The microscopic origin of the Quantum Hall Effect

Abstract

Topology is key in describing unconventional quantum phases of matter and devising robust quantum technology. Exactly how topology mixes with quantum mechanics remains largely unclear, as testified by the lack of a unifying microscopic theory for the ever-expanding and still puzzling transport behavior of electrons in the Quantum Hall Effect. Here we formulate a microscopic theory able to quantitatively describe the large wealth of Quantum Hall physics starting from one basic assumption, that the topological constraint in actual space leads to a superposition of states in the associated angular space. This allows us to identify the mechanism underlying quantum topology, single-particle wavefunction regularity in 3D, while many-body physics and disorder play no fundamental role. Our findings introduce a new far-reaching perspective in analyzing topological quantum systems and applications, such as topological quantum computing.

Figures (11)

• Figure 1: Theory versus experiment for the IQHE. Prediction based on Eq.(\ref{['fermions']}) and corresponding Eq.(\ref{['Gxz_fermionic']}) pitted against measurements from Ref.Paalanen1982 (top panel, violet and orange curves, respectively). The absence of gaps signals the spin-dominated regime of $p_{max}=0$. Note the spin-splitting acting on the odd values for $\nu>3$ (green values). Bottom panel, the governing electron spectrum. Values of $\nu>10$ are not clear in data and are not discussed.
• Figure 2: Theory versus experiment for the FQHE. (a) Prediction based on Eqs.(\ref{['fermions']}) and (\ref{['bosons']}) and corresponding Eq.(\ref{['Gxz_fermionic']}) (violet curve) pitted against sample Hall resistance measurements taken from Ref.Willett1987 (orange curve) for $\nu>1$ ($0<\nu^{-1}<1$). (b) Observed normalized $R_{xx}$. (c) Gap-center states $\nu_{gc}$ and gaps $\nu_g$ are referenced back to the fermionic spectrum ($p_{max}=1$) and (c) to the corresponding topologically prohibited states of the bosonic spectrum ($p'_{max}=4$). For clarity, the comparison is halted for high values of $\nu>5$. (e), (f) Overall qualitative comparison and (g) sample expanded view of comparison down to the very minute details of $R_{xz}$.
• Figure 3: Evidence of bosonic spectrum. (a)-(d) Analysis of results at $T$=150mK for $2<\nu<3$ from Ref.Willett1987 using Eq.(\ref{['Gxz_fermionic']}). (e) Experiment (orange line) for $T$=25mK (Fig.2 of Ref.Willett1987) and theory (violet line) using Eq.(\ref{['Gxz_bosonic']}) with complete boson condensation ($\eta=0.99$). Note the emergence of the plateau at $\nu=5/2$ at 25mK that is a topologically prohibited state at 150mK. (f) Observed $R_{xx}$ and predicted (g) fermionic and (h) bosonic spectra.
• Figure 4: FQHE in Graphene. (a) Hall conductance measurements taken from Fig.3c of Ref.Schmitz2020 for $1/2 < \nu < 3$ (orange curve) compared to prediction based on Eqs.(\ref{['fermions']}) and (\ref{['bosons']}) and corresponding Eq.(\ref{['Gxz_fermionic']}) (violet curve) with $p_{max}=1$. (b) Observed normalized $G_{xx}$. States are referenced back to the fermionic spectrum (c) and corresponding topologically prohibited states of the (d) bosonic spectrum ($p'_{max}=2$). (e)-(h) Analogous comparison for the extreme quantum regime $2/7<\nu<1$ indicating a higher $p_{max}=3$ (see text).
• Figure S1: Quantization in topological space. (a) Classical Topology and the invariant number of times $W_\Gamma$ the curve $\Gamma$ winds around the forbidden infinite curve $a$. (b) Single and double-valuedness in the wavefunction $\psi$ introduces a quantum topological invariant in the form of the accumulated phase $\Delta \theta_\Gamma$ along a closed path $\Gamma$. (c) Path $\Gamma$, represented by the segment in actual space from $P$ to $P' \equiv P$, is characterized by the superposition of different associated paths in corresponding angular spaces $\theta_1$, $\theta_2$, and $\theta_3$ for the case of $j=3$. Shaded regions represent the three different angular topological quanta $\delta (\Delta \theta_\Gamma) (j=3,j_z=1)=2\pi (3/1)$, the fractional $\delta (\Delta \theta_\Gamma)(3,2)=2\pi (3/2)$, and $\delta (\Delta \theta_\Gamma)(3,3)=2\pi (3/3)$ (see Eq.(\ref{['anglequanta']})).