One-dimensional L evy Quasicrystal Pallabi Chatterjee1and Ranjan Modak1 1Department of Physics Indian Institute of Technology Tirupati Tirupati India 517619

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One-dimensional L´evy Quasicrystal

Pallabi Chatterjee1, ∗and Ranjan Modak1, †

1Department of Physics, Indian Institute of Technology Tirupati, Tirupati, India 517619

Space-fractional quantum mechanics (SFQM) is a generalization of the standard quantum me-

chanics when the Brownian trajectories in Feynman path integrals are replaced by L´evy ﬂights.

We introduce L´evy quasicrystal by discretizing the space-fractional Schr¨odinger equation using the

Gr¨unwald-Letnikov derivatives and adding on-site quasiperiodic potential. The discretized version of

the usual Schr¨odinger equation maps to the Aubry-Andr´e Hamiltonian, which supports localization-

delocalization transition even in one dimension. We ﬁnd the similarities between L´evy quasicrystal

and the Aubry-Andr´e (AA) model with power-law hopping, and show that the L´evy quasicrystal

supports a delocalization-localization transition as one tunes the quasiperiodic potential strength

and shows the coexistence of localized and delocalized states separated by mobility edge. Hence, a

possible realization of SFQM in optical experiments should be a new experimental platform to test

the predictions of AA models in the presence of power-law hopping.

I. INTRODUCTION

In usual quantum mechanics, the typical energy-

momentum relation for a particle of mass mis given by

E=P2/2m. In general, one can have a situation where

the energy-momentum relation is given by E∝Pα1,

where αcan be a fraction. This kind of situation can

be described by the Space Fractional Quantum Mechan-

ics(SFQM), which was introduced by Laskin2–7. SFQM

is a natural generalization of the standard quantum me-

chanics that arises when the Brownian trajectories in

Feynman path integrals are replaced by L´evy ﬂights. The

classical L´evy ﬂight is a stochastic process that, in one

dimension is described by a jump length probability den-

sity function (PDF) of the form, Πα(x)∝1/|x|α+1 for

|x| → ∞, with α∈(0,2]5, where αis known as L´evy

index.

There have been numerous applications of classical

L´evy ﬂights. Especially, in the context of successfully

predicting the anomalous scaling of dynamical correla-

tions of conserved quantities in one-dimensional (1D)

Hamiltonian systems, the L´evy scaling for the spread-

ing of local energy perturbation has been predicted, as

well as diverging thermal conductivity (via GreenKubo

formula)8–11. Also, in order to understand the motion

of the particle in a rotating ﬂow 12,13 or even the trav-

eling behavior of animals14–17, the complex dynamics of

real-life ﬁnancial markets18, L´evy description has been

extremely useful. It has been shown recently in Ref.5,

one can discretize Space fractional Schr¨odinger equation,

introduce a system which is referred to as L´evy Crys-

tal. Given the extraordinary advancements of ultra-cold

experiments in the last two decades, L´evy Crystal is a po-

tential candidate for an experimentally accessible realiza-

tion of SFQM in a condensed-matter environment19–21,

however, it has not been explicitly demonstrated how

that can be achieved in Ref.5. Moreover, fractional quan-

tum mechanics plays a very crucial role in optical systems

22–27. In this context, there have been recent theoretical

works on domain walls in fractional media28, fractional

diﬀraction in the context of parity-time symmetric poten-

FIG. 1. Here we show the phase diagram of the L´evy qua-

sicrystal. Black circle represents α= 0.5 and ∆ = 0.75. In the

main text, we show the results corresponding to this point.

tials 29–31, and on optical solitons32–54, that bought lots

of attention. The experimental realization of an optical

system representing the fractional Schr¨odinger equation

has been reported very recently in Ref.55. Also, it is

important to point out that in the context of Sisyphus

cooling, fractional equations arise from the dynamics of

cold atoms, an eﬀect that is well-studied56,57.

On the other hand, the disorder is ubique in nature,

and in the condensed matter system, it plays a very im-

portant role. In the one-dimensional(1D) system, the dis-

order has an extreme consequence, even a tiny bit of dis-

order is suﬃcient to localize all the single-particle states.

This phenomenon is famously known as Anderson local-

ization 58–60. In recent days, there has been a plethora of

work to understand the interplay of interaction and dis-

order, and that cause a delocalization-localization tran-

sition in the many-body Fock space, which is referred to

as many-body localization (MBL) transition20,61–65. In

the experimental point of view, realizing true disorder

in ultra-cold experiments is non-trivial, hence one of the

obvious candidates is to replace it with the quasiperiodic

arXiv:2210.10772v5 [cond-mat.stat-mech] 28 Sep 2023

potential. In contrast to the true disorder, quasiperiodic

potential can cause a delocalization-localization transi-

tion in 1D even in the absence of interaction, and this

Hamiltonian is known as Aubry-Andr´e (AA) model66.

While the nature of the localized phase observed due to

quasiperiodic potentials and true disorders are the same

(in both cases, eigenstates are exponentially localized),

there are diﬀerences in critical properties associated with

the localization-delocalization transition. In both cases,

the localization length ξdiverges at the transition fol-

lowing as ξ∼δ−ν, where δis the distance to the critical

point in the parameter space and νis the localization

length exponent. In true disorder-driven localization,

there is a rigorous bound on the localization length ex-

ponent ν, i.e., it must satisfy ν≥2/d criteria to ensure

the stability of the transition67. However, in quasiperi-

odic models, such criteria do not apply. For most of the

one-dimensional quasi-periodic models, νis close to 168.

In recent days, there have been numerous studies, both

theoretically and experimentally, involving a variant of

the AA model that shows the coexistence of both local-

ized and delocalized states separated by a mobility edge

69–83.

In this work, one of the main aims is to address the

question of what happens to the fate of Anderson lo-

calization in the context of SFQM in the presence of

quasiperiodic potential, we call it L´evy quasicrystal. In-

terestingly, a similar question has been asked in a recent

study26, but a detailed theoretical understanding of the

phase diagram of the diﬀerent phases was lacking there.

Also, the L´evy-ﬂight models have been used extensively

in understanding real-life ﬁnancial markets, the traveling

behavior of humans, and even biological systems. If one

wants to model such a system and take into account cor-

related random events, our L´evy quasicrystal model can

be an extremely suitable candidate for such cases. We

demonstrate the phase diagram as a function of αand the

strength of the quasiperiodic potential ∆, which is shown

in Fig. 1. While we ﬁnd that like the Aubry-Andr´e model

(which is the discretized version of the usual Schr¨odinger

equation), there exists a completely delocalized (DL) and

Anderson localized (AL) phase, but on top of that there

is a parameter region where these two types of states co-

exist, and they are separated by the mobility edge (ME).

Those ME phases can also be characterized as diﬀerent

psphases (where s > 0 can be any positive integer) based

on the fraction of delocalized states in the spectrum. We

also compare our model with a potential experimentally

realizable AA model with power-law hopping.

The paper is organized as follows. In Sec. II, we intro-

duce the model of L´evy quasicrystal. Next, we discuss

the analytical prediction of the mobility edge in Sec. III.

In Sec. IV, we show our numerical results. Sec. V shows

the comparison between L´evy quasicrystal and the eﬀec-

tive power-law hopping model and ﬁnally, we summarize

our results in Sec. VI.

II. MODEL

The one-dimentional space-fractional Schr¨odinger

equation is given by,

H|ψ⟩=DαPα|ψ⟩+V|ψ⟩=E|ψ⟩,(1)

where Dα∈Ris a constant (note that D2is equivalent

of inverse of mass term in usual Schr¨odinger equation),

Pαis the α-th power of the momentum operator, Vis

the potential energy operator and |ψ⟩is the eigenstate

with eigenvalue E2,3,5.

The position space representation of α-th power of

the momentum operator is ⟨x|Pα|ψ⟩=−ℏαDα

|x|ψ(x)4,5,

where Dα

|x|is the Riesz Fractional Derivative of order α.

While usually to get the ﬁnite ﬁrst moment of the L´evy

process, αis taken within the limit α∈(1,2] 4, but given

there exist, biological models, where the L´evy parameter

is taken to be 0 < α < 115, also in a very recent optical

experiment, the parameter range α < 1 of the fractional

Schr¨odinger equation has been realized55 in the temporal

domain, that motivates us to even explore the extended

parameter regime, and we use the same position space

representation of α-th power of the momentum opera-

tor for α∈(0,2] to get the discretized version. Riesz

Fractional Derivative of order α(note that we consider

α= 1 point separately in the appendix), can be written

as5,84(note that the expression of the Riesz Fractional

Derivative is taken from Ref.84),

Dα

|x|=−1

2 cosαπ

2(I−α

++I−α

−),(2)

where I−α

±are given by (approximating Gr¨unwald-

Letnikov operators)5,84,

I−α

±ψ(x) = lim

a→0

aα

∞

n=0

(−1)nα

nψ[x∓an],(3)

for 0 < α ≤1.

I−α

±ψ(x) = lim

a→0

aα

∞

n=0

(−1)nα

nψ[x∓(n−1)a],(4)

for 1 < α ≤2. Where astands for a small positive step

length. Here, α

n=Γ(α+1)

Γ(n+1)Γ(α−n+1) , where Γ(.) is the

usual Gamma function. It is straightforward to check

that for α= 2 Riesz Fractional Derivative becomes the

standard second-order derivative.

In this work, we consider the potential to be space-

dependent (quasiperiodic potential), hence the model can

be represented by the equation,

⟨xl|H|ψ⟩=⟨xl|DαPα|ψ⟩+ ∆ cos(2πβl +ϕ)ψ(xl).(5)

xlis equally spaced grid points i.e xl=la,a=lattice

constant, a > 0 and l∈Z.βis an irrational number.

We choose β=√5−1

2, and ϕis a random number

chosen between [0,2π]. Note that in the previous

study5the potential energy term was considered to be

periodic in lattice spacing a, the construction of the

space-fractional Schr¨odinger equation via Riesz Frac-

tional Derivative is not only limited to periodic potential.

Replacing the exact momentum operator by its dis-

cretized version5,85 and using Eqn. (4), one gets for

1< α ≤2,

⟨xl|DαPα|ψ⟩:= t0

∞

n=0

(−1)nα

n

(ψ[xl+ (n−1)a] + ψ[xl−(n−1)a]),(6)

where t0=Dαℏα

aαcos(απ

2).

Then, ⟨xl|DαPα|ψ⟩becomes,

⟨xl|DαPα|ψ⟩:=

∞

n̸=0

(−1)(n+1)α

n+ 1

ψ(xl+na) + t0

2[ψ(xl−a) + ψ(xl+a)] −αt0ψ(xl).(7)

Next, we drop the constant diagonal term αt0(as it

will create only a shift to the energy level).

Eqn:(5) becomes,

⟨xl|H|ψ⟩=∞

n̸=0

t(n)ψ(xl+na) + ∆ cos(2πβl +ϕ)ψ(xl),(8)

where t(n) is the hopping parameter for 1 < α ≤2,

t(n) = t0

2[(−1)|n|+1α

|n|+ 1+δ|n|,1],(9)

where δn,m is the Kronecker delta.

Now, one can repeat similar calculations for 0 < α < 1,

and gets the following equation,

⟨xl|DαPα|ψ⟩:= t0

∞

n=0

(−1)nα

n

(ψ[xl+na] + ψ[xl−na]).(10)

Then, ⟨xl|DαPα|ψ⟩becomes86,

⟨xl|DαPα|ψ⟩=

∞

n̸=0

(−1)nα

nψ(xl+na) + t0ψ(xl).(11)

Once again, subtracting the constant diagonal terms,

one gets,

⟨xl|H|ψ⟩=

∞

n̸=0

t(n)ψ(xl+na) + ∆ cos(2πβl +ϕ)ψ(xl),(12)

where t(n), the hopping parameter for 0 < α < 1 is

given by,

t(n) = t0

2[(−1)|n|α

|n|].(13)

Hence in general our model can be written as,

H=X

j,n̸=0

(t(n)c†

jcj+n+ H.c.) + ∆ X

cos(2πβj +ϕ)nj,

(14)

where, c†

j(cj) is the fermionic creation (annihilation)

operator at site j,nj=c†

jcjis the number operator,

t(n) = −1

2[(−1)|n|α

|n|] 0 < α < 1,

2δ|n|,1α= 1,

2[(−1)|n|+1α

|n|+ 1+δ|n|,1] 1 < α ≤2.

(15)

We do average over ϕto obtain better statistics and for

all of our calculations, we have used periodic boundary

conditions.

For α= 2, the Hamiltonian His the same as

the Aubry-Andr´e (AA) Hamiltonian, which supports a

delocalization-localization transition as one tunes ∆. In

the thermodynamic limit, ∆ = 2 corresponds to the tran-

sition point66 between localized and delocalized phases

and for ∆ <2 (∆ >2), all the eigenstates of the model

are delocalized (localized).

III. ANALYTICAL PREDICTION

Unlike the AA model, the Hamiltonian we are inter-

ested in here Eqn. (14), which has higher order hopping

terms. It has been shown in Ref.87, that in the pres-

ence of incommensurate potential with long-range hop-

ping, there exists a mobility edge. One example of such a

model is the exponential hopping model (where the hop-

ping amplitudes fall oﬀ exponentially with increasing the

range nof hopping as te−p|n|). This exponential model

is self-dual87. One can obtain the mobility edge line, us-

ing the self-duality condition of the exponential hopping

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One-dimensionalL´evyQuasicrystalPallabiChatterjee1,∗andRanjanModak1,†1DepartmentofPhysics,IndianInstituteofTechnologyTirupati,Tirupati,India517619Space-fractionalquantummechanics(SFQM)isageneralizationofthestandardquantumme-chanicswhentheBrowniantrajectoriesinFeynmanpathintegralsarereplacedbyL´evyfl...

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One-dimensional L evy Quasicrystal Pallabi Chatterjee1and Ranjan Modak1 1Department of Physics Indian Institute of Technology Tirupati Tirupati India 517619

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