Majorana Anyon Composites in Magneto-Photoluminescence Spectra of Natural Quantum Hall Puddles

In magneto-photoluminescence (magneto-PL) spectra of quasi two-dimensional islands (quantum dots) having seven electrons and Wigner–Seitz radius rs~1.5, we revealed a suppression of magnetic field (B) dispersion, paramagnetic shifts, and jumps of the energy of the emission components for filling factors ν > 1 (B < 10 T). Additionally, we observed B-hysteresis of the jumps and a dependence of all these anomalous features on rs. Using a theoretical description of the magneto-PL spectra and an analysis of the electronic structure of these dots based on the single-particle Fock–Darwin spectrum and many-particle configuration-interaction calculations, we show that these observations can be described by the rs-dependent formation of the anyon (magneto-electron) composites (ACs) involving single-particle states having non-zero angular momentum and that the anyon states observed involve Majorana modes (MMs), including zero-B modes having an equal number of vortexes and anti-vortexes, which can be considered as Majorana anyons. We show that the paramagnetic shift corresponds to a destruction of the equilibrium self-formed ν~5/2 AC by the external magnetic field and that the jumps and their hysteresis can be described in terms of Majorana qubit states controlled by B and rs. Our results show a critical role of quantum confinement in the formation of magneto-electrons and implies the liquid-crystal nature of fractional quantum Hall effect states, the Majorana anyon origin of the states having even ν, i.e., composite fermions, which provide new opportunities for topological quantum computing.


Introduction
Superconducting (dissipationless) transport corresponding to zero electrical resistance of the materials, discovered in 1913 [1], gave a quantum mechanical illustration of Newton's first law of motion, which states that moving material objects will conserve a constant velocity if no external force acts on them, and which represents an obvious paradox for our imagination, since natural movements observed in everyday life are subject to a frictional force and are slowed down. Thus, in superconducting materials, the persistent currents loops (PCLs) can be thought to be perpetual and laboratory measurements show their lifetime to be at least 100,000 years [2]. In an external magnetic field (B e ), the persistent vortex currents induce a perfect diamagnetism, i.e., the expulsion of B e from the superconductor (SC), known as the Meissner effect [3]. PCLs in SCs provide the quantization of the magnetic flux [4] in multiples of φ 0 * = h/2e 0 , where e 0 is the electron charge, which was documented in the measurements of the magnetization of micrometer size SC cylinders [5,6]. Single φ 0 * vortexes, also called Abrikosov vortexes or fluxons, arranged in regular lattices, are generated in major SCs above the critical field B e1 [7,8]. The vortexes consist PCLs tens of nanometers in size, whose density is proportional to B e , and they merge at a critical field Nevertheless, since we observed e ν ems in QDs having ν < 1, one can expect their formation in QDs having ν > 1, as for FQHE states [24]. These states correspond to partial (total) filling of the highest (lower) LLs and involve CFs, which are anyons [16], in the highest LL and IQHE es in the lower one.
A much investigated ν > 1 FQHE state is the ν = 5/2 state, which, according to the conventional description, corresponds to the half-filled second LL and fully filled lowest one. The CFs in the second LL show non-Abelian anyon properties [25], which implies that they have half-flux Vs with a zero-energy excited state, known as a Majorana zero mode or simply a Majorana mode (MM) [26][27][28][29], and thus are a perspective for the realization of TQC [30,31].
In em, the MM is a particle-antiparticle pair, consisting of Vs having an opposite, anti-parallel direction/orientation. It can be naturally generated in a state having non-zero angular momentum l z , for which the e-wave-function has "anti-phase spatial splitting". In such states, only half of the ψ i can generate V along the B e direction, which thus is a half-V having a MM excited state. For the em the MM adds two more Vs resulting in a two-times smaller charge, the signature of which was observed for the 5/2 state [32]. Moreover, since a MM adds zero magnetic flux, it can possibly be generated at zero B e .
Here, we analyze a few such dots in detail using measurements of the B e -dispersion of their PL spectra components. In the measurements, we have revealed several anomalies, which include suppression of the dispersion, paramagnetic shifts, jumps (up to~2 meV), and their hysteresis. We found that the appearance of these anomalies depends on the dot size and the direction of change of B e . We analyzed the experimental data using a theoretical description of PL spectra and the electronic structure of such dots based on single-particle Fock-Darwin (FD) states and a many-particle configuration-interaction (CI) approach, which allows one to explain the observed anomalies by the self-formation of em-MM-anyon composites corresponding to ν 0~5 /2, their collapse and the emergence of the anti-em-MM composites induced by the magnetic field. Our analysis has shown that the formation of the ems corresponds to the generation of Vs by single and pair non-interacting es occupying quantum-confined states. This allows one to describe the FQHE in terms of liquid-crystal states involving Majorana anyons and opens new routes for the realization of topological quantum computing.

Natural Quantum Hall Puddles
The details of the growth procedure and structural and emission properties of the InP/GaInP 2 QDs were described previously [34]. The dots have a flat lens shape (aspect ratio 10) and a lateral size D~50-180 nm. Their shape reveals a small elongation and asymmetry, which in most cases can be described as a combination of a ∼5% elliptical distortion (D /D ⊥ = 1.05) and a 10% change of R ⊥ . The specific structural property of this QD system [35] is an atomic ordering of the GaInP 2 matrix material, which results in composite core-shell structure consisting of InP QD having zinc blende crystal structure surrounded by a few atomically ordered GaInP 2 domains having rhombohedral crystal structure and size 10-50 nm. In this composite, the domains generate strong piezo-electric fields resulting in e doping (up to 20) and B ν (up to 15 T) forming natural quantum Hall puddles.
Here, we studied three of such puddles denoted D01m, D07m and D09m. They have a number of electrons N = 6 and 7 and D~75 nm resulted in the electron density n~3 × 10 11 and r s~1 .3, which corresponds to a weak Wigner localization regime.

Single Dot Magneto-Photoluminescence Spectroscopy Measurements
We measured magneto-PL spectra of single puddles using a home-made near-field scanning optical microscope (NSOM) having spatial resolution up to 25 nm operating at 10 K and magnetic fields of up to 10 T. We used home-made tapered fiber probes coated with Al, having an aperture size of 50-300 nm in a collection-illumination mode. The spectra were excited by the 514.5 nm Ar-laser line (Edmundoptics, Cherry Hill, NJ, USA) and measured using a CCD (multi-channel) detector (Horiba, Piscataway, NJ, USA) together with a 280 mm focal length monochromator (Horiba, Piscataway, NJ, USA. The excitation power measured before fiber coupler was~5 µW, which provided a power density of~0.5 W/cm 2 . The spectral resolution of the system is 0.2-0.4 meV. σ + and σ − circular polarization were measured using λ/4 plate (Newport, Irvine, CA, USA) and linear polarizer (Newport, Irvine, CA, USA). In the spectra, we measured the shifts of PL peaks related to occupied e-shells versus B e , which were compared with the theoretical calculations and used to extract the charge of single particle states of the puddles.

Analysis of the Data
The shape of the PL spectra, i.e., the number of spectral components, their position and full-width-at-half-maxima, were analyzed using a multi-peak fitting procedure from graphic software OriginPro (version 20.0, Northampton, MA, USA) The values of effective (screened) quantum confinementhω 0 * (see Appendix A.3) and N were measured from the splitting and the number of anti-Stokes peaks in the PL spectra, respectively. For N, the measurements of the shifts of anti-Stokes components in magnetic field were used as well. The D values were estimated from the charge density distributions (CDD) calculated from the unscreened quantum confinement valuehω 0 =hω 0 * /0.7 and N values using CI approach [16,34] and from NSOM scanning experiments. Using N and D values we calculate n and r s . Note that these values are related to the photo-excited state having N* = N + 1 electrons.

Theory and Calculations
The theoretical description of the PL spectra of quantum Hall puddles in a magnetic field based on the Fock-Darwin (FD) spectrum is presented in Appendix A. The description involves a general model (Appendix A.1), an analysis of the single-particle states of InP/GaInP 2 QDs involved in PL transitions using the 8 band k·p method, i.e., non-interacting electrons and holes (Appendix A.2), and the analysis of the effect of the Coulomb interaction on single-particle states using the Hartree-Fock (HF) method (Appendix A.3). The resulting formulas of the model, i.e., B e -dispersion of PL peaks, are Equations (A4c) and (A5) in Appendix A.4.
The ground states (GSs) of the puddles in a magnetic field are analyzed in Appendix B. Total energy and total angular momentum values were calculated using the multi-particle CI method (Appendix B.1) and the single-particle FD spectrum, including the vortex contribution (Appendix B.2). Both methods revealed GSs having fractional ν, suggesting em-composite (em-C) formation.
The parameters of em-Cs, which are composition ν n + (ν n ) S , where n is number and S is a total spin, and charge e N * are specified in Appendix C.1 (see Equations (A6)-(A8)). There, a mismatch of the emission energy of em-Cs is analyzed and corresponding quantities, which are ∆ ν jumps and related PL shift (E s em-C ), are described (Equations (A9) and (A10)). The fitting of the Fock-Darwin spectrum for the measurements of ν and details of the construction of the V structure of em-Cs are described in Appendices C.2 and C.3, respectively.

Summary of Measured Parameters
In Table 1, we summarize the initial zero-field parameters of the puddles, which are the number of the electrons in photo-excited state N* = N + 1, D, r s ,hω 0 * /hω 0 , and the final parameters of em-Cs formed in these puddles obtained from the measurements and the analysis of the data, which are two intrinsic magnetic fields B ec and B ν (see below), and ν 0 (ν 0 + ) S and ν n (ν n + ) S values of the equilibrium and B e -induced em-Cs, respectively. In the notation of the compositions in Table 1, we include reduced charge value e N * as a superscript.  Figure 1a compares the PL spectra of the dots/puddles studied at zero internal field B (see below), plotted in Stokes energy units. The inserts show CI CDDs of the dots and a contour plot of spatially resolved PL spectra near the center of D07m dot. Figure 1b shows circularly polarized spectra of the D07m dot measured at B e = 0, 1, 2, . . . , 10 T.

QD
N* ħω0 * /ħω0 (meV)  Figure 1a compares the PL spectra of the dots/puddles studied at zero internal field B (see below), plotted in Stokes energy units. The inserts show CI CDDs of the dots and a contour plot of spatially resolved PL spectra near the center of D07m dot. Figure 1b shows circularly polarized spectra of the D07m dot measured at Be = 0, 1, 2, …, 10 T.

Shell Structure and Built-In Magnetic Field of Quantum Hall Puddles
CDDs show a decrease of the size related to a decrease/increase of D/ħω0 (see Table  1). The CDDs are elongated along x and their landscape has the same topology consisting from two minima of ~D/6 size separated along x by ~D/3, which indicates molecular structure (see Appendix B.1). The left insert shows the measured size of the emission area of the D07m dot of ~90 × 60 nm (average size of 75 nm), in agreement with the corre sponding CDD size.
In Figure 1a, it is seen that the spectra of the dots studied consist of the main peak denoted by s and two weaker anti-Stokes ones denoted by p and d. These three peaks are related to three occupied e-shells of the QD in the photo-excited state, as described in Appendix A.1. The peaks have a full-width-at-half-maximum ~3 meV and varying splitting corresponding to ħω0 * (see Table 1), related to the size variation. The D09m dot also has a few times larger relative intensity of p-and d-components. There is an increase of the ratio of the s-p to p-d splitting related to the size increase.  CDDs show a decrease of the size related to a decrease/increase of D/hω 0 (see Table 1). The CDDs are elongated along x and their landscape has the same topology consisting from two minima of~D/6 size separated along x by~D/3, which indicates molecular structure (see Appendix B.1). The left insert shows the measured size of the emission area of the D07m dot of~90 × 60 nm (average size of 75 nm), in agreement with the corre sponding CDD size.
In Figure 1a, it is seen that the spectra of the dots studied consist of the main peak denoted by s and two weaker anti-Stokes ones denoted by p and d. These three peaks are related to three occupied e-shells of the QD in the photo-excited state, as described in Appendix A.1. The peaks have a full-width-at-half-maximum~3 meV and varying splitting corresponding tohω 0 * (see Table 1), related to the size variation. The D09m dot also has a few times larger relative intensity of pand d-components. There is an increase of the ratio of the s-p to p-d splitting related to the size increase. In Figure 1b, the PL spectra of the D07m dot at zero B e reveals nearly a two-times stronger intensity of the σ − -component, i.e., they are σ − -polarized, which indicates B ν . At B ec = 3 T, the σ − -polarization disappears and the spectra acquire a σ + -polarization at larger B e . This indicates B ν~− B ec (see Table 1), where B ec is a compensating field and B ν has a direction opposite to B e . Thus, in the range from 0 T to B ec , the internal field B = B e + B ν is negative and decreases from B ν to zero. In this range, which we denote B − ↓ or B e a , B e < |B ν | and the PL spectra have an anomalous B e -shift (see below). Such an anomalous shift is a direct signature of B ν , which allows one to detect it independently from polarization measurements. For larger fields, i.e., B e > |B ν |, B becomes positive and increases versus B e . In this range, which we denote by B + ↑ or B e n , normal B e -shifts are observed, but B is lower B e on B ec .
There is an anomalous B e a -range, indicating B ν in D01m and D09m dots too.

Shell Structure in Magnetic Field
In Figure 2a-c, we present unpolarized spectra of the dots, measured at B e = 0, 1, 2, . . . 10 T. The spectra were measured under an increase of the field from 0 to 10 T, denoted as B e ↑ and shown in the lower part of the graphs, and a decrease from 10 to 0 T, denoted as B e ↓ and shown in the upper parts of the graphs. For this range, the B e n and B e a ranges are the B + ↓ and B − ↑-ranges, respectively. stronger intensity of the σ − -component, i.e., they are σ − -polarized, which indicates Bν. At Bec = 3 T, the σ − -polarization disappears and the spectra acquire a σ + -polarization at larger Be. This indicates Bν~−Bec (see Table 1), where Bec is a compensating field and Bν has a direction opposite to Be. Thus, in the range from 0 T to Bec, the internal field B = Be + Bν is negative and decreases from Bν to zero. In this range, which we denote B − ↓ or Be a , Be < |Bν| and the PL spectra have an anomalous Be-shift (see below). Such an anomalous shift is a direct signature of Bν, which allows one to detect it independently from polarization measurements. For larger fields, i.e., Be > |Bν|, B becomes positive and increases versus Be. In this range, which we denote by B + ↑ or Be n , normal Be-shifts are observed, but B is lower Be on Bec.
There is an anomalous Be a -range, indicating Bν in D01m and D09m dots too.

Shell Structure in Magnetic Field
In Figure 2a-c, we present unpolarized spectra of the dots, measured at Be = 0, 1, 2, … 10 T. The spectra were measured under an increase of the field from 0 to 10 T, denoted as Be↑ and shown in the lower part of the graphs, and a decrease from 10 to 0 T, denoted as Be↓ and shown in the upper parts of the graphs. For this range, the Be n and Be a ranges are the B + ↓ and B − ↑-ranges, respectively.
The spectra having B = 0 T and dividing the Be a -and Be n -ranges (Bec = 3 and 2 T was found in D01m and D09m dots, respectively) are shown by thick solid lines. In Figure 2ac, the peak maxima are connected by straight lines, which allows one to quali tatively trace their shift.
At all Be values, the s-peak is dominant, and changes of the spectral shape are caused by the changes of the intensity and the position of the p-and d-peaks at Be > 4 T. These are: the peak merging at Be = 7 T in the D01m dot; the few-times intensity increase of the dpeak at Be = 9 T and the appearance of the additional x-peak between the s-and p-peaks at Be = 7 T in D07m dot; a few-times intensity decrease of the p-and d-peak at Be = 8 T in the D09m dot.  and D09m (c) QDs measured in the field range B e = 0-10 T (thick solid curves are for B = 0 T, shortdot curves are for |B e | < |B ec |). Solid, dashed and short-dashed lines (thinner for |B e | < |B ec |) connecting the maxima of s-, pand d-peaks, respectively, are drawn to outline their shift and bends; a star symbol denotes neighboring dot, which in (c) is the single electron dot S11 studied in Ref. [15]. Vertical arrows at the left show direction of field increase.
The spectra having B = 0 T and dividing the B e a -and B e n -ranges (B ec = 3 and 2 T was found in D01m and D09m dots, respectively) are shown by thick solid lines. In Figure 2a-c, the peak maxima are connected by straight lines, which allows one to quali tatively trace their shift.
At all B e values, the s-peak is dominant, and changes of the spectral shape are caused by the changes of the intensity and the position of the pand d-peaks at B e > 4 T. These are: the peak merging at B e = 7 T in the D01m dot; the few-times intensity increase of the d-peak at B e = 9 T and the appearance of the additional x-peak between the sand p-peaks at B e = 7 T in D07m dot; a few-times intensity decrease of the pand d-peak at B e = 8 T in the D09m dot.
The peak shift, further denoted as s, pand d-shifts, in the B e n -range is mostly positive, i.e., increases with field increase, and at B e = 10 T reaches average values of~2,~5, and 3 meV, respectively. In the B e a -range, the shifts are negative (paramagnetic), i.e., "anomalous", and have values from 0 to 2 meV. In both ranges, the shifts reveal a few bends (see spectra near B e~3 T and 6 T).

Anomalous Shifts and Jumps Induced by Magneto-Electrons
In Figure 3a lated shifts are presented in the Be n -range, i.e., for B ≥ 0 T. The peaks are denoted b quantum numbers of the Fock-Darwin (FD) spectrum kl (see Appendix A.2).
The s-shift data show that the bends seen in the spectra in Figure 2a-c are having amplitude 1-2 meV over ΔBe~1 T (see dashed ovals in Figure 2a-c). Their ap ance is different in different dots and B/Be ranges, revealing size dependence and B asymmetry (hysteresis). In the D01m dot, one step is observed in both B − ↓and B − ↑-r at B = −1 T, which is thus symmetric. In the D07m dot, two asymmetric steps are obse one in the B − ↓and the other in the B + ↓-ranges at B = −2 and 5 T, respectively, a mirror steps appear in the ranges of reversed direction of field change. In the D09m four asymmetric steps are observed in the B − ↓and B + ↑-ranges at B = −2, 3, 5 and 7.
The calculated s-shift shows a weak, nearly linear dispersion with slope ~0.25 m and in the regions where the jumps are absent the experimental slope is the same calculated one.
The p-shift follows a dispersion of 10-state up to Bc~5 T and at larger field the s saturated (for D01m dot) or becomes negative (for D07m and D09m) approximately ing the 02-04 states, i.e., shows bending related to level crossing (see Figure A2c). T shift has a ~30% smaller slope than the 01-state (~1 meV/T).
The d-shift follows the 11-state for Be < Bc for all dots and the 03-, 04-and 05-sta B > Bc and the D01m, D07m and D09m dots, respectively, revealing a level crossing s to the p-shift. For the D01m dot, the d-shift is strongly suppressed compared to t state.  The s-shift data show that the bends seen in the spectra in Figure 2a-c are jumps having amplitude 1-2 meV over ∆B e~1 T (see dashed ovals in Figure 2a-c). Their appearance is different in different dots and B/B e ranges, revealing size dependence and B e ↑-B e ↓ asymmetry (hysteresis). In the D01m dot, one step is observed in both B − ↓and B − ↑ranges at B = −1 T, which is thus symmetric. In the D07m dot, two asymmetric steps are observed, one in the B − ↓and the other in the B + ↓-ranges at B = −2 and 5 T, respectively, and no mirror steps appear in the ranges of reversed direction of field change. In the D09m dot, four asymmetric steps are observed in the B − ↓and B + ↑-ranges at B = −2, 3, 5 and 7.
The calculated s-shift shows a weak, nearly linear dispersion with slope~0.25 meV/T and in the regions where the jumps are absent the experimental slope is the same as the calculated one.
The p-shift follows a dispersion of 10-state up to B c~5 T and at larger field the shift is saturated (for D01m dot) or becomes negative (for D07m and D09m) approximately reaching the 02-04 states, i.e., shows bending related to level crossing (see Figure A2c). The p-shift has a~30% smaller slope than the 01-state (~1 meV/T).
The d-shift follows the 11-state for B e < B c for all dots and the 03-, 04and 05-states for B > B c and the D01m, D07m and D09m dots, respectively, revealing a level crossing similar to the p-shift. For the D01m dot, the d-shift is strongly suppressed compared to the 11-state.
In the B e a range, the experimental data points do not coincide with mirror counterparts in the B e n -range, i.e., the shifts have a B e a /B e n asymmetry. This is most clearly seen for the d-peak in the D01m dot.
The appearance of the B e a -range is direct evidence for the existence of the ems in this range. For the B e n -range, the existence of the ems follows from the observation of the jumps of the s-peak and the suppression of the B e -dispersion of the pand d-peaks for B e < B c . The former are ∆ ν steps and the latter is direct evidence for charge reduction of the e in the single-particle state (see Appendix C.2).
Since the B has opposite directions in the B e a and B e n ranges, the Vs and ems of the former are anti-Vs (aVs) and anti-ems (aems) in the latter (see Appendix C.1). The ems at zero B e are self-organized and can be considered to be equilibrium ems, which we will denote as S-ems. Thus, the shifts of the peaks in the B a e -range are related to the destruction of S-ems by the external field, which implies that the internal field generated by S-ems can be B ν = B ec and the B decrease versus B e increase in this case is non-linear in the B e a -range, as observed. The ems in the B e n range are induced by the B e and can be considered as stimulated, further denoted as B-ems.

Majorana Anyon Composites in Fock-Darwin Spectrum
In Figure 4a-c, we present the results of the e-em-FD spectrum fit to the experimental data. The data and fit are presented in the reduced field ν B -1 and energy ω/ω 0 units (see Figure A2c) and the e-em-FD-states (e-em-FDSs) are labeled as e ν l , where l is angular momentum of the state (see Appendix C.1). In the figures, we also show total charge values e N * (upper plots) and constructed V-structures (inserts). Total charge values are shown for photo-exited and initial states and were used to calculate the ∆ ν and E S em-C . The experimental data for B e ↓ and B e ↑ measurements are plotted in the same graph and they reveal a B e ↑-B e ↓ asymmetry of the s-peak jumps as discussed above, and the same for In addition, a comparison of the plots shows that the ν B −1 -dispersion of the FDSs is smoothed out as D increases (see the shallowing of 2e 1 level "depth" at ν B −1~0 .5), which shows a suppression of the quantum confinement effect.
The data in Figure 4a-c show that the ems are observed in the p-and d-single particle states forming B-em-Cs over the entire B e n -range measured. The composites correspond to a set of n GSs |ν nS1 + > N 1/νB given in Table 1. In all B-em-Cs, only the 2e s-state is em-free and has ν = 2 (e 2 0 -state) for n = 1 and 2 and ν = 1 (2e 1 0 -state) for the n=3 composites. The rest are B-em states, which have ν = 1 and 2/3 for 2e p y /p x -states and ν = 1/2, 1/3, 14 and 1/8 for 1e 10(p x )-, 02(d y )-, 20(d x )-and 03(f y )-states, depending on B, N * and D.
The n = 1 structures in the inserts in Figure 4a-c show that in the D01m dot B-em M -C has two single-state Majorana ems (em M ) (see Appendix C.1) e 1 -1 (0) and e 1/4 −2 (0) and one two-state em M , formed by s-e and e 2/3 1 states, in which the V of the former is compensated by the aV of the latter. Note, however, that this compensation is not complete since two s-es in the ν = 2 state generate two-times smaller flux. In the D07m dot, the n = 1 em M -C has the same number of Vs N V since the extra e generates e 1/2 2 (0) em M in a d x -state taking two Vs from the d y -state. In the D09m dot, MMs e 1/2 −2 (2) and e 1/2 2 (2) are generated in d xand d y -states increasing the N V on four Vs/aVs. For n = 2 and n = 3, em M -C are formed by the addition of Vs and elimination of aVs, resulting in an increase/decrease of NV/νnS + . Thus, we observed the expected NV increase, with an increase of D and B, and the minimum NV value of 9 has n = 1 em M -C in D01m, while the maximum of 21 has n = 3 em-C in D09m. The charge values obtained are in the range eN = (3−6)e0, which is thus up to two times smaller than that without ems, i.e., (7)(8)e0. The reduced charge for n = 1 corresponds to a zero field Bν generated by the composite, i.e., ν1S + = ∞, and thus can be considered as Majorana anyon/em-C (em M -C). This em M -C has eN * = 4.91, 5.66, and 4.91 in the dots D01m, D07m and D09m, respectively, which shows that the increase of eN of D07m relative to D01m due to an extra e is compensated in D09m by its decrease due to the increase of the NV.

Charge Hysteresis and Majorana Modes
The Δν values calculated show an absence of the steps in the D01m and D07m dots and their presence in the D09m dot (see the corresponding curves in Figure 4a-c). Zero Δν (no steps) occurs when the photo-excited (topmost) em states and the initial states of the neighboring em-Cs have the same ν and NV, respectively (see Equation (A9)). This is the case for the smallest D01m dot for both Be↑ and Be↓ measurements.
For the intermediate size D07m dot, this took place only in the Be↑ data, and in the Be↓ data a negative jump occurs at νB −1~0 .27 (ν~7/2). After the jump, the low-energy position of the s-peak is maintained under a further field decrease down to zero. This shows that two em M -Cs exist in the initial state in the D07m dot, which are 3(∞) and 19/7(∞) em M -Cs (see V-structure in the lower right insert in Figure 4b), among which the latter is a MM For n = 2 and n = 3, em M -C are formed by the addition of Vs and elimination of aVs, resulting in an increase/decrease of N V /ν nS + . Thus, we observed the expected N V increase, with an increase of D and B, and the minimum N V value of 9 has n = 1 em M -C in D01m, while the maximum of 21 has n = 3 em-C in D09m.
The charge values obtained are in the range e N = (3−6)e 0 , which is thus up to two times smaller than that without ems, i.e., (7-8)e 0 . The reduced charge for n = 1 corresponds to a zero field B ν generated by the composite, i.e., ν 1S + = ∞, and thus can be considered as Majorana anyon/em-C (em M -C). This em M -C has e N * = 4.91, 5.66, and 4.91 in the dots D01m, D07m and D09m, respectively, which shows that the increase of e N of D07m relative to D01m due to an extra e is compensated in D09m by its decrease due to the increase of the N V .

Charge Hysteresis and Majorana Modes
The ∆ ν values calculated show an absence of the steps in the D01m and D07m dots and their presence in the D09m dot (see the corresponding curves in Figure 4a-c). Zero ∆ ν (no steps) occurs when the photo-excited (topmost) em states and the initial states of the neighboring em-Cs have the same ν and N V , respectively (see Equation (A9)). This is the case for the smallest D01m dot for both B e ↑ and B e ↓ measurements.
For the intermediate size D07m dot, this took place only in the B e ↑ data, and in the B e ↓ data a negative jump occurs at ν B −1~0 .27 (ν~7/2). After the jump, the low-energy position of the s-peak is maintained under a further field decrease down to zero. This shows that two em M -Cs exist in the initial state in the D07m dot, which are 3(∞) and 19/7(∞) em M -Cs (see V-structure in the lower right insert in Figure 4b), among which the latter is a MM of the former appearing because of the generation of an e 1 −1 (0) em M . The B e ↓-jump is thus related to a charge increase in the initial state due to N V reduction in the 3(∞)-em M -C and is reproduced well by E em-C s (B). Thus, B e ↑-B e ↓ measurements reveal charge hysteresis (CH). CH is accompanied by B e ↑-B e ↓ p-peak splitting at ν B −1 , B = 0, which can indicate generation of additional Vs in the p-state.
In the largest dot D09m the ν l values of the four topmost em states are 1/4 2 , 1/8 −2 , 1/2 1 , and 1/3 −3 , which are different and thus give three ∆ ν steps at ν = 7/2, 2 and 5/4, the first and the third of which are negative, as is seen in Figure 4c. The number and position of these steps match well to that of the B e ↑ experiment, but the sign of the first and third jumps is positive instead. This implies a MM in the corresponding initial state, substituting Vs of the photo-excited em. Analysis has shown that this is the e 1 −1 (0) em M (the same as in the D07m dot) forming the e 2/5 −1 (1) em from the e 2/3 −1 (1) one. The calculated E s em-C accounting for this MM shows good agreement. The B e ↓ data show two CH starting at ν~5/4 (CH 1 ) and 2 (CH 2 ), respectively. In the CH 2 em-C, there is an increase of p-peak energy, indicating the generation of additional Vs.

Collapse of S-Magneto-Electron-Composites
The structure of S-em-Cs and the ν 0 + (B ν ) values can be derived from the crossing of the positions of the pand d-peaks at B e = 0 and em-FDSs of the B n -range. Thus, a position of the topmost d-peak corresponds to a d x -state at ν~5/2 and S = 3/2 in the D01m dot and to d y -state at ν~5/2 and S = 0 in the D07m and D09m dots. Note that while for the D01m dot the total spin of the mirror aem Sand B-composites is the same (3/2), for the D07m and D09m dots it is different (0 versus 1 and 2). Note also that the ν 0 + value of all dots (see Table 1) are nearly the same, i.e.,~5/2 (for dots D01m and D07m ν 0 + = 5/2 + 1/8 and 5/2 + 1/6, respectively), and the corresponding B ν values are larger than B ec on~1 T.
The preference of the self-formation of the S-em-Cs follows from their smaller charge compared to the B-em-Cs; it is 3.75e 0 , 3.05e 0 and 2.08e 0 for the S-em-Cs versus 3.91e 0 , 5.41e 0 and 3.85e 0 for the B-em-Cs for the D01m, D07m, and D09m dots, respectively. Charge reduction is provided by the extra Vs in the p-states (see the corresponding V-structures in the left inserts in Figure 4a-c) and thus there can be some mechanism for selecting the V direction in these states. These preferential Vs create e *1/2 −1 (2) and e *1/6 1 (4) S-ems (we add a superscript * to distinguish S-ems from B-ems).
In the D01m dot, a symmetric positive s-jump at ν B −1~0 .1 related to S-em-C destruction indicates the generation of aVs/Vs decreasing/increasing B ν needed for formation of n = 1 B-em-C/S-em-C in B e ↑/B e ↓ measurements. A transition to S-em-C for the B e ↓ data is accompanied by a red shift of the p-peak, indicating emission of the p y -component, which is not seen for B-em-Cs. Its appearance implies a redistribution of the hole density (see (Appendix A.3) under V formation and the instability of B-em-C.
In the D07m and D09m dots, negative B e ↑ s-jumps correspond to the annihilation of MMs under a decrease of B and their absence in the B e ↓ data indicate a suppression of the MM under the collapse of n = 1 B-em-Cs. The signatures of S-em-C instability in these dots is a B e ↑-B e ↓ splitting of the p-peak, which is stronger in D07m dot.

Majorana Anyons
A subset of the GSs measured is different from that obtained in the exact quantum mechanical CI calculations in Appendix B.1. For D09m dot, they are |∞ 1 , 3 2 , 4/3 2 > 8 1.1 (see Table 1) and |8 0 , 7/2 1 , 7/3 0 , 2 1 > 8 0.6 (see Figure A2a,b), respectively, showing a reduction of the number of B-em-C GSs compared to e-GSs due to a suppression of the level crossing (see Appendix C.2). The CI set includes fractional ν GSs, which, however, do not imply a generation of Vs by single electrons, i.e., fractional charge, and reflect only a matching of the number of Vs to the total angular momentum value [36,37]. Moreover, the observed B-em-Cs have a variety of MM states, including d-type MMs, em M and em M -Cs, which is unexpected and very important accounting for the limited experimental data on MMs (mainly for the p-type-states) in FQHE [25], one-dimensional hybrid superconductor topological structures [38,39] and in topological superconductor [40]. Thus, our observations give high impact to the physics of anyon/Majorana states and to the perspectives of their use for TQC. Note that from five em-Cs observed there is only one MM-free, which is a ν 2,1 + = 5/2 composite in the D07m dot, and, thus, MMs seen are intrinsic features of the confined electrons for ν > 1, in contrast to ν < 1 molecular states [16]. In the context of em M and em M -Cs, a SC analog of such states has been reported in the compound EuFe 2 (As 1−x P x ) 2 [41].
The ∆ ν steps observed demonstrate detection of fractional charge variations related to MMs using PL spectra. Moreover, the observed CHs are related to the states with and without MM, i.e., to a MM qubit (see Appendix D), and for the D07m and D09m dots these are zero-field qubits. This two-level qubit can be used in conventional schemes of quantum computing and/or quantized information bit of classical Boolean logics [42]. The data show that the appearance of the ∆ ν steps depends on D, N and B, or more generally on N V and r s , and is suppressed with r s decrease. This gives the opportunity for engineering of MM qubits. The CH is related to the suppression of aV generation under a decrease of the field and evidently reflects some fundamental interaction between em Vs and a sign of the magnetic field variation, which needs further investigation.

Magneto-Electrons and Fractional Quantum Hall Effect States
Our experimental results presented here and in the previous studies [15,16] show that the fractionally charged em-anyons emerge in the quantum confined (gapped) e-states, which allows superconducting-type sub-e PCLs/Vs generated by a single or a pair of es, creating ems. This allows consider a description of the FQHE in terms of a liquid crystal (LC), rather than as an incompressible liquid, in which specific ν nS + -B-em-C is a unit cell of the crystal. The nucleation of the ν nS + -B-em-C LC is provided by a quasi-ordering of potential fluctuations (PF) [43] of the corresponding size as was suggested by us in [15]. The experimental evidence of such quasi-ordering is the observation of the Wigner-crystaltype network patterns of the localized electron states for ν = 1, observed in a scanning electrometer probe experiment in quantum Hall 2D-e structures [44].
Within the LC description, the conductivity plateaus are quantum-confined-type L S plateaus (see Figure A2b,d) rather than a set of states above the mobility edge of the disorder-broadened LLs. The L S plateaus strongly overlap, as follows from the E L S curves in Figure A2e, which is consistent with the models describing the temperature-dependence of the width of IQHE and FQHE plateaus [45], revealing a Lorentzian broadening of LLs. Moreover, LC description does not involve e-e interaction as shown in Appendix B.
We can suggest that the em M Majorana anyons revealed are the CFs of the quasiparticle description of the FQHE and they naturally explain the zero internal field of CFs, leading to a termination of the conductance plateaus at even fractional filling factors. The description of CFs using LC implies B-em-C consisting of e 1/2 0 and e 1/2 −1 (0) ems, in which the latter contribute to the conductivity. The S-em-C of such type can explain the 1 /2 and 1 /4 fractional quantization of the conductivity of the holes and electrons in Ge [46] and GaAs [47] nanowires, respectively, in a zero magnetic field and the "0.25 anomaly" of the conductivity of a point quantum contact [48,49].
The signatures of em M s in these experiments using confined geometries, i.e., probing of submicron areas, imply that the corresponding PFs are naturally formed in 2D-e structures and that their short-versus long-range order arrangement determine the formation the corresponding LCs and FQHE conductivity plateaus. Thus, in the extremely pure, high mobility structures in which FQHE plateaus emerge, the imperfection density is in a dilute regime (~1%), which provides a wide spectrum and long-range order of the PFs [43], induced by Sand B-em-Cs. The PFs depth in 2D-e-structures is~10 meV [50], which is an order of magnitude smaller than in InP/GaInP 2 QDs, and we can expect that the FQHE LC states will include em-Cs having size~50 nm, adopting only sand p-ems, and assume extended imperfections having a size of at least a few nanometers.

Majorana Anyons and Topological Quantum Computing
The proposal of the topologically protected qubit involving the ν = 5/2 state [31] is based on Laughlin's quasi-particle description of FQHE states [14], in which this state is assigned to a non-Abelian topological phase characterized by a Pfaffian wave function [51,52]. This state is considered as a set of degenerate p-wave paired fermions having charge e 0 /4. The qubit is formed by a zero-energy Majorana excitation of a fused fermion having charge e 0 /2. We can suggest that the em analog of the e 0 /4 fermion is e 2 0 + e 1/8 −1 (2) B-em-C, the e 0 /2 state is e 2 0 + e 1/8 −1 (2) + e 1/8 1 (2) B-em-C, formed by addition of two es, and MM is e 2 0 + e 1/8 −1 (2) + e 1/8 1 (0) B-em-C. It follows from our results that it may be possible to realize and measure such B-em-Cs in InP/GaInP 2 using appropriate N and D, and to check the formation of the corresponding LC states in 2D-e structures using high-spatial-resolution optical [16] and electrometrical [53] measurements. Moreover, it may be possible to realize similar states using S-ems, thus creating magnetic-field-free TQCs.

Conclusions
We have used the magneto-PL spectra of InP/GaInP 2 QDs having seven electrons and a Wigner-Seitz radius~1.3 to demonstrate the self-formation of fractionally charged anion(magneto-electron) composites having ν~5/2 and Majorana modes/anyons. We observed the destruction of these composites and the appearance of anti-em composites induced by a magnetic field. Using a theoretical description of magneto-PL spectra and an analysis of the electronic structure of these dots based on single-particle Fock-Darwin spectrum and many-particle configuration interaction calculations, we show the formation of fractionally charged anion states based on the mechanism in which non-interacting quantum confined electrons generate sub-electron persistent current loops (vortexes). This implies the liquid-crystal nature of the fractional quantum Hall effect states, in which Majorana anyons are spontaneously formed, and opens new perspectives for the realization of topological quantum computing. Data Availability Statement: All data needed to evaluate the conclusions in the paper are present in the paper.

Conflicts of Interest:
All authors declare that they have no competing interest.

Appendix A. Theory of Photo-Luminescence Specra of Quabtum Hall Pudlles
Appendix A. 1

. General Model
The PL spectrum of a QD having N electrons arises from the radiative recombination of the photo-excited hole (h) state in the valence band with N* = N + 1 es in the conduction band, which together form the photo-excited state (PS) of the QD. This PS is formed after absorption of the incoming photon by QD in initial state (IS) creating an e-h pair, as shown in Figure A1a. Accounting for the fact that IS can have fractionally charged ems, we assume that all the interactions leading to formation of such a state are included in the formation of the e-h pair and PS state and do not affect photo-excitation and radiative recombination processes. This implies that, for an IS having ems, the generated and recombined e-h states have fractional charge. The energy of the photon of the exciting laser is ħωph~E0 + Egr_QD + ΔEexc, where E0 is a band gap energy of the material, Egr_QD is the ground state energy of N * e-h QD, and Eexc is the excess excitation energy (~500 meV). After photon absorption, the photo-excited e-h pair relaxes (loosing Eexc by phonon emission) to the ground PS state, which is the initial state of the PL process (ISPL). The many-particle Hamiltonian of the PS-ISPL state can be written, using composite indexes i for a single particle state | ⟩ and accounting for the Coulomb interaction [54], as The operators and (ℎ and ℎ ) create and annihilate an electron (a hole) in a single particle (SP) state | ⟩ (| ⟩), where there are N * electrons. The first two terms in Equation (A1) are the kinetic energy of the electrons and the hole, quantized by ħω0 and ħωh, respectively, where ω0 and ωh are the harmonic frequencies of e and h in the parabolic confinement potential. The third and the fourth terms are scattering terms described by corresponding two-body matrix elements of e-e [55,56] and e-h [56,57] Coulomb interactions, respectively. The energy of the photon of the exciting laser ishω ph~E0 + E gr_QD + ∆E exc , where E 0 is a band gap energy of the material, E gr_QD is the ground state energy of N * e-h QD, and E exc is the excess excitation energy (~500 meV). After photon absorption, the photo-excited e-h pair relaxes (loosing E exc by phonon emission) to the ground PS state, which is the initial state of the PL process (IS PL ). The many-particle Hamiltonian of the PS-IS PL state can be written, using composite indexes i for a single particle state |i and accounting for the Coulomb interaction [54], as The operators c i and c † i (h n and h † m ) create and annihilate an electron (a hole) in a single particle (SP) state |i (|n ), where there are N * electrons. The first two terms in Equation (A1) are the kinetic energy of the electrons and the hole, quantized byhω 0 andhω h , respectively, where ω 0 and ω h are the harmonic frequencies of e and h in the parabolic confinement potential. The third and the fourth terms are scattering terms described by corresponding two-body matrix elements of e-e [55,56] and e-h [56,57] Coulomb interactions, respectively.
For the final PL state (FS PL ), which is the IS of the QD (see Figure A1c), the Hamiltonian H IS does not have the fourth term of Equation (A1) and there are N electrons and no hole. The PL spectrum, i.e., the dependence of the emission intensity on the photon frequency I PL (ω), is determined by Fermi's Golden Rule: where |v i and v f are the initial and final states of the QD having corresponding energies E i and E f . The inter-band polarization operator P = ∑ kn k|n c k h n removes an e-h pair from the PS state |v i . The matrix element (overlap integral) k|n determines the selection rules. Since there is only a single valence band hole, we can define a purely electronic PS state v k i = ∑ n k|n h n v i [54]. The operator P create a hole in PS P |v i = ∑ k c k v k i and leaves the QD in the excited state IS*-FS PL * (see Figure A1b), having energyˇ N k . This state can be degenerate with a few other IS* configurations, which can be mixed by Coulomb exchange leading to a splitting of the PL transition [54]. Such a contribution is important in relatively small excitonic QDs [58], which are not considered here.
The contribution of the IS* state to the PL spectrum is described by a difference ∆ k = v k f * − v k f and it is important in dots having a molecular electron arrangement, i.e., r s > 2 [16].
Accounting for a low temperature (hω 0 ,hω h < kT) and low pumping rate (N ph~1 ), generating only one e-h pair in the dot in IS PL (see Figure A1a), the PL spectrum, can be written as: where is the many-particle orbital energy of i-electron in PS(IS), N * i = ε N * i − ε N * 1 is difference in the kinetic (confinement) energy between the higher and the lowest states,ˇ N k is the energy of the v k f * state and γ i (ω) is a Lorentzian spectral function of the corresponding transition, describing the final width of the spectral lines and replacing of the delta function used in Equation (A2).
The PL spectrum thus has three contributions: a "zero" energy s-line (first term) accompanied by the anti-Stokes (ASCs) and Stokes (SCs) components (second and third terms, respectively). The SCs arise from the "molecular" contribution to the electron distribution, i.e., center-of-mass and relative electron vibrational modes and, in general, include combinations of translational, breathing, and rotational modes, having frequencieš N k h ∼ω 0 ,~1.8ω 0 , and~0.2ω 0 , respectively [33].
We also analyzed SPSs accounting for the atomic crystal structure of InP/GaInP 2 dots using the 8-band k·p method [35]. In Figure A1d we show the probability density distributions of the 1s, 1p x , and 1p y states calculated for a circular dot having size 100 nm. The calculations givehω 0 = 4 meV andhω h = 2 meV. This shows that the spatial extent of e and h states in these QDs is nearly the same (despite a two-times difference in the confinement energy) and that their wave-functions show FD type sand p-shell spatial structure. For the h, however, the spatial extent is two times larger than that of FD states, which is a result of an inter-valence-band mixing induced by a piezo-electric field. The piezo-electric field also leads to a~10 nm separation of the e and h along the x-direction, which is [−110]. This makes transitions involving x-type electron shells allowed.

Appendix A.3. Effect of Coulomb Interaction on Confined Electrons States
The e-h Coulomb interaction between the N * es and single h (the fourth term in Equation (A1)) mixes single-band h states (intra-band mixing) allowing PL transitions for all es in QD [34] as shown in Figure A1a. Accounting for the mixing, we can approximate the distribution of the density of the h in the lowest state by a nearly uniform positive charge distribution having weak modulations induced by spatial variations of the occupied electron shells and extending over the whole QD size. Thus, we can neglect the effect of the e-h interaction on ε N * i and include only e-e interaction in analysis of PS. In Figure A1e, we present the electron density distributions (EDDs) of the occupied PS shells of a QD having N = 9,hω 0 = 7.5 meV (D~75 nm), total spin S z = 3/2 and a non-circular dot shape (analog of D07m) calculated by the Hartree-Fock (HF) method. The densities have minor shape deviations from those of the single-particle ones in Figure A1d, except (1d-2s) x shell hybridization together with slight asymmetry of the states having different spin (see s-state) and spatially split parts for non-zero angular momentum states (see spin down p x -state), which appear from the non-circular shape. The e-e interaction, however, gives a renormalization of the shell splitting, resulting in ε N * i /ε e i~0 .7. This shows that the electron states in InP/GaInP 2 QDs can be approximated by FDSs having screened hω 0 * = ε N * i ≈∆E sp ≈∆E sd /2, where ∆E sp and ∆E sd is s-p and s-d splitting.

Appendix A.4. Magnetic Field Dispersion of Photoluminescence Spectra
The shift of s-line in magnetic field is where for the parabolic potential, for which the energy of center-of-mass motionhω 0 does not depend on N [59], the Coulomb energy term is The formation of e ν ems corresponds to a reduction of the charge from e 0 to e 0 /ν and results in a decrease of V ee (B). Since N * ·V ee (B) <hω h (B), the effect on E s (B) is expected to be weak.
For the ASCs the B-shift is , where B ν accounts for internal field generated by Vs to experimental data, as was shown by us for dots having r s > 2 and ν < 1 in Refs. [15,16].

Appendix B. Ground States of Quantum Hall Puddles
Appendix B.1. Total Angular Momentum Plateaus of Confined Electrons in Magnetic Field In Figure A2a,b we show the total energy E S tot and total angular momentum L S z , respectively, versus B calculated using CI for N = 8 andhω 0 = 5.8 meV, corresponding to the PS of the QD D09m. We used reduced units ν LS −1 = L S /L 0 = −L z Sz /L 0 , where L 0 = N(N − 1)/2 [16,60] and ν B −1 = B/B 0 , where B 0 is a field corresponding to ν = 1, and B(ν B −1 ) ranges up to~B 0 (1). The ν LS −1 (ν B −1 ) curves are presented for all total spin S z = S states from 0 to 4 and for S > 1 the curves are truncated for ν −1 > 0.25. The inset in Figure A2a shows E S tot for N = 7 and S from 1 2 to 5/2 related to the IS of the QD D09m. The upper inserts in Figure A2b show the calculated zero field CDDs for N = 8 and 7, and the lower ones show the HF EDD of the 10-p x state with a few V circles adjacent to each other, which represent a tentative description of the em structure. E S tot and L z for N = 6 (the IS of D01m) were calculated in Ref. [16].
The CDDs give D ≈ 85 nm and r s ≈ 1.5 (see Table 1). They have two minima having size~15 nm and separated by~30 nm along the x-axis, which can be assigned to (7,1) and (6,1) isomers in a molecular arrangement in the limit of r s >> 1 [61], for N = 8 and 7, respectively. For N = 7, the CDD show enhanced density (clustering of electrons) along the y direction.
The E S tot (ν B −1 ) energies in Figure A2a and in the insert are nearly constant up to ν B −1~0 .6 and then increase linearly with B increase. The former corresponds to weak fields (ω c < ω 0 ), for which the kinetic and Coulomb energy, related to quantum confinement, E conf = 16hω 0 * , and center-of-mass motion, E c.m ≈ r s E conf , are dominant. The latter corresponds to strong fields (ω c > ω 0 ,) when the cyclotron energy of electrons, E cycl = 4hω c , becomes dominant [16].
As can be seen from the comparison of E S tot (ν B −1 ) curves, the boundary between weak and strong fields is S dependent and increases from ν B −1~0 .5 for S = 0 to ν B −1~1 for S = 4.
The analysis of the curves also give B 0 = 7.2 T, which is a field at a crossing point between the E 3 tot (ν B −1 ) and E 4 tot (ν B −1 ) curves. This value is equal to B 0 = A·(h/e)(4N/πD 2 ), where a geometrical factor A = 1.25.

Appendix B.2. Single-Particle Fock-Darwin Ground States and Vortexes
In Figure A2c we present the FD spectrum E k,l (ν B −1 )/hω 0 (hω 0 =hω 0 * = 3.5 meV), which is related to D09m, plotted up to 3hω 0 and ν B −1 = 1.15, which includes levels up to 13 and 07. Using this spectrum, we calculated the total angular momentum L FD S and energies of the corresponding states E L S versus ν B −1 for N = 8 and S = 1, which are shown in Figure A2d,e, respectively. For the calculation of the E L S energy, we use the expression E L S = 16·hω 0 ·r s + ∑ N E kl L(S) , where the first term is the Coulomb energy and E kl L(S) are FDSs constituents of the L(S) state. In Figure A2d, we also show L FD,V 1 = L V + L FD 1 (see Appendix C.1), which is also shown in Figure A2b. In Figure A2c we present the FD spectrum Ek,l(νB −1 )/ħω0 (ħω0 = ħω0 * = 3.5 meV), which is related to D09m, plotted up to 3ħω0 and νB −1 = 1.15, which includes levels up to 13 and 07. Using this spectrum, we calculated the total angular momentum LFD S and energies of the corresponding states EL S versus νB −1 for N = 8 and S = 1, which are shown in Figures  B1d and B1e, respectively. For the calculation of the EL S energy, we use the expression EL S = 16•ħω0•rs+ ∑NEkl L(S) , where the first term is the Coulomb energy and Ekl L(S) are FDSs constituents of the L(S) state. In Figure A2d, we also show LFD,V 1 = LV + LFD 1 (see Appendix C.1), which is also shown in Figure A2b.  (7), dashed curve is the LFD,V 1 function (see also (d). The inserts show calculated total CCD (upper) for at N = 8 (left) and N = 7 (right) at B=0 T and HF EDD of 1px single particle state (lower) with Vs (circles with arrow, showing circulation direction), corresponding to e 1/2 l em (left) and its MM (right). The CCD and EDD size is 100 × 100 nm 2 . FD spectrum (dark and light color curves are occupied and unoccupied levels, respectively) Ek,l FD (νB −1 )/ħω0 * calculated for ħω0 * = 3.5 meV (c) together with total angular momentum (thick curve) LFD 1 (νB −1 ) (d) and energies (thin dot curves) EL 1 (νB −1 ) of N = 8 and S = 1 (see thick lines outlining topmost levels in (c)) (e). Thick solid lines in (e) outline minima of EL 1 (νB −1 ) and vertical arrows mark corresponding GSs. Thin line in (d) is LFD,V 1 function.
The topmost S = 1 FDSs in Figure A2d show three crossings at νB −1~0 .35, 0.57 and 0.74. The last crossing corresponds to the transition between the weak and strong field of Figure  A2a, in which all electrons occupy the lowest Landau level. As can be found from an analysis of the FD spectrum in Figure A2c, this crossing shifts to larger fields with step ΔνB −1~0 .1 when S increases, which is in good agreement with the E S tot crossings and, thus, allows one to explain this shift using a single-particle description. The topmost S = 1 FDSs in Figure A2d show three crossings at ν B −1~0 .35, 0.57 and 0.74. The last crossing corresponds to the transition between the weak and strong field of Figure A2a, in which all electrons occupy the lowest Landau level. As can be found from an analysis of the FD spectrum in Figure A2c, this crossing shifts to larger fields with step ∆ν B −1~0 .1 when S increases, which is in good agreement with the E S tot crossings and, thus, allows one to explain this shift using a single-particle description.
Between the crossings, a set {l z } N is conserved resulting in the L FD S plateaus (see Figure A2d). Within a specific L FD S -plateau, the energy of the corresponding state E L S is the lowest and has a minimum, which gives modulations of the total energy and a set of GSs (see Figure A2e), like that in Figure A2a. Thus, the plateaus and discrete GSs of electrons in magnetic field arise in QDs due to quantum confinement and they appear without accounting for the e-e interaction. The L FD S plateaus, however, have one-and-a-half-times larger period and two times smaller amplitudes than L S ones, which implies a contribution of additional non-zero l z states, missing in the FD spectrum. It is naturally assumed that these states are the Vs, which add l z = 1 each. Since at ν = 1 the number of Vs N V = N*, the contribution of Vs to L S can be approximated by a stepped function L V , in which l z = 1 step is added at ν B −1~k /N*, where k is integer. A L FD,V 1 function, accounting for their contribution (L FD,V S = L FD S + L V ), has increased period amplitude (see Figure A2d) and approximately matches L S (see Figure A2b). Thus, accounting for Vs and single-particle states seems to be adequate for the description of the ems states.

Appendix C. Fractionally Charged Ground States
Appendix C.1.

Magneto-Electron Composites and Majorana Modes
The fractional ν nS implies the formation of ems. Based on our previous studies [15,16], we suggest that for the p x state and ν = 1/2 the em structure consists of two vortexes (Vs) generated at the center of the left half of the EDD at x~−50 nm, which corresponds to a clockwise phase change of the electron wave-function (see right lower insert in Figure A2b). We denote this em as e 1 1/2 , where we add a subscript l = l z in the em notation, which further will be e l ν . For a paired e-state em is e 2ν l . The adjacent right insert shows corresponding MM, i.e., e 1 1/4 em, in which single V and anti-V (aV) are added. The aV is generated in the right half of p x -state at x~+50 nm, which has anticlockwise phase change of electron wave-function. The e 1 1/4 em can also have a configuration consisting of two Vs and two aVs, which exist at zero B and, thus, can be considered to be a Majorana magneto-electron/anyon (em M ). To distinguish it from a MM, we add the number Vs of the "host" em to the MM-type ems notation. Using such a notation, one can see that e 1 1/4 (0) is em M and e 1 1/4 (2) is a MM. Note that there exists e 1 1/2 (0) em M , which implies that e 1 1/4 (0) em is the MM of e 1 1/2 (0) em M . For the p y -state, the Vs/aVs generated are located along y at distances ±50 nm and the Vs of e 1 1/2 and e -1 1/2 ems do not overlap. The same can be generally valid for other l z states and, thus, different l z states can form ems simultaneously and independently creating a multi-em state, i.e., an em-composite (em-C).
The classification of the em-Cs can be done by listing the values of ν l , which are the νs of e l ν ems, mapping of Vs positions as shown in the insert in Figure A2b and specification of the total composition value ν nS , which we set up as where ν Nie is an integer value related to Ni electrons, N em = N − N i is the number of ems and N V = ∑(ν l ) −1 is the total number of em's Vs. For MM em-Cs having a number of aVs N aV ≤ N V , there is an additional, effective composition value ν iS + , which is where N V + = N V − 2N aV is the number of uncompensating Vs determining B ν . The reduced charge of the em-C is e N * = e N /e 0 = (N i + ∑ n l ν l )/e 0 , where n l is the SPS occupation equal to 1 or 2. A change of ν nS of the PS and IS em-Cs at the boundaries of the L S plateaus can give ∆ ν steps in the dispersion of the s-peak, which can be written as: where b is a coupling coefficient, assumed to be unity, and ( ∆e N e N ) 1,2 = (∆N i + n p l ν p l + ∑ ∆n l ∆ν l ) 1,2 and the superscript p denotes a photo-excited e, is a relative charge between PS and IS of adjacent em-C 1 and em-C 2 composites. The steps are positive(negative) when N V in IS(PS) of em-C 2 is larger than the same in em-C 1 and vice versa. In our dots (N = 7, k = 3 and hω 0~6 meV), ∆E C s (0)~20 meV (as discussed above in Appendix B.1) and for single e 1/2 em, i.e., two Vs N V difference, the expected step amplitude is ∆ ν~1 meV. Equation (A4c) can thus be written as E s em−C (B) ≈ N·V ee (B) + hω h (B) + ∆ ν . (A10)

Appendix C.2. Fock-Darwin Spectrum Fit
The ems composition of the single particle e-states were obtained using a fitting procedure. In the procedure we first extracted the experimental pand d-FD states values from the data in Figure 3a-c using the subtraction of the s-shift from the pand d-shifts, respectively. Second, we fitted calculated FD energies to the experimental data by adjusting the cyclotron frequency ω c , which give ν l values. Third, we plotted the experimental and calculated FD shifts and denoted them by e ν l (see Figure 4a-c), together with s-shift and the calculated ∆ ν and E s em-C values. The results are plotted using reduced units, as in Figure A2c. This allows a unified description of the e-em-FD spectrum of different dots. The fitted curves include an adjustment of the zero field pand d peak positions for dots D07m and D09m.
The fit accounts for a decrease of the B-dispersion due to a 1/ν-times decrease of ω c , which in the limit 1/ν→0 comes out to be dispersionless, i.e., a zero shift, eliminating the level crossing. Thus, for the states having negative l z (see the FD spectrum in Figure A2c), the ems lead to an increase, while for zero and positive l z to a decrease of the energy, towards a zero field value. For negative l z this can be traced for e 1/2 -2 and e 1/4 -2 B-ems of D01m in Figure 4a.

Appendix C.3. Construction of the em-C Structure
The em-C structure is represented by the Vs overlaid on the CDD of the dot. To plot it, we used the ν l values obtained and put (ν l ) −1 arrowed circles of individual Vs in the region occupied by corresponding HF EDD (see inserts in Figure A2b) adjusting their positions to avoid overlap with Vs of other FDSs. The condition prohibiting V overlap restricts total number of Vs and was used to check the values of ν nS obtained from the fit. The arrows of the circles contour line show the V direction which is clockwise for Vs and anticlockwise for and Vs/aVs and are color coded as is shown in the inserts in Figure 4a-c.

Appendix D. Majorana Qubit
In Figure A3, we present the observation of the mixed 3(∞) and 9/7(∞) em M -C states in the magneto-PL spectra of dot D07m, which correspond to the spectra measured at time delays <4 min, before equilibrium 3(∞) state will be stabilized. The mixed states have a position of the s-peak between 0 (3) and 1 (19/7) qubit states as is seen in Figure A3 for a 2 min delay. Nanomaterials 2022, 12, x FOR PEER REVIEW 20 of 22 Figure A3. Spectra of D07m QD measured at Be↓ = 5T at time delay 0, 2 and 4 minutes (solid curves) together with the spectra measured at Be↑ = 5T (dotted curve). Figure A3. Spectra of D07m QD measured at B e ↓ = 5T at time delay 0, 2 and 4 minutes (solid curves) together with the spectra measured at B e ↑ = 5T (dotted curve).