Drawing phase diagrams using deep
convolutional neural networks  !# "" #$Deep Learning and Physics 2018%  "
J. Phys. Soc. Jpn., 85, 123706 (2016)
J. Phys. Soc. Jpn., 86, 044708 (2017)
J. Phys. Soc. Jpn., 86, 113704 (2017)

outline• Analyze the wave function using deep
convolutional neural network à phase diagram forquantum phase transition• What is Anderson metal-insulator transition(delocalization-localization transition)?
• Deep 3d convolutional neural network approach(image recognition)• Various types of Anderson transitions• Quantum percolation transitions• Application to random topological insulator: real vs.
k-space analysis
2Anderson localization• Wave propagation in random medium àconstructive and destructive interferences (wave canbe electron, microwave, light, sound, matter waves)àlocalization of wave (P.W. Anderson, ‘58)

Ubiquitous phenomena in many fields of physics
3Anderson localization• Eigenfunctions are exponentially localized

its distribution can then be performed yielding results entirely consistent withthose for the Q1D localisation length (Slevin, Markos et al. 2001B Slevin, Markos etal. 2003).

⎛ | x − x0 | ⎞Ψ(x) = a(x)exp ⎜ −ξ ⎟⎠⎝| ψ (x, y) |2

Figure 4: A plot of the wavefunction intensity near the band centre in a realization of the 2D Anderson

Red-green, 80%. Red-green-blue, 99%

model with box distributed random potential (W = 5) and lattice size 1000 × 1000. The colour indicates

Slevin –Ohtsuki ‘12

the logarithm of the wavefunction intensity (decreasing from red to blue). The plotted points support
99% of the electron probability. Approximately 80% of the electron probability is supported on the3d delocalized-localized w.f.

Metal (delocalized)

Insulator (localized)
5Anderson Model
Non-interacting, random HamiltonianH = ∑ ε j j j + ∑ Vj ', j j ' j ,jj ', jW
W− < εj <22

In magnetic fields
U(1) matrix

In the presence of spin-orbit interaction

H = ∑ ε j j, σj,σj, σ + ∑ Vj ',σ ', j,σ j ', σ ' j, σj ', j,σ ,σ 'SU(2) matrix

6

Lyapunov exponent and quasi 1D
localization length
Transfer matrix

Eψ n = H nψ n + Vn,n +1ψ n +1 + Vn,n−1ψ n−1⎛ ψ n +1 ⎞⎛ ψn ⎞⎜⎟ = Tn ⎜⎟VψVψ⎝ n +1,n n ⎠⎝ n,n−1 n−1 ⎠n-1 nn+1

⎛n ⎞
ψ n ≈ exp ⎜ −⎟⎝ ξq1D (L) ⎠Finite size scaling analysis⎧ L2−d (delocalized, metal)⎪⎪1/νΓ c (critical)
L/ξq1d(L)= Γ(L) = F(L / ξ ) = f (L (W − Wc )) = ⎨⎪ L / ξ (localized, insulator)⎪⎩LLWcmetalinsulator

Slevin-Ohtsuki 2012

Limit of transfer matrix• If the matrix elements between sites on n-thlayer and n+1th layer, Vn,n+1 = ⟨n|H|n+1⟩, is notinvertible, we can not use transfer matrix.
– This happens in the case of bcc and fcc lattices,quantum percolation and of localization on fractallattice.

• To draw a phase diagram in W-E space,– Fix E, change W, do FSS and obtain Wc(E).
– Then change E to E’ and fix it, change W, FSS, Wc(E’).
– …..

Use 3D image recognition• 3D convolutional neural network• Train the neural network for a simple case,and use the trained neural network to analyze
the more complex situations/systems.
– Simple model: Anderson model of localization at
the band center E=0.àused for training theneural network.
– Application to• |E|>0• In the presence of magnetic field• Random lattice (ex. quantum percolation)

Models• Anderson modelH = ∑ ε j j j + ∑ Vj ', j j ' j ,jj ', jW
W− < εj <22• Quantum percolation! = ∑ $% &,% |* + ⟩⟨*|, $%+,% =0 or 1
Sites/bonds are occupied withprobability p. For p>pc , a cluster
connecting left side to right appears.
https://en.wikipedia.org/wiki/Percolation_theory#/media/
File:Bond_percolation_p_51.png

3D CNNCONVOLUTIONCONVOLUTIONPOOLINGCONVOLUTIONCONVOLUTIONPOOLING
14<W<1617<W<19Diagonalize
Anderson model

Size 404040
Select E=04000 eigen
functions formetal and ins.

CONVOLUTIONCONVOLUTIONPOOLINGDENSEDeep (9 weight layer) CNN

3D Anderson transitioncu



BTO

3 UVU

transfer matrix + FSS

Mano-Ohtsuki, JPSJ ‘17

Wave function of Anderson model vs. quantum
percolation
LÁSZLÓ UJFALUSI AND IMRE VARGA

PHYSICAL REVIEW B 90, 174203 (2014)

Andersonmodelquantum
percolation

Ujfalusi and Varga, PRB ‘14

FIG. 3. (Color online) First row: Eigenvectors of the Anderson model at E = 0 (a) on the metallic side at W = 14, (b) close to criticality
W = 16.5, and (c) on the insulating side at W = 20. Second row: eigenvectors of the quantum percolation model at energy E = 0.1, (d) on
the metallic side at p = 0.5, (e) close to criticality, p = 0.4535, and (f) on the insulating side at p = 0.4. Box sizes correspond to (a),(d)Phase diagram for quantum percolation=metal

=metal=insulator

Site percolation; T. Mano and TO: J. Phys. Soc. Jpn. 86, 113704 (2017)(white dashed line,Ujfalusi andVarga, ‘14)

るわけである．
exp − 2Wx 2 一度このモデルで学習して作ったニューラルネットワークは，今度は他のモデルで得ら, コーシー分布
P (Vx ) = π(V 2W+W
1. 2ポテンシャル
Vx の確率分布を上記の箱型分布 P (Vx )) にしV#x$

一度このモデルで学習して作ったニューラルネットワークは，今度は他のモデル
Vx21
ウス分布 P (Vx ) = √exp − 2W 2 , コーシー分れた波動関数が金属的であるか，絶縁体的であるかを判定できるようになる．演習問題
ルネットワークは，今度は他のモデルで得ら2πW 2れた波動関数が金属的であるか，絶縁体的であるかを判定できるようになる．演たもの．
を十分勉強すれば応用問題も解けるようになるのと似ている．ここでは，以下の“応用問あるかを判定できるようになる．演習問題
が，格子がランダムなもの．例えば，サイトが確率
psを十分勉強すれば応用問題も解けるようになるのと似ている．ここでは，以下の2. ポテンシャルは一定であるが，格子がランダムなもの
題”を考える．
るのと似ている．ここでは，以下の“応用問るパーコレーションクラスター上での電子の局在ー非局題”を考える．
で占有されている，いわゆるパーコレーションクラス

Change the model and draw phase
diagrams• Modify the potential distribution!
"
1 *2 W．古典的にはパーコレーション閾値
p(≈
0.3116)よ!
"c1. ポテンシャル
Vの確率分布を上記の箱型分布P(V)=Θ−
1 |Vx|W から，ガ在問題がこの状況に当たる．古典的にはパーコレーxxW2
1. ポテンシャル Vx の確率分布を上記の箱型分布
P (Vx ) = W Θ 2 − |Vx| か#$ #2$!W" √ 12VxW
Vったクラスターが形成され電気が流れ始め，金属ー絶11Wx
り大きなpで系全体に広がったクラスターが形成さ
P−(V)
=exp−
,
コーシー分布P(V)=√s2 exp2)2 ) にしxウス分布xP(V)=−,コーシー分布
P(V=
箱型分布 P (Vx ) =ウス分布
Θ|Vから，ガ22Wπ(V+W22 +Wxxx|22πW2Wπ(VxW22πWx$縁体転移が起こる．一方，量子力学的に振る舞う電子W
量子力学的に振る舞う電子の場合，たとえクラスターが, コーシー分布たもの．
P (Vx ) = π(Vたもの．
2 +W 2 ) にし2x系全体に広がっていても，電子がクラスター上で局2. ポテンシャルは一定であるが，格子がランダムなもの．例えば，サイトが2. ポテンシャルは一定であるが，格子がランダムなもの．例えば，サイトが確率ps
電子がクラスター上で局在している限り絶縁体のまま• Applying the magnetic fields, which weakensである．よって電流が流れ始める条件は p > p (≥で占有されている，いわゆるパーコレーションクラスター上での電子の局在で占有されている，いわゆるパーコレーションクラスター上での電子の局在ー非局ランダムなもの．例えば，サイトが確率始める条件は
pthe
ppq sは量子パーコs > plocalizationq (≥ pc ) である．dueto thebreaking of time
レーション閾値と呼ばれる．
在問題がこの状況に当たる．古典的にはパーコレーション閾値 p (≈ 0.3*2 ．古典的にはパーコレーション閾値在問題がこの状況に当たる
pc (≈ 0.3116) よーションクラスター上での電子の局在ー非局．
reversal symmetry
3. 上記の問題に磁場をかけることで時間反転対称性をり大きな p で系全体に広がったクラスターが形成され電気が流れ始め，金s*2qc

り大きな pps で系全体に広がったクラスターが形成され電気が流れ始め，金属ー絶%%にはパーコレーション閾値
(≈ 0.3116)
よ†ことで時間反転対称性を破ったもの．この場合，最– Modifycthetransferterm,近接格子へのトランスファー縁体転移が起こる．一方，量子力学的に振る舞う電子の場合，たとえクラス⟨x,x′ ⟩ cx cx′ を⟨縁体転移が起こる．一方，量子力学的に振る舞う電子の場合，たとえクラスターが%%スターが形成され電気が流れ始め，金属ー絶†tx,x′ = exp(iθx,x′ ) で, θx,x′ はランダムな磁場から生ァー
tx,x′ c†x cx′ と変形する．
⟨x,x′ ⟩ cx cx′ を⟨x,x′ ⟩系全体に広がっていても，電子がクラスター上で局在している限り絶縁体系全体に広がっていても，電子がクラスター上で局在している限り絶縁体のままに振る舞う電子の場合，たとえクラスターが様乱数とする．
である．よって電流が流れ始める条件はps > pq (≥ pc ) である．pq は量子はランダムな磁場から生じているとして，[0, 2π]の一である．よって電流が流れ始める条件は ps > pq (≥ pc ) である．pq は量子パーコ18スター上で局在している限り絶縁体のまま
sレーション閾値と呼ばれる．

Anderson model, box

Anderson model+B

Anderson model, Gauss

Anderson model, Cauchy

Quantum percolation

Quantum percolation+B19Mano-Ohtsuki, JPSJ ’17, JPS Mtg ‘18

So far, real space wave functions
have been studies.
• What happens if we train the neural network
using wave functions in k-space,ψ(k)=Σ eikr ψ(r)• Application to the analysis of topologicalmaterials.
– Difficult to define the topological number in thepresent of randomness.
– k-space representation is more natural.
20

Applications to 3D topological materials(3Dtopological insulator and 3D Weyl semimetal)• topological materials and ordinary material/
vacuum have different topology à surfacestates• focus on the existence of surface states,
determine whether the system is topologicalor not.

Surface states of strong TI in the presence ofrandomness

Fermi arc states of Weyl semimetalin the presence of randomness

0it0rlement with Minimum
vx,y,z random model
at a site for(x, y,3Dz), topological insulator⎤r) + vx,y,z000⎥0(m + 3r) + vx,y,z00⎥⎥⎦
00−(m + 3r) + vx,y,z0000−(m + 3r) + vx,y,z

given by'E(k) = ± m(k)2 + t2 (sin2 kx + sin2 ky + sin2 kz )-4 < m0/m2 < -2 < m0/m2 <0t /m2= 2
Slab geometry< m0/m2surface
Dirac cone-π0

Weak TI

π-π0

Strong TI

π-π0π

Ordinary insulatorelectrons (especially in theagainst disorder has been
y of graphene [5,6]. As abackward scattering [7], theHowever, in the presence
ac cones) they do localize
ngs [8]. Does this mean thatn the presence of arbitrarily
simply collapses on theged potential disorder thatthe conductance in the slab geometry characterizes thenature of the corresponding phase [e.g., Fig. 2(a)], thephase boundaries are marked by a peak of the conductancein the bulk geometry [e.g., Fig. 2(b)]. Under the breakingof translational invariance by disorder, standard techniques[4] for calculating the topological invariants fail. Yet, theabove behaviors of the conductance clearly distinguishdifferent topological phases, providing us with sufficientcomparing the result with that bytransfer matrix method

rdered WTI [9,10] seemmuch different. Our global
1 finds its way also in thisestablished by a combinatwo-terminal conductancegth in the transfer matrix
ation, the 3D disordered Z2
d as an Wilson-Dirac-type
an effective (k-dependent)P
1 ! cosk! Þ [11],!¼x;y;z ð
The topological nature of
tio of two mass parameters
se with Z2 (one strong and1; 000Þ appears
1 "2 "3 Þ ¼ ð
the regime of parameters
n a WTI phase withtransfer matrix
method873

DM’phaseDiagramT2R1.txt’
2.5

FIG. 1 (color6 online). The ‘‘global phase diagram’’ of the2disordered Z2 topological insulator in the (m0 =m2 , W=m2 ) plane
determined by5 the behavior of two-terminal conductance. Solid
1.5lines on the phaseboundaries are guides
to the eyes. Dotted lines
4WTISTI
indicate the valueof parameters relevant in Figs. 2 and 3. 1The(111)3metallic (M) phase lies in the intermediate range OIof disorder
0.5
2strength, typically
10 & W=m2 & 25 in the parameter range of0
m0 =m2 shown in this-2figure.
-1.5-1-0.500.5

m0/m2machinelearning method

TO, JPSJ (‘17)

Can we distinguish surface states, STI and WTI?
K-space

Real space|ψ(r)|2|ψ(k)|2STI

STI|ψ(r)|2WTI|ψ(k)|2WTI

27

Application of convolutional
neural network to K-space w.f.
Real space

8K-space, preliminary
3DM

7’phaseDiagramT2R1.txt’

625
43

Metal

2.5

1.5

WTI
(111)

STI

1OI

0.5

2-2-1.5-1-0.5

00.5

0

WTI

STI

lor map of the intensity 0× POI +1× PW001 +2× PSW111 +3× PDM for the case of= 1 obtained from the CNN trained with diﬀerent parameters, t = 2, m2,z = 0.5.

OICannot distinguish STI and WTI.

change the parameters and obtain the diﬀerent set of eigenfunctions, but conCan distinguish STI and WTI.
he previously trained CNN. We consider the case t = 2, m2,z = 1, where (in thel randomness) the OI phase appears at m0 > 0, the STI phase at 0 > m0 > −2,phase with weak index (111) at −2 > m0 > −4 [11, 12].

28

Machine learning in condensed
matter physics• Now getting popular and seem to be well
accepted.
• Use of 3D image recognition à various metalinsulator transitions– Anderson transition in the presence/absence ofmagnetic fields• Fourier space image recognition: sometimes
better to analyze the wave function.
– Ex. Disordered topological insulators.
29