Disorder-induced strong-field strong-localization in 2D systems

TL;DR

The paper addresses disorder-driven localization in the lowest Landau level under strong magnetic fields, motivated by STM observations of FQHL, WC, and an amorphous solid (AS) in bilayer graphene. It develops a disorder-only heuristic framework using the self-consistent Born approximation to relate the LLL broadening $Γ_{LLL}$ to the cyclotron energy $ħω_c$ and a zero-field broadening $Γ_0$, yielding the scaling $ν_c \sim (Γ_0/(ħω_c))^{1/2}$ (up to constants) and a complementary IRM-based form $ν_c ∼ Γ_0/(ħω_c)$; it also links $ν_c$ to impurity density via $n_i$ and $B$, and presents qualitative phase diagrams showing disorder competing with FQHE and WC to produce an AS. The work emphasizes that disorder is essential to account for the sample-dependent insulating crossover and makes testable predictions, such as a dirtier sample raising $ν_c$, while acknowledging that a full interacting theory would modify but not overturn the central disorder-driven picture. Overall, the paper reframes the LLL phase competition as a disorder-controlled crossover, offering a complementary perspective to purely energetics-based WC vs FQHL studies and highlighting implications for interpreting high-field 2D electron systems.

Abstract

A recent STM experiment in 2D bilayer graphene [Y.-C. Tsui, et al., Nature 628, 287 (2024)], under a strong perpendicular magnetic field, has made a direct observation of the existence of three distinct filling-factor-dependent quantum phases in the lowest Landau level: the incompressible fractional quantum Hall liquid, a crystalline compressible hexagonal Wigner crystal with long-range order and rotational symmetry-breaking, and a random localized solid phase with no spatial order. We argue that the spatially random localized phase at low filling is the recently proposed disorder-dominated strongly localized amorphous "Anderson solid" phase [A. Babber, et al., arXiv:2601.03521], which appears generically at a sample-dependent filling factor.

Figures (4)

• Figure 1: Schematic for the zero temperature phase diagram of the localized and delocalized states in IQHE, demonstrating floating as the disorder broadening increases relative to $\hbar \omega_c$. At low disorder, the Landau levels are located at half integers and float to higher fillings with increasing disorder. Note that we consider a fixed magnetic field keeping $\hbar \omega_c$ constant, and change the filling factor or chemical potential by increasing density. Note also that the floating or the levitation of the individual Landau levels slows down with increasing Landau level number, and the disorder-induced localization due to floating begins at the lowest Landau level, moving up in Landau level as disorder increases.
• Figure 2: (a) Density of states (DOS) for a 2D electron gas in a magnetic field, calculated within the self-consistent Born approximation (SCBA). The Landau levels, centered at energies $E_N$, exhibit semi-elliptic broadening with a spectral width $\Gamma$ and are separated by the cyclotron energy $\hbar\omega_c$. (b) Diagrammatic representation of the SCBA. The first line illustrates the self-consistency condition, where $\Sigma_N$ is constructed from a single impurity scattering line (dashed line, characterized by concentration $n_i$ and short-range potential $u_0$) dressing the full propagator $G_N$. The second line depicts the Dyson equation relating the full Green's function $G_N$ (thick line) to the bare Green's function $G_N^{(0)}$ (thin line) and the self-energy $\Sigma_N$.
• Figure 3: The schematic finite-temperature ($T\ll \hbar \omega_c$) phase diagram (in log-log scale) is presented in the plane of disorder strength versus inverse filling factor. The regions shaded in red and blue correspond to the integer and fractional quantum Hall insulator phases, respectively, where $\sigma_{xx}=0$ and $\sigma_{xy}/(e^2/h)$ is an integer and fraction, respectively. The upper dashed line marks the disorder-induced metal-insulator transition, defined by $\Gamma=E_F$. Below this upper dashed line, the white region represents a metallic phase without quantized Hall conductivity plateaus. Above the dashed line, the white region corresponds to an insulating phase where both $\sigma_{xx}$ and $\sigma_{xy}$ vanish. The gray curves within the quantum Hall regime indicate the delocalized states at the center of each Landau level, which are broadened by finite temperature, extending into the metallic phase.
• Figure 4: The schematic $T=0$ phase diagram for the lowest LL regime ($1/\nu >2$) is presented in the plane of disorder strength versus inverse filling factor. The upper dashed line marks the disorder-induced metal-insulator transition, defined by $\Gamma=E_F$. At relatively small disorder and close to $\nu=1/2$, the effective magnetic field for the 2-flux composite fermions is small, and we have delocalized composite fermi liquid (CFL). CFL can be destroyed by disorder into electron fermi liquid and eventually to Anderson insulator when disorder is sufficiently strong. Moving away from $\nu=1/2$, the CFL transitions into FQH states, analogous to the metal-to-IQHE transitions shown in Fig. \ref{['fig:phase']}. At low filling factors (large $1/\nu$), a pinned WC emerges, confining FQH states to narrow ranges around fractional fillings. With finite disorder and decreasing $\nu$, the pinned WC eventually crosses over into Wigner glass and Anderson solid phases.