Supplementary Figures Supplementary Figure 1 | Sem Images and Illustration of a Fpi with a Center Ohmic Contact

surrounded by an isolation-gate. The regime of this device can be tuned by varying the voltage on the isolation gate í µí± !"#$%&!#' ; it can be tune to either the pure AB regime, by setting í µí± !"#$%&!#' = 0, or to an intermediate regime by setting a low enough voltage resulting in depletion of the electron-gas below it. We note that a pure CD regime cannot be reached with this device.


Supplementary Note 2. Capacitive model of the FPI A capacitive model for the FPI
The FPI setup is depicted in Supplementary Figures 5 as an electrical circuit. By means of two QPCs, a quantum Hall (QH) strip is separated into (weakly coupled) three parts: left and right leads, and the FPI in the middle which can be regarded as a quantum dot (QD). As in the main text, we consider the case of filling factor 1 < ! < 2. The FPI consists of an outer edge channel (with a chiral onedimensional model), and an interior compressible puddle in the bulk 2 , separated by an incompressible = 1 electron gas.
We consider a capacitive model, described by an equivalent electrostatic circuit, depicted in Supplementary Figure 5, which is widely accepted to describe our setup 1,3,4 . It consists of three effective capacitors, !"#! , !"#$ , !" , describing the electrostatic energy due to charging at the edge channel, at the bulk, and due to edge-bulk interaction, respectively. The charge distribution at the edge and the bulk is dictated, in principle, by the varied perpendicular magnetic field , and the modulation gate !" , and is subject to minimization of the electrostatic energy.
We first consider an initial tuning of the FPI. The initial value of the magnetic field is ! . The system's ground state is described by the charge ↓ = ↓ with ↓ electrons occupying the LLL, and the charge ↑ = ↑ with ↑ electrons located at the compressible bulk puddle. The edge of the FPI has the initial boundary of the incompressible ν = 1 area , dictated by the chemical potential of the leads. We will argue below that this boundary may be modified by varying the magnetic field and the modulation gate voltage !" . In order to determine the ground state configuration ( ↓ , ↑ ) for the ( , !" ) plane, we first calculate the total electrostatic energy.
We note that even if the number of electron occupying the = 1 level in the FPI remains unchanged (i.e. no transfer of electrons to/from the bulk puddle or the leads), charging of the edge can be induced by varying the magnetic field = − ! . The excess charge on the edge is then: The compressible puddle at the center of the FPI, serves as an effective reservoir: it may take out or give away charge from/to the edge, then minimizing the electrostatic energy. The excess charge in the bulk is expressed as: For a fixed ↑ , the variation > 0 leads to charge accumulation in the bulk of the = 1 Landau . As a matter of fact, the charge in the bulk of the FPI is the sum of two contributions: the electrons of which the incompressible = 1 liquid is comprised of, and the electrons forming the compressible puddle in the middle. The total electrostatic energy of the electric circuit is expressed as: Here, In terms of the AB phase Supplementary Equation 3 reads: We will disregard the last two terms since they do not depend on the number of the electrons, and hence play no role in determining the ground state; evidently, dropping them results in Eq. 4 in the main text. The parameter = ! !"#! ! ! |!| ! effectively accounts for changes of the area by MG voltage variations !" , while keeping ↓ unchanged, and the parameter is defined as !"#! + !"#$ .
We emphasize that the presence of an Ohmic-contact renormalizes the values of capacitances !"#! and !"#$ upwards, while the value of !" is kept unchanged, thus reducing the charging energy.
When we additionally account for the effect of the Ohmic contact through the mutual capacitance, !" ( !" ), between the bulk (or the edge) and the Ohmic contact, !"#$ renormalizes to !"#$ + !" (this is due to the parallel connection of !"#$ with !" ). The effective value of is then scaled down, pushing the regime of behavior of the FPI more towards the AB regime.

Relation to previous works
In the work by Halperin et al. 1 the energy of the system was formulated as follows: The first two terms in this expression are physically (up to the coefficients) equivalent to the first and third terms in Supplementary Equation 3 & 4; namely: Now physically the interpretation of the third term in the energy above differs from that of the first term in our energy; !" ⋅ ! ! represents an effective interaction between the edge and the bulk, while ! !" ! !"! ! represents an effective total charging energy. The relation between the different coefficients reads: And we can identify at ease that Δ = ξ, where Δ ∈ [0,1] is the parameter that denotes the regime in Halperin et al.'s model 1 .

Calculation of the conductance
In this section, we derive an expression of the conductance through a FPI. For a fixed ↑ , the total transmission through the FPI is expressed as !! ↑ = !! ↑ ! with the total transmission amplitude , considering the summation over all possible number of roundtrips of an interfering electron. Here ! ( ! ) and ! ( ! ) are the reflection and transmission amplitudes of the left (right) QPC, respectively. This total transmission is simplified as where !,! = 1 − !,! = !,! ! = 1 − !,! ! . In the single particle picture, !"! is determined by the ratio of the highest energy level ( = − Two comments are due here: Supplementary Equation 9 coincides with a single-particle analysis of transmission through the FPI. There are two facets of many-body physics that are neglected here. ) over the energy as: This conductance is related with , which is the average over the different possible values of ↑ with the corresponding probabilities: This equation is used for drawing the Figure 4b and Supplemental Figures 8c and 9b.

Charge stability diagram
In order to formally express the charge stability diagram we first optimize the energy in Supplementary Equation 4 with respect to both !"# and ↑ at the same, resulting in: which describe the vectors of the charge-stability diagram. Similar to a Bravais lattice, these vectors may be spanned by two: so that the lattice is given by !, . Hence the reciprocal lattice is given by the four equations ! • ! = 2 !" which results in the: These are indeed the underlying AB and CD frequencies, as anticipated.
Now it is clear from Supplementary Equation 14 above that, the vectors spanning the charge stability diagram do not depend on the regime. Nevertheless, the different regimes' charge stability diagram (CSD) differs by the shape of the unit-cells, as we show in the different asymptotic cases in the following section. The full CSD including the structures of the unit-cells is obtained by finding ( ↓ , ↑ ) which minimize the energy (Supplementary Equation 4). In this way, Supplementary   Figures 7 & 8a, discussed in what follows, are simulated.

The Aharonov-Bohm and Coulomb-dominated regime
First we discuss the two distinct regimes of behavior of the FPI: the Aharonov-Bohm (AB) regime and the Coulomb-dominated (CD) regime 1,4 . These are distinguished by the slope of their equi-phase lines in the 2D conductance as function of magnetic field and modulation gate voltage !" (the so-called pajama patterns). In the AB regime, the equi-phase lines have a negative slope; when the magnetic field increases, the phase remains invariant as the modulation gate voltage (and thus the area) decreases. On the other hand, in the CD regime, equi-phase lines follow a non-negative slope.
The different regimes can be retrieved by examining the asymptotic limits of the energy given in Supplementary Equation 3; we shall discuss, for each of the asymptotic limits, its physical interpretation and its charge stability diagram.
(i) A truly non-interacting system, expressed in terms of the effective capacitances: Supplementary Figure 6a). In this regime, ! = 0, !" = 0, and ! corresponds to the level spacing of the single-particle energies, a scale which is not included in Supplementary Equation 3. ! is then determined by the slope of the confining potential at the Fermi level, as well as the magnetic field , and the area .
Since ! = 0, the charge imbalance in the bulk does not play a role in determining the ground state configuration so that the CSD describes the charge state of the edge solely.

Conductance
In the high transmission limit (namely, ≲ 1), and assuming Γ !"#$ = 0, we may approximate: This expression is used for the theoretical graphs in Fig. 5 in the main text.
Most generally, for any setting of our two external knobs and !" , the system optimizes its energy by setting !"# & ↑ (or equivalently ↓ & ↑ ), according to the energy given in Supplementary Equation 4. The high transmission limit is translated into relaxing the assumption of quantization of ↓ , which is made in Supplementary Note 2. In this way we can retrieve the total phase in a straightforward fashion by optimizing the energy with respect to !"# , for a given value of ↑ ∈ [0, ±1, ±2, … ]: setting now !"# !"! !"# = 0 we get: Now, once plugged into the total phase we get: or: Now the first two terms are continuously-varying variables which describe the interference phase of electrons in the lowest Landau level for a constant ↑ in the compressible puddle; while the third term describes the phase-jump which occurs once ↑ changes by ± (as further explained below).
We stress here that the transmission does not affect the value of , but only the shape of the unitcells in the CSD as well as the clarity of its lattice-like, via SP states broadening, as shown in Supplementary Figure 5 with both experimental data and theoretical.

Phase-jump lines
As seen in Supplementary Figure 6a, the lines along which ↑ increments by ± are sets of broken lines ('zig-zags'). These zig-zags consist of two types of segments; the first (dark-blue) describes a processes of varying the number of electrons in the compressible puddle ↑ , ↓ → ↑ ± , ↓ ; and the second (light-blue) describes 'reshuffling' of the charge between the edge and the bulk ↑ , ↓ → ↑ ± , ↓ ∓ .
Nonetheless, for higher transmission probabilities, these zig-zags become straight lines. These results can be easily obtained be rewriting the energy in the following form: Since at the high transmission limit we consider ↓ to be a continuous variable, it is evident that the first term in this energy is zero at all times (since ↓ appears only within it. Thus the energy effectively depends on the second term only, which is clearly minimized by setting ↑ = !" −