Low and high-energy localization landscapes for tight-binding Hamiltonians in 2D lattices Luis A. Razo-López1Geoﬀroy J. Aubry1Marcel Filoche2 3and Fabrice Mortessagne1

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Low and high-energy localization landscapes for tight-binding Hamiltonians in

2D lattices

Luis A. Razo-López,1Geoﬀroy J. Aubry,1Marcel Filoche,2, 3 and Fabrice Mortessagne1, ∗

1Université Côte d’Azur, CNRS, Institut de Physique de Nice (INPHYNI), France

2Laboratoire de Physique de la Matière Condenséee, CNRS,

École Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France

3Institut Langevin, ESPCI, PSL University, CNRS, Paris, France

(Dated: October 4, 2022)

Localization of electronic wave functions in modern two-dimensional (2D) materials such as

graphene can impact drastically their transport and magnetic properties. The recent localization

landscape (LL) theory has brought many tools and theoretical results to understand such local-

ization phenomena in the continuous setting, but with very few extensions so far to the discrete

realm or to tight-binding Hamiltonians. In this paper, we show how this approach can be extended

to almost all known 2D lattices, and propose a systematic way of designing LL even for higher

dimension. We demonstrate in detail how this LL theory works and predicts accurately not only

the location, but also the energies of localized eigenfunctions in the low and high energy regimes for

the honeycomb and hexagonal lattices, making it a highly promising tool for investigating the role

of disorder in these materials.

The promises of the expected electronic properties of

new 2D materials often face the reality of genuine ma-

terials where disorder can be diﬃcult to avoid [1]. Its

inﬂuence might be large enough to switch the behav-

ior of a material from metal to insulator [2], a transi-

tion which can be related to Anderson localization. The

concept of Anderson localization, initially introduced in

tight-binding models in the context of condensed mat-

ter physics [3], has been applied since in the continuous

setting to all types of waves, being quantum [4], classi-

cal [5–9] or even gravitational [10]. In this setting, the

recent theory of the localization landscape (LL) [11] has

brought new insights and methods to address the wave lo-

calization properties in systems such as gases of ultracold

atoms [12], disordered semiconductor alloys [13], or en-

zymes [14], and have been successfully extended to Dirac

fermions [15] and non scalar ﬁeld theory [16]. In this

letter, we show that the whole machinery of the LL can

be generalized to tight-binding systems for most known

1D and 2D lattices, allowing us to broaden the range of

predictivity of this fruitful approach.

Tight-binding models are commonly used to study per-

fect [17] as well as disordered lattices [18–20]. The tight-

binding Hamiltonian ˆ

Hwith on-site disorder and nearest-

neighbor coupling is deﬁned as

(ˆ

Hψ)n=−tX

m∈hni

(ψm−ψn)+(Vn−bnt)ψn(1)

where ψ≡(ψn)n∈[[1,N]] is the wave function deﬁned on

the sites of the lattice (numbered from 1 to Nhere), Vnis

the on-site potential at site n,−tis the coupling constant

between neighboring sites (assumed to be constant here),

hniindicates the ensemble of nearest neighbors of site n,

and bnis its cardinal. In the following, one will assume

that thas value 1, thus setting the energy unit, and that

Vn=W νnwhere νnis an i.i.d. random variable with

uniform law in [−0.5,0.5],Wbeing therefore the disorder

strength for a given lattice.

Responsible for the remarkable properties of graphene,

the honeycomb lattice will be the ﬁrst paradigmatic

structure that we study. Figures 1(a) and 1(e) show

this lattice and its celebrated dispersion relation in the

tight-binding approximation, respectively [21]. We solve

the Schrödinger equation for the Hamiltonian deﬁned in

Eq. (1) on the honeycomb lattice, with the on-site poten-

tial depicted in Fig. 1(b). In Fig. 1(c) are displayed the

ﬁrst four eigenstates which, as expected, exhibit a ﬁnite

spatial extension typical of Anderson-localized modes.

On the other end of the spectrum, Fig. 1(g) illustrates

a feature that has no continuous counterpart: the exis-

tence of high-energy localized modes (the last four eigen-

states are displayed in the example). This phenomenon

is well known for instance in the case of 3D Anderson

localization on a cubic lattice at low disorder strength,

in which the spectrum of the Hamiltonian is symmetric

in the range [−6−W/2; 6 + W/2] and exhibits a transi-

tion (the mobility edge) between localized and delocalized

states at both ends [22].

In the following, we show how to build the two dis-

crete localization landscapes displayed in Fig. 1(d) and

1(i). These landscapes allow us to accurately predict the

location of the localized modes near the two band edges

(low and high energy), as well as their energies, without

solving Eq. (1). We then generalize this method to the

most common lattices encountered in 2D materials.

Let us ﬁrst summarize the salient features of the

LL theory in the continuous setting. For any positive

deﬁnite Hamiltonian Hin such setting, the localization

landscape uis deﬁned as the solution to

Hu= 1 .(2)

One of the main results of the LL theory is that the

arXiv:2210.00806v1 [cond-mat.dis-nn] 3 Oct 2022

~a1

~a2

(a) A

φ1

φ2

φ3

φ4

ψ1

ψ2

ψ3

ψ4

(c)

(h)

(d)

1/u

ψN

ψN−1

ψN−2

ψN−3

(g)

(i)

1/u?

(b)

ΓM K Γ

Path in k-space

−2

Energy

0.0 0.2 0.4

DOS

(e) (f)

W= 0

W= 3

W=−3

-0.5

-0.25

0.25

0.5

−0.4

−0.2

0.0

0.2

0.4

0.7

0.8

0.9

1.1

≥1.2

−0.4

−0.2

0.0

0.2

0.4

0.7

0.8

0.9

1.1

≥1.2

−0.4

−0.2

0.0

0.2

0.4

FIG. 1. (a) The honeycomb lattice; (b) Plot of the random potential Vn/W =νn; (c) Eigenmodes with the four lowest

eigenvalues of a honeycomb lattices with on-site disorder, N= 964 sites and W= 3; (d) Inverse of the localization landscape

calculated for the system as in (c) where the four lowest minima are numbered; (e) Band structure of the honeycomb lattice;

(f) density of state of the honeycomb lattice without and with disorder; (g) Eigenmodes with the four highest eigenvalues of

Eq. (1); (h) Eigenmodes with the four lowest eigenvalues of the inverted Hamiltonian; (i) Inverse of the dual landscape.

quantity 1/u—which has the dimension of an energy—

acts as an eﬀective potential conﬁning in its wells the

localized states at low energy [23]. Moreover, the energy

of the local fundamental state inside each well was found

to be almost proportional to the value of the potential

1/u at its minimum inside the well [24],

E≈1 + d

4min(1/u),(3)

where dis the embedding dimension of the system.

In the case of a tight-binding Hamiltonian, Lyra et al.

[25] have studied a 1D chain with nearest-neighbor cou-

pling and have shown that the positions of the local-

ized modes are given by two diﬀerent localization land-

scapes. The low energy LL is obtained by solving the

analog of Eq. (2) in the discrete setting, i.e., ˆ

Hu=1

(u≡(un)n∈[[1,N]],1is a vector ﬁlled with 1) with the

same boundary conditions than the eigenvalue problem.

Another LL, called the dual localization landscape (DLL),

gives the position of the envelope of the highly oscillat-

ing, high-energy, wave functions. More recently, Wang

and Zhang [26] have proved mathematically that the re-

ciprocals of these discrete LL and DLL act indeed as

eﬀective conﬁning potentials in a tight-binding system at

both low- and high-energy regimes, respectively.

Figure 1(d) shows the reciprocal of the LL, 1/u≡

(1/un)n∈[[1,N]] computed on the honeycomb lattice with

the on-site disorder depicted in Fig. 1(b). Note that a

shift ˆ

H → ˆ

H+Vshift has been performed in (1) to ensure

a positive deﬁnite Hamiltonian, see Supplemental Mate-

rial A. As already observed for continuous systems, the

role of eﬀective conﬁning potential played by 1/uis re-

vealed through its basins, labelled following their depth

in Fig. 1(d). Indeed, one can observe the correspondence

between the deepest wells of 1/uand the positions of

the ﬁrst eigenmodes plotted in Fig. 1(c). As analyzed by

Arnold et al. [24] in the continuous setting, two almost-

equal eigenvalues can lead to a diﬀerent ordering in the

values of the minima of 1/u, thus inducing a mismatch

in the correspondence. This eﬀect, which does not aﬀect

the ability of the LL to predict the position of localized

modes, is visible in Fig. 1(c) and (d) with the ﬁrst and

fourth eigenstates and minima, and has been analyzed in

detail in the Supplemental Material B. Finally, we have

quantitatively tested that, regardless of the lattice, the

tight-binding LL eﬃciently pinpoint the localized modes

(see Supplemental Material C).

The symmetry of the honeycomb lattice allows us a

straightforward derivation of the landscape governing the

high-energy localized states, namely the DLL. Indeed,

the tight binding Hamiltonian (1) can be decomposed

into ˆ

H=ˆ

H0+ˆ

V, where ˆ

H0stands for the uniform hon-

eycomb lattice with zero on-site energy, and ˆ

Vaccounts

for the disordered on-site potential. The unperturbed

part of the Hamiltonian displays the usual chiral sym-

metry for a bipartite lattice, Σzˆ

H0Σz=−ˆ

H0, where

the Pauli-like matrix Σzacts on the sublattice degree

of freedom: it keeps the amplitudes on the Asites ﬁxed

but inverts those on the Bsites (Σz=PA−PB, the

diﬀerence between the respective projectors on the two

sub-lattices). Due the diagonal nature of the disordered

potential, the complete Hamiltonian obeys the symme-

try Σzˆ

H0+ˆ

VΣz=−ˆ

H0−ˆ

V. The latter property

is exempliﬁed in Fig. 1(e) and (f): unlike the DOS of

the uniform lattice, the DOS of a given realization of the

disordered system is not symmetric with respect to the

origin, but the DOS obtained by inverting the sign of

all on-site energies is the exact symmetric of the original

situation.

Let us call φ≡(φn)n∈[[1,N]] the eigenstates of the

inverted Hamiltonian ordered by increasing eigenval-

ues. The low-energy states of the inverted Hamiltonian

now correspond to the high-energy states of the origi-

nal Hamiltonian through φ= Σzψ. Since the high-

energy eigenstates oscillate with a period equal to the

nearest-neighbor distance, the new low-energy states ap-

pear as “demodulated” versions of their high-energy coun-

terparts, see Fig. 1(g) and (h). We can now therefore use

the localization landscape for the inverted system, but

with u?being now the solution to ˆ

H?u?= 1 with

(ˆ

H?φ)n=tX

m∈hni

(φm−φn)+(Vshift −Vn)φn.(4)

In the example of Fig. 1(i), one can clearly see how

the deepest wells pinpoint the locations of the localized

states. Beyond the honeycomb lattice, this spectrum in-

version strategy can be deployed for others lattices with

symmetric band structure, as the 1D dimer chain or the

2D Lieb lattice [see Supplemental Material Table S2].

As mentioned in the introduction, the localization

landscape also provides accurate estimates of the local-

ized eigenvalues in the continuous setting [24]. However,

the generalization of the simple Eq. (3) to tight-binding

Hamiltonians has never been studied systematically, nor

its extension to the higher part of the spectrum. We

plot on Fig. 2(a) (resp. 2(b)) the lowest (resp. highest)

eigenvalues of Eq. (1) versus the local minimum values

of the eﬀective potential 1/u(resp. 1/u?) at the posi-

tion of localized eigenstates for a honeycomb lattice with

N= 2135 sites and for a given disorder W= 3. Each

scatter plots corresponds to 100 realizations of the disor-

dered potential. In both cases, a direct proportionality

is clearly observed for the lowest part of the plots, with

Pearson coeﬃcients of the linear regression close to 0.99.

A general study of the quality of the proportionality is

presented in Supplemental Material B. Note also that in

order to obtain this proportionality (which is more than

a simple linear dependency), one has to choose Vshift so

that the shifted potential has a minimum value close to

zero (see Supplemental Material A).

These observations indicate that the discrete low-

energy localization landscape performs as well as its con-

tinuous analog in predicting energy and spatial distri-

bution of localized modes without resolving an eigen-

value equation. Moreover, the high-energy DLL also ex-

hibits the same properties. For both range of energy,

LL and DLL provide a good estimate of the integrated

density of states, as shown in Supplemental Material D.

All these results are not restricted to the honeycomb lat-

0.0 0.5 1.0

hIPRi

1.0

1.2

1.4

1.6

s(Vshift −E, min(1/u?))

d= 1

d= 2

(d)(b)

1.0

1.2

1.4

1.6

s(E, min(1/u))

d= 1

d= 2

(c)(a)

Chain

Dimer

ChainSec

Square

Honeycomb

Lieb

Hexagonal

Kagome

tts

min(1/u)

1.0

1.5

2.0

E(units of t)

ρ= 0.989

s= 1.481

W= 3

0.5 1.0 1.5

min(1/u?)

1.0

1.5

2.0

Vshift −E(units of t)

ρ= 0.987

s= 1.488

FIG. 2. (a) Proportionality between min (1/u)and Efor the

lowest energies. The blue dots correspond to the 3% states

of lowest energy for 100 diﬀerent conﬁgurations, the orange

dots to the 3-5% tier, and the pink dots to the 5-7% tier,

respectively. The black line corresponds to a linear ﬁt of the

pink dots, the slope sand the Pearson coeﬃcient ρbeing

given in the frame. (b) Proportionality between min (1/u?)

and (Vshift −E)for the highest energies. Similar plot to (a),

but for the states of highest energy. (c) Slope sfor the low-

energy states, for diﬀerent lattices in 1D (squares) and 2D

(circles). Each symbol corresponds to a disorder strength W,

but instead of reporting Won the horizontal axis, we chose to

use the average value of the IPR which is a better comparison

parameter across diﬀerent lattices. The dashed horizontal

lines show the limits expected in the continuous case from

Eq. (3). The black circle corresponds to the case displayed in

(a). (d) Similar plot to (c) for the highest-energy states.

tice. Figures 2(c) and 2(d) show that the proportional-

ity is obtained for a large variety of “canonical” lattices

(1D: chain, dimer chain, chain with second-neighbor cou-

pling; 2D: square, honeycomb, Lieb, hexagonal, Kagome,

tts) and in a wide range of strength disorder. Note that

to quantify the latter unequivocally for diﬀerent lattices,

the parameter Wis not the best suited. Indeed, for a

given value of W, the relative weight of the potential

term in (1) compared to the kinetic term depends on

the connectivity of the discrete Laplacian Pm(ψm−ψn).

The number of edges of the graph on which this operator

is relying is given by the number bnof nearest-neighbor

couplings, which itself depends on the elementary mo-

tif of each given lattice. Therefore, we use a less con-

tingent quantity, namely the inverse participation ratio

(IPR) deﬁned for a given eigenvector ψ(j)=Pnψ(j)

n|ni

as IPRj=Pn



ψ(j)

n



4/Pn



ψ(j)

n



22

. More precisely,

in Fig. 2, we consider the ﬁrst (c) and last (d) 3% of the

eigenstates to compute the slopes, that are plotted ver-

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Lowandhigh-energylocalizationlandscapesfortight-bindingHamiltoniansin2DlatticesLuisA.Razo-López,1GeoroyJ.Aubry,1MarcelFiloche,2,3andFabriceMortessagne1,1UniversitéCôted'Azur,CNRS,InstitutdePhysiquedeNice(INPHYNI),France2LaboratoiredePhysiquedelaMatièreCondenséee,CNRS,ÉcolePolytechnique,InstitutPol...

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Low and high-energy localization landscapes for tight-binding Hamiltonians in 2D lattices Luis A. Razo-López1Geoﬀroy J. Aubry1Marcel Filoche2 3and Fabrice Mortessagne1

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