = e − z i / ( e − z j ) X y i = 1 probability j i Universal approximation theorem for multilayer NN F : R n → R m Expression of any nonlinear function Cybbenko MathCon 2(’89) Hornik etal. Neural Network 2(‘89) ‣ Supervised Learning … Tune parameters by minimizing the “distance” btw output and correct label ⇣ ⌘ W ( i ) j,k → W ( i ) ∂ L / ∂ W ( i ) j,k − η j,k Loss = Cross entropy + L2 regularization: (#data) (#class) y ( k ) log y ( k ) X X L ( w ) = − ˆ ( x j ; w ) / (#data) j j j =1 k =1 (#layers) (improves the generalization power) X | W ( i ) | 2 . + λ 15 i =1
Classification by Artificial Neural Network ‣ Image -> Probability RGB Probability/Confidence per pixel Krizhevsky etal. ILSVRC 2012 16
Classification by Artificial Neural Network ‣ Quasiparticle dist. -> Probability H | Ψ n > = E n | Ψ n > Take first excitation state ↑ ( r ) | 2 + | ψ e P ( r ) = | ψ e ↓ ( r ) | 2 Probability/Confidence ↑ ( r ) | 2 + | ψ h + | ψ h ↓ ( r ) | 2 Phase Expected behavior Z 2 -> Edge localized Trivial -> Bulk localized Thermal -> Extended metal “Training” requires knowledge on disorder phase boundary. Possible to avoid it by statistical average! 17
“Statistical recovery” of translational symmetry cf.) Recovery of TRS, Inv. Fulga et al. (‘14) “Sector” by symmetry Some representation e.g.) wave func. in clean limit Learn in clean phase, classify dirty phase. averaged 18 under disorder
▸ Introduction Objective of Machine Learning Application to Physics ▸ Method and Hamiltonian Problem set up Classification by Artificial Neural Network ▸ Result and Discussion 19
Result I: Ternary Classification t=1, Δ p =3, Δ s =2, Lx = Ly = 14 ▸ Output of the NN M Z 2 Triv. Phase boundary reproduced at W=0 - Accuracy>90% for test at μ ∈ [0,10]. - Small window of ThM at μ ~3.5 detected. Consistency with Transfer Matrix at W>0 by Transfer - ThM-Z2-ThM transition at μ ~3.5. Matrix - Close boundaries of Z2-Thm, ThM-Triv. Phase/training data at W=0 - Confusion(gray) at W~15 improved by increasing disorder average. 20
Fails without statistical symmetry recovery t=1, Δ p =3, Δ s =2, Lx = Ly = 14 ▸ Output for Single-shot P(r) M Z 2 Triv. Phase/training data at W=0 21
Result II: Binary Classification t=1, Δ p =3, Δ s =0, Lx = Ly = 14 ▸ Output of the NN Z 2 Triv. Consistency with TM - Accuracy>95% for test at μ ∈ [0,10], W=0. - Z2-triv phase boundary reproduced. - Z2-Z2 boundary for confusion at μ ~0 Confused region above Z2 phase - Output convergence below 0.75. Detection of metallic phase. Phase/training data at W=0 - Shrink of Z2 phase due to finite-size e ff ect. 22
Summary and Future works Quantum phase diagram of class DIII by new method Extension of phase boundary from clean limit Consistency with TM (and NCI) Higher precision by increasing samples Inclusion of higher moments Application to many-body system with disorder Further classification within the unknown phase 23
Supplement 1:Transfer Matrix … … ▸ Localization length in quasi-1D system … … … … Localization length ▸ Finite-size scaling MacKinnon & Kramer (1983), Yamakage et al. (2012) critical exp.
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