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Imaging the quantum melting of Wigner crystal with electronic quasicrystal order

Wigner crystal, as the most fundamental exemplification where the many-body interaction forges the electrons into a solid, experiences an intriguing quantum melting where diverse intermediate phases are predicted to emerge near the quantum critical point. Indications of exotic Wigner orders like bubble phase, liquid-solid phase, and anisotropic Wigner phase have been established by optical or transport measurements. However, the direct visualization of lattice-scale melting order, which is of paramount importance to unequivocally uncover the melting nature, remains challenging and lacking. Noting that Wigner crystals have been achieved in the fractionally filled moire superlattice recently, here, via scanning tunneling microscope, we image the quantum melting of Wigner solid realized by further varying the moire superstructure in monolayer YbCl3/graphene heterostructure. The Wigner solid is constructed on the two-dimensional ensemble of interfacial electron-hole pairs derived from charge transfer. The interplay between certain moire potential and ionic potential leads to the quantum melting of Wigner solid evidenced by the emergence of electron-liquid characteristic, verifying the theoretical predictions. Particularly, akin to the classical quasicrystal made of atoms, a dodecagonal quasicrystal made of electrons, i.e., the Wigner quasicrytal, is visualized at the quantum melting point. In stark contrast to the incompressible Wigner solid, the Wigner quasicrytal hosts considerable liquefied nature unraveled by the interference ripples caused by scattering. By virtue of the two-dimensional charge transfer interface composed of monolayer heavy electron material and graphene, our discovery not only enriches the exploration and understanding of quantum solid-liquid melting, but also paves the way to directly probe the quantum critical order of correlated many-body system.

In a dilute many-electron system, strong Coulomb correlation can localize and freeze electrons into periodic arrays forming a Wigner solid, in order to minimize the electrostatic energy which exceeds the motional quantum fluctuation 2 . The common strategy to quench the electron kinetic energy is to apply a magnetic field and enforce electrons filled into the Landau cyclotron orbits, whereby the close affinity between Wigner crystal and integer/fractional quantum hall states has been uncovered 22 . In the past several years, a novel strategy regarding moiré physics arises as filling electrons inside the moiré trapping potential efficiently suppresses the kinetic energy and enhances the comparative Coulomb correlation. Hence a series of fascinating states have been discovered: magic-angle superconductivity 23 , correlated insulating state 24 , quantum anomalous Hall state 25 , and Wigner crystal settled in moiré lattices 17,19 .
In electron solid immersed in a uniform neutralizing background, increasing the electron density leads to comparable kinetic energy competing with the electrostatic energy, resulting in the quantum melting of Wigner crystal [3][4][5][6] . Enigmatic strongly correlated orders are theoretically and experimentally reported to appear at the quantum critical melting point, manifested as the solid-liquid hybrid, nematic/hexatic Fermi liquids, and the hybrid of striped and bubbled orders 7-12,21,26 . Evidently, putting the mutually repulsive electrons in a composite electrostatic potential consisting of both the ionic and moiré potential, in replacement of the uniform electrostatic background, would greatly intertwine different physical interactions and totally reshuffle the game, leading to a Coulomb frustration regime 21,27 . An enlightening example illustrating the final arrangement of electrons in the composite scenario is the direct visualization of the Wigner crystal where the electrons are settled according to the dominating moiré landscape 17 . The moiré superstructure imposes a deep and intricate impact on filled tenuous electrons or interlayer excitons due to the similar spatial scale of the moiré unit cell and interelectron distance. As a result, varying the moiré parameter could also give rise to corresponding delocalized orders 28-31 besides the Wigner solid phase. What's more, embedding and constraining the electron solids in discrete lattice sites of another periodic spatial structure incommensurately is predicted to lead to the emergence of electronic quasicrystal 32,33 . Therefore, with the multifaceted involvement of electron correlation, ionic lattice, and moiré periodicity, the waning and waxing of the three parameters would introduce new complexity into the quantum phase diagram, appealing for the exploration of novel states subjected to competitive/collaborative mechanisms.
In this work, we fabricate monolayer-YbCl3/graphene heterostructures via molecular beam epitaxy (MBE). The Wigner solid states, which stem from the twodimensional electron-hole pairs of equilibrium induced by interfacial charge transfer, are visualized by scanning tunneling microscope (STM) at 5 K. The electron-hole pairs featuring Rydberg-like wavefunctions are identified by their spatial distribution normal to the surface. Various Wigner orders, including the melting phase of Wigner quasicrystal which is the electron counterpart of classic quasicrystal made of atoms 34 , are observed in different stacking domains, indicating the intriguing interplay between Coulomb interaction, ionic lattice, and moiré periodicity. The rare-earth halide YbCl3 is a representative strongly correlated material of 4f electrons 35 . Fig. 1a illustrates its atomic structure where the honeycomb layer of Ybions is sandwiched between two layers of triangularly arranged Cl-ions. Single layer YbCl3 is epitaxially grown on the substrate of highly oriented pyrolytic graphite (HOPG) with high quality as shown by the nanoscale morphology in Fig. 1b. Details of epitaxial growth and corresponding characterization are given in Methods and section A of Supplementary Information (SI). Fig. 1c shows an atomic scale STM image of YbCl3 with typical atomic defects where the triangular array of protrusions represents the topmost Cl-lattice. The absence of a single protrusion leaving an individual hole in the image is identified as a Cl-defect (marked by the dashed red circle), and the triangle depression, upon which the trio of protrusions remains, is assigned to a vacancy of Ybion beneath the topmost Cl-layer (marked by the dashed black circle).

Charge transfer states inside YbCl3 band gap
To investigate the electronic structure, tunneling spectra of YbCl3 are obtained.
The top panel in Fig. 1d shows the dI/dV spectrum (blue curve) taken at a relatively large tip-sample distance (the small distance case is described down below), unveiling the insulating nature by the full gap with vanished tunneling conductance around the Fermi level. Density functional calculations of sole YbCl3 monolayer produce density of states (violet and orange curves displayed in Fig. 1d) that nicely agree with the dI/dV spectrum. The conduction band (the states detected from 0.8 to 2.0 eV) is the upper Hubbard band derived from the unoccupied Yb-4f orbitals while the valence band (states detected below -0.8 eV) stems from the occupied Cl-3p orbitals. The lower Hubbard bands of occupied Yb-4f orbitals subjected to a significant Coulomb correlation are buried deep below the Fermi level at −9 eV to −5 eV (see Fig. S1) which exceeds the accessible energy range of tunneling spectroscopy. As for now, no state is observed within the full gap range from -0.8 V to 0.8 V. Henceforth, we use these two explicit values to denote the boundary of the insulating gap and define the bias voltages from -0.8 V to 0.8 V as in-gap range (IGR), while other voltages beyond the full gap belong to out-of-gap range (OGR).
Surprisingly, unexpected non-zero differential conductance (coined as in-gap states later in the text) arises as several humps added to a parabola-like background within the IGR (bottom panel of Fig. 1d) when dipping the tip closer to the surface.
Whereas, no density of states shall exist in IGR according to the DFT computation. The in-gap dI/dV curve is quite distinct from the typical V-shape spectrum of graphene, excluding the simple explanation of tunneling into the graphene states directly through the YbCl3 insulating barrier. The shape of the in-gap differential conductance is highly dependent on the orbital-selective tunneling matrix determined by the setpoint and tip apex, suggesting the multiorbital nature of the in-gap states (see details in section E of SI).
Moreover, many peculiarities of the in-gap states are observed as shown in Fig.1e, f. Firstly, superlattices with different lattice constants arise and become particularly prominent as the bias approaches and enters IGR. The periodic corrugation resolved in IGR (right panel of Fig. 1e, f) is much stronger than that resolved in OGR (left panel of Fig. 1e, f). The differential conductance mapping (Fig. S2) illustrates the evolution of superlattice with bias and further proves that the superlattices are formed by the in-gap states. Secondly, the topmost Cl-and Yb-defects, though clearly resolved in OGR, become completely invisible as varying the bias voltage from OGR to IGR (Fig. 1e, f and Fig. S2), indicating the defect-immunity of in-gap states (the origin of the superlattices and the reason of their immunity towards ionic defects will be discussed later). Lastly, despite that all the in-gap states of different domains share the common features of superlattice and defect-immunity, their electronic properties are extremely domain-dependent. Domains with different superlattices, which show the same height in the image taken at OGR bias (left panel of Fig. 1f), exhibit a conspicuous height difference in the image taken at IGR bias (right panel of Fig. 1f, more data is given in well-defined superlattice constant are discovered experimentally as displayed in Fig. 3 and Fig. S4. It is natural to assign these superlattices to moiré patterns at first sight, which is verified to be true for some cases (see crystal potential. In addition, as the defect-immunity is based on the incompressbility of Wigner solid, it will be lifted as soon as the electron solid begins to melt, which will be discussed in the next chapter.

Discussion and outlook
The amount of charge transfer is around 0.21 e/nm 2 from the computational estimation (see Methods). In other words, the average interelectron distance is a few nanometers, exactly the same order of magnitude compared to the observed superlattice constant, making the various competing parameters comparable and leading to a rich phase diagram. The critical melting distance between two neighboring electrons in an ideally clean two-dimensional system is predicted to be rs=37, in the unit of a0 (a0 is the Bohr radius so that rs is roughly 2 nm) 2 . However, rs varies from 5 to 60 in realistic twodimensional electrons 7 which is close to the interelectron distance in our system. We

MBE growth
The YbCl3 monolayer was grown on cleaved HOPG substrate by MBE at a base pressure of ~ 1×10 -9 mbar. The commercial anhydrous YbCl3 powder was evaporated at 680 o C and the substrate was kept at 340 o C during the growth. The process of growth was monitored by reflective high-energy electron diffraction (RHEED). The YbCl3 monolayer coverage reaches roughly 60% after 30 minutes of deposition (see section A of SI). In this work, we focus on a sample with full coverage of YbCl3 monolayer which is realized after 50 minutes of deposition.

STM measurements
Experiments were performed with a home-built hybrid STM and MBE system, and the STM measurements were operated at a temperature of 5 K. The sample was directly transferred into STM to perform in situ measurement after finishing the growth. Scanning tunneling spectroscopy was performed using a lock-in amplifier technique with a modulation frequency of 963 Hz and a root-mean-square modulation voltage between 5 mV and 20 mV depending on the spectral range of interest.

DFT calculation
The density functional theory calculations for YbCl3 monolayer are carried out using the full-potential augmented plane wave plus local orbital code (Wien2k) 47 . We use the experimental lattice parameters of a=b=6.64 Å and optimize the atomic positions. The plane wave cutoff energy of 14 Ry is set for the interstitial wave functions, and 11*11*1 k-mesh is used for integration over the Brillouin zone. The muffin-tin sphere radii are chosen to be 2.5 and 2.0 Bohr for Yb and Cl atoms, respectively. We perform LSDA+U+SOC calculations with the typical values Hubbard U=8.5 eV and Hund exchange JH=1.0 eV for Yb 4f electrons 48 , and the SOC effect is included by the secondvariational method with scalar relativistic wave functions.

Height-bias characteristic
We perform the height-bias characteristic measurement by recording the tip position during ramping the bias voltage in constant-current mode, where a feedback system raises the tip height to maintain the current as more tunneling channels are opened by the increased bias. The start point of the tip position determined by the initial setup condition is considered as the zero-height point. The salient steplike feature is commonly recognized as the fingerprint of Rydberg-like series seeing that the tipsample distance is exponentially sensitive to the energy threshold of quantum states.

Current-height characteristic
Before we take the current-height characteristic, the setup condition described in Fig.  2e is adopted so that the tip is positioned over the sample at a relatively close distance in order to ensure that the interfacial charge transfer states can be accessed by the tip. Then disable the feedback to fix the tip at current position which acts as the zero-height start point in the curve recorded later. Next step is to retract the tip and, in the meantime, record the tunneling current so the current-height curve is obtained. The bias can be set to the chosen value in the backward process of the tip, therefore the bias-dependent current-height characteristic can be acquired.

Work function and interfacial charge transfer
The work function of YbCl3 monolayer is calculated to be 8.2 eV, which is larger than that of 4.6 eV for graphene, suggesting a charge transfer across the YbCl3/graphene interface. In our computation, YbCl3 monolayer is slightly electron doped when contacting to graphene, and the electron doping level is estimated to be 0.21 e/nm 2 by Bader charge analysis 49 . In realistic system, the electrostatic field of the various twodimensional Wigner crystals formed in the interface would alter the work function difference and the consequential amount of charge transfer accordingly.

Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

Competing interests
The authors declare no competing interests.

A. Characterization of surface morphology at different growth stage.
The coverage of YbCl3 monolayer reaches roughly 60% after 30 mins deposition at the growth condition described in "Methods" section. The corresponding surface structure is characterized by RHEED and STM. As shown in the STM image of Fig.  S7a, an YbCl3 monolayer island of high quality with a height of 4 Å is formed on the HOPG substrate. The lattice constant of epitaxial YbCl3 single layer is observed to be 6.64 Å, which is calibrated via the direct comparison between the lattice structure of YbCl3 monolayer and HOPG substrate. The main text mentions that we focus on a sample of full coverage realized by 50 mins deposition. Fig. S7b shows the surface characterization of a sample of 70 mins deposition, which exhibits the same fully covered morphology of YbCl3 monolayer, suggesting that prolonging the deposition duration at the same growth condition cannot lead to the formation of a second layer.

B. Energy levels configuration at the charge transfer interface.
In the main text, Fig. 2b, c illustrates two configurations with binding energy of small size and large size, respectively. The electron (hole) series, whose principal quantum number ascends (descends) as increasing energy, lie above (below) the Fermi level in Fig. 2b. In Fig. 2c, the large binding energy leads to a different configuration where some hole (electron) series are pushed to energy positions above (below) the Fermi level. To find out the exact energy configuration of charge transfer states, we perform the bias-dependent current-height characteristics measurement. Controlling the tip to repeat the same path of retraction at different tip-sample voltage, the biasdependent current-height curves are obtained. The results are shown in Fig. S8. The tunneling current is contributed by the integral of quantum states lying in between the Fermi level and the value of bias, which helps to decipher the tunneling components inside the energy range of integration. As shown in Fig. S8b, the current-height curve recorded at 0.8 V shows a composite shape containing both the monotonic decaying of small n (for example, n=1) state and the oscillating behavior of large n states. As lowering the bias close to the Fermi level till 0.1 V, the trend of monotonic decay, which corresponds to the portion with smaller n, is gradually reduced and replaced by the trend of a drastic oscillation with two nodes, indicating an exclusive portion of n=3 or n>3 states. Thus, smaller n states occupy higher energy positions and larger n states occupy lower energy positions in the positive bias range. This result demonstrates that a holesequence of Rydberg-like series lie above the Fermi level, of which the n descends with increasing energy. Vice versa, the ascending n of electron-sequence is also observed below the Fermi level (Fig. S8c). Thus, the configuration with large binding energy described in Fig. 2c (Fig. S8a) is confirmed to be our case. The large binding energy observed here can be ascribed to two reasons. Firstly, the remarkable localization of forbital polarons efficiently enhances the Coulomb correlation. Secondly, the metallic nature of underlying graphene renders a much tighter binding due to the high mobility of graphene hole states.

C. Tunneling scheme for charge transfer states
We report the interfacial charge transfer states which appear as the in-gap states in tunneling spectrum featuring the Rydberg-like spatial distribution. Since charge transfer is a rather common phenomenon which occurs for almost all heterostructures, one would expect that such interfacial in-gap states are supposed to be universal in STM measurement. However, the relevant report is very seldom. From the standpoint of the tunneling mechanism, we think that the exact energy configuration of charge transfer states, i.e., the relative position of electron/hole levels with respect to the Fermi level, plays a crucial role in determining whether the charge transfer ground state can be detected by the tunneling probe or not. In Fig. S8, the energy configuration of Fig. 2c is experimentally pinned down where some electron series lie below the Fermi level and some hole series lie above the Fermi level. We think that this energy configuration is not merely an observation, but also a necessary requirement for producing the tunneling conductance by charge transfer states.
I. Energy configuration of charge transfer states inaccessible to tunneling probe. The small binding energy case as depicted in Fig. 2b is a more common one for interfacial charge transfer states, where the transferred electron states lie above the Fermi level with the n =1 state occupied by an electron in the ground state (similarly, the hole states occupy energy positions below the Fermi level). After taking the tunneling mechanism into account, however, the charge transfer states in this case are invisible to tunneling probe as they do not participate directly in the tunneling process. When setting the bias voltage to a positive value, the tip Fermi level is higher than that of the sample, and the electrons tunnel from the tip into the unoccupied states above the Fermi level of the sample where the transferred electron series are situated in Fig. 2b. While the electron-like series of different n are excitation levels of the transferred electron which belongs to the electron-hole pair, they are not empty states to accept the electrons tunneling in, and thus they will not directly show up in the tunneling spectrum. Analogously, the hole series blow the Fermi level also do not contribute to the tunneling current, making the charge transfer states in this case invisible to tunneling probe. Nevertheless, the charge transfer states shall make a difference in the tunneling process, which manifests as a side effect. In terms of the dipole field derived from interlayer electron-hole pairs, the intermediate state after the injection of tunneling electron would be renormalized and turn out to be a three-particle state consisting of two electrons and a single hole, that is, the trion state 1 . In other words, the tip electron tunnels through an intermediate state where the tunneling electron is trapped by the dipole field of interlayer electron-hole pairs. Since the trion state formed after the electron injection is not the ground state, the tunneling electron would leave the trion state soon for the unoccupied graphene state, contributing to the tunneling current and leading to a tunneling peak in the conductance spectrum. Compared to the exciton, the typical binding energy of trion state is greatly reduced. Consequentially, such conductance peak induced by the trion state is probably located at the energy position tens of millivolts 2 below the conduction band minimum, with the charge-transferred Rydberg series invisible to the tunneling process. II. Energy configuration of charge transfer states accessible to tunneling probe. As for the large binding energy case illustrated in Fig. 2c, things are different as the unique energy configuration renders the charge transfer states accessible via tunneling process. Since the hole series are situated above the Fermi level, they can accept the tunneling electrons at posive bias voltage, and vice versa for the electron series below the Fermi level. At positive bias, the tunneling electron annihilates the hole state and destroys an interlayer electron-hole pair. Since that the interlayer electronhole pairs are energetically favored ground state, the tunneling electron which annihilates the electron-hole pair will leave the hole state rapidly owing to the fast interlayer charge transfer 3 , restoring the interlayer electron-hole pair and giving rise to tunneling conductance. As long as the time of the fleeing of excess tunneling charge and the restoring of electron-hole pairs is much shorter compared to the tunneling lifetime, the effect of tunneling charge is pertubative and the pair annihilation is insiginificant. As a result, the charge transfer states in this case directly participate in the tunneling process and show up in the tunneling spectrum. In the measurement shown in Fig. S8, the electron tunnels out from the occupied charge transferred states of the sample at negative bias, displaying an ascending n sequence of electron series. When tunneling into the empty states of the sample at positive bias, the n sequence switches to the descending hole series immediately. Such sudden change with respect to the bias polarity is consistent with the tunneling regime where switching the polarity reverses the direction of electron tunneling at once. This result indicates that all the electron-like charge transfer series lying below the Fermi level are occupied (so do the hole-like states above the Fermi level), ensuring that there are electrons that could tunnel out from the occupied electron-like charge transfer states at negative bias, and could tunnel into the empty holes at positive bias. A corollary of Fig. S8 is that at least several charge transfer states of different n are occupied by electrons/holes, resulting in the bound state of multiple electrons and holes. We are not sure if each n state is doubly occupied or not, but the strong Coulomb repulsion is likely to favor a single occupation. The bound electron-hole states must contain multiple particles no matter what the internal electronic configuration is, generally speaking, making the multiparticle bound states more or less resemble Helium orbitals or Lithium orbitals. We attribute the unusual multiparticle bound states to the ordering and gathering process of transferred charges during the formation of Wigner lattice. Thus, very likely, more than one electron is localized at each Wigner site, which is consistent with the Wigner bubble phase 4 .

D. Current-height characteristic for intrinsic band of YbCl3 monolayer.
The unusual oscillation observed in the current-height characteristic is a representative feature of charge transfer states, proving the Rydberg-like nature. When increasing the tip-sample distance to probe the intrinsic band states of YbCl3 monolayer, as expected, a common monotonic decay is observed in the current-height characteristic. Fig. S9 shows the results, the tip-sample distance is gradually increased by changing the setpoint bias from IGR to OGR while maintaining the setpoint current in constantcurrent mode.

E. Multiorbital nature of tunneling process
When probing the charge transfer states, the tunneling process occurs between the tip states and the Rydberg-like series, while the latter contain many states of different n as well as of different spatial distribution normal to the interface. As revealed by the current-height characteristic in Fig. 2, the unusual shape of the current-height curve can be decomposed into several Rydberg-like components. Each component dominates different spatial area along the out-of-interface direction. Changing the tip-interface distance is supposed to alter the relative ratio of Rydberg components which participate in the tunneling process, leading to a distinct dI/dV spectrum. The result is shown in Fig. S10, where the current-height curve taken at 0.6 V in Fig. S10a shows a typical multiorbital nonmonotonic oscillation, featuring a hump around 0.2 to 0.3 nm and an undulation at the tail around 0.5 to 0.6 nm. Fixing the tip at different positions (marked as colored arrows) to perform dI/dV spectroscopy will lead to disparate tunneling spectra. Fig. S10b shows the corresponding dI/dV curves, which indicate the alternation of probed Rydberg components and explain the extreme setpoint sensitivity of dI/dV spectrum for charge transfer states. Similarly, changes in the tip geometry and anisotropy are also going to alter the orbital-selective tunneling matrix, resulting in a disparate shape of dI/dV curve.

F. Steplike feature observed in height-bias characteristic taken at negative bias.
The Rydberg-like charge transfer states occupy the IGR in energy, rendering a steplike behavior in the height-bias characteristic across this energy range. While Fig.  2d illustrates the height-bias curve taken at positive bias, here, we show that the steplike feature is also present in the height-bias curve taken at negative bias (Fig. S11).

G. Moiré periodicity
The possible moiré patterns formed in the heterostructure can be geometrically constructed by two beating spatial frequencies belonging to YbCl3 lattice and underlying graphene lattice, respectively. Based on the comprehensive mathematical model 5 which involves not only the first order but also higher-order spatial frequencies of two coinciding lattices, the analytical expression of possible moiré wavelength and the relative angle with respect to the YbCl3 lattice are obtained. We use 1 and 2 to denote the two reciprocal vectors of graphene lattice and 1 and 2 to denote the two reciprocal vectors of YbCl3 lattice, so where a = 2.46 Å and b = 6.64 Å are the graphene lattice constant and YbCl3 lattice constant respectively. Thus, the higher-order spatial lattice frequencies, which correspond to the higher-order spots in the reciprocal lattice, can be represented as the composition of the multiples of two reciprocal vectors: where (m, n) and (r, s) tuples consisting of integers indicate the order of spatial frequencies of graphene and YbCl3 lattice. Letting be the reciprocal vector generated by the difference of ( , ) and ( , ) , so we have that = ( , ) − ( , ).
Thus, varying the order tuple of (m, n) and (r, s) leads to the calculated of different kinds, which gives the wavelength and orientation of the final moiré periodicity. However, there is no need to go through all the combinations of (m, n) and (r, s) because, on the one hand, the experimentally discovered superstructure has wavelength larger than the lattice constant of YbCl3, which brings about the constraint upon the reciprocal vector This constraint condition greatly reduces the possible choices of (m, n) and (r, s). On the other hand, we shall focus on the low-order spatial frequencies with small integers in (m, n) and (r, s) since that the amplitude of higher-order spatial frequency is negligible. As a result, we find that the possible moiré periodicities generated by six low-order combinations listed down below contain all the non-commensurate superlattices discovered in experiment where ( , ) − ( , ) points out the order of spatial frequencies participating in generating the moiré periodicity. Among the combinations of formula (7-12), cases (7-10) involve the lowest order spatial frequencies noting that (1,0) is the reciprocal vector of graphene lattice, while formula (11, 12) is the doubling frequency of formula (7, 8) respectively. Mapping the involved vectors into cartesian coordinate system and varying the relative orientation between graphene and YbCl3 lattice allow us to establish the analytical functional relationship between the moiré wavelength and θ, where θ is the counterclockwise rotated angle modulo 60ºbetween moiré lattice and YbCl3 lattice. The function curve and datapoints of discovered superlattices are plotted in Fig. S12a, from which we can see that the superlattices marked as black boxes conform well to the moiré periodicities. In comparison, Fig. S12b plots the calculated datapoints of relevant commensurate superlattices with respect to YbCl3 lattice together with the experimental datapoints. We can see that all the superlattices marked as red triangles, which are inconsistent with moiré periodicities (as shown in Fig. S12a), coincide with the commensurate-to-YbCl3 superlattices. The two plots clearly illustrate that the observed superlattices can either be categorized into moiré type or commensurate-to-YbCl3 type.    Fig. 1f, but at a larger setup current I=100 pA (the setpoint of each map is indicated). The height difference between two domains observed in the bottom panel of a is about 1.5 Å, which is bigger than that of 0.8 Å in the right panel of Fig. 1f whose setup current is I=10 pA. The dependency of setpoint also suggests the multiorbital nature (see details in section E of SI). b, Images of another intersection of two domains with different superlattices. Similarly, no height difference is present in the image taken at OGR, while an ~3 Å height difference develops at IGR.  Most defects imaged at IGR do not coincide with the defects resolved at OGR (which is the case for the defects marked by red circles), implying that they are bottommost Cl-defects. and their zero-current points are upshifted with the factors indicated respectively. The shape evolution coincides with the hole-series whose n descends as increasing energy. c, the current-height characteristic curves obtained at different negative bias voltages, of which the shape evolution suggests electron-series whose n ascends as increasing energy. Similarly, the multiplicator and offset for each line are indicated.   The horizontal axis of θ is the relative angle between the moiré lattice and YbCl3 lattice, the vertical axis represents the ratio of the superlattice constant to YbCl3 lattice constant. The colored curves and legends indicate the order of spatial frequencies that generate the moiré periodicity. b, a plot illustrating all the standard θ and superlattice constant of the relevant commensurate-to-YbCl3 superperiodicites. The blue circles represent the calculated coordinate points for standard commensurate (CM) periodicities. In comparison, all the data points for the experimentally observed Wigner lattices are indicated as solid boxes and triangles in a and b. The black boxes are the superlattices ascribed to moiré periodicities while the red triangles are those commensurate-to-YbCl3 (CM) superlattices. Among the data points, the periodicity of (7,4) commensurate one is also very close to the (0,1)g-(1,2)h moiré order (orange curve in the left panel), implying that the (7,4) Wigner lattice might originate from the collaboration of the ionic potential and moiré potential. The (0,2) commensurate one has small difference in periodicity with respect to neighboring moiré patterns (the pink and purple curves in the left panel), however, the high spatial resolution and precisely fulfilled commensurability relation (see in Fig. 3d) clarify its commensurate nature.