Flat-bands in translated and twisted bilayer Penrose quasicrystals

Correlated phases in Moiré materials together with the flat-bands in twisted systems play a central role to explain superconductivity in the new twisted bilayer graphene. In this paper, flat-bands are shown to exist in both translated and twisted bilayer of quasicrystals. Such flat-bands arise for different displacements and twisting angles of two-coupled Penrose lattices where Moiré patterns are also shown. Moiré patterns analyzed in this work have at least two inverted worms showing an interference pattern going along the five-fold axes of the pentagon. In order to analyze the behavior of the flat band, our study has been done for fixed interference worm directions but increasing the worm interference density, and for fixed worm interference density but increasing the number of worm directions. In case of rotations, the Moiré patterns that occurs for special angles such as π/5, 2π/5, 3π/5, 4π/5 and π are discussed in detail because they clearly show flat-bands along with quasicrystalline electronic states at the Fermi level.


Introduction
The recent experimental results of robust superconducting states in twisted bilayer graphene [1], open new ideas to visualize an explanation of the copper oxides superconductors which remain as an open question.The experiments exhibits both unconventional superconducting phases [1,2] and Mott insulating states [3], the superconducting phase shows the existence of narrow bands created by the Moiré pattern arisen from twisting two graphene sheets till a magic angle of 1.1 • .Near the magic angles of these Moiré materials, flat-bands appear when the Fermi velocity goes to zero together with an increase Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
of the electron-electron interaction, enhancing the possibility of finding different correlated phases.The features of the observed superconductivity in twisted bilayer graphene are very similar to those of high-T C superconductors [4], including the strong electron-correlation and the narrow bandwidth.
Several studies have been done to analyze electronic correlation, one of the simplest models which consider the strong electron-electron interaction is the Hubbard Hamiltonian [5], this model includes implicitly all kind of effects to obtain electron pairing through the parameters of the Hamiltonian.The renewed interest in this Hamiltonian comes due to the fact that it contains the basic ideas to investigate the electron or hole pair states at narrow bands, and the dynamics of these pairs who are believed to be fundamental to explain unconventional superconductivity.In previous work [6][7][8], we considered widely electron-pairing in different lattices within the Hubbard Hamiltonian in real space allowing us also to analyze pairing even in complex and non-periodic lattices like the one-dimensional Fibonacci lattice [9,10].
The physics of flat-band in twisted systems has been of tremendous interest in the last years.However, there are many open questions concerning their effects even in the oneparticle limit.For example, flat-bands induce antiferromagnetic and ferromagnetic domains at weak electron-electron interaction [11], transport in bilayer materials with angle disorder is weakly affected [12], and the appearance of quasicrystalline electronic states far away from the Fermi level for a 30 • twisted bilayer graphene [13,14] or in 45 • twisted square lattices [14].Shortly after the discovery of quasicrystals, the Moiré patterns were observed in two identical Penrose tilings translated one with respect to the other [15], unfortunately not too much discussion was done regarding the electronic states.
The tight-binding Hamiltonian together with the Wannier wave functions is the most useful Hamiltonian to study the one-body problem not only in periodic lattices but also in quasiperiodic ones.Quasicrystals display a non periodic arrangement of atoms but their structure is not random either, since Bragg peaks are shown in the diffraction pattern even though have non crystalline symmetry, i.e. the Bloch theorem no longer holds.The electronic states of quasicrystals shows a remarkable rich behavior [16], including critical states [17,18], fractally confined states [19], unconventional conduction properties [20][21][22], fractal superconductivity [23] and conventional superconductivity [24].
In this work, we show an analysis of the electronic effects of translated and twisted bilayer quasicrystals on the flat-band, together with a detailed discussion of the wave function behavior when the Moiré patterns appear.This analysis has been done for fixed interference worm directions but increasing the worm interference density, and for fixed worm interference density but increasing the number of worm directions.

Model
Using a tight-binding model the Penrose lattice has already been studied in detail where important results have been found, for example, the electronic density of states (DOS) is symmetric due to the bipartite character of this quasicrystalline lattice and shows a Van-Hove singularity for E = 0 (see figure 1(a)).The zero-energy eigenstates are confined in a fractal distribution region (fractally confined) [19,25].Therefore, the wave functions have amplitude only in the majority sublattice and no amplitude on the minority sublattice [26].
The bilayer Penrose lattice, a decagonal quasicrystal, is modeled using the tight-binding Hamiltonian in the following form: where H B , H T are the Hamiltonians for the bottom and top layers respectively and H ⊥ = H † ⊥ is the inter-plane Hamiltonian.The plane Hamiltonians H B , H T are given by: where t 0 is the hopping parameter in a Penrose lattice, |i, B⟩ is a localized function at site i in the bottom plane and |m, T⟩ in the top plane, the sums take into account only nearest neighbors.The inter-plane Hamiltonian is defined by [27]: here, t is a constant, r i,m is the distance from site i in the bottom plane to site m in the top plane, d is the distance between planes and λ is a decaying factor.In this paper, we consider λ big enough that the hopping is t if the bottom layer site is exactly below a top layer site, and the hopping is zero otherwise.The case of two-coupled Penrose lattices without a translation or a rotation have also been considered as a reference, the electronic DOS is shown in figure 1(b), where two Van-Hove singularities can be observed.In this calculation the hopping parameters are equal, i.e. t 0 = t = −1.0.Moiré patterns are not present and the wave function can be observed in figure 2 for E = 0, showing a critical behavior in both, the bottom layer and the top layer.Figure 3, shows the behavior of the wave function which is fractally confined for E = 1.0 or −1.0 in both, the bottom and the top layer.

Results and discussion
Moiré patterns in quasicrystals can be obtained through a displacement or a rotation of two-coupled Penrose lattices.The Moiré pattern in bilayer Penrose lattices consists on destructive interference of four type of verticess, K site, Q site, D site and composite site obtained by a K site and two Q sites, which will be called the KQQ site [28] (see figure 4).This decagon is a cartwheel within the Jack configuration [29].
For a displacement or a rotation between the two-coupled Penrose lattice, the Moiré pattern looks like worm lines [30] having lines and intersections which destructively interfere, and patches of constructive interference where the two Penrose  lattices perfectly match (figure 5).The constructive interference shows the translational symmetry of the Penrose lattice.The destructive interference shows in general a 5-grid, where each curve (worm) of each grid is perpendicular to the 5-fold axes of the pentagon.In each grid, every worm (see figure 6) is separated to the adjacent curves by a distance L or S, where the sequence of these distances is the Fibonacci sequence.confined states with energy E = 0, and are completely contained in a finite region.These states appear also in the twocoupled Penrose lattice but with energies E = ±1.0(bonding and antibonding state).The worm interference density leads to a frustration in the formation of these confined states.In the case of rotations or twisted bilayer Penrose lattices, the Moiré patterns arise for different twisted angles, in particular, results are shown for the following special angles π/5, 2π/5, 3π/5, 4π/5 and π, see figure 8.The behavior of the wave function is localized for E = 0 in both, the bottom and top layers.For E = ±1.0, the wave function is null for the destructive interference region and fractally confined for the constructive interference area, in both layers.Figure 8, also shows the electronic DOS for different twisted angles of the bilayer Penrose lattices.In all these electronic DOS we can observe a peak for E = 0 indicating the existence of flat-bands, the Fermi velocity goes to zero enhancing the possibility of finding different correlated phases, like the superconductive one.In order to understand the appearance of flat-bands when the Moiré patterns arise, we will analyze in detail a single worm and a pair of worms (Moiré patterns of figure 5), see figure 9.
Different angles can lead to different worm interference directions as figure 8 shows.The interference patterns lay along the 5-fold axes directions.The number of interference directions also affect the DOS.We can see, in figure 8 for the electronic DOS, that increasing the number of direction increases the peak of the DOS at E = 0. Also, the peaks at E = ±1.0  are affected by the frustration of the localized states due to the appearance of the interference worms.

Single worm
In figure 10, we can observe that a worm has a repetitive structure similar to connected hexagons in series.These hexagons can be long or short, but the hopping are the same.
Renaming the vertices of the above hexagons in the cell i by A i , B i , C i , D i , E i as is described in figure 11 and the localized eigenstates by In a real space basis, the Hamiltonian matrix applied to the above eigenstates gives: In a reciprocal space using the orthonormal basis, defined as: where k = 0, 2π  where . The basis change disconnect the system in N systems of 5 equivalent sites for any given value of k.An eigenstate of the Hamiltonian has the form: where ψ(X) ∈ R for X ∈ {A, B, C, D, E}.

Double worm
In order to make the interference pattern shown in figure 9, the two worms that show the Moiré pattern, we need a worm and   his inverted worm on top.The inversion symmetry of this system gives us two solutions, the symmetrical and the antisymmetrical.The structure of the wave function and its equivalent Hamiltonian is shown in figures 12 and 13 for the symmetrical and the antisymmetrical solution, respectively.In both cases, we will have solutions with E = 0 for any value of k.
For the symmetrical solution, we have that nodes A are connected twice with nodes B and the node E is connected with node D. The solution with energy equal zero has the form  β .In the simplest case, we can construct a Moiré pattern that has the same inversion symmetry described above, as follows, starting with two inverted worms that identically grow the Penrose quasicrystal at both sides, then we couple two of these Penrose lattices using the worm as center.This quasicrystal will look exactly like two Penrose lattice on top of each other except for a line of destructive interference worms as is shown in the figure 9. We can no longer use the same basis along the worms because they are not periodic anymore, but we can use the inversion symmetry argument to find the symmetrical and antisymmetrical solutions.These systems will behave like a two coupled identical Penrose lattices but the worm will be replaced with an edge, where there are hopping of value 2t between A and B like sites in the symmetrical case.There is no hopping between A and B like sites in the antisymmetrical case.The interference caused by the worms can be seen as hopping impurities in a semi-infinite two Penrose lattices.It is known that hopping impurities can lead to bound states [31] and this is exactly the phenomenon that we are having at the interference of worms.

Conclusion
We have established the existence of flat-bands for both, the translated and the twisted bilayer of quasicrystals.These results enhance the possibility to find different correlated phases in quasicrystals like the superconductivity one.The appearance of flat-bands that result from the Moiré patterns, were analyzed for both displacements and rotations of the twocoupled Penrose lattices, in particular, the electron localization is studied in detail.In the translated case the destructive interference shows in general a 5-grid.The behavior of the wave function for E = 0 and E = 1.0 was shown along with the electronic DOS.The density of interference worms can lead to a frustration of the localized states, decreasing the states with energy E = ±1.0.In the rotation case, the quasicrystalline Moiré patterns occurs for the following special angles π/5, 2π/5, 3π/5, 4π/5 and π.The behavior of the electronic DOS for the above rotation angles were shown, where it can be observed a central peak, indicating the existence of flat-bands.The number of directions of the interference worms is changing which have an effect in the states with energy E = 0, the more directions the more states.

Figure 1 .
Figure 1.Electronic density of states for one (a) and two (b) Penrose layers.

Figure 2 .
Figure 2. Wave function for E = 0 in both the bottom and top Penrose layers.

Figure 3 .
Figure 3. Wave function for E = 1.0 or −1.0 in both the bottom and top Penrose layers.
The worms are constituted by D and Q sites which follow a Fibonacci sequence of the type: LSLLSLSLLSLLSLSLLSLSL . . ., where L is the large distance and represents two D hexagons, and S is the short distance and represents one D hexagon.Each element of the Fibonacci sequence is separated by a Q hexagon.The KQQ site works as a rotation center where different worms intersect.Figures 5(a)-(d), shows the Moiré patterns for different displacement between the two-coupled Penrose lattices, together with their respective DOS.The wave function associated to each case is shown in figures 7(a)-(d) for E = 0 and E = 1.0 respectively.In a single layer Penrose lattice there are strictly

Figure 4 .
Figure 4. Four type of vertexes of a Moiré pattern in Penrose lattices.

Figure 5 (
a) shows that the electronic DOS has bigger peaks at energies E = ±1 than those in figure 5(b).

Figure 5 (
b) has bigger peaks than those in figure 5(c) and finally the pattern in figure 5(d) has almost no peaks at all.

Figure 7
shows that there are no localized states with energy E = ±1.0along the interference worms, and the pattern in figure 7(d) has almost no localized states at all.In other hand, the central peak at E = 0 is mildly affected by the density of patterns with a change of approximately 16% between those patterns in figures 5(a) and (d).

Figure 5 .
Figure 5. Moiré patterns with interfering worms in four directions formed by translation of two Penrose quasicrystals and their respective electronic density of states.For details of the worms see figures 6 and 9.

Figure 6 .
Figure 6.Structure of a worm with Fibonacci sequence.

Figure 7 .
Figure 7. Wave function for E = 0 (left) and E = 1.0 (right) with different density of interfering worms in both bottom and top Penrose layers.

Figure 8 .
Figure 8. Electronic density of states for various rotation angles of the bilayer Penrose quasicrystals.The Moiré patterns with interfering worms in different directions are also include.

Figure 9 .
Figure 9.A Penrose Moiré pattern having inversion symmetry and destructive interference, a zoom obtained from the worms at figure 5.

Figure 10 .
Figure 10.Structure of the sites of a single worm.

Figure 11 .
Figure 11.Estructure of a worm in real (a) and in reciprocal space (b).

Figure 12 .
Figure 12.(a) Symmetrical solution wave function in reciprocal space and (b) its equivalent Hamiltonian.

Figure 13 .
Figure 13.(a) Antisymmetrical solution wave function in reciprocal space and (b) its equivalent Hamiltonian.