Design and characterization of electronic fractals
Abstract
The dimensionality of a quantum system plays a decisive role in its electronic spectral and transport properties. In 3D, electrons behave as a non-interacting Fermi liquid, whereas in 1D interactions are relevant. On the other hand, in 2D exotic phenomena such as charge fractionalization may occur. However, very little is known about electrons in fractional dimensions. Here, we design and characterize an electronic Sierpiński triangle fractal in real and reciprocal space by confining the surface-state electrons of Cu(111) with adsorbed CO molecules. We observe single-electron wave functions in real space with a fractal dimension of 1.58 as well as a subdivision of the wave function in self-similar parts. These results open the path to fractal electronics in a systematic and controlled manner.
SReferences Supplementary Materials ^{†}^{†}thanks: Both authors contributed equally. ^{†}^{†}thanks: Both authors contributed equally.
Introduction: Fractals, structures with self-repeating patterns at any length scale and a non-integer dimension Mandelbrot , are pervasive in nature and emerge in a wide variety of research areas, such as polymers Newkome , porous systems Yu , stretchable electronics Fan , electrical storage Dubal ; Thekkekara2017 and disordered solids Soukoulis . On the quantum level, however, fractal structures and fractal behavior are much less common. Typical examples are the Hofstadter butterfly and the quantum Hall resistivity curve in a 2D electron gas Hofstadter ; Dean ; Ponomarenko ; Hunt ; Smet ; Goerbig . In disordered electronic systems, the wave functions at the transition from a localized to delocalized regime exhibit a multi-fractal behavior Evers . Electronic fractals, i.e. structures in which electrons are confined to a fractal geometry, have been studied from a theoretical perspective only. In the 80s, renormalization group (RG) methods were used to show that fractal confinement must result in recurrent patterns in the energy spectrum Domany ; Ghez ; Andrade1989 ; Andrade1991 ; Kappertz ; Wang . In the spatial domain, both extended and localized electronic states were predicted. More recent simulations of quantum transport in fractals revealed that the conductance fluctuations are related to the fractal dimension Katsnelson , and that the conductance in a Sierpiński fractal shows scale-invariant properties Chakrabarti ; Liu ; Lin . Experimentally, molecular fractal structures have been made with self-assembly of nanoscale Sierpiński hexagons Newkome and triangles with DNA Rothemund , and small aromatic molecules Shang ; Li . However, well-defined fractal structures that confine electrons have not been reported. Moreover, until recently, the experimental techniques did not allow for a full characterization of energy-resolved wave functions.
Here, we report a first step in this direction by showing how electronic fractals can be constructed and characterized: the electrons that reside on a Cu(111) surface are confined in a Sierpiński geometry that we constructed by controlled atomic manipulation of CO molecules on the Cu(111) surface. Similar approaches in a scanning tunneling microscope (STM) have been used before to create electronic structures ”on demand” such as molecular graphene Manoharan , the electronic Lieb lattice Drost2017 ; Slot2017 , the checkerboard and a stripe-shaped lattice Otte , and the quasiperiodic Penrose tiling Collins2017 . We characterize the first three generations of an electronic Sierpiński triangle by scanning tunneling microscopy and spectroscopy (STM/STS), acquiring the electronic local density of states (LDOS) in the entire energy domain at specific spatial positions and at variable energies over the entire Sierpiński structure (so-called wave-function maps), and show that the wave-function magnitudes themselves are fractal, with a Hausdorff dimension of 1.58 defined by the Sierpiński geometry. The self-similarity in real-space manifests itself in the subdivision of a fully bonding wave function delocalized over the third-generation Sierpiński triangle in self-similar first-generation parts at higher energy. Fourier filtering of the third-generation wave function in momentum space results in the second- and finally first-generation wave function, again highlighting the self-similarity of the geometric electronic fractal. The results are corroborated by muffin-tin and tight-binding calculations based on artificial atomic s-orbitals that are coupled in the Sierpiński geometry. These calculations also show another fractal feature, namely the existence of self-similar patterns in the energy-level spectrum.
Design of an electronic Sierpiński triangle fractal on a Cu(111) surface: The Sierpiński triangle with Hausdorff dimension is presented in Fig. 1A Sierpinski . We define atomic sites at the corners and in the center of the bright blue triangles as shown in Fig. 1B for the first generation Shang ; Oftadeh : has three inequivalent atomic sites that differ by connectivity. A triangle of generation consists of three triangles , sharing the red corner sites. The surface-state electrons of Cu(111) are confined to the atomic sites by adsorbed CO molecules, acting as repulsive scatterers. Fig. 1C shows the experimental realization of the first three generations of the Sierpiński triangle and Fig. 1D shows the relation with the atomic sites. The distance between neighboring sites is 1.1 nm, such that the electronic structure of the fractal will emerge in an experimentally suitable energy range Manoharan .
Differential conductance spectra: Figure 1E presents the experimental LDOS at the red, blue and green atomic sites in the G(3) Sierpiński triangle (cf. open circles in Fig. 1C). The differential conductance () spectra were normalized by the average spectrum taken on the bare Cu(111) surface, similar to Ref. Manoharan . The electrons on the Cu(111) surface occupy states above V. We focus on the bias window between V and V. Around V the LDOS on the red, green and blue sites is nearly equal, whereas slightly above V, the red sites show a remarked minimum, while the green and blue sites show a considerably higher LDOS. At V, the blue sites show a minimum, while the red and green sites exhibit a pronounced maximum in the LDOS. At V, the blue sites show a larger peak in the differential conductance, while the green and red sites exhibit a smaller peak. The experimental LDOS is in good agreement with both the tight-binding (see Fig. 1F) and muffin-tin simulations (see SI). This finding corroborates that our design leads to the desired confinement of the 2D electron gas to the atomic sites of the Sierpiński geometry. In addition, it allows us to characterize the wave functions of the chosen Sierpiński geometry in detail.
Spectrally and spatially resolved wave functions:
Figure 2 shows experimental wave-function maps obtained at different bias voltages and a comparison with simulations using a tight-binding and muffin-tin model. In a thought experiment, we will discuss how electrons can be transported between a source at the bottom left corner, and a drain positioned at the center of the right edge. At a bias voltage of V, the red (R), green (G) and blue (B) sites all have a high LDOS, and this also holds between the sites. Hence, from a chemical perspective, this wave function has strong bonding character. There is an excellent connectivity between source and drain along (R-B-G-B-R)-pathways, where the atomic sites have high amplitudes and are all in phase. At V, the red sites that connect the triangles have a low amplitude: the wave function of the triangle partitions into 9 parts, each corresponding to a Sierpiński triangle. The connectivity between source and drain along (R-B-G-B-R)-pathways suffers from the lower amplitude on the red sites. At V, the LDOS shows a marked minimum on the blue sites and a peak at the green and red sites (except the red corner sites). From the tight-binding calculation, we find that the wave function changes sign between R and G in the (R-B-G-B-R)-chains with nodes on the blue sites. From a chemical perspective, this is similar to a non-bonding molecular orbital. It is clear that the connectivity between the source and drain is broken, and that electrons have to perform NNN-hopping between the red and green sites to be able to reach the drain. At V, all blue sites in the Sierpiński structure have a high amplitude, whereas the red and green sites exhibit a low amplitude. Again, the connectivity between the source and drain is suppressed.
Fractal dimension of the Sierpiński wave functions:
In order to determine whether the electronic wave functions inside the Sierpiński structure inherit the fractal dimension of the lattice, we determine the dimension of the wave-function maps at different energies. We calculate the box-counting dimension Bouligand (also called Minkowski-Bouligand dimension) for both the experimental and simulated muffin-tin LDOS maps using Slot2017 . It can be clearly seen that the box-counting dimension of the Sierpiński triangle is around the theoretical Hausdorff dimension 1.58 (orange solid line), while the square lattice has a dimension close to 2 (blue solid line). Applying the same procedure to the muffin-tin simulated maps results in very similar fractal dimensions. , with N the number of circles needed to cover the contributing LDOS and r the radius of these circles. The method is presented in Fig. 3A and more details are given in the SI. Fig. 3B shows the box-counting dimension (dark orange) for the experimental wave-function maps acquired at different energies (see e.g. Fig. 2). For comparison, we also show the dimension obtained from the wave-function maps of a square lattice (dark blue), realized in the same way and measured in the same energy window
Fourier analysis:
The energy () vs. momentum () relation of the Sierpiński electrons was studied by transforming the real space wave-function maps to momentum space. In Fig. 4A,E-G, we recognize that the high-intensity maxima occur at higher k-values with increasing bias voltage. For instance, the most pronounced maxima at mV, mV, mV, and mV are positioned at nm (red dot), nm (green), nm (purple), and nm (orange), respectively (uncertainty 0.05 nm). Note that the maxima at nm and nm (brown) are visible at all energies, as they correspond to the next-nearest-neighbor (NNN) and nearest-neighbor (NN) distance between not only the electronic sites, but also between the CO molecules themselves. We plotted the above values as well as intermediate values in Fig. 4H. We observe that the values clearly follow the free surface-state electron parabola with and Burgi . However, only discrete values of the momentum are allowed, demonstrating the confinement of the electrons in the Sierpiński geometry. We now show how the self-similarity of the wave-function maps is reflected in momentum space. The Fourier-transformed wave-function map at mV (Fig. 4A) exhibits remarked maxima at nm (turquoise), nm (red, most pronounced), and nm (yellow). These maxima correspond to the NN-distances between the artificial atomic sites (see Fig. 1), the side of a triangle, and the side of a triangle in real space, respectively. We can transform parts of the Fourier map back into real space. The data inside the turquoise circle recover the full Sierpiński triangle, as shown in Fig. 4B. Transforming the values inside the red circle, however, results in a Sierpiński triangle of generation 2, while the size is retained (see Fig. 4C). Analogously, transforming the data inside the yellow circle yields a first-generation Sierpiński triangle (Fig. 4D). The self-similar features of consecutive generations of the Sierpiński triangle are thus inherently encoded in momentum space.
Conclusion: We have shown that it is possible to design several generations of an electronic Sierpiński fractal by accurate positioning of CO molecules on a Cu(111) surface. The surface-state electrons that reside in the structure define artificial atomic sites, and the coupling between these leads to wave functions that exhibit fractal properties in several respects. Although we have built an artificial fractal using the CO/Cu(111) platform in an STM, the rational concept of artificial-atomic building blocks coupled in a fractal geometry can be transferred to real semiconductor and metallic systems. In 2D systems with strong interactions, for instance, the electron charge becomes fractionalized in the presence of a strong perpendicular magnetic field. In the Sierpiński fractals that we have realized, interactions are still absent, but the single-particle wave functions already show a fractal nature, inherited from the underlying Sierpiński geometry. Our work paves the way to study many-body interactions, spin-orbit coupling, and the effects of external magnetic fields in real-material electronic fractals, possibly unveiling even more exotic types of electron fractionalization.
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subsection
Supplementary Materials
Design and characterization of electronic fractals
S. N. Kempkes,* M. R. Slot,* S. E. Freeney, S. J. M. Zevenhuizen,
D. Vanmaekelbergh, I. Swart, and C. Morais Smith
I Methods
i.1 Scanning tunneling microscope (STM) experiments
The STM and STS experiments were performed in a Scienta Omicron LT-STM system at a temperature of K and a base pressure around mbar. A clean Cu(111) crystal, prepared by multiple cycles of Ar sputtering and annealing, was cooled down in the STM head. Carbon monoxide was leaked into the chamber at mbar for three minutes and adsorbed at the cold Cu(111) surface. A Cu-coated tungsten tip was used for both the assembly and the characterization of the fractal. The CO manipulation was performed in feedback at nA and mV, comparable to previously reported values \citeSMeyer2, Celotta2, and was partly automated using an in-house developed program. STM imaging was performed in constant-current mode. A standard lock-in amplifier was used to acquire differential conductance spectra ( = 973Hz, modulation amplitude 5mV r.m.s.) and maps ( = 273Hz, modulation amplitude 10mV r.m.s.) in constant-height mode. The Fourier analyses were performed using the software Gwyddion.
i.2 Tight-binding calculations
The atomic sites in the first three generations of the Sierpiński triangle are modeled as -orbitals, for which electron hopping between nearest- and next-nearest neighbor sites is defined. The parameters used are eV for the on-site energy, eV for the NN-hopping and for the NNN-hopping, similar to the values reported in Ref. \citeSManoharan2. Furthermore, we included an overlap integral between nearest neighbors and solved the generalized eigenvalue equation , where is the overlap-integral matrix. The local density of states is calculated at each specific atomic site and a Lorentzian energy-level broadening of eV is included to account for bulk scattering. For the simulation of the LDOS maps, the same energy-level broadening was used and the LDOS at each site was multiplied with a Gaussian wave function of width , where nm is the distance between two neighboring sites.
i.3 Muffin-tin calculations
The surface-state electrons of Cu(111) are considered to form a 2D electron gas confined between the CO molecules, which are modeled as disks with a repulsive potential of eV and radius . The Schrödinger equation is solved for this particular potential landscape, and a Lorentzian broadening of eV is used to account for the bulk scattering.
i.4 Box-counting method
The box-counting method is a useful tool to determine the fractal dimension of a certain image, but has to be handled with care. In particular, as was shown in Ref. \citeSForoutan2, the size of the boxes needs to be chosen within a certain radius. More specifically, the largest box should not be more than 25% of the entire image and the smallest box is chosen to be the point at which the slope starts to deviate from the linear regime in the vs. plot. Redundant features such as the background Friedel oscillations were removed by applying a mask. Furthermore, the wave-function maps are not binary, and therefore it is necessary to specify the threshold value above which the pixels are part of the fractal set. The threshold is a certain percentage of the maximum amplitude of the wave-function map at a specific energy. The error introduced by the choice of the threshold is accounted for by performing the calculation procedure for several threshold percentages: 30%, 45%, and 60% for the top, center and bottom of the error bar for the experimental wave-function maps of the Sierpiński and square lattice, and 60%, 75%, and 90% for the top, center and bottom of the error bar for the simulated LDOS maps of the Sierpiński and square lattice. Since the difference in amplitude amongst pixels is less prominent in the experiment than in the calculated muffin-tin, the error due to the choice of the threshold is smaller for the experimental wave-function maps.
Ii Supplementary Information
ii.1 Design of the Sierpiński triangle
The behavior of electrons confined to zero, one, two or three dimensions has been widely described. Previous research, in particular from the experimental point of view, has mainly been limited to these integer dimensions. Fractional dimensions, which manifest themselves in fractal structures, could open a new range of possibilities. A suitable platform to create electronic fractals is constituted by the CO/Cu(111)-system. Here, surface-state electrons of the Cu(111) substrate are confined to a desired geometry by carbon monoxide molecules, acting as repulsive scatterers. Previously, this method was successfully employed for the 2D honeycomb lattice \citeSManoharan2, the Lieb lattice \citeSSlot20172, and quasicrystals \citeSCollins20172, making it an evident candidate for the realization of 2D (i.e. flat) fractal types.
Geometry
The fractal of our choice is a Sierpiński triangle, commensurate with the triangular symmetry of the Cu(111) surface on which the CO molecules are positioned. In contrast to the basic Sierpiński triangle (shown in Fig. 1A \citeSSierpinski2), we chose a Sierpiński triangle with a honeycomb basis \citeSOftadeh2, Shang2. Fig. S1A shows the geometry of the Sierpiński triangle with honeycomb basis for the first three generations . The triangle is characterized by three inequivalent artificial atomic sites: red (connectivity 1, i.e. number of nearest-neighbor sites =1), green (=2), and black (=3). (Note that slightly different colors are used in the SI than in the paper to allow for a more in-depth analysis.)
Three triangles are interconnected to form a triangle. This changes the connectivity and/or environment of the initial red, green, and black sites. For instance, we distinguish between a ’red’ corner site (still =1) and a ’maroon’ site which connects the triangles (initially red site, but now with =2). Similarly, there is a ’black’ (still =2) and a ’navy’ site (still =2, but with slightly different neighbors).
Three triangles, or nine triangles, are interconnected to form a triangle. Here, a similar distinction is made for the no longer equivalent sites. Whereas the differences are only very subtle for most sites, it is important to distinguish between the ’salmon’ sites connecting the triangles (position of an initial red site, but with =2) and the ’red’ sites with their original character at the very corners of the triangle.
The corresponding positions of the CO molecules, which form the anti-configuration of the configuration of the electronic fractal, are shown in Fig. S1B.
Size
The energies at which the electronic states of the fractal emerge depend on the size - i.e. the degree of confinement - of the ”artificial atom” sites to which the Cu(111) surface-state electrons are confined, as reported by Gomes et al. \citeSManoharan2. On the one hand, the features need to appear above the onset of the Cu(111) surface state at eV. On the other hand, the contribution of the bulk states is less pronounced if the energy window is chosen below eV.
For these reasons, we choose the same size of the artificial atom sites as the undoped honeycomb lattice with next-neighbor distances nm in Ref. \citeSManoharan2, which has features in the range eV.
ii.2 First generations: Experimental results
ii.2.1 Energy resolution
The resolution of the differential conductance spectra and maps is mainly limited by the used modulation amplitude of mV r.m.s. and the CO-induced coupling between the Cu(111) surface state and bulk states. To compare the tight-binding and muffin-tin results with the experiment, we use a linewidth meV, which is the same as was used for the Lieb lattice geometry in Ref. \citeSSlot20172. Note that as the number of wave functions scales with the number of artificial atom sites, the individual wave functions in Sierpiński triangles of higher generations become too close in energy to be resolved experimentally.
ii.2.2 Differential-conductance spectra
Differential-conductance spectra were acquired above the Sierpiński triangle and normalized by the average spectrum above clean Cu(111), following the procedure by Gomes et al. \citeSManoharan2. This normalization cancels contributions due to the tip and the slope of the Cu(111) surface state. In Fig. S2, the normalization is shown for three different tips, characterized by significantly different averaged spectra on clean Cu(111) (black dashed lines in Fig. S2A,C,E). With each tip, we acquired several spectra over red corner sites, as shown in Fig. S2A,C,E for a Sierpiński triangle. These spectra were divided by the Cu(111) spectrum (Fig. S2B,D,F). Similar features are observed for all different tips, corroborating the reproducibility of the normalized differential conductance spectra.
In Fig. S3A and SS3B, we present a typical normalized differential-conductance spectrum on an isolated artificial atom site and on a plain triangle, the building block of the Sierpiński triangle. Furthermore, Fig. S3C-E shows typical differential-conductance spectra over the red, green and black sites (defined in Fig. S1A) of a , , and Sierpiński triangle. Each Sierpiński triangle was built and measured at least twice with different tips. We observe that the spectrum on the isolated artificial atom site already resembles the spectra over red sites in the Sierpiński triangles, despite the reduced connectivity (Fig. S3A). The red corner sites of the plain triangle (=1) also show a behavior similar to the red Sierpiński sites, which have the same connectivity =1 (Fig. S3B). Similarly, the spectrum over the black center site of the plain triangle () resembles the spectra over black sites in the Sierpiński triangles (also ). The red, green, and black Sierpiński sites defined in Fig. S1A retain their connectivity for all generations, leading to spectra with similar features for - (Fig. S3C-E). When the connectivity of a site is changed in a higher-generation triangle, this is reflected in the spectrum, as shown for the spectrum over a ’salmon’ site () compared to a ’red’ site () in (Fig. S3E).
In Fig. S4A, a green dashed line is defined along which a series of differential-conductance spectra was taken over the Sierpiński triangle. The contour plot in Fig. S4B shows the amplitude of the normalized differential-conductance spectra as a function of the location along this line and the bias voltage. The main features expected for red, black and green sites are reproduced along the line. (For clarity, red and maroon as well as black and navy sites have not been distinguished in Fig. S4A.)
ii.2.3 Wave-function maps
An overview of the wave-function maps at bias voltages V, V, V, and V above the , , and Sierpiński triangle is shown in Fig. S5. At each particular bias voltage, the main features (as discussed in the main text) are similar for each generation.
ii.3 First generations: Muffin-tin model
Now, we consider two different theoretical approaches to study the electronic structure of the Sierpiński fractal shown above. First, we concentrate on the muffin-tin approximation, which has shown to provide an accurate description of the CO/Cu(111)-system \citeSManoharan2, Slot20172, Park2, Qiu2. The surface-state electrons of Cu(111) form a 2D electron gas with an effective electron mass \citeSBurgi2. The band bottom is defined by the onset of the surface state at eV \citeSKroger2. In the muffin-tin calculations, the Schrödinger equation is solved for this 2D electron gas with CO molecules modeled as disks with a repulsive potential and effective radius . This leads to a Hamiltonian
(S1) |
where nm \citeSLi2, Ropo2 and eV for , while zero otherwise. A broadening of meV is included in the spectra and maps to account for the hybridization with bulk states, as described in Sec. II.2.1.
In order to take into account the triangular shape geometry of the Sierpiński lattice, the calculations of the LDOS were carried out on a rectangular box and a triangular box. Due to the broadening of the LDOS, the results are almost identical for the two boundaries. We use the rectangular box for the displayed results.
Figure S6 shows the LDOS spectra simulated using muffin-tin calculations, adopting the color coding defined in Fig. S1A. Figure S7 shows the wave-function maps for the three generations and at energies corresponding to interesting peaks in the LDOS (at energies V, V and V). The results are in excellent agreement with the measured differential-conductance spectra and wave-function maps.
ii.4 First generations: Tight-binding calculations
The second theoretical approach we describe is the tight-binding model. In the tight-binding approach, each artificial atom site is modeled by an -orbital, which couples to the neighboring sites with nearest-neighbor (NN) hopping and next-nearest-neighbor (NNN) hopping (see Fig. S1A). The Hamiltonian reads
(S2) |
where are the annihilation (creation) operators for the electrons, is the on-site energy of the site-localized orbital, denotes the sum over NN and the sum over NNN sites. Since the configuration has a honeycomb basis with the sizes of the undoped honeycomb lattice created by Gomes et al. \citeSManoharan2, we consider similar values for the NN hopping amplitude eV, a NNN hopping and an on-site energy eV. We also include an orbital overlap between the nearest neighbors, and therefore solve the generalized eigenvalue problem. This choice of parameters yields the best agreement with the experiment and the muffin-tin simulations.
Note that the fractal is a non-periodic structure, and therefore it is not possible to identify a band spectrum. In particular, due to scale-invariance the energy levels of are related to the levels of , and it was shown that the energy spectrum shows self-similar features and has highly degenerate energy levels \citeSDomany2. We will comment on this property in more detail in Sec. SII.5.
We solve the tight-binding eigensystem and calculate the LDOS as a function of energy via
(S3) |
where we sum over the number of states for each lattice position r. In order to accommodate the broadening of the spectrum due to the repulsive scatterers, we approximate the delta function by a Lorentzian with a broadening meV. In Fig. S8, we present the LDOS as a function of energy for the , , and Sierpiński triangle. The results are in good agreement with the experiment and muffin-tin calculations (see Figs. S3, S6). In particular, we observe the features around eV, eV, eV, and eV, as discussed in the main text. We note a particle-hole asymmetry, as the right peaks around eV are lower than the left peaks around eV.
Furthermore, we compare the wave-function maps of the experiment and muffin-tin with the tight-binding by approximating the orbital wave-functions as Gaussian functions. At each lattice site , we model the -orbitals
(S4) |
where the LDOS acts as an amplitude for the Gaussian wave function, is the normalization constant and is the standard deviation, which is set by comparing with the experimental and muffin-tin wave-function maps. The resulting figures for are shown in Fig. S9 and exhibit a good correspondence with the experimental and muffin-tin maps.
ii.5 Higher generations: Tight-binding calculations
One of the interesting features of fractals is self-similarity at different length scales. In the 1980s, multiple groups used this recursiveness to derive the DOS for these fractals and observed localized and extended states \citeSDomany2, Ghez2, Andrade19892, Andrade19912, Kappertz2, Wang2. One specific feature is that also the DOS obeys a certain recursion relation between different generations, which is a universal feature of the fractal structure. We now investigate the self-similarity in the DOS for the Sierpiński fractal that is under consideration in this paper and explicitly show this repetition of the DOS using a tight-binding model.
We follow Ref. \citeSDomany2 in the discussion below. In order to construct an iteration scheme for higher-order generations, we first need to take care of the corner sites with connectivity , since these sites are different than the other sites in the lattice. Therefore, we mirror the image in the plane, and connect the corner sites to each other as a kind of periodic boundary conditions, see Fig. S10. Now, we can distinguish two different sites: sites with connectivity and sites with .
The corresponding Hamiltonian of this system can be separated into two subspaces and , where the sites of are part of a hexagon, and of connect the hexagons. The iteration scheme is then configured as follows: at each step, a hexagon replaces a site with , e.g. when we go from the zeroth generation (7 sites) to the first generation (17 sites), we first remove the two sites with , and insert a hexagon in that place (see Fig. S10). A scheme is constructed in detail below when we go from the first to the zeroth generation, and then reversing the process.
In the first generation, we want to project out the sites that are part of the two hexagons. When we only consider NN-hopping , we have
(S5) |
where
(S6) |
and . As a convenient choice, we assumed the on-site energy is 0. In the first step, we project out the wave functions such that . We focus only on the lower, upward pointing triangle, but a similar procedure can be done for the total. The energy of this lattice is bound between . In this case, we obtain
(S7) |
where we introduced the dimensionless on-site energy . The effective Hamiltonian describes three sites (1, 2 and 3 in Fig. S10) connected with hopping and energy . The factor in arises because each individual site (1, 2 and 3) was connected to two hexagons before the projection, and therefore the diagonal element is twice the diagonal element of . To complete this step of the iteration scheme, we want to recast the on-site energy back to 0, then we have changed nothing with respect to the original Hamiltonian (only a NN hopping) and can repeat this procedure multiple times. Therefore, we equate , which is equivalent to . This redefined parameter now describes the energies at one lower generation.
However, since this decimation leaves us with only three sites, whereas actually four sites should remain (one site in the center connecting the other sites), there is a final decimation step that needs to be included. This step can be written down in a similar manner as before, by decimating a 4x4 matrix into a 3x3 one as follows: we start with a 4x4 Hamiltonian, where
(S8) |
and . The effective Hamiltonian is
(S9) |
resulting in . Finally, we equate the two expressions for to find , which is the actual energy at a lower generation. Once the energy for a low generation is known, it can be used to find the energy for a higher generation, and this starts an iterative cycle. Each energy eigenvalue in a lower generation gives rise to new eigenvalues via this iteration.