Optical response of the tightbinding model on the Fibonacci chain Hiroki Iijima Yuta Murakami and Akihisa Koga Department of Physics Tokyo Institute of Technology Meguro Tokyo 152-8551 Japan

2025-05-02 0 0 3.94MB 7 页 10玖币

侵权投诉

Optical response of the tightbinding model on the Fibonacci chain

Hiroki Iijima, Yuta Murakami, and Akihisa Koga

Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan

(Dated: October 6, 2022)

We theoretically study the optical conductivity of the tightbinding model which has two types

of the hopping integrals arranged in the Fibonacci sequence. Due to the lack of the translational

symmetry, many peak structures appear in the optical conductivity as well as the density of states.

When the ratio of two hopping integrals is large, the self-similar structure appears in the optical

conductivity. This implies that the optical response between the high-energy bands is related to that

within the low-energy bands, which should originate from critical behavior in the wave functions.

The eﬀects of disorders on the optical conductivity are also analyzed in order to show the absence

of the self-similarity in the tightbinding model with the random sequence.

I. INTRODUCTION

Quasicrystal has been attracting much interests since

the ﬁrst discovery of the quasicrystalline phase in the Al-

Mn alloy [1]. Experimental and theoretical eﬀorts have

been made to explore and understand physical proper-

ties inherent in quasicrystals [2–12]. Among them, the

Au-Al-Yb alloy with Tsai-type clusters [13] exhibits in-

teresting properties at low temperatures. In the qua-

sicrystal Au51Al34Yb15, quantum critical behavior ap-

pears, while heavy fermion behavior appears in the ap-

proximant Au51Al35Yb14 [14]. This distinct behavior

stimulates theoretical investigations on the quasiperiodic

structures in correlated electron systems [15–23]. The op-

tical response characteristic of the quasicrystals have also

been observed. In the Al-Cu-Fe quasicrystal, the linear-ω

dependence in the optical conductivity, which is distinct

from the conventional impurity scattering, has been ob-

served [24]. Furthermore, the direction dependent opti-

cal conductivity has been reported in the Al-Co-Cu qua-

sicrytal [25]. These studies suggest that the exotic opti-

cal properties arise from the quasiperiodic structure. Al-

though the low-frequency behavior has theoretically been

examined for quasiperiodic systems [26–29], the optical

response at ﬁnite frequencies has not been discussed in

detail. An important point is that the spatial features of

the initial and ﬁnal states play a crucial role for the op-

tical process. It is known that spatially extended states

characterized by the momenta are realized in the peri-

odic system, while critical states are, in general, realized

in the quasiperiodic one [30–33]. Therefore, it is instruc-

tive to clarify the optical response inherent in quasiperi-

odic systems by comparing them to systems with distinct

properties of eigenstates.

Motivated by this, we treat the tightbinding model

which has two types of the hopping integrals arranged in

the Fibonacci sequence, as a simple model. It is known

that each eigenstate for the system shows critical behav-

ior [31–35], which are characterized by multifractal prop-

erties and the power law decay of the amplitude of the

wave functions in the real space. This eigenstate prop-

erty is distinct from those for the periodic systems where

the spatially extended states are realized. We then dis-

cuss the optical response in the tightbinding model on

the Fibonacci chain, examining the matrix elements of

the current operator, which play a crucial role for the

optical conductivity.

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

troduce the tightbinding model on the Fibonacci chain

and derive the expression of the optical conductivity in

this system. In Sec. III, we discuss the optical response

inherent in the Fibonacci chain, comparing with that in

the approximants. The eﬀect of the disorders is also ad-

dressed. A summary is given in the last section.

II. MODEL AND METHOD

We consider the tightbinding model to study the op-

tical conductivity inherent in the Fibonacci chain. The

Hamiltonian is given as

H(t) = −X

vne−iqLnA(t)ˆc†

nˆcn+1 + H.c.,(1)

where ˆcn(ˆc†

n) is the annihilation (creation) operator of

a spinless fermion at the nth site, vn(Ln) denotes the

hopping integral (lattice spacing) between the nth and

(n+ 1)th sites, and qis the charge of the fermion. The

site-independent vector potential A(t) leads to the uni-

form electric ﬁeld E(t) = −∂tA(t).

Here, we introduce the Fibonacci sequence Sinto the

Hamiltonian. It is known that the Fibonacci sequence is

generated by means of the substitution rule for two letters

Land S:L→LS and S→L. Applying the substitution

rule to the initial sequence S1=Siteratively, we obtain

sequences {L, LS, LSL, LSLLS, ···}. The ith sequence

Siis composed of Filetters, where Fiis the Fibonacci

number. We deal with the tightbinding model with the

total number of sites N=Fiand the hopping integral is

given as vn=vL(vS) if the nth letter of the Fibonacci

sequence Siis L(S).

In the paper, we study the linear optical response for

the tightbinding model on the Fibonacci chain. The op-

tical conductivity is given as σ(ω) = J(ω)/E(ω), where

J(ω) and E(ω) are the Fourier components of the current

J(t) and electric ﬁeld E(t). The current operator is split

arXiv:2210.02158v1 [cond-mat.mes-hall] 5 Oct 2022

into two parts as ˆ

J(t) = ˆ

j1+ˆ

j2(t), with

j1=−iq X

vnLn(ˆc†

nˆcn+1 −ˆc†

n+1ˆcn),(2)

j2(t) = −q2X

vnL2

n(ˆc†

nˆcn+1 + ˆc†

n+1ˆcn)A(t).(3)

By means of the Kubo formula, the optical conductivity

is expressed as

σ(ω) = 1

a,b

f(Eb)−f(Ea)

i(ω+iδ)|hb|ˆ

j|ai|2

ω−(Eb−Ea) + iδ

−1

q2f(Ea)

i(ω+iδ)ha|X

vnL2

n(ˆc†

nˆcn+1 + ˆc†

n+1ˆcn)|ai,

(4)

where |aiis a single electron eigenstate of the model with

A(t) = 0, Eais the corresponding energy, f(x) is the

Fermi distribution function, and δis inﬁnitesimal. The

ﬁrst term represents the optical transition, and the other

is the so-called Drude part. For the sake of simplicity,

we set Ln= 1 in order to focus on the eﬀects of the

Fibonacci structure in the hopping integrals [26,27].

In the following, we mainly consider the system with

N=F19 = 4181 under the periodic boundary condition.

This system is large enough to take the quasiperiodic

structure into account. In fact, we have conﬁrmed that

the obtained results are essentially the same as those with

N= 17711. Here, setting the Fermi energy at E= 0 and

vLas the unit of energy, we discuss how the quasiperiodic

structure aﬀects the optical linear response.

III. RESULT

We study the optical response in the tightbinding

model with the Fibonacci structure. First, we brieﬂy

discuss the one-particle states in the model without the

external electric ﬁeld. Figure 1(a) shows the density of

states (DOS) for the tightbinding model on the Fibonacci

chain with vS= 2.5. Many delta-function peaks appear

in the DOS, which is known to be purely singular contin-

uous like the Cantor set [33,34,36,37]. To plot a delta-

function peak in DOS, we practically use the Gaussian

with a small width in Figure 1. We also use the same

strategy to plot delta-function peaks in the following.

The energy levels are roughly classiﬁed into lower (L),

middle (M), and upper (U) bands [see Fig. 1]. When one

focuses on the M band, there exist three smaller bands,

suggesting the nested structure in the DOS [38]. In fact,

the DOS scaled by Ris in a good agreement with the orig-

inal one, where R=EU

max/EM

max and Eα

max is the max-

imum energy in the α(= U, M) band. It is also known

that single-particle states of this model are critical and

the decay for each wave function is slower than expo-

nential one [31–35]. In contrast to the Fibonacci case,

the smooth DOS should appear in the periodic system.

°3°2°1 0 1 2 3

DOS

(a)

°3°2°1 0 1 2 3

DOS

(b)

°3.142 3.142

°3

L M U

<latexit sha1_base64="i/+os+hmv7F06/gQ67Ugqh9VTlQ=">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</latexit>

⇡

<latexit sha1_base64="iDBT0SzcBx/DQCkHMowmbpauwVs=">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</latexit>

⇡

FIG. 1. Density of states in the tightbinding model on the

Fibonacci chain (a) and the approximant (b) with vS= 2.5

and A(t) = 0. Inset in (b) shows the dispersion relation of

the model for the approximant. The vertical axes are in an

arbitrary unit.

For comparison, we consider the tightbinding model on

the approximant, where the shorter Fibonacci sequence

S4(= LSL) is periodically arranged. This is the mini-

mal approximant with three bands in common with the

Fibonacci chain, as shown in Fig. 1(b). Since each single-

particle state in the approximant is characterized by the

momentum, its wave function is spatially extended, in

contrast to the Fibonacci case. When one considers the

approximant with the longer sequence, the corresponding

DOS approaches the Fibonacci one and critical behavior

should appear in the wave function.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

OpticalresponseofthetightbindingmodelontheFibonaccichainHirokiIijima,YutaMurakami,andAkihisaKogaDepartmentofPhysics,TokyoInstituteofTechnology,Meguro,Tokyo152-8551,Japan(Dated:October6,2022)Wetheoreticallystudytheopticalconductivityofthetightbindingmodelwhichhastwotypesofthehoppingintegralsarrangedi...

展开>> 收起<<

Optical response of the tightbinding model on the Fibonacci chain Hiroki Iijima Yuta Murakami and Akihisa Koga Department of Physics Tokyo Institute of Technology Meguro Tokyo 152-8551 Japan.pdf

共7页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Optical response of the tightbinding model on the Fibonacci chain Hiroki Iijima Yuta Murakami and Akihisa Koga Department of Physics Tokyo Institute of Technology Meguro Tokyo 152-8551 Japan

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: