**4. Comparison between theory and experiment**

As mentioned previously, scientists are currently unable to understand the entire physics behind the fractional quantum Hall effect. Experimenters constantly conduct novel analyses to gain an insight into this non-trivial phenomenon. As a result, many measuring techniques have been developed—the local compressibility measurements or experiments in Hall-bar and two-terminal geometries. Selected results, for all structures considered in this chapter, are presented in **Figure 1**. It is also worth emphasizing that experiments carried out in monolayer and bilayer graphene samples are exceptional. For example, it is possible to modify the carrier density (with a lateral gate voltage) in a fixed magnetic field strength (**Figure 1b** and **c**).

**Figure 1a** presents a very famous transport measurement conducted on a typical GaAs/ AlGaAs quantum well sample (and published in the Pan et al. paper [19]). Its uniqueness is owed to the well-developed plateaus in the longitudinal resistance for LLL fillings which are impossible to obtain within the quasi-particle approach (at least without implementing residual and unclear interactions between CFs) [16, 17]. The appearance of these incompres‐ sible states can be, however, explained by the non-local theory of quantum Hall effects. Consider a multi-loop cyclotron trajectory of a particle (as *M <sup>N</sup>* coordinate) that represents a square of the system braid group generator. It is probable that additional loops, belonging to this path, fit to the separation of higher order (not nearest) neighbors. In this case the incom‐ pressible Hall-like state can also be formed—which was confirmed by Pan's experiment—but it cannot be captured by Jain's model. The latter is owed to the fact that every flux quanta pinned to an electron is a convenient model of a supplementary loop in the cyclotron trajectory only when it fits perfectly to the inter-particle distance.

A surprising form of the basic set in bilayer graphene samples—where *ν* =1 / 2 is the most prominent incompressible state—can be observed in transport measurements depicted in **Figure 1b**. We have already mentioned that this feature follows immediately from the necessity to embrace (by the multi-loop cyclotron path) a flux supplied by the additional layer (surface). One loop is, thus, wasted on the utilization and falls out from the commensurability conditions —the obtained hierarchy contains even-denominator filling factors. However, the exchange trajectory (representing a braid group generator) consists of an odd number of half loops that protects the antisymmetric character of a particle statistics.

A careful reader probably has already noticed that an application of the non-local theory of Hall systems to structures with a greater (than two) number of layers is straightforward. The most important modification concerns the commensurability conditions for *q*-looped cyclotron trajectories (when only multi-loop exchanges are accessible and generate the braid group describing the system). Each additional layer of atoms supplies an additional surface (and magnetic field flux) that needs to be embraced by this path. The most energetically efficient trajectory seems to be achieved when every added plane is utilized by a single loop. Dimen‐ sions of these loops are not raised and they are not included, at least explicitly, in the com‐ mensurability conditions. As a result, only remaining *q* −*γ* ones (where *γ* stands for the number of supplementary layers) must share the total flux per particle, *BS* /(*N* −*εN*0), which is

As mentioned previously, scientists are currently unable to understand the entire physics behind the fractional quantum Hall effect. Experimenters constantly conduct novel analyses to gain an insight into this non-trivial phenomenon. As a result, many measuring techniques have been developed—the local compressibility measurements or experiments in Hall-bar and two-terminal geometries. Selected results, for all structures considered in this chapter, are presented in **Figure 1**. It is also worth emphasizing that experiments carried out in monolayer and bilayer graphene samples are exceptional. For example, it is possible to modify the carrier density (with a lateral gate voltage) in a fixed magnetic field strength (**Figure 1b** and **c**).

**Figure 1a** presents a very famous transport measurement conducted on a typical GaAs/ AlGaAs quantum well sample (and published in the Pan et al. paper [19]). Its uniqueness is owed to the well-developed plateaus in the longitudinal resistance for LLL fillings which are impossible to obtain within the quasi-particle approach (at least without implementing residual and unclear interactions between CFs) [16, 17]. The appearance of these incompres‐ sible states can be, however, explained by the non-local theory of quantum Hall effects. Consider a multi-loop cyclotron trajectory of a particle (as *M <sup>N</sup>* coordinate) that represents a square of the system braid group generator. It is probable that additional loops, belonging to this path, fit to the separation of higher order (not nearest) neighbors. In this case the incom‐ pressible Hall-like state can also be formed—which was confirmed by Pan's experiment—but it cannot be captured by Jain's model. The latter is owed to the fact that every flux quanta pinned to an electron is a convenient model of a supplementary loop in the cyclotron trajectory

A surprising form of the basic set in bilayer graphene samples—where *ν* =1 / 2 is the most prominent incompressible state—can be observed in transport measurements depicted in **Figure 1b**. We have already mentioned that this feature follows immediately from the necessity to embrace (by the multi-loop cyclotron path) a flux supplied by the additional layer (surface). One loop is, thus, wasted on the utilization and falls out from the commensurability conditions —the obtained hierarchy contains even-denominator filling factors. However, the exchange

experienced by the single-loop path.

24 Recent Advances in Graphene Research

**4. Comparison between theory and experiment**

only when it fits perfectly to the inter-particle distance.

Finally, another interesting feature is worth to be noted—the remarkable stability of incom‐ pressible Hall-like states for *ν* =7 / 3, 8 / 3 and 5/2. The unexpected robustness of related plateaus makes "7/3 and 8/3 states unlikely to be the analogues of the 1/3, 2/3 Laughlin correlated states" [40]. This conclusion seems to agree with the non-local theory predictions. The braid group describing the system for these fillings (connected with the paired IQHE and the single-loop FQHE) is generated with loopless exchanges. Thus, the Laughlin correlations are described by a *p* = 1 power in the Jastrow polynomial, despite the fractional quantization of a transverse resistivity. Hence, the robustness of these states is expected to exceed one for the ordinary FQHE states, which was confirmed experimentally (**Figure 1c** and **d**).

**Figure 1.** (a) Typical semiconductor quantum well. A longitudinal resistance as a function of external magnetic field. The figure is based on the Pan et al. paper [19]. (b) Bilayer graphene. A longitudinal resistance as a function of a mag‐ netic field and a filling factor. *Rxx* ≈0.1 Ω is colored dark red and *Rxx* ≈4 *k*Ω is colored bright yellow. The figure is based on the Ki et al. paper [23]. (c) Monolayer graphene. A longitudinal resistance as a function of a filling factor. The figure is based on Amet et al.'s [53] paper. (d) Typical semiconductor quantum well. A longitudinal resistance as a function of external magnetic field. The figure is based on the Choi et al.'s [40] paper. Important filling factors (e.g. falling out of the CF hierarchy or corresponding to the single-loop FQHE) are highlighted.
