Melting of quantum Hall Wigner and bubble crystals

TL;DR

The work addresses the challenge of predicting the melting temperature $T_m$ of quantum Hall bubble/Wigner crystals, showing that defect-mediated melting can quantitatively capture the solid-liquid boundaries observed in ultraclean GaAs/AlGaAs 2DEGs. The authors combine Landau-level–projected Hartree–Fock elasticity with the full KTHNY-Halperin-Nelson-Young melting criterion and its renormalization-group flow to produce $T_m$ predictions that agree with Corbino-transport measurements across Landau levels $N=2$–$5$ and spin branches. By fitting a dislocation-core energy parameter $\\alpha$ and a screening strength $\\gamma$, they extract microscopic defect energetics and many-body screening effects directly from bulk transport data, establishing a predictive link between LL physics and finite-temperature phase boundaries in strongly interacting electronic solids. The results validate defect-mediated melting as a robust framework for quantum Hall solids and suggest bulk transport as a quantitative probe of screening and topological defect energetics, with potential extension to generalized Wigner crystals in moiré Chern-band systems as well as other 2D electronic crystals. $K_R(T_m^-)=16\\pi$ marks the critical condition in the RG treatment, connecting microscopic elastic constants to macroscopic phase behavior.

Abstract

A two-dimensional crystal melts via the proliferation and unbinding of topological defects, yet quantitatively predicting the melting temperature $T_m$ in real systems is challenging. Here we resolve this discrepancy in quantum Hall electron bubble phases by combining Corbino-geometry transport experiment in an ultraclean GaAs/AlGaAs quantum well for Landau levels 2 to 5 with Hartree--Fock elasticity and the full Kosterlitz--Thouless--Halperin--Nelson--Young melting criterion including the finite-temperature renormalization-group calculation. The theoretically obtained $T_m$ quantitatively captures the measured solid-liquid phase transition boundaries across all probed ranges, validating the bubble-crystal interpretation and establishing defect--mediated melting as a predictive framework for strongly interacting electronic solids. This agreement further supports using bulk transport to probe the energetics of topological defects and screening in quantum Hall physics, and the approach is readily extendable to other electronic crystals, including the generalized Wigner crystal in moiré Chern bands.

Figures (5)

• Figure 1: (a) Longitudinal conductance as a function of out-of-plane magnetic field B at T = 40 mK, N=2 to 4 Landau levels ($4 < \nu < 10$). Different bubble states are characterized by previous convention and illustrated with shaded stripes of different colors representing varying Landau levels. (b) Temperature-dependent magnetoresistance around R4a state. Blue triangles mark dips corresponding to the insulating R4a bubble state, the red triangle marks the transition from dip features to peak features, signaling the SLT of R4a state. (c) Longitudinal conductance vs $T$ at filling factors corresponding to maximum transition temperature. Peaks in curves are assigned to be SLT temperatures $T_m$ of bubble phases.
• Figure 2: Top panel: $T_m$ versus filling factor $\nu$ in Landau levels $N=2$--$5$. Filled circles show experimental $T_m$ values, and open squares show KTHNY--RG results. Colors label the LL index; shaded area marks the corresponding intervals $\nu\in[2N,2N+2]$. Bottom panel: $\gamma$ used in the calculation, solid (dashed) lines are for the lower (higher) bands in each LL.
• Figure 3: Heat maps of the longitudinal conductance as a function of temperature $T$ and filling factor $\nu$. Overlaid symbols mark the melting temperatures $T_m(\nu)$ for the indicated $M$-electron bubble phases obtained from theory ($\alpha$ and $\gamma$ are listed in the legends). (a) for $2$LL, (b) for $3$LL.
• Figure S1: Hartree--Fock cohesive energy of bubble crystals, $E_{\rm coh}(\nu^\ast)$, as a function of partial filling $\nu^\ast$ for Landau levels $n=2$ ((a1),(a2)), $n=3$ ((b1),(b2)), $n=4$ ((c1),(c2)), and $n=5$ ((d1),(d2)). Each curve corresponds to an $M$-electron bubble phase (triangular lattice), as labeled in the legends. Calculations are performed with $\beta=0$ and the screening parameter $\gamma$ below: (a1)$=0.00$, (a2)$=0.01$, (b1)$=0.01$, (b2)$=0.02$, (c1)$=0.02$, (c2)$=0.05$, (d1)$=0.02$, (d2)$=0.05$. For each $\nu^\ast$, the lowest $E_{\rm coh}$ identifies the energetically favored electron-solid morphology at zero temperature.
• Figure S2: Color maps of the longitudinal conductance as a function of temperature $T$ and filling factor $\nu$, showing re-entrant insulating regions associated with bubble-crystal phases at LL $N=4$ and $5$. Overlaid symbols mark the melting temperatures $T_m(\nu)$ for the indicated $M$-electron bubble phases, obtained from the KTHNY--RG melting criterion using Landau-level--projected HF elastic moduli (model parameters ). $\alpha$ and $\gamma$ are listed in the legends.