Deep Learning Topological Phases of Random Systems

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1 Deep Learning Topological Phases of Random Systems 2017/6/5 Osaka Univ. in Deep Learning and Physics Physics division Sophia University Tomi Ohtsuki with Tomoki Ohtsuki, NTT data mathemahcal systems J. Phys. Soc. Jpn. 85, (2016)à2D, open access J. Phys. Soc. Jpn. 86, (2017)à3D, open access

2 Classifica;on of wave func;ons in random systems We oqen visualize wave funchons to see what is going on in random materials. These wave funchons in random systems show specific features à Regard ψ(r) 2 (instead of cats and dogs) as an image Use mulhlayer (four-weight layer) convoluhonal neural network (CNN) to classify the pazern of ψ(r) 2 and draw the phase diagram of quantum phase transihon 2D topological insulator (edge states)-anderson insulator transihon (exponenhally localized in the bulk) 2D quantum magnon Hall Use the existence of edge states in topological insulator (cf. Hashimoto-Kimura: Phys. Rev. B, 16, Hashimoto-Kimura-Wu: arxiv: ) 3D strong/weak topological insulator-ordinary insulator transihon

3 Supervised training Training stage Prepare thousands of eigenstates for a phase, teach the CNN that the eigenfunchon belongs to this phase. Minimize cross entropy, -Σ i p i log p i =-Σ log p i 90% for training, 10% for teshng: when sahsfied with the score, stop. PredicHon stage Diagonalize the Hamiltonian again with a set of parameters {x}, and let the CNN judge ψ(r) 2. Then CNN outputs probabilihes: the state corresponding to parameters {x} belongs to a phase with probability p i (i=label for quantum phase such as TI, AI, metal.). Tools: Caffe, Keras+tensorflow, Keras+theano hidden layers, iterate if necessary

4 2d topological insulator(chern insulator) Haldane, PRL. 61, 2015 ( 88) for quantum Hall effect without Landau levels. Qi-Wu-Zhang model, PRB. ( 06). Disorder effect by CNN, TO JPSJ 16, interachon effect, CNN, Zhang-Kim PRL 17 2d topological insulator (edge states + localized bulk) àanderson insulator (localized bulk) -W/2<v(x)<W/2 W: strength of random potenhal ψ(r) 2 small W large W

6 Results Probability of states near E=0 to be topological Probability of states away from E=0 to be topological Topological ins. Anderson ins. Strength of randomness energy

Applica;on to quantum magnon Hall effect Effect of disorder on quantum magnon Hall in spin ice model (Xu-Ohtsuki-Shindou, PRB 16) H b H on + H nn + H nnn, H on 1 { (D+d j )S ( b 2 j b j 2 b j) + (HZ

7 Applica;on to quantum magnon Hall effect Effect of disorder on quantum magnon Hall in spin ice model (Xu-Ohtsuki-Shindou, PRB 16) H b H on + H nn + H nnn, H on 1 { (D+d j )S ( b 2 j b j 2 b j) + (HZ +h j )b j b } j j A j B + H.c., H nn m=1,2 { (D + d j )S ( b 2 j b j b j) + (HZ +h j )b j b } j Holstein-Primakoff trans. j A i= j±δ m,i B JS 4 ( b j b i 2b i b i 2b j b j H = i, j 1 i j 3 [Ŝ i Ŝ j 3(Ŝ i n ij )(Ŝ j n ij )] D i A (Ŝx ) 2 (Ŝy ) 2 i D HZ j B j i A,B Ŝ z i. H nnn + 6i( 1) m b j b i + H.c.), α=a,b m=1,2 j α J α,m S ( b j 4 b i b i b i b j b j i= j±e m + 3( 1) m b j b i + H.c.), ( 1 μm

8 delocalized magnon Anderson localized magnon topological edge magnon

9 Procedure use the training of 2D electron Chern insulator to draw a rough phase diagram use this rough phase diagram to determine the training regions in magnon systems draw the more precise phase diagram

10 E Draw a rough phase diagram from the results of CI-AI training randomness TI deloc AI Then guess the typical regions of delocalized, topological edge and localized (AI), train CNN, draw the phase diagram. deloc

11 Use the already trained CNN to draw phase diagrams with different set of parameters H z =15.8, D=2, JS=1.0. J S=0.35 H z =15, D=1.8, JS=0.9. J S=0.4

12 two-weight-layer CNN, Keras+TF, ES four-weight-layer CNN, Keras+TF, ES six-weight-layer CNN, Keras+TF, ES four-weight-layer CNN, Keras+Theano, ES

13 Comparing the methods of drawing phase diagram Finite size scaling (Slevin and Ohtsuki, NJP 14) Define a nondimensional quanhty Λ(L, E, W,..) such as conductance. Plot Λ(L, E, W,..) as a funchon of E, W, etc. with different system sizes L. Analyze Λ(L, E, W,..). Scaling invariant point is the phase boundary. Precise eshmate of the phase boundaries. CriHcal exponents. Machine learning method Simple analysis. [python train.py;] python test.py; python dataarrange.py; python plot.py Wider applicability. Once trained, can draw phase diagrams for different parameters. DetecHon of states on the phase boundary. Only rough eshmate of the phase boundaries. Too many tuning parameters like number of hidden layers, convoluhon size, pooling size, bias, padding,.

14 Outlook Several preprints have been submized to arxiv in 2016, refereeing processes took a bit, but they are now published. Though shll restricted to standard simplified models, machine learning methods seem to be gradually accepted among solid state physics community. arxiv (Nat. Phys. 17) : 2D Ising model arxiv (Science 17): Heisenberg model arxiv: (Nat. Phys. 17): Kitaev chain, Ising, disordered quantum spin chain arxiv (JPSJ 17): 2D Ising model, the most beauhful arxiv : 2D Hubbard model, G(E, x, y) arxiv : 3D Hubbard model, temperature transihon arxiv : Chern insulator, bulk (PRL 17) arxiv (PRB 17): DFT with machine learning

Kouki Nakata. University of Basel. KN, S. K. Kim (UCLA), J. Klinovaja, D. Loss (2017) arxiv:

Kouki Nakata. University of Basel. KN, S. K. Kim (UCLA), J. Klinovaja, D. Loss (2017) arxiv: Magnon Transport Both in Ferromagnetic and Antiferromagnetic Insulating Magnets Kouki Nakata University of Basel KN, S. K. Kim (UCLA), J. Klinovaja, D. Loss (2017) arxiv:1707.07427 See also review article