Emergence of quasiperiodic behavior in transport and hybridization properties of clean lattice systems Cecilie Glittum1Antonio Strkalj1and Claudio Castelnovo1

2025-04-29 0 0 9.55MB 12 页 10玖币

侵权投诉

Emergence of quasiperiodic behavior in transport and hybridization properties of

clean lattice systems

Cecilie Glittum,1Antonio ˇ

Strkalj,1and Claudio Castelnovo1

1T.C.M. Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom

(Dated: January 4, 2023)

Quasiperiodic behaviour is mostly known to occur in systems with enforced quasiperiodicity or

randomness, in either the lattice structure or the potential, as well as in periodically driven systems.

Here, we present instead a rarer setting where quasiperiodic behaviour emerges in clean, non-driven

lattice systems. We illustrate this through two examples of experimental relevance, namely an

inﬁnite tight-binding chain with a gated segment, and a hopping particle coupled to static Ising

degrees of freedom. We show how the quasiperiodic behaviour manifests in the number of states

that are localised by the geometry of the system, with corresponding eﬀects on transport and

hybridisation properties.

I. INTRODUCTION

Some of the most striking discoveries in physics occur

when simple systems are found to exhibit unexpectedly

complex behaviour. These are often prime examples of

the beauty and elegance of mathematical modelling that

succeeds in describing the world around us. Examples

include the chaotic motion of the double pendulum [1, 2]

and the quantum Hall eﬀect of a 2D electron gas in a

strong applied ﬁeld [3].

Without claim of drawing a comparison in importance,

we present here a simple result that ﬁts well in this cate-

gory. In one of its simplest forms, our ﬁnding shows that

the number of states localised [4] on a gated portion of

an inﬁnite tight-binding chain forms a quasiperiodic se-

quence in the number of gated sites. Equivalently, the

number of resonance peaks in the conduction along the

chain also forms a quasiperiodic sequence. This happens

for both negative and positive gate potential, within the

bounds of the tight-binding band. We provide both an

exact solution to the problem as well as a simple, elegant

and intuitive (albeit only eﬀective) derivation of the same

solution.

Our results are of direct relevance to several ex-

perimental settings – thinking for example about cold

atoms [5, 6], trapped ions [7, 8], quantum dot arrays [9],

superconducting cubits [10] and other quantum simula-

tors [11] – and can be readily veriﬁed in a laboratory.

Moreover, in addition to this simplest formulation of the

problem, we show that a similar phenomenology is ex-

pected to play a role in other settings, for instance when

particles hopping on a lattice are coupled to underly-

ing spin degrees of freedom, as in the Falicov-Kimball

model [12] and in systems that exhibit disorder-free lo-

calisation [13].

Quasiperiodic systems have received signiﬁcant atten-

tion of late (see for instance Refs. 14–28). In most cases

however the quasiperiodicity is built into the system, e.g.,

via quasiperiodic modulation of on-site potentials and

hopping amplitudes, or induced by incommensurate pe-

riodic driving; then, properties deriving from it are stud-

ied. Examples where quasiperiodicity is not seeded but

rather emerges from periodic constituents are rare (most

notably this was uncovered to result from the interplay

of lattice ﬁlling fractions and interactions in Refs. 29

and 30; however, see also Ref. 31 for an example of a

quasicrystalline potential emerging from collective light

scattering). In our setting, the quasiperiodic behaviour

emerges in a clean system made of non-driven periodic

components – something that does not happen often in

nature. We envision that the simple mechanism discussed

in this work is likely to operate in interesting ways in

other condensed matter systems, and possibly in higher

dimensions as well.

II. GATED CHAIN

We study a tight-binding chain of lattice spacing a= 1

and hopping amplitude t= 1, with a segment of length

Lgated at potential, i.e., on-site energy, V0. The rest

of the system (i.e., the left and right leads) is kept at

the reference potential V= 0. The Hamiltonian of the

system is given by

H=−tX

ihc†

i+1ci+ h.c.i+X

Vic†

ici,(1)

where Viis ﬁnite on the gated segment and vanishes on

the leads. The system is illustrated in Fig. 1(a).

Due to the mismatch in on-site potential between the

gated segment and the leads, some states localise on the

gated segment, while other states are extended over the

whole system. Our goal is to investigate the nature of

these localised states, and more precisely how their num-

ber changes with the system parameters Land V0. As

we detail below, the number of states Wannier-Stark-

localised on the gated segment forms a quasiperiodic se-

quence as a function of the discrete length Lof the seg-

ment; see Fig. 1(b).

By choosing an appropriate ansatz for the wave func-

tion, we look for eigenstates of the Schr¨odinger equa-

tion that are exponentially decaying on the leads (i.e.,

localised on the gated segment); see App. A for details

and App. F for examples of wave function proﬁles. We

arXiv:2210.02480v2 [cond-mat.mes-hall] 3 Jan 2023

gated segment

V = V0

right lead

V = 0

left lead

V = 0

……

tttttttttttttt

(a)

(b)

10 20 30 40

conductance IPR Eq. (4)

V0=1.45

V0=0.33

Nloc - (1-α)LNloc - (1-α)L

FIG. 1. (a) Illustration of the gated chain. The solid black

dots illustrate the inﬁnite leads, at potential V= 0. The solid

pink dots illustrate the gated segment of length Lat potential

V0, here for L= 5. (b) Number of states localised in the gated

region Nloc as a function of its length L. We subtracted the

linear slope (1 −α)Lto emphasise the quasiperiodic nature

of the ﬂuctuations following from sampling at integer L. Nu-

merically obtained data from conductance simulations (black

dots) and IPR (blue squares) overlap and agree perfectly with

the analytical prediction in Eq. (4) (black line).

ﬁnd that the number of such states corresponds to the

number of solutions of either of the following equations:

−earccosh[V0/2−cos k]=cos k−kL+1

2

cos −kL+1

2,(2)

−earccosh[V0/2−cos k]=sin k−kL+1

2

sin −kL+1

2.(3)

The task can be conveniently re-cast into counting the

number of qn=nπ/L with n∈ {1, . . . , L}larger than

απ ≡arccos V0

2−1, which is the point in the Brillouin

zone where the dispersion of the isolated inﬁnite gated

segment intersects the band edge of the isolated leads;

see Fig. 2(a). Knowing qnand α, one can show that the

number of states localised on the gated segment is given

Nloc =L−ﬂoor (αL),(4)

which is a linearly growing function of Lwith slope 1−α,

and it exhibits quasiperiodic ﬂuctuations whenever αis

irrational.

We note in passing that the same result can be ob-

tained using Levinson’s theorem [32], appropriately mod-

iﬁed for a lattice model, from the phase diﬀerence be-

tween the states at the bottom and top of the band (see

App. B).

The quasiperiodicity can be distinctly seen in Fig. 1(b),

where we also verify the perfect agreement between the

analytical result, Eq. (4), and the numerical calculation

of Nloc using both the inverse participation ratio (IPR)

and transport measurements. Albeit simple, we ﬁnd this

result surprising since the leads and the gated segment

are periodic, and there is no additional quasiperiodicity

or randomness present in the system.

A. Simple counting argument

It is tempting to interpret the above qnas the wave vec-

tor of a tight-binding chain of length Lwith open bound-

ary conditions. The argument appears then to be count-

ing the number of discrete eigenenergies of the gated seg-

ment, whose dispersion in the limit of inﬁnite length is

Egated(k) = V0−2 cos k, that fall outside the continuous

band of the leads, of dispersion Eleads(k) = −2 cos k.

The two dispersions intersect at k=±απ, as illus-

trated in Fig. 2(a), and one can see that the discrete

energies of the gated segment fall outside the band of

the leads if qn> απ. Recalling that n= 1,2, . . . , L,

one can straightforwardly count such states and obtain

L−ﬂoor (αL) for the number of states localised on the

gated segment, which coincides with the exact solution,

Eq. (4).

This is however only an intuitive, and somewhat in-

correct, picture which happens coincidentally to give the

correct answer. In the exact calculation, the qn’s are sim-

ply a tool to count the number of solutions and have no

direct physical interpretation.

B. Probing localised states

One way to probe the localised states is by computing

the IPR of the eigenstates of the Hamiltonian (1),

IPR(En) = Pj|ψj(En)|4

Pj|ψj(En)|2.(5)

The IPR is known to scale as (i) O(1/Ltot) for states

extended over the whole system of length Ltot = 2Lleads+

L, where Lleads is the number of sites in a single lead; (ii)

O(1/(2Lleads)) for states localised on the leads (when one

allows for a small matrix element due to tunnelling across

the gated segment); and (iii) O(1/L) for states localised

only on the gated segment. We use exact diagonalisation

and compare the results for two diﬀerent sizes of leads,

5·103and 104sites, and for diﬀerent gate potentials. The

results for a gated segment with L= 50 sites are shown

in Fig. 2(b), for V0= 0.33 and V0= 1.45 (recall that we

work in units where t= 1). We see that all the states

with energy E > 2 have an IPR that is independent of the

size of the leads, meaning that these states are localised

on the gated segment. The aforementioned states cannot

hybridise with the states belonging to the leads as their

-4

-2

-��

α�

gated segment

leads

(a)

-2 -1 0 2 3

-2 -1 1 2

10-4

10-3

10-2

10-1

IPR

(b)

G [e2/h]

-1 0 1 2

(c)

V0=0.33 Lleads

104

5∙103

V0=1.45 Lleads

104

5∙103

FIG. 2. (a) Dispersion of the inﬁnite leads (black solid line) and of the gated segment in the L→ ∞ limit (pink dashed line).

The two dispersions overlap for k∈[−απ, απ]. We also show the energies of an isolated gated segment with L= 9 plotted as

a function of qn=πn/L;n= 1,2,...,L (pink solid dots). Counting the number of discretised energies falling outside of the

bandwidth of the leads gives the number of states localised on the gated segment, which in this case is 4 states. (b) IPR for a

gated segment of length L= 50 with V0= 0.33 (left) and V0= 1.45 (right), comparing two diﬀerent lead sizes, Lleads = 5 ·103

and Lleads = 104. States with E > 2 (highlighted by a green rectangle) are localised inside the gated segment and their IPR is

independent of the lead size. (c) Conductance through the whole system as a function of energy of the incident particle. Here

we used L= 9 and V0= 1.45. We remind that potentials and energies are expressed in units of the hopping amplitude t= 1.

energy falls outside their bandwidth. On the other hand,

states falling within the bandwidth of the leads hybridise

and delocalise.

The steplike behaviour seen in Fig. 2(b) is a numerical

artefact of the diagonalisation algorithm. When the en-

ergy of the eigenstates is smaller than −2 +V0, the parti-

cle cannot propagate across the barrier; however, a small

tunnelling amplitude exists for ﬁnite values of Lsuch that

the eigenstates found by diagonalisation are even and odd

superpositions of the states on the left and on the right

lead, and the IPR is approximately 1/(2Lleads) (which

in the ﬁgure is indistinguishable from 1/(2Lleads +L),

the IPR of delocalised states). As we continue to lower

the energy of the eigenstates however, the tunnelling am-

plitude is correspondingly suppressed; when it falls be-

low numerical accuracy, the diagonalisation routine is no

longer able to ﬁnd even and odd superpositions of states

on the left and on the right lead, but rather sees the left

and right lead states as eigenstates of the system. This

results in an IPR ∼1/Lleads and in the step increase ob-

served when we consider the lowest energy eigenstates.

Another way to directly probe the aforementioned lo-

calised states is via energy-dependent transport, i.e., via

the conductance

G(E) = e2

hT(E)T∗(E),(6)

where eis the charge of the particle, Eis its incident

energy, and Tis the transmission coeﬃcient. The above

conductance is observed to have an oscillatory behavior

in energy, with maxima reaching the value e2/h, and the

distance between them determined by the length L(see

App. D). In Fig. 2(c) we show the numerical result from

the simulation of a system consisting of two identical and

inﬁnite leads and a gated segment of L= 9 sites (see

App. C for a plot of the conductance as a function of V0

for ﬁxed L). The peaks in conductance occur once Eis

in resonance with the energies of the gated segment that

hybridise with the leads, i.e., the energies that lie within

the bandwidth of the leads. By counting the number of

peaks Npeaks, whose maxima reach the value e2/h, we can

extract the number of states that localise on the gated

segment as Nloc =L−Npeaks. The result for the gated

chain discussed before is shown in Fig. 1(b). We observe

perfect agreement with Nloc obtained by IPR, as well as

with the analytical prediction (4). Interestingly, a sim-

ple analytical calculation of conductance which assumes

that the leads do not aﬀect the dispersion of the gated

segment gives the same expression for Nloc as in Eq. (4);

see App. D.

III. FERROMAGNETIC SEGMENT IN AN

ANTIFERROMAGNETIC CHAIN

Similar quasiperiodic behaviour occurs also in related

systems of relevance to other experimental settings. Con-

sider for instance the case of an Ising chain where + and

−correspond to an on-site energy ±Wfor a tight-binding

particle along the same chain, with hopping amplitude

t= 1 (where without loss of generality we set W > 0).

Speciﬁcally, take an inﬁnite antiferromagnetic (AFM)

chain with a ferromagnetic (FM) insertion + + . . . + of

length L(see Fig. 3(a) for a precise illustration of how

we deﬁne the inserted FM segment).

The dispersion of the AFM leads is given by

EAFM(k) = ±pW2+ 2 (1 + cos k),(7)

which exhibits a gap near zero energy, in contrast with

the case considered earlier. The dispersion of the FM

segment is instead given by the usual relation

EFM(k) = W−2 cos k. (8)

Both dispersions are illustrated in Fig. 3(a). Note that

the FM segment overlaps with both bands of the leads

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：

EmergenceofquasiperiodicbehaviorintransportandhybridizationpropertiesofcleanlatticesystemsCecilieGlittum,1AntonioStrkalj,1andClaudioCastelnovo11T.C.M.Group,CavendishLaboratory,JJThomsonAvenue,CambridgeCB30HE,UnitedKingdom(Dated:January4,2023)Quasiperiodicbehaviourismostlyknowntooccurinsystemswithen...

展开>> 收起<<

Emergence of quasiperiodic behavior in transport and hybridization properties of clean lattice systems Cecilie Glittum1Antonio Strkalj1and Claudio Castelnovo1.pdf

共12页,预览3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！

Emergence of quasiperiodic behavior in transport and hybridization properties of clean lattice systems Cecilie Glittum1Antonio Strkalj1and Claudio Castelnovo1

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: