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arXiv:2508.21000 · Physical review. B./Physical review. B

Quantum melting a Wigner crystal into Hall liquids

Aidan P. Reddy, Liang Fu

Published Feb 24, 2026• 0 citations
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Abstract

Recent experiments have shown that, counterintuitively, applying a magnetic field to a Wigner crystal can induce quantum Hall effects. In this Letter, we use variational Monte Carlo to show that a magnetic field can melt zero-field Wigner crystals into integer quantum Hall liquids. This melting originates from quantum oscillations in the liquid's ground state energy, which develops downward cusps at integer filling factors due to incompressibility. Our calculations establish a range of densities in which this quantum melting transition occurs.

DOI
10.1103/xt9w-8dg7

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•Jan 20, 2018•0 citations
Recent experiments have shown that, counterintuitively, applying a magnetic field to a Wigner crystal can induce quantum Hall effects. In this Letter, we use variational Monte Carlo to show that a magnetic field can melt zero-field Wigner crystals into integer quantum Hall liquids. This melting originates from quantum oscillations in the liquid's ground state energy, which develops downward cusps at integer filling factors due to incompressibility. Our calculations establish a range of densities in which this quantum melting transition occurs.
We generalize a path integral Monte Carlo formula to investigate the phase boundary between the fractional quantum Hall liquid and the Wigner crystal for a two-dimensional electron (fermion) system in the presence of strong magnetic fields and at low temperatures. At very low temperature, the state of the system is found to be a solid at filling factors $\ensuremath{\nu}=1∕7$, 1∕91∕91∕9, and 1∕111∕111∕11, while the state remains liquidlike at $\ensuremath{\nu}=1∕3$ and 1∕51∕51∕5. The temperature-density phase diagrams are obtained at low filling factors. At $\ensuremath{\nu}=1∕7$ and a fixed electron density, the calculated melting temperature is $\ensuremath{\sim}220\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$, in good agreement with some experimental results. The nature of the phase transition is suggested to be continuous.
The two-dimensional Wigner crystals are studied with the variational quantum Monte Carlo method. The close relationship between the ground-state wave function and the collective excitations in the system is illustrated, and used to guide the construction of the ground-state wave function of the strongly correlated solid. Exchange, correlation, and magnetic-field effects all give rise to distinct physical phenomena. In the absence of any external magnetic field, interesting spin orderings are observed in the ground state of the electron crystal in various two-dimensional lattices. In particular, two-dimensional bipartite lattices are shown not to lead necessarily to an antiferromagnetic ground state. In the quantum Hall effect regime, a strong magnetic field introduces energy and length scales. The magnetic field quenches the kinetic energy and poses constraints on how the electrons may correlate with each other. Care is taken to ensure the appropriate translational properties of the wave function when the system is in a uniform magnetic field. We have examined the exchange, intra-Landau-level correlation as well as Landau-level-mixing effects with various variational wave functions. We also determine their dependences on the experimental parameters such as the carrier effective mass at a modulation-doped semiconductor heterojunction. Our results, when combined with some recent calculations for the energy of the fractional quantum Hall liquid including Landau-level-mixing, show quantitatively that in going from n doping to p doping in GaAs/Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterojunction systems, the crossover filling factor from the fractional quantum Hall liquid to the Wigner crystal changes from filling factor \ensuremath{\nu}\ensuremath{\sim}1/5 to \ensuremath{\nu}\ensuremath{\sim}1/3. This lends strong support to the claim that the observed reentrant insulating phases around \ensuremath{\nu}=1/5 for n-doped and around \ensuremath{\nu}=1/3 for p-doped high-mobility samples are primarily caused by electron-electron interaction effects. We discuss the possible implications of our theoretical results for some recent experiments carried out in the quantum Hall regime in search of the electron solid.
The generalized Wigner crystal (GWC) is a novel quantum phase of matter driven by further-range interaction at fractional fillings of a lattice. The role of further-range interaction as the driver for the incompressible state is akin to the Wigner crystal. On the other hand, the significant role of commensurate filling is akin to the Mott insulator. Recent progress in simulator platforms presents unprecedented opportunities to investigate quantum melting in the strongly interacting regime through synergy between theory and experiments. However, the earlier theory literature presents diverging predictions. We study the quantum freezing of GWC through large-scale density matrix renormalization group simulations of a triangular lattice extended Hubbard model. We find a single first-order phase transition between the Fermi liquid and the sqrt[3]×sqrt[3] GWC state. The GWC state shows long-range antiferromagnetic 120° Néel order. Our results present the simplest answers to the question of the quantum phase transition into the GWC phase and the properties of the GWC phase.
The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the non-perturbative fixed phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the ν−κ\nu-\kappaν−κ plane, where ν\nuν is the filling factor and κ\kappaκ is the strength of Landau level mixing. The phase boundary is seen to exhibit a striking ν\nuν dependence, with the states away from the magic filling factors ν=n/(2pn+1)\nu=n/(2pn+1)ν=n/(2pn+1) being much more susceptible to crystallization due to Landau level mixing than those at ν=n/(2pn+1)\nu=n/(2pn+1)ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n-doped and p-doped GaAs quantum wells, and, in particular, the existence of an insulating state for ν<1/3\nu<1/3ν<1/3 and also for 1/3<ν<2/51/3 <\nu< 2/51/3<ν<2/5 in low density p-doped systems. We predict that in the vicinity of ν=1/5\nu=1/5ν=1/5 and ν=2/9\nu=2/9ν=2/9, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal but rather into a strongly correlated crystal of composite fermions carrying two vortices.