# Numerical Investigation of Localization in Two-Dimensional Quasiperiodic
Mosaic Lattice

Hui-Hui Wang  School of Physics and Materials Science, Guangzhou University,
510006 Guangzhou, China  Huangpu Research and Graduate School of Guangzhou
University, 510700 Guangzhou, China  Si-Si Wang  School of Physics and
Materials Science, Guangzhou University, 510006 Guangzhou, China  School of
Mathematics and Information Science, Guangzhou University, 510006 Guangzhou,
China  Yan Yu  SKLSM, Institute of Semiconductors, Chinese Academy of
Sciences, P.O. Box 912, Beijing 100083, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Biao Zhang
School of Physics and Materials Science, Guangzhou University, 510006
Guangzhou, China  Huangpu Research and Graduate School of Guangzhou
University, 510700 Guangzhou, China  Yi-Ming Dai  School of Physics and
Materials Science, Guangzhou University, 510006 Guangzhou, China  Hao-Can Chen
School of Physics and Materials Science, Guangzhou University, 510006
Guangzhou, China  Yi-Cai Zhang  School of Physics and Materials Science,
Guangzhou University, 510006 Guangzhou, China  Yan-Yang Zhang
<EMAIL_ADDRESS>School of Physics and Materials Science, Guangzhou
University, 510006 Guangzhou, China  Huangpu Research and Graduate School of
Guangzhou University, 510700 Guangzhou, China  School of Mathematics and
Information Science, Guangzhou University, 510006 Guangzhou, China

###### Abstract

A one-dimensional lattice model with mosaic quasiperiodic potential is found
to exhibit interesting localization properties, e.g., clear mobility edges [Y.
Wang et al., Phys. Rev. Lett. 125, 196604 (2020)]. We generalize this mosaic
quasiperiodic model to a two-dimensional version, and numerically investigate
its localization properties: the phase diagram from the fractal dimension of
the wavefunction, the statistical and scaling properties of the conductance.
Compared with disordered systems, our model shares many common features but
also exhibits some different characteristics in the same dimensionality and
the same universality class. For example, the sharp peak at $g\sim 0$ of the
critical distribution and the large $g$ limit of the universal scaling
function $\beta$ resemble those behaviors of three-dimensional disordered
systems.

##  I 1\. Introduction

Anderson localization of wavefunctions in disordered systems is a subtle
consequence of quantum interferenceAderson1958 ; LocalizationReview2 ;
LocalizationReview1 ; MarkosReview ; Evers . It is known that electrons in
lower dimensions are more prone to localization in the presence of even weak
disorderAbrahams79 ; LocalizationReview2 ; LocalizationReview1 ; Evers .
Theoretical progressesTheory02 ; Theory03 ; Theory04 ; Theory05 ; Theory06 ;
Theory07  and experimental realizationsExperiment01 ; Experiment02 ;
Experiment03 ; Experiment04 ; Experiment05  of the delocalization-localization
transition (or the metal-insulator transition, MIT) in disordered materials
are still exciting topics until today. On the other hand, quasiperiodicity is
a delicate structure between order and disorderQuasiCrystalRev1 ;
QuasiCrystalRev2 . The most typical case is the Aubry-André-Harper (AAH)
model, a one-dimensional (1D) lattice with nearest hopping and with a
quasiperiodic potential incommensurate with the underlying lattice
structureHarper ; AA . For this 1D model, there exist delocalized states at
finite potential magnitude, which is impossible for a completely disordered
counterpartAbrahams79 ; MacKinnon . When the quasiperiodic potential magnitude
is strong enough, all states will be localized.

Recently the study of quasiperiodic systems has attracted a lot of attentions
due to inspiring progresses of experimental realizationsQuasiCrystalRev1 ;
QuasiCrystalRev2 ; Experiment2D ; AAHExperiment1 ; AAHExperiment2 ;
AAHExperiment3 ; AAHExperiment4 , and its relation with the many body
localizationAAHMBL . New analytical methods based on Green’s functions are
developed, offering distinct insights into their physical groundsAnalytical01
; Analytical02 . In the presence of pairing interaction, surprisingly, it was
found that the quasiperiodic potential can enhance superconductivity
remarkablyEnhancedSC1 ; EnhancedSC2 , suggesting more unexpected phenomena of
quasiperiodicity on quantum states. Even in the single particle picture, the
quasiperiodic potential results in rich phenomena in different models. Several
nontrivial variations of the AAH model have been investigated recently, for
instance, generalizations to a dimerized chainRoy  or two coupled
chainsCoupledChains , with an unbounded quasiperiodic potentialYiCaiZhang2022
or with a relative phase in the quasiperiodic potentialAAH2023 , and higher
dimensionsAAModel3D . Novel transport phases and topological phases are found
when hoppings are long rangedPowerLawHopping ; LongRangeAAH ; YWang  or
quasiperiodic as wellOffDiagonalMosaic . Besides these, two-dimensional (2D)
quasiperiodic systems provide richer phenomena of
localizationMoireLocalization ; Localization2D2020 , topologyShuChen2022 ,
flat bandFlatBand2022 , and many-body effectsExperiment2D ; ManyBody2022 .

The original 1D AAH model possesses a self duality for the transformation
between the real and momentum spaces. This leads to the absence of mobility
edges, which means that all eigenstates are either localized or delocalized,
depending only on the strength of the potential. Among attempts to break the
self duality, a nontrivial modification is the 1D quasiperiodic mosaic
lattice, where the quasiperiodic potential only appears at sites with equally
spaced distanceWangY . This is an analytically solvable model that exhibits
mobility edges separating localized and delocalized states for a fixed
potential strength.

In this manuscript, we generalize this quasiperiodic mosaic lattice to a two-
dimensional (2D) version on the square lattice and study its localization
properties. Different from its 1D counterpart, the phase boundary between
localization and delocalization is highly fractal. By varying the initial
phase of the quasiperiodic potential, statistical properties of the
conductance are studied. In the end, we summarize all numerical results of
conductance in a universal scaling function $\beta(g)$, which shows some
different features from those in disordered 2D systems. Interestingly, for
this 2D quasiperiodic model, some of these properties are similar to those in
three-dimensional (3D) disordered systems.

##  II 2\. Model and Methods

Figure 1: Illustration of our 2D quasiperiodic mosaic lattice model with
$\kappa=2$. Only solid red sites have nonzero quasiperiodic potential. Each
site is represented by its coordinate $(i_{x},i_{y})$ and the nearest hopping
is $t$.

Our 2D model is defined on a square lattice with the nearest hopping
Hamiltonian

$\displaystyle\mathcal{H}$
$\displaystyle=\sum_{i_{x},i_{y}}t(c^{\dagger}_{i_{x},i_{y}}c_{i_{x}+1,i_{y}}+c^{\dagger}_{i_{x},i_{y}}c_{i_{x},i_{y}+1})+\mathrm{H.c.}$
(1)
$\displaystyle+2\sum_{\bm{i_{x},i_{y}}}V(i_{x},i_{y})c^{\dagger}_{i_{x},i_{y}}c_{i_{x},i_{y}},$

where $c^{\dagger}_{i_{x},i_{y}}$ ($c_{i_{x},i_{y}}$) creates (annihilate) an
electron at the site with integral coordinate $(i_{x},i_{y})$. In the
following, the hopping integral $t=1$ will be used as the energy unit, and the
lattice constant $a=1$ will be used as the length unit.

The quasiperiodic mosaic potential $V(i_{x},i_{y})$ is a generalization of the
1D version asWangY ; Devakul

$\displaystyle V(i_{x},i_{y})$ $\displaystyle=\begin{cases}\lambda
F(i_{x},i_{y}),\enspace i_{x}=m\kappa\text{ and }i_{y}=n\kappa,\\\ \qquad
0\qquad,\qquad\text{otherwise}\end{cases}$ (2a) $\displaystyle F(i_{x},i_{y})$
$\displaystyle=\cos\big{[}2\pi(\omega{i_{x}}+\theta_{x})]+\cos[2\pi(\omega{i_{y}}+\theta_{y})\big{]},$
(2b)

where $\lambda\geqslant 0$ is the amplitude of the potential. The irrational
number $\omega$ defines the quasiperiodicity, and the integer $\kappa$ defines
the mosaic period as illustrated in Figure 1. When $\kappa=1$ it returns to
the original 2D AAH model without mosaic, and only models with $\kappa>1$ can
be called mosaic. Recently an endoepitaxial growth of mosaic heterostructures
has been experimentally realized on monolayer 2D atomic crystalsMosaic2DExp .
In this manuscript, except in Figure 2 (a), we always adopt
$\omega=(\sqrt{5}-1)/2$ and $\kappa=2$,WangY  . The real numbers
$0\leqslant\theta_{x},\theta_{y}<1$ are phase offsets of the potential
profile. With other model parameters ($\lambda$, $\omega$ and $\kappa$) fixed,
different pairs of $(\theta_{x},\theta_{y})$ corresponds to different
realizations of an ensemble, similar to different realizations of a disordered
ensembleDevakul . This will be useful when one needs statistical properties of
this model, for example, mean values and statistical fluctuations.

Since analytical treatments for a 2D model are more difficult than those in
1D, we will rely on numerical methods, which will be briefly introduced in the
following. All of our following calculations are performed in FORTRAN codes,
along with Intel Math Kernel Library.

For a square shaped finite sample with $L\times L$ lattice sites, the inverse
participation ratio (IPR) of the $m$-th normalized eigenstate
$|\Psi_{m,j}\rangle$ is defined asEvers

$\text{IPR}(L,m)=\sum_{j=1}^{L\times L}|\Psi_{m,j}|^{4}.$ (3)

Then the localization of the state can be characterized by the fractal
dimension of the wavefunction as

$\mathrm{\Gamma}(L,m)=-\frac{\ln\big{[}\text{IPR}(L,m)\big{]}}{\ln L}.$ (4)

In the thermodynamic limit $L\to\infty$, the state is extended (localized) if
$\lim\limits_{L\to\infty}\Gamma(L)\to 2$
($\lim\limits_{L\to\infty}\Gamma(L)\to 0$).

The localization properties can also be studied from quantum transports, by
attaching two leads to the sample with quasiperiodic potential. At zero
temperature, the two-terminal conductance $G_{T}$ at Fermi energy $E$ is
proportional to the transmission (Landauer formula)Landauer , and can be
expressed asDatta ; Datta2

$G_{T}(E)=2\frac{e^{2}}{h}g_{T}=2\frac{e^{2}}{h}\mathrm{Tr}\left[\Gamma_{L}(E)G^{r}(E)\Gamma_{R}(E)G^{a}(E)\right],$
(5)

where the prefactor 2 accounts for the spin degeneracy. Here
$G^{r/a}(E)\equiv\left(E\pm-H-\Sigma^{r/a}_{L}-\Sigma^{r/a}_{R}\right)^{-1}$
is the dressed retarded/advanced Green’s function of the central sample, and
$\Gamma_{L(R)}=i(\Sigma_{L(R)}^{r}-\Sigma_{L(R)}^{a})$ with
$\Sigma_{L(R)}^{r/a}(E)$ being retarded/advanced self energies due to the left
(right) lead, respectively. In order to diminish the interface scattering, we
take both leads to be lattices identical to the sample with vanishing
potential $V$.

Based on the dimensionless conductance $g_{T}$ defined in Equation(5), the
appropriate variable for size scaling is the intrinsic conductance $g$
expressed as Braun1997 ; Slevin2001

$\frac{1}{g}=\frac{1}{g_{T}}-\frac{1}{N_{c}},$ (6)

with $N_{c}$ the number of active channels at Fermi energy when the potential
is absent. The second term $\frac{1}{N_{c}}$ is used to deduct the effect of
contact resistance so that the intrinsic transport property of the sample can
be manifested. This intrinsic conductance of square shaped ($L\times L$)
samples can be used to evaluate the scaling function $\beta=\frac{d\langle\ln
g\rangle}{d\ln L}$ of the metal-insulator transition, where
$\langle\cdots\rangle$ stands for averaging over an appropriate ensemble
Slevin2001 ; LocalizationGraphene ; XRWang2019 . An increase (decrease) of
$\ln g$ with increasing $\ln L$ indicates a metal (insulator) phase.

##  III 3\. Results

Figure 2: Fractal dimension $\Gamma$ of eigenstates as functions of
corresponding eigenvalues $E$ and quasiperiodic potential strength $\lambda$,
for a fixed sample size $L=200$. (a) Non-mosaic with $\kappa=1$. (b) Mosaic
with $\kappa=2$.

###  III.1 3.1. Phase diagram from eigenstates

First let us have a global view of the localization property of this model. In
Figure 2, we present the fractal dimension $\Gamma$ of eigenstates as
functions of potential magnitude $\lambda$ and corresponding eigenenergy $E$,
for an isolated $200\times 200$ sample with a definite potential configuration
$\theta_{x}=\theta_{y}=0$. For comparison, we present the cases of $\kappa=1$
(non-mosaic) and $\kappa=2$ (mosaic) lattice in panels (a) and (b)
respectively. With increasing $\lambda$, the band is broadened outwards and
localized states ($\Gamma\sim 0$, blue color) appear. However, there is a
remarkable difference between these two cases. For the non-mosaic case [Figure
2 (a)], all states on the energy spectrum transit into localization
simultaneously around $\lambda\sim 1$. In other words, there is no mobility
edge for this case. For the mosaic case, on the other hand, there are both
localized (blue color) and delocalized (orange color) states after
$\lambda\gtrsim 0.6$. This distinction between mosaic and non-mosaic models
are similar to that in 1D WangY . Also similar to the mosaic lattice in
1DWangY , extended states ($\Gamma\sim 2$, yellow color) mostly distribute
around the band center. Although their fraction among all states decreases
with increasing $\lambda$, their existence can survive at large $\lambda$. In
the following, we will focus on the mosaic lattice only.

Figure 3: (a) $\Gamma$ as a function of $E$ for $\lambda=2.5$, which is just
the cross section shown as the red dotted line in Figure 2 (b). (b) $\Gamma$
as a function of $E$ for a 1D mosaic lattice, with $\lambda=2.5$, $\kappa=2$
and $N_{x}=3000$. (c) The $\log-\log$ plots of box counting parameters
$\delta$ versus $N$ for panel (a) (red) and panel (b) (blue), whose slope
gives the Hausdorff dimension.

To see more details of this phase diagram in Figure 2 (b) for the 2D mosaic
lattice, we plot the profile of $\Gamma(E)$ along the red dashed line at
$\lambda=2.5$ in Figure 3 (a). For comparison, a typical counterpart of the 1D
model is plotted in Figure 3 (b). Compared to the 1D case, the first obvious
feature of the 2D model is strong fluctuations even around adjacent
eigenenergies. For example, in the delocalization region $E\lesssim 3$,
$\Gamma$ for most eigenstates fluctuate rapidly between $(1.35,1.8)$, and some
of them even drop around $\Gamma=1$. Furthermore, the transition between
localization and delocalization tends to be a fractal region. In 1D, on the
contrary, the profile consists of smooth curves with sudden jumps at
transitions. Similar to that shown Figure 3 (a), fractal transitional region
of transports is also predicted for the topological transition of a 2D
incommensurate bilayer, which is a quasiperiodic structure as wellYuYan2022 .

To characterize the difference between Figure 3 (a) and (b) quantitatively, we
calculate the Hausdorff dimension $d_{H}$ of the point set
$\big{\\{}(\Gamma_{i},E_{i})\big{\\}}$ ($i$ is the index of eigenstates) shown
in these 2D panels, by using the standard box-counting (BC) method Guarner ;
XYang . This algorithm counts the number $N$ of squares with size $\delta$
which are necessary to continuously cover the graph of points
$(\Gamma_{i},E_{i})$ rescaled to a unit square. In the intermediate region of
$\delta$ (“the scaling region”) where the scaling relation
$N\sim\delta^{-d_{H}}$ holds, the slope of the log-log plot of $\delta-N$ is
the estimate of the Hausdorff dimension $d_{H}$ Veen ; vanVeen . The results
corresponding to Figure 2 (a) and (b) are shown as red and blue plots in panel
(c) respectively. As expected, $d_{H}$ for the 1D model is close to $1$,
reflecting the fact that $\Gamma(E)$ consists of simple smooth curves. For the
2D model however, $d_{H}\sim 5/3$ suggests that $\Gamma(E)$ is indeed highly
fractal. We have checked that (but not shown here) for a very small potential
magnitude, say, $\lambda=0.3$, the relative fluctuation of $\Gamma(E)$ of the
2D model can be smaller, but its Hausdorff dimension ($d_{H}=1.68$) is still
distinctly larger than that of the 1D model ($d_{H}=0.87$).

###  III.2 3.2. Transport

The above results were from eigenstates of an isolated sample. Now let us
scrutinize transport properties obtained by attaching conducting leads to the
sample.

Figure 4: The typical value of the intrinsic conductance
$\exp\big{(}\langle\ln g\rangle\big{)}$ as a function of sample size $L$, for
different model parameters $\lambda$ and $E$. The sample is square shaped and
the average $\langle\cdots\rangle$ is over 150,000 realizations of
$(\theta_{x},\theta_{y})$.

The localization (delocalization) can be characterized by the increasing
(decreasing) of $g$ [Equation(6)] with growing sample size
$L$LocalizationReview1 ; Evers . Similar to the case of disordered
systemsSlevin2001 , the conductance should be ensemble averaged before size
scaling, to diminish the effect of sharp coherent fluctuations. To this end,
we choose $\theta_{x}$ and $\theta_{y}$ to be random variables uniformly
distributed within $(0,1)$, and define $\langle\cdots\rangle$ to be the
arithmetic average over different realizations of $(\theta_{x},\theta_{y})$.
In the following calculations, all averages are over 150,000 realizations.
Since $g$ is not a self-averaging quantity and $\ln g$ is “better distributed”
than $g$ itself (especially in the case of localization, as will be seen in
the following), it is numerically more preferable to extract information from
$\langle\ln g\rangle$ (or equivalently the typical value
$g^{\text{typ}}\equiv\exp\big{(}\langle\ln g\rangle\big{)}$) rather than the
mean value $\langle g\rangle$Slevin2001 ; Evers ; MarkosReview . In Figure 4,
we present the typical values of conductance $g^{\text{typ}}$ as functions of
the sample size $L$, for different model parameters. There are typical
delocalization and localization states with increasing or decreasing $g(L)$
dependence respectively, and also critical states ($\lambda=3.1$ and $E=2.5$)
between them.

Besides the averaged quantity, the distribution of the conductance also
provides insightful information of localizationEvers ; MarkosReview . In Figs.
5 and 6, we present distributions of $g$ and $\ln g$ respectively, for four
typical regimes. In Figure 5 (a) with extremely weak disorder, the conductance
has a very small relative deviation $\Delta g/g$. The distribution profile is
rather irregular with multiple peaks. This is not surprising because the
quasiperiodic potential is not “random-like” enough. Moreover, the energy
scale associated with the quasiperiodic potential has not dominated over the
level separation of the finite sample yet, so that the transmission can be
very sensitive to the competition between these two factors, resulting in
multiple peaks of the distribution. With a stronger potential but still in the
delocalized state as shown in Figure 5 (b), the distribution is smoothened to
be a perfect Gaussian profile. This is identical to what happens in a
delocalized state with disorderMarkosReview . From Figure 6 (a) and (b) we can
see that in the delocalized regime, the distribution profile of $\ln g$ is
almost identical to that of $g$.

At the critical state between localization and delocalization presented in
Figure 5 (c), the distribution profile is largely deformed. There is an
obvious nonanalytic point at $g=1$, which is also the common feature at the
MIT in disordered systemsMarkos2DAndo ; MarkosReview . Another noticeable
feature is the sharp peak near $g=0$. From the distribution of $\ln g$ shown
in 6 (c) it can be confirmed that this is a peak _close to_ $g=0$, instead of
one _at_ $g=0$ Markos2DAndo . Interestingly, such a peak close to $g=0$ is
also found at the MIT of a three-dimensional (3D) orthogonal
systemMarkosReview , but is absent at that of a 2D symplectic system (the only
example to exhibit bulk MIT in 2D)Markos2DAndo ; Marko  and at the plateau-
plateau transition of a 2D unitary system (quantum Hall effect)Distribution2D
; Evers . In one word, for this 2D quasiperiodic model, the statistical
distribution of the MIT exhibits similar behavior to that of a 3D disordered
system.

In the localized phase shown in Figure 5 (d), the conductance displays a
single-peak distribution highly concentrated around $g=0$, which is also
similar to that in the localized phase of disordered
systemsDistributionSoukoulis ; DistributionSuslov . In disordered systems with
localization, it was found that the quantity $\ln g$ is “better distributed”
as a partial Gaussian distribution terminated around $\ln
g=0$DistributionSoukoulis ; DistributionSuslov . The distribution of $\ln g$
for our model at strong localization is presented in Figure 6 (d). One can see
that, although it is not a typical Gaussian shape but a clear termination
around $\ln g$ still persists.

Figure 5:  Statistical distribution of $g$ with sample size $L=150$. (a) and
(b): the delocalized states with $\beta>0$. (c) The critical state with
$\beta=0$. (d) The localized state with $\beta<0$. The statistics is over
150,000 realizations of $(\theta_{x},\theta_{y})$. Figure 6:  Similar to
Figure 5 but for the quantity $\ln g$.

###  III.3 3.3. Scaling function

For a certain set of model parameters $\\{\lambda,\omega\\}$, and after a
polynomial fitting of data points $(\langle\ln g\rangle,\ln L)$ shown in
Figure 4, one can obtain a numerical evaluation of the scaling function
$\beta=\frac{d\langle\ln g\rangle}{d\ln L}$. Results of $\beta$ as a function
of $\langle\ln g\rangle$ for a wide range of model parameters are shown in
Figure 7. Within our best capability of calculation, i.e., 150,000
configurations at size $300\times 300$, there are still strong fluctuations,
especially around the transition region, $\beta\sim 0$, which is consistent
with what we have seen from Figure 3 (a) and Figure 5 (c). Nevertheless, some
useful information can still be drawn. Firstly, all data points, similar to
the case of disordered systemsSlevin2001 , tend to collapse around a universal
curve. We manage to fit these data with an appropriate nonlinear function.
Numerically, among common functions we find that the Boltzmann function
provides the best smooth curve that fits these data with the expression

$\displaystyle\beta(\ln g)$ $\displaystyle=$ $\displaystyle
A_{1}+\frac{\displaystyle A_{1}-A_{2}}{\displaystyle 1+\exp\big{[}(\ln
g-x_{0})/\Delta\big{]}},$ (7) $\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle-9.387,\qquad A_{2}=0.755,$ $\displaystyle x_{0}$
$\displaystyle=$ $\displaystyle-5.038,\qquad\Delta=1.550$

which is plotted as the blue dashed curve. This is a monotonic function with a
single zero point at $\ln g_{c}=-1.130$ where the MIT occurs. We stress again
that this Boltzmann function is just an empirical choice instead of a
principle-based analytical result, and we merely use it to extract some useful
quantities in a numerically convenient way. For example, the slope of
$\beta\big{(}\ln g\big{)}$ at its zero gives the inverse of $\nu$, the
critical exponent characterizing the the divergence of the localization
(correlation) length near the critical pointLocalizationReview1 ; Slevin2001 .
From Equation(III.3) we have an estimate $\nu\sim 2.22$. This value is larger
than those of MITs in 3D, and that of MIT in 2D symplectic systemsSlevin2001 ;
DNShengCriticalExponents , but is close to that of the plateau-plateau
transition of 2D quantum Hall effects (unitary system)QHEnu1 ; QHEnu2 ; QHEnu3
; QHEnu4 . Another important feature is the saturation of $\beta\sim 1$ at
large $\ln g$ limit. For disordered systems, it has been known that the
$\epsilon$ expansion in the large conductance limit gives $\beta\rightarrow
D-2$, for all three universality classesEpsilonExpansion1 ; EpsilonExpansion2
; LocalizationReview2 ; Slevin2017 . In this sense, the scaling behavior of
our 2D quasiperiodic model in the weak potential (therefore large conductance)
limit seems to resemble that of the 3D disordered model.

Figure 7: The scaling function $\beta(g)=d\langle\ln g\rangle/d\ln L$ as a
function of $\langle\ln g\rangle$ in the two-dimensional quasiperiodic mosaic
lattice. Dots are numerical differential (see text) results from data shown in
Figure 4. The dashed curve is a numerical fitting of these dots with the
expression Equation(III.3).

##  IV 4\. Summary and Discussion

In this paper, we numerically investigate the localization properties of the
2D quasiperiodic mosaic lattice model. We find some properties similar to, and
also some different from those of 2D disordered systems.

Similar to the 1D counterpart, there exists localization-delocalization
transition when varying the energy and the strength of the quasiperiodic
potential. However the transition region is fractal like, contrary to clear
phase boundaries in 1D. This model shares many common statistical features of
the conductance $g$, for example, Gaussian-like distribution in the
delocalized phase, high peak near zero in the localized phase, and the
existence of non-analytical behavior of the distribution function at the
critical point between two phases. However, this critical distribution at the
small $g$ limit shows a sharp peak around zero, which is a particular feature
of the 3D disordered systems. The scaling function $\beta$ is also a universal
function of $g$, but its large $g$ limit approaches 1, which is also the
feature of 3D disordered systems. This may suggest a novel role of the spatial
dimensionality of quasiperiodic systems, which will be studied in the future.
QuasiCrystalRev1 ; QuasiCrystalRev2 .

##  V Acknowledgements

This work was supported by National Natural Science Foundation of China under
Grant Nos. 12104108, 11774336 and 11874127, the Joint Fund with Guangzhou
Municipality under Nos. 202201020198 and 202201020137, and the Starting
Research Fund from Guangzhou University under Grant Nos. RQ2020082, RQ 2020083
and 62104360.

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# Pedagogical Demonstrations and Pragmatic Learning in Artificial Tutor-
Learner Interactions

Hugo Caselles-Dupré, Mohamed Chetouani, Olivier Sigaud  Sorbonne Université,
CNRS, Institut des Systèmes Intelligents et de Robotique (ISIR)

When demonstrating a task, human tutors pedagogically modify their behavior by
either “showing” the task rather than just “doing” it (exaggerating on
relevant parts of the demonstration) or by giving demonstrations that best
disambiguate the communicated goal [6, 9]. Analogously, human learners
pragmatically infer the communicative intent of the tutor: they interpret what
the tutor is trying to teach them and deduce relevant information for
learning[4, 5, 7]. Without such mechanisms, traditional Learning from
Demonstration (LfD) algorithms will consider such demonstrations as sub-
optimal [8]. In this paper, we investigate the implementation of such
mechanisms in a tutor-learner setup where both participants are artificial
agents in an environment with multiple goals. Using pedagogy from the tutor
and pragmatism from the learner, we show substantial improvements over
standard learning from demonstrations.

More specifically, we make the following contributions: (i) we adapt
pedagogical tutors and pragmatic learners ideas from developmental psychology
to AI in a collaborative training setup ; (ii) we introduce pedagogy teaching
and pragmatic learning mechanisms that are general enough to be applied to any
artificial agent policy learning scenario (multi-arm bandit, evolutionary
strategies, reinforcement learning) ; (iii) we provide an analysis of the
benefit of adopting such approaches in the context of teachable and autotelic
agents.

##  I Experimental tutor-learner setup

Our collaborative tutor-learner setup is depicted in Figure 1. It is inspired
from the squeaking balls environment presented in [4]. In our variant, there
is a bucket of purple, orange and pink balls to choose from. Purple balls are
more numerous than any other balls, an information that the agents have. The
agents must pick two balls consecutively, and upon their choice, they can
obtain three possible outcomes. If the picked balls are (orange, orange) or
(pink, orange), goal 1 is reached. If the picked balls are (orange, pink),
goal 1 and goal 2 are achieved. Otherwise, no goal is reached and nothing
happens. The achievement of the goals are communicated to the agent.

Figure 1: Our ”Draw two balls” environment. An agent draws two balls from the
box consecutively, and obtains a squeaking sound if goal 1 is achieved, a
beeping sound if goal 2 is achieved, and nothing otherwise. Notice that
picking the orange ball and then the pink ball reaches both goal 1 and goal 2,
a source of goal ambiguity for demonstrators.

###  I-A Policy learning

The tutor and the learner are both represented by their respective goal-
conditioned policies $\pi_{T}(.|g_{t})$ and $\pi_{L}(.|g_{l})$. These policies
might be learned using multi-armed bandits, reinforcement learning or
evolutionary strategies. The tutor’s policy is already learned when it
provides demonstrations to the learner.

In our experiments, a policy is a probability distribution for the first ball
pick (purple, pink or orange), plus three policies conditioned on the first
pick provided the probability distribution of picking the second ball given
the first. There are three possible outcomes: no goals achieved, goal 1 and
goal 2.

###  I-B Learning from demonstrations: training loop

At each iteration, the tutor chooses a desired goal $g_{d}$ it wants the
learner to achieve it on its own. It then presents a demonstration $D(g_{d})$
for the chosen goal to the learner, which makes a prediction $\widehat{g_{d}}$
of the associated goal based on the demonstration. Then the learner plays and
explores the environment using its policy conditioned on the predicted goal
$\pi_{L}(.|\widehat{g_{d}})$ and obtains a trajectory
$traj_{L}(\widehat{g_{d}})$ resulting in an achieved goal $g_{a}$. Finally,
the learner is rewarded if $g_{d}=\widehat{g_{d}}$ and if
$\widehat{g_{d}}=g_{a}$. It updates its policy using these two feedback
signals.

Such autotelic (capable of generating and pursuing its own goals) and
teachable (benefits from teaching via natural social interactions) agents
present abilities that previous research argue are key steps to endow agents
with the necessary capabilities to reach general intelligence [10, 1, 3, 2].

##  II Adding pedagogy and pragmatism

We now present how the tutor can add pedagogy to its teaching signals to
better convey important information about the goals, and how the learner can
pragmatically interpret this implicit information in the teaching signals to
learn better.

###  II-A Teaching with pedagogy

Psychological research [6] has shown that human tutors often emphasize and
exaggerate certain crucial aspects of the task to better demonstrate it, not
with words but with actions. The key insight is that helpful tutors do not
merely select optimal demonstrations of a task, but rather choose
demonstrations that best disambiguate a goal from other goals.

For the tutor, goal disambiguation results from using a goal prediction
reward. When the pedagogical tutor learns to solve the task, it modifies its
policies to better achieve the goal, like a naive tutor, but also when it
produces a trajectory with which it can predict the pursued goal. This way,
the learner will better predict the right goal using the pedagogical
demonstration.

###  II-B Learning with pragmatism

If the tutor demonstrations are biased towards actions that are unlikely for a
naive prior, the learner can detect this pedagogical intention and deduce
relevant information from it [4]. In our case, the purple balls being more
frequent, if the tutor does not pick them often, the learner can detect this
strong sampling bias and deduce that picking the purple balls is not relevant
for goal achievement.

We implement this using a prior mean probability of demonstration $D_{mean}$
using the probability of picking each colored ball. If the presented
demonstration probabilities are inferior to $D_{mean}$ five times in a row,
then strong sampling bias is detected and the pragmatic learner increases its
probability of picking the purple ball for achieving no goal.

##  III Results

(a) Naive tutor policy for goal 1.

(b) Pedagogical tutor policy for goal 1.

Figure 2: Pedagogical and naive tutor policies for goal 1. Left: action
probability for the first pick. Right: action probability knowing the first
pick is respectively purple, orange and pink.

We first analyze the tutors’ policies. In Fig. 2, we present the learned tutor
policies for goal 1, which best illustrates the difference between a naive and
a pedagogical tutor. Contrary to the naive tutor, the pedagogical tutor
specifically avoids the demonstrations (orange, orange) because it does not
allow to directly detect goal 1 from the first pick. Moreover, it avoids the
ambiguous demonstration (orange, pink), which not only reaches goal 1 but also
goal 2. This pedagogical tutor policy maximally avoids ambiguity in
demonstrations.

(a) Predictability: can learners predict goals given demonstrations?

(b) Reachability: can learners reach pursued goals?

Figure 3: Predictability (goal prediction accuracy) and reachability (goal
reaching accuracy) results for all four possible combinations of experimental
setups: literal/pragmatic learner and naive/pedagogical tutor.

We then experiment with pedagogical/naive tutors and pragmatic/literal
learners with the training loop presented in Sec.I-A, and present our results
on Fig.3. When a learner is trained with a pedagogical tutor, it consistently
learns faster, and can predict the goals from all demonstrations, whereas it
is not capable of disambiguating goal 1 from goal 2 with the (orange, pink)
demonstration from a naive tutor. Moreover, a learner benefits from pragmatism
if the tutor is pedagogical, resulting in the best tutor-learner combination.

##  IV Conclusion and Future work

In this short paper, we present preliminary results on implementing tutor
pedagogy and learner pragmatism for a more efficient collaborative learning
from demonstrations. Using a toy environment, we provide proof-of-concept for
those pedagogical and pragmatic mechanisms. Future work will focus on
implementing those mechanisms in more complex environments, using
reinforcement learning as the core learning algorithm.

## Acknowledgement

This work was performed using HPC resources from GENCI-IDRIS (Grant
2022-A0131013011).

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