Conned states in the tight-binding model on the hexagonal golden-mean tiling Toranosuke Matsubara1 Akihisa Koga1and Sam Coates2

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Conﬁned states in the tight-binding model on the

hexagonal golden-mean tiling

Toranosuke Matsubara1, Akihisa Koga1and Sam Coates2

1Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan

2Department of Materials Science and Technology, Tokyo University of Science, Katsushika,

Tokyo 125-8585, Japan

E-mail: matsubara@stat.phys.titech.ac.jp

Abstract. We study the tight-binding model with two distinct hoppings (tL, tS) on the two-

dimensional hexagonal golden-mean tiling and examine the conﬁned states with E= 0, where

Eis the eigenenergy. Some conﬁned states found in the case tL=tSare exact eigenstates even

for the system with tL6=tS, where their amplitudes are smoothly changed. By contrast, the

other states are no longer eigenstates of the system with tL6=tS. This may imply the existence

of macroscopically degenerate states which are characteristic of the system with tL=tS, and

that a discontinuity appears in the number of the conﬁned states in the thermodynamic limit.

1. Introduction

Quasicrystals have attracted much interest since the discovery of the quasicrystalline phase of

Al-Mn alloy [1]. Among them, electron correlations in quasicrystals have actively been discussed

after the observation of quantum critical behavior in Au-Al-Yb [2]. Recently, long-range

correlations have been observed – such as superconductivity in the Al-Zn-Mg quasicrystal [3]

and the ferromagnetically ordered states in the Au-Ga-Gd and Au-Ga-Tb quasicrystals [4].

These experiments have necessarily stimulated further theoretical investigations on electron

correlations in quasiperiodic tilings [5–14].

A simple example of such investigations is the study of magnetically ordered states in the

Hubbard model on the bipartite quasiperiodic tilings, i.e., the Penrose [15, 16], Ammann-

Beenker [17–19], and Socolar dodecagonal tilings [20]. Non–interacting systems on the above

tilings have a common feature in the density of states, namely, macroscopically degenerate states

at E= 0, so-called conﬁned states. These play an essential role for stabilizing the magnetically

ordered states, in particular, in the weak coupling regime. Therefore, to understand magnetic

properties of quasiperiodic tilings, it is important to examine their conﬁned states under the

tight-binding model [16, 19–23].

Recently, the hexagonal golden-mean tiling has been introduced [24], with a section of the

tiling and its constituent tiles shown in Fig. 1. This tiling is composed of large rhombuses,

parallelograms, and small rhombuses, and, one of its important features is the existence of two

length scales which is in contrast to the Penrose, Ammann-Beenker, and Socolar dodecagonal

tilings. In our previous paper [25], we have considered the vertex model on the hexagonal golden-

mean tiling, where the hopping integral on each edge is assumed to be equivalent. However, it

arXiv:2210.15108v1 [cond-mat.str-el] 27 Oct 2022

(a)

(b)

LPS

Figure 1. (a) Hexagonal golden-mean tiling [24]. Blue thin and red bold lines indicate the

hopping integrals tLand tSdeﬁned on the long and short length edges. Solid circles indicate

the F0, F4, and C0vertices. (b) Large rhombus, parallelogram, and small rhombus.

is also instructive to clarify conﬁned state properties in the tight-binding model with distinct

hoppings.

The paper is organized as follows: in Sec. 2, we introduce the tight-binding model on the

hexagonal golden-mean tiling and show the density of states. In Sec. 3, we discuss conﬁned state

properties in the model and compare our results to our previous work and the Penrose tiling. A

summary is given in the last section.

2. Tight-binding model on the hexagonal golden-mean tiling

Here we brieﬂy summarise relevant properties of the hexagonal golden-mean tiling as introduced

in the original paper [24]. The tiling can be generated using deﬂation rules for eight distinct

directed tiles, and there are 32 allowed vertex conﬁgurations. Important for further discussion

are the F0, F4, and C0vertices. The F0vertex is located at the centre of six adjacent large

rhombuses and locally has 6-fold rotational symmetry, while the F4, and C0vertices locally have

3-fold rotational symmetry.

As the tiling is multi-length-scale, we can introduce two kinds of hopping integrals in the

tight-binding model. The tight-binding model with two distinct hoppings is given as

H=−tLX

(ij)

(c†

icj+ H.c.)−tSX

hiji

(c†

icj+ H.c.),(1)

where ci(c†

i) annihilates (creates) an electron at the ith site. tL(tS) denotes the transfer integral

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摘要：

Connedstatesinthetight-bindingmodelonthehexagonalgolden-meantilingToranosukeMatsubara1,AkihisaKoga1andSamCoates21DepartmentofPhysics,TokyoInstituteofTechnology,Meguro,Tokyo152-8551,Japan2DepartmentofMaterialsScienceandTechnology,TokyoUniversityofScience,Katsushika,Tokyo125-8585,JapanE-mail:matsubar...

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Conned states in the tight-binding model on the hexagonal golden-mean tiling Toranosuke Matsubara1 Akihisa Koga1and Sam Coates2

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