Activity Report 2020 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

High precision study of the Anderson transition
TOMI OHTSUKI1
KEITH SLEVIN2
1) Dept. Phys., Sophia University, Chiyoda-ku, Tokyo 102-8554, Japan
2) Dept. Phys., Osaka University, Toyonaka, Osaka 560-0043, Japan(Dated: May 3, 2021)

INTRODUCTION

More than six decades have passed sinceAnderson’s seminal paper on localisation was
published[1]. During the subsequent decades, there
have been numerous important discoveries including weak localisation, universal conductance fluctuations, and the scaling theory of localisation[2].
Localisation phenomena are observed not only inelectron systems, but also in optical [3–9], acoustic [10–12], and cold atom systems [13–18].
Since the proposal of the scaling theory of
localisation[19–23], determining the critical behavior of the localisation-delocalisation transition,
which is usually referred to as the Anderson transition (AT), has continued to attract considerable
attention. Depending on the symmetry of the
Hamiltonian, systems are classified into orthogonal, unitary and symplectic symmetry classes, cf.
the classification of random matrices[24, 25]. This
classification was extended to include three classes
with chiral symmetry[26, 27], and four classes withparticle-hole symmetry[28]. These ten classes havealso proven useful in the discussion of whether or
not there is a topological phase and, if there is,what type of topological phase is realized, given
the symmetry class and the dimensionality of thesystem.[29, 30]
In this activity report, we review the numericalapproach for the study of the Anderson transition,
and also another type of transition, the metal tosemimetal transition. We emphasize the importance of the concept of universality class, and thescaling analysis of high precision numerical data.
The rest of this report is organized as follows. In
the next section, we explain the method, followedby the results for our recent high precision studies of three dimensional (3D) Anderson transitions
with and without time reversal symmetry. We thenreview the recent progress on the novel symmetry
classes with chiral and particle-hole symmetries aswell as the extension to non-Hermitian systems.
We conclude this report by discussing the metal

to semimetal transition and the scaling behavior ofthe density of states.

METHODS

We start with the Anderson’s model of
localisation[1],X
X
H=
Ei |ii hi| −|ii Vi,j hj| .
(1)
i

hiji

where |ii is an orbital localised on site i of a 3D
cubic lattice. The first sum is over all sites on thelattice and the second sum is over pairs of nearest neighbours. The unit of energy is the nearest neighbour transfer energy V = |Vi,j |, which we
set to unity V = 1. The orbital energies Ei are
assumed to be identically and independently distributed with a uniform distribution
1/W , |Ei | ≤ W /2 ,p (Ei ) =(2)0,otherwise .
The parameter W determines the strength of disorder. The Hamiltonian commutes with the complex
conjugation operator, i.e. a time reversal operator that squares to +1, and this model belongs tothe orthogonal symmetry class [21, 22, 24, 25] (seeTable I).
We can extend the Anderson model by including Peierls phases Vi,j = exp(iθi,j ), which describe
magnetic fields, in the nearest neighbour hoppings.
Here θi,j with i > j are randomly and uniformly
distributed between [0, 2π] and and θi,j = −θj,i .
We call this the U(1) model. In this case the Hamiltonian does not commute with a time reversal operator, and the model belongs to the unitary symmetry class.

Symmetry classification

We can further extend the model to include
spin/orbital degree of freedom by modifying Vi,j
and Ei and can realize different symmetry classes.

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Activity Report 2020 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo
Class

Symbol TRS PHS CS

Unitary
Orthogonal
Symplectic

A
AI
AII

No
1
-1

No No
No No
No No

Chiral Unitary
Chiral Orthogonal
Chiral Symplectic

AIII
BDI
CII

No
1
-1

No Yes
1 Yes-1 Yes

D
C
DIII
CI

No
No
-1
1

1
-1
1
-1

BdG

where ψx is the wavefunction on the slice at position x,(ψx )y,z = hx, y, z |Ψi .

(8)

Mx is the transfer matrix defined by,


No
No
Yes
Yes

Mx =

TABLE I. Classification according to TRS, PHS and
CS. For TRS (PHS), 1 means C T = C (C 0T = C 0 )whereas -1 means C T = −C (C 0T = −C 0 ). BdG means
Bogoliubov-de Gennes class.

−1−1 
Vx,x+1(E − Hx ) −Vx,x+1.
Vx+1,x
0N(9)

For the Anderson model where we set Vi,j = −1,
we have

 


ψx+1
Hx − E 1N
ψx=. (10)−ψx−1N 0N−ψx−1
Hx is the following sub-matrix of the Hamiltonian

Using the unitary operators C, C 0 and P , we canclassify the Hamiltonian according to whether itsatisfies the following symmetries[29],H = CH ∗ C −1 ,(3)for time reversal symmetry (TRS), andH = −C 0 H ∗ C 0−1 .

(4)

for particle-hole symmetry (PHS). Systems withTRS and PHS are further classified according towhether C or C 0 is symmetric or antisymmetric.
We also have chiral symmetry(Hx )y,z,y0 ,z0 = hx, y, z| H| x, y 0 , z 0 i .

(11)

0N and 1N are the N × N (N = L2 ) zero and unit
matrices, respectively. The boundary conditions in
the transverse directions influence some of the critical behavior[36]. In this report, we impose periodic
boundary conditions in the transverse directions.
We set the energy at the band centre, i.e. E = 0.
We note that the transfer matrix must satisfy thefollowing relation
MxT ΣMx = Σ ,

(12)

where
H = −P HP −1 , P 2 = 1 .

(5)Note that chiral symmetry is automatic when we
have both TRS and PHS.
The classification[29] is summarized in Table I.

Σ=

0N −i1N
i1N 0N
.

(13)

The wave-function amplitudes on the first twolayers are related to the wave-function amplitudes
on the last two layers as follows

The transfer matrix method
One of the ways to study the Anderson transition
with high precision is to calculate the quasi-one
dimensional localisation length and perform a finite
size scaling analysis [31–35].
We consider a system with a square cross section
L × L, which we divide into layers labelled by theirx coordinate. Then the Schrödinger equation for astate vector |Ψi and energy E
H |Ψi = E |Ψi ,is expressed in the following form




ψx+1
ψx
= Mx
,
Vx+1,x ψxVx,x−1 ψx−1

(6)

(7)

ψLx +1−ψLx

= MLx · · · M1

ψ1
−ψ0


,

(14)

which involves the product of Lx independently
and identically distributed random matrices M =MLx · · · M1 .
The following limiting matrix exists[37],ln M T M
.
Lx →∞
2Lx

Ω = lim(15)

The limit depends on the particular sequence ofrandom matrices, but the eigenvalues {γi } of Ω are
the same for all sequences (“all” means with probability one). These values are called Lyapunov exponents. From Eq. (12) these eigenvalues occur in28

Activity Report 2020 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

pairs of opposite sign. It is usual to number themas follows
γ1 > · · · > γN > γN +1 = −γN > · · · > γ2N = −γ1 .
(16)This ± symmetry can be derived from Eq. (12).
To estimate the Lyapunov exponents we startwith a 2N × 2N orthogonal matrix, truncate thematrix product at a very large but finite Lx , andperform a QR decomposition of the result
QR = M Q0 .

(17)

Here, Q0 and Q are a 2N × 2N orthogonal matrix
and R is a 2N × 2N upper triangular matrix withpositive diagonal elements. We then defineγ̃i =1ln Ri,i .
Lx(18)

In the limit of infinite length
γi = lim γ̃i .
Lx →∞

(19)

For sufficiently large Lx , the {γ̃i } may be used toestimate the Lyapunov exponents.
This method requires the simulation of a single
very long sample. While this method has been employed very successfully in numerous simulations
over the preceding decades, the calculations are inherently serial and do not allow us to take advantage of massively parallel computers.

where h· · · i is the sample average. To generate such
matrices, we have found that the following procedure works well. We start with Q0 given by the
2N -dimensional unit matrix,
Q0 = 12N(22)

Q0 R = Mq · · · M1 Q0 .

(23)

and calculate

The matrix R is then discarded and we set Q0 = Q0 .
This procedure is then repeated a sufficient numberof times.
For a given L, we have found that, when a sufficient number of randomizing multiplications are
performed, the distribution of γ̃N becomes independent of the number of such multiplications[39].
We assess this by applying the KolmogorovSmirnov test to the data for γ̃N with different
numbers of randomizing multiplications. For sufficiently large number of randomizing multiplications we find that the Kolmogorov-Smirnov test
is unable to distinguish the distribution of γ̃Nobtained[39].

Fitting

Once we calculate γN , which is a function of thecross section size L and the strength of disorder W ,we defineΓ(W, L) = Lγ ,(24)

Parallel transfer matrix method

and assume the scaling form,
An alternative way is to simulate an ensembleof much shorter samples and consider an ensemble
average. While for simplicity we consider cubeswith Lx = L, the method is also applicable whenLx 6= L.
We note here that the matrix Q0 should be a
2N × 2N random matrix with orthogonal columns
sampled from a probability distribution that is invariant under convolution with the transfer matrix
distribution, i.e., with a distribution that is invariant under the operation[38]0Q R = Mx Q .

(20)For such a distribution, it immediately follows that[39]γN = hγ̃N i ,(21)

Γ(W, L) = F (φ1 , φ2 , φ3 , · · · ).

(25)

Each scaling variable φi on the right hand side has
a power law dependence on L
φ1 ≡ u1 (w)L1/ν ,φ2 ≡ u2 (w)L−y ,0φ3 ≡ u3 (w)L−y ,··· .
Here 1/ν (> 0) is the scaling dimension of the relevant scaling variable and −y is the scaling dimension of the least irrelevant scaling variable;· · · < −y 0 < −y (< 0).
In this report, we use the disorder strength W totune the system through the transition. We denotethe critical disorder where the transition occurs as

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Activity Report 2020 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Wc . We define w ≡ (W − Wc )/Wc . The functions uj (w) (j = 1, 2, · · · ) obey u1 (w = 0) = 0 anduj (w = 0) 6= 0 (j = 2, 3, · · · ). When w is sufficiently small, the u’s can be expanded in powers ofw
ui (w) ≡miXbi,j wj

are 3 WD classes (see Table I). Using System B
and simulating the Anderson and U(1) models, wehave determined the critical exponents of the 3D
orthogonal and unitary universality classes. The
results are summarized in Fig. 1 and Table II.

(26)
3.0

j=0

with i = 1, 2, · · · , b1,0 = 0, and bj,0 6= 0 (j =2, 3, · · · )[35, 40]. When the φ’s are sufficiently
small, the universal scaling function F can also be
expanded[41]. We keep only the relevant scaling
variable φ1 and the least irrelevant scaling variable
φ2 , while assuming the other irrelevant scaling variables to be zero, φ3 = · · · = 0. This should be reasonable for w sufficiently small and L sufficiently
large. We expand F in terms of φ1 and φ2 as
F =

n1 X
n2
X

aj1 ,j2 φj11 φj22 .

2.5

G

2.0

1.5

1.0

15.5

16.5

16.0

17.0

W

17.5

(a) 3D class AI(27)

j1 =0 j2 =0

To remove the ambiguity of fitting parameters, we
set a1,0 = a0,1 = 1. The parameter a0,0 is sometimes related to quantities of interest such as multifractal exponents[42], and we write it as Γc .
Whether the fit is plausible is determined by calculating the goodness of fit probability. When the
fit is plausible, we check the stability of the fit
against changes of the range of data being fitted
and of the orders of the polynomial expansions.
Fig. 1 shows an example of simulation data Γ andthe results of the fitting.
The confidence intervals of the fitting parameters
are determined by Monte Carlo simulations. From
the fitting function Fi = F (Wi , Li ) for the i-th data
(1 ≤ i ≤ ND , ND the number of data points), we
produce an ensemble of synthetic data sets with
Γ̃i = Fi + σi where σi is a random number, thevariance is the same as that of i-th data and themean zero. By fitting the synthetic data sets {Γ̃i }’s
we obtain the distribution of the critical parameterssuch as Wc and ν, and estimate their confidence
intervals.

NUMERICAL STUDIES OF THE
ANDERSON TRANSITION
Wigner-Dyson classes

For the Wigner-Dyson (WD) classes, neither theparticle-hole (PHS) or chiral (CS) is present. There

2.5

2.0

1.5

1.0
17.5

18.0

18.5

W

19.0

19.5

20.0

(b) 3D class A

FIG. 1. Γ as a function of the disorder strength W forvarious cross section size L. (a) 3D class AI, where L =12.18, 24, 32, 48 and 64. L = 96 is underway using new
System B. (b) 3D class A, where L = 4, 6, 8, 12, 16, 20
and 24. In the delocalised (localised) phase, Γ decreases(increases) with L. (a) is taken from [39] and (b) from[43].

The calculation time is proportional to the time
for the QR decomposition and the number of transfer matrix multiplications. The dimension of transfer matrix is proportional to L(d−1) , hence QR decomposition takes time proportional to L3(d−1) , dbeing the space dimension. In addition, to obtainthe same precision for larger L, we need to increase
the length Lx proportional to L. The calculation
time, therefore, is proportional to L3d−2 . This
rapid increase of calculation time for higher dimen-

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Activity Report 2020 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo
class

ν

Γc

Wc

ND NP p-y

3D AI 1.572[1.566,1.577] 1.7372[1.7359,1.7384] 16.543[16.541,16.545] 117 7 0.5 –3D A 1.443[1.437,1.449] 1.805[1.803,1.808]
1.805[1.803,1.808]
171 10 0.4 -3.1[-3.9,-2.4]
TABLE II. Results of the finite size scaling fits. Numbers are taken from Refs. [39, 43]. The systems sizes are
L = 24, 32, 48 and 64 for 3D orthogonal class (Anderson model, class AI), whereas they are L = 4, 6, 8, 12, 16, 20
and 24 for 3D unitary class (U(1) model, class A). The precision is expressed by 95% confidence intervals, ND
is the number of data points, NP the number of fitting parameters, and p the goodness of fit probability. For
the 3D orthogonal case, irrelevant scaling variables are not necessary. This is because we use only L ≥ 24 where
corrections to scaling are smaller than the precision of the data [35].

sion, at first sight, gives the impression that higher
dimensional simulation is almost impossible. However, the corrections to scaling are smaller in higher
dimensions, and the critical behavior in dimensions
higher than three has also been studied[43–45]. On
the other hand, the critical behavior of the quantum Hall transition, which occurs in two dimensions, is still controversial due to the very slowly
converging corrections to scaling[46].

Distribution of Kondo temperature

At the critical point, eigenstates exhibit multifractality. This is reflected in the fluctuations of
the local density of states. As a result, when we
consider magnetic impurities, the Kondo temperature TK has a broad distribution. It has been predicted that the distribution of TK has a power law
tail at small TK with a universal exponent whose
value is related to the multifractal exponent η [47].
Using the kernel polynomial method [48], we calculated the local density of states at the Andersontransition, and determined the distribution of TK .
The massively parallel calculations on System B enabled us to reach the small TK needed to check the
analytic prediction [49]. Experimental verification
may be possible by comparing with measurementsof the temperature dependence of the magnetic susceptibility.

Beyond the Wigner-Dyson classes

Systems with CS/PHS symmetries have attracted much attention recently, because manyof them are topological insulators or topologicalsuperconductors[29]. We note that these unconventional universality classes, i.e. classes other thanWigner-Dyson, are realized only at E = 0.
One way to realize a model with chiral symmetry

is to set all the orbital energies Ei = 0, and consider random hopping, Vi,j . Setting E = 0, we maythen vary the strength of the randomness of the
hopping and study the Anderson transition. Thisapproach, however, has proven to be difficult tohandle numerically because an unphysically large
disorder in the hopping is needed to cause an Anderson transition[50].
Fixing the hopping and changing the diagonal
disorder is easier for numerical calculations. Thiscan be realized by considering, for example, the following Hamiltonian, which corresponds to 3D classCI (see Table I). It is a two-orbital cubic latticemodel,XX
H≡|i, di[H](i,d|j,d0 ) hj, d0 |i,j d,d0

=

Xn



Ei + ∆ |i, aihi, a| − |i, bihi, b|i+ tk |i, aihi, b| + |i, bihi, a|
X X

+ t⊥
|i + eµ , di + |i − eµ , di hi, d|µ=x,y d=a,b+ t0k |i + ez , aihi, a| − |i + ez , bihi, b| + h.c.

o
.

(28)
Here d, d0 = a, b denotes the orbital index, i ≡(ix , iy , iz ) with ex = (1, 0, 0), ey = (0, 1, 0) andez = (0, 0, 1) is the site index on the 3D cubic
lattice. The orbital energies at two different lattice sites have no correlation; Ei Ej = δi,j W 2 /12.
The model has a particle-hole symmetry (PHP−1 =−H) as well as the time-reversal symmetry (H∗ =H) with [P](i,d|j,d0 ) ≡ (−1)ix +iy δi,j [σy ]d,d0 withthe 2 by 2 Pauli matrices, σx , σy and σz . Since
PT = −P, the Hamiltonian has a set of doubly
degenerate real-valued eigenstates at zero energy,
which results in the degeneracy of the Lyapunov
exponents at E = 0; the degeneracy is protectedby the particle-hole symmetry. For simplicity, we
set ∆ = tk = t0k = t⊥ = 1. As in the previ-

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Activity Report 2020 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyoous subsection, the localisation length of the zeroenergy eigenstates along the z-direction (λz ) is calculated via the transfer matrix method. The periodic boundary condition is imposed along x and y
directions. The density of states (DOS) of H withfinite disorder strength W is calculated in terms ofkernel polynomial expansion (KPE) method [48].
Due to the particle-hole symmetry, the calculated
DOS is symmetric about E = 0, while the DOS
at E = 0 remains finite at the critical point[51].
The exponent of 3D class CI thus estimated isν = 1.16 ± 0.02.
We can construct a similar Hamiltonian for the
3D class BDI, which is a model for disordered
nodal-line Weyl semimetal [51].

(a) 3D Anderson model

SUMMARY AND CONCLUDINGREMARKSphase diagram

In this review, we focused on the Anderson transition at E = 0 (center of the band). Once weknow the critical disorder Wc ≈ 16.54 at E = 0(see Table II), we can prepare thousands of delocalised (W < Wc ) and localised (W > Wc ) wave
functions. Then we can let a convolutional neural network (CNN) learn the features of these wave
functions[52, 53], and draw the phase diagram in
W − E parameter plane[54, 55].
Fig. 2(a) is an example. We take the training regions indicated by the arrows in Fig. 2(a), and let
the neural network calculate the probability thatthe states are delocalised in the rest of the parameter region. We can also draw the phase diagram forquantum percolation, where sites are present withprobability p and absent with probability 1 − p,and all the orbital energies are set to zero, Ei = 0.
From the training in the Anderson model where
the transfer matrix method is applicable, we candraw the phase diagram for the quantum percolation problem where the transfer matrix method
is not applicable. Fig. 2(b) is the phase diagram
for the quantum percolation problem drawn by theneural network trained using the Anderson model.

Beyond Hermitian classes

The symmetry classification according to TRS
and PHS can be extended to non-Hermitian (NH)(b) 3D quantum percolation

FIG. 2. Phase diagram of 3D Anderson transitions.
(a) is for Anderson model. The green arrows in (a) indicate the regions where the CNN is trained, whereas
the white dashed line and crosses indicate the phaseboundary estimated by other methods [56, 57]. (b) isfor site-type quantum percolation. The white dashedline is from the estimates by Ref.[58], whereas the green
horizontal dashed line indicates the classical percolation threshold. Taken from Refs.[54, 55].

systems [59]. For example, for TRS for Hermitian systems we have H = CH ∗ C −1 = CH T C −1
[Eq. (3)], but for non-Hermitian systems there are
two possibilities,
H = CH ∗ C −1 , H = CH T C −1 .

(29)

The latter symmetry is straightforwardly realized
for the Anderson model [Eq. (1)] by making theorbital energies Ej random complex numbers. We
then have H = H T but H 6= H ∗ . The system iscalled NH class AI† . Making the Ej complex in
the U(1) model realizes NH class A. The criticalbehaviors have been shown to be different from theHermitian classes AI and A[60, 61] via the finite

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Activity Report 2020 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

size scaling analyses.
These two classes are only a part of the 38 symmetry classes[59] in non-Hermitian disordered systems. The critical behavior for these classes is aninteresting topic left for the future.

Density of states scaling

So far, we have discussed the Anderson transition. There are other transitions in the same symmetry classes. For example, the semimetal to metaltransition[62] occurs in 3D Dirac and Weyl systems,
where the systems remains semimetal up to certain strength of disorder, then undergoes semimetal
to metal transition. (Further increase of disorderleads to an Anderson transition.) Though the symmetry classes are AII (Dirac) and A (Weyl), thecritical behaviors described by the scaling of thedensity of states[63, 64] is different from the Anderson transition, i.e., both the exponent ν and the
dynamical exponent z differ from those of the Anderson transition in the same symmetry class anddimensionality.

Experiments

Interpreting the critical behavior found experimentally in doped semiconductors remains difficult
because the role of the electron-electron interaction, which may be relevant in the renormalisation group sense, is not well understood. In experiments, the critical exponent s, which describes
how the zero temperature conductivity vanishes as
the critical point is approached from the metallic side, is measured. This exponent is related
to the critical exponent ν by Wegner’s relations = (d − 2)ν[19]. As yet there is no agreement
between theory and experiment and understanding
the critical behaviour at the metal-insulator transition in doped semiconductors remains an open
problem[65, 66].
The quantum kicked rotor with suitable quasiperiodic modulation exhibits an Anderson transition in the the same universality class as that inAnderson’s model of localisation in 3D. This experimental realisation of the quantum kicked rotor
has provided an alternative avenue for experimental investigation of the Anderson transition.[13–15].
The value ν = 1.63 ± 005 [15] found in these experiments agrees well with our numerical estimate.

Light waves [3, 4, 7–9] and acoustic waves [10–12] also localize and interaction effects play less ofa role. For small loss, such systems are describedby the 3D class AI. When the loss is not negligible,
the system might show the critical behavior in 3Dclass AI† . Quantitative studies and detailed comparisons between theory and experiment are interesting topics left for the future.
Acknowledgement The work was partly supported by JSPS KAKENHI Grants 19H00658.

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