Cn -symmetric quasi-periodic Chern insulators

Chern insulators have been recently extended to quasicrystal systems, namely the quasi-periodic Chern insulators (QCIs). Here we study the topological properties and topological response of QCIs on two-dimensional singular surfaces. Such singular QCIs with arbitrary n− fold rotational symmetry (i.e. C n -symmetric QCIs) can be constructed by ‘cutting and gluing’ unit sectors on the Dürer’s pentagonal tiling. Chiral edge states and real-space Chern number can well characterize the topological properties of C n -symmetric QCIs. Intriguingly, we numerically identify the emergence of charge fractionalization in unit of e/10 around the singular center in Cn -symmetric QCIs though their bulk densities are inhomogeneous. In addition, we explore the phase transitions of these C n -symmetric QCIs.


Introduction
The discovery of topological states, for example, the integer quantum Hall states in two dimensional electronic gas with the strong magnetic field [1], opens up a new field in classifying various quantum phases beyond the Landau's symmetry-breaking paradigm.By introducing the staggered fluxes through plaquettes, integer quantum Hall states can be achieved on lattice models in the absence of the external magnetic field, such as the famous Haldane model [2].Topological states in Haldane-model-like systems have been successively proposed [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].These systems are dubbed as Chern insulators (CIs) because their topological nature can be characterized by the non-zero Chern number [19].Recently, CIs have been extended to aperiodic systems without translational symmetry [20,21].Particularly, quasi-periodic CIs (QCIs) have been constructed in Dürer's pentagonal tiling by imposing staggered fluxes through the plaquettes [21].The topological properties of QCIs are characterized by chiral edge state and quantized topological invariant.In contrast to the Chern number well defined by integrating the Berry curvature over the first Brillouin zone in periodic systems, topological invariant of QCIs cannot be calculated directly in the reciprocal space on account of lack of translational symmetry.Fortunately, the real-space Chern number was proposed to characterize the topological properties of CIs without translational symmetry.There are several scenarios to calculate the real-space Chern number, including the Kitaev formula [22], the local Chern marker (LCM) [23], the Bott index [24], etc.
Intriguing phenomena have been reported when CIs were proposed in singular lattices with arbitrary n-fold rotational symmetry [25][26][27][28][29], such as defect-core states, fractional charge and multiple branches of edge excitations.These singular lattices can be constructed by 'cutting and gluing' unit sectors with a disk geometry.The topological nature of singular CI states is identified by the real-space Chern number due to the absence of translational symmetry in finite lattice systems, similar to the QCIs [21].Some schemes to calculate the real-space Chern number have been proposed to identify the curved CIs [26], i.e. the Kitaev's formula and the LCM.Here, the curved CIs indicates CIs with arbitrary n−fold rotational symmetry and isolated singularity or disclination appears around the center.Besides, as a topological response of singular CIs, fractional charge emerges around the singularities.For example, the charge fractionalization in unit of e/4 has been reported in the curved-Haldane and curved-Kagomé models [25,27,29,30].How to construct and characterize the topological properties of curved-QCIs and whether charge fractionalization emerges in curved-QCIs, remain open.
In this paper, we study singular QCIs and explore their topological properties.Starting from the QCIs established in Dürer's pentagonal tiling, we successfully construct singular QCIs with arbitrary n-fold rotational symmetry (also dubbed as C n -symmetric QCIs) by the 'cutting and gluing' method.We characterize the topological nature of C n -symmetric QCIs based on the chiral edge states and the real-space Chern number.More intriguingly, charge fractionalization in unit of e/10 around the singular center of the C n -symmetric QCIs is observed, even though the many-particle densities in the bulk are inhomogeneous.Topological phase transitions of the designed n-fold QCI are further discussed.Our studies shed light on the topological characterization and topological response of C n -symmetric QCIs and provide a route to search more topological materials in curved lattices.

Models and single particle states
The prototype of the QCI we adopted is a C 5 -symmetric QCI in the Dürer's tiling [21] with staggered fluxes threading the plaquettes as illustrated in figure 1(a).The real-space Hamiltonian is given by, Here, a † r (a r ) creates (annihilates) a fermion at vertex r, ⟨rr ′ ⟩ runs over all the nearest neighbor (NN) vertexes, and ♢, ⟨rr ′ ⟩ ′ indicates the next-nearest-neighbor (NNN) vertexes in diamonds with hopping potential t ′ .ϕ r ′ r is the phase difference between the NN vertexes with amplitude ϕ.We set the NN hopping potential t as unit, t ′ = −1.3 and ϕ = 0.2π, unless specified.
Here, we study the curved Dürer-QCI with arbitrary n-fold rotational symmetry.These curved quasicrystals can be constructed by 'cutting and gluing' unit sectors (highlighted in figure 1(a)) and a Dürer's tiling in the plane.This similar 'cutting and gluing' process has been implemented to explore the topological properties of various curved-topological states [25-28, 30, 31].For simplicity, we map the curved quasicrystal into a plane (figures 1(b)-(d)) by preserving the length of the ith vertex, i.e.
) denotes the ith vertex position in a curved surface (a mapped disk) and we choose the center of the quasicrystal as the coordinate origin.This mapping process preserves the rotational symmetry [26] as well.Consequently, we can investigate the topological characteristic of the singular QCIs with arbitrary n−fold rotational symmetry.
The present Hamiltonian (equation ( 1)) hosts n-fold rotational symmetry, and therefore the angular quantum number L is a good quantum number.We can obtain the single-particle energy spectra arranged in various angular momentum sectors (as illustrated in figure 2) by exact diagonalization.There are chiral edge states and defect-core states in the gap (details shown in in figure 2).Particle densities for edge states and defect-core states of singular QCIs are shown in figure 3.These edge states propagate along the edge robustly and the defect-core states are localized around the center of tilings because of the emergence of defects around the center of singular QCIs.The in-gap chiral edge states manifest the existence of topology of these C n -symmetric QCIs, in analogy to the previously reported C 5 -symmetric QCI in disk geometry [21].We can establish the direct link between the arbitrary C n -symmetric QCI and the prototype C 5 -symmetric QCI, similar to the relationship between CIs and C n -symmetric CIs [26,27].

Topological properties
The in-gap states are mainly localized near boundaries based on the real-space distributions of densities shown in figure 3, i.e. the chiral edge states localized along the edge and defect-core states localized around the center.These space distributions can reveal the existence of topological property in these C n -symmetric QCIs.To characterize their topological properties, we calculate their real-space Chern number based on the Kitaev's formula.The Kitaev's formula is expressed as [22], where 'A' , 'B' , and 'C' mark three distinct neighboring regions arranged in the counterclockwise order in the bulk (which are shown in figure 7(a) in the appendix).P ij is the element of the projection operator P defined  up to the fermi energy E F , i.e.
, where ϕ n (r i ) = ⟨r i |ϕ n ⟩ is the real-space single-particle state at the ith vertex with energy E n .
On the basis of the Kitaev's formula, one can find the number C is related to the chosen Fermi energy, i.e.C ≡ C(E F ).The number C of C n -symmetric QCIs as a function of the Fermi energy E F is displayed in figure 4. Apparently, the real-space Chern number plateaus emerge when the Fermi energy E F locates in the bulk gap with C = +1, which manifests the existence of CI states in these singular QCIs.We also find that the sizes of the real-space Chern number plateaus are the same for various C n -symmetric QCIs, including the C 5 -symmetric QCI [21].Except for the real-space Chern number plateaus, there are noninteger number C in the metallic state which just correspond to the nonquantized Hall conductivity and cannot identify topological properties because the Chern number as a topological invariant should be an integer.In addition, we can characterize the topological properties based on the LCM (details shown in the appendix A.1.)

Charge fractionalization
Charge fractionalization has been proposed in fractional quantum Hall and fractional Chern insulator states.Moreover, as a topological response of the topological state, some non-interactive systems also host fractional charge, such as the SSH model [32,33], the Kekulé graphene [34], the topological states in singular surfaces or lattices [25,27,28,30,31] and the higher-order topological insulators [35][36][37][38].Whether the fractional charge also exists in the curved QCIs?
Inspired by the calculation of the excessive charge of C n -symmetric CIs, the fractional charge (in units of e) around the center of C n -symmetric QCIs is, where r max is the upper limit of the summation, ρ r (C n ) is the many-particle density of C n -symmetric CIs along the radial direction r, and N r is the number of sites at radius r.The many-particle density is defined as [25,27,28,30,31] where ϕ n (r) is the nth single-particle state.
Here we consider a series of curved QCIs with radius r = 18 and fixed radius r max = 10 to exclude the finite-size effect.The many-particle densities of C n -symmetric QCIs displayed in figure 5(a) are highly inhomogeneous, in analogy to that of the C 5 -symmetric QCI in Dürer's tiling [21].Especially, the many-particle densities around the center are distinct from each other.More importantly, the difference of many-particle densities between the curved-QCIs and the C 5 -symmetric QCI is, e * = [(5 − n) mod 10] /10. ( The corresponding excessive charges around the center is shown in figure 5(b), i.e. which reveals the excessive charge is exactly quantized in unit of e/10.Therefore, we identify the existence of charge fractionalization in C n -symmetric QCIs.Such fractional charge in unit of e/4 was also observed in the curved-Haldane model and the curved-Kagome model [25,27,30], further manifesting the similarity between the curved-QCIs and the curved-CIs.Notably, this fractional charge emerges in noninteracting systems, instead of interacting systems.Unlike quasiparticle excitations in fractional quantum Hall states or fractional Chern insulators, this fractional charge appeared in singular systems is bound to topological defects in a classical field and without emergent dynamical excitations [30].Therefore, the present charge fractionalization is closely related to the singularities around the center of systems.An reasonable explanation to the e/4 fractional charge in the curved-CIs is the excessive berry phase with a fictitious flux which comes from the isolated disclination in CIs with non-zero Chern numbers [25,30].The fictitious flux pierces the defect core and finally leads to the charge fractionalization.At present, we are still unclear how to connect the fictitious flux and the e/10 fractional charge in the curved-QCIs and will leave the open problem for future studies.On the other hand, Wen-Zee term [39] has been extended to lattice systems with discrete rotation symmetries to explain the fractional charge as well [28].How to establish the relationship between fractional charge in QCIs and the Wen-Zee term remains to be further discussed.

Topological phase transition
Topological phases have been studied on torus geometry [2,9,[12][13][14][15][16][17]40] and various phases can be identified by the well-defined Chern number in momentum space.In disk geometry, CI phases have been explored based on the edge-state spectra and the local density of states [41].However, it is a tedious process to search CI phases in large parameter spaces.Fortunately, the real-space Chern number is proposed to identify various CI phases in curved CIs [26], QCIs [21], and amorphous CIs [20].
Based on the Kitaev's formula [22], we investigate the topological phase transition of curved QCIs by tuning the staggered fluxes (ϕ) and the NNN hopping potentials (t ′ ) in diamonds.Figure 6 displays the topological phase diagram of the C 6 -symmetric QCI.Three phase regions emerge marked with Chern number C = 0 and C = ±1, respectively.Here, the Fermi energy is chosen the 288th energy in the C 6 -symmetric QCI.The phase diagrams of other curved-QCIs are very similar, including the C 5 -symmetric QCI (figure 10 in appendix A.2), which manifests the rotational symmetries cannot commonly induce topological phase transitions.The same case occurs in the C n -symmetric CI [26], which is another similar feature between the curved-QCIs and curved-CIs.

Summary and discussion
We have systemically studied the topology of the C n -symmetric QCIs by 'cutting and gluing' unit sectors with a Dürer tilling.The topological properties of the curved QCIs are well characterized by the chiral edge states and the nontrivial real-space Chern number.Similar to the conventional C n -symmetric CIs, the quantized charge fractionalization (equation ( 5)) near the singular center in C n -symmetric QCIs is also found in unit of e/10 even though the many-particle densities are highly inhomogeneous.In addition, we explore the topological phase transitions of C n -symmetric QCIs and find a similar feature between the curved-QCIs and the curved-CIs.
Our work reveals the topological nature of C n -symmetric QCIs and provides directions for the future research in topological matter physics.For example, it is very interesting to investigate the curved-CI systems with higher Chern number.Furthermore, other topological states have been proposed in quasicrystals [38,[42][43][44][45][46][47][48][49][50], such as the quantum spin Hall state, the topological crystalline insulator, the higher-order topological insulator, etc.It may be interesting to construct these curved topological insulators and identify their topological properties.It is very meaningful to identify the C n -symmetric fractional Chern insulators which host novel features [51,52], including many branches edge excitations and geometry-dependent wave functions.Other topological states with defects have been proposed [53][54][55][56].Identifying these topological states [57,58] with defects and exploring charge fractionalization will be very attractive and significant as well.

A.1. Real-space Chern number
To calculate the real-space Chern number based on Kitaev's formula, the bulk of systems should be divide into three regions arranged in the counterclockwise order and these three regions are marked with 'A' , 'B' and 'C' .Here, we take a C 6 symmetric quasicrystal as an example and we divide its bulk into three parts marked with various colors (shown in figure 7(a)).
LCMs can be used to characterize the topological nature of the curved QCIs as well.Following previous works [23], the definition of LCMs is expressed as, where is the matrix element of the projection operator P, and P(r i , r j ) = ∑ E λ <EF ⟨r i |ψ λ ⟩⟨ψ λ |r j ⟩.Q is the complementary projector (i.e.Q = Î − P ) with the matrix element Q(r i , r j ) = ∑ E λ >EF ⟨r i |ψ λ ⟩⟨ψ λ |r j ⟩.And r i = (x i , y i ) is the position of the ith vertex.From equation (6), one can find the LCM depends on the position of the vertex.
The bulk LCMs are inhomogeneous which differs from the CIs in lattice models.The averaged bulk LCMs can be as a real-space Chern number to identify various CIs which is defined as We select the region D with radius r D = 10.706shown in figure 7(b).The averaged Chern number C D = 0.999, which is very close to integer.However, this averaged Chern number is not stable C D ∼ 1 ± 0.05, depending on the selected region D, which has been reported in previous work [21].
Based on the definition of the LCMs (equation ( 6)), we can obtain the LCMs on every site in a unit section of arbitrary C n -symmetric QCIs.Here, we show the LCMs in a unit sector of C 4 -, C 5 -, C 6 -and C 8 -symmetric QCIs in figure 8.We numerically find the bulk LCMs satisfy C n (r) = 5/n × C 5 (r).And in other CIs, the bulk LCMs satisfy the similar behavior, i.e.C n (r) = N/n × C N (r), where the perfect CI models  in disk geometry with C N rotational symmetry.Here, we consider checkerboard model and Haldane model.We numerically find the bulk LCMs (in figure 9) are homogeneous with C n (r) = 4/n × C 4 (r) (for checkerboard model, detail can refer to [26]) and C n (r) = 6/n × C 6 (r) (for the Haldane model).

A.2. Topological phase transition in curved QCIs
We have displayed topological phase transitions of the C 6 -symmetric QCI in the main text.Here, we present phase diagrams of QCIs with other rotational symmetries, including the C 5 -symmetric QCI (details in figure 10).We find the phase regions are roughly similar for various C n -symmetric QCIs.Because of the finite-size effect and different rotational symmetry, a little slight differences emerge in these phase diagrams.The similar features of topological phase transitions in the curved CIs have been reported [26] as well.

Figure 1 .
Figure 1.Schematic curved Dürer-QCI lattice model.(a) Prototype of the QCI model in the Dürer's tiling with 5-fold rotational symmetry.The selected gauge is explicitly shown by the arrows, which introduce an additional phase factor ±ϕ for part of the nearest-neighbor hopping process.The arbitrary Cn-symmetric QCI can be constructed by 'cutting and gluing' unit sectors (highlighted with gray) in a quasicrystal disk.(b)-(d) Curved Dürer-QCI models are mapped into a disk geometry with (b) 4-, (c) 6-, and (d) 8-fold rotational symmetry, respectively.

Figure 2 .
Figure 2. The single-particle energy spectra of Cn-symmetric QCIs depending on the angular momentum L. In (a)-(e), n = 4, 6, 7, 8 and 10, respectively.The edge states and several defect-core states are respectively colored with blue and red emerge in the gap.Here, quasicrystals are with 196n vertexes.

Figure 4 .
Figure 4. Number C as a function of the Fermi energy EF in curved QCIs with (a) 4-, (b) 6-, (c) 8-and (d) 10-fold rotational symmetry.There is a Chern number C∼ + 1 plateau which suggests the existence of the CI state in these Cn-symmetric CIs.

Figure 5 .
Figure 5. (a) Many-particle densities of Cn-symmetric QCIs filling with N = 48n free spinless fermions.The lattice radius is near 18, and rmax is the upper limit of summation (in equation (3)).(b) Charge fractionalization in the unit of e/10 as a function of n in Cn-symmetric QCIs.

Figure 6 .
Figure 6.Topological phase diagrams of the C6-symmetric QCI by tuning the staggered flux parameter (ϕ) and the hopping potential (t ′ ).We mark every phase region using the real-space Chern number C = 0 or ±1 calculated from the Kitaev formula.

Figure 7 .
Figure 7. (a).Schematic illustration of three neighboring regions in counterclockwise order marked with 'A' , 'B' and 'C' , which is used to calculate the real-space Chern number by Kitaev's formula.(b).LCMs of the QCI and the selected bulk region circled by the red line.

Figure 9 .
Figure 9. LCMs in (a) the checkerboard CI model and (b) the Haldane model.These two CIs are with Chern number C = +1.