Honeycomb Crystal in a Mass-Asymmetric Electron-Hole Bilayer

Shengdu Chai; Wangqian Miao; Yuqiang Li; Lei Wang; Xi Dai; X. C. Xie

cond-mat.str-elOct 12, 2026

Honeycomb Crystal in a Mass-Asymmetric Electron-Hole Bilayer

Authors: Shengdu Chai, Wangqian Miao, Yuqiang Li, Lei Wang, Xi Dai, X. C. Xie

Organizations: Interdisciplinary Center for Theoretical Physics and Information Sciences (ICTPIS), Fudan University, Shanghai 200433, China · Shanghai Artificial Intelligence Laboratory, Shanghai 200232, China · Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China · Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China ·

Abstract

A two-dimensional electron gas in a uniform neutralizing background forms a triangular Wigner crystal at low density. In a strongly mass-asymmetric electron-hole bilayer, localized heavy holes instead provide a nonuniform positive background that supports an electron crystal at densities two orders of magnitude higher than for a homogeneous electron gas in the same material. Here we show that correlated mobile electrons can overturn the preference for triangular order, stabilizing a honeycomb electron-hole crystal. Using the powerful real-space neural-network variational Monte Carlo in the Born-Oppenheimer limit with fixed holes, we find that increasing the density drives a transition from a triangular electron-hole crystal to a honeycomb crystal, which eventually melts into itinerant electrons coexisting with the honeycomb or the triangular hole lattice at fixed interlayer separation. Relative to triangular order, honeycomb order emerges when gains in electronic kinetic energy and electron-hole attraction outweigh its higher intralayer Coulomb repulsion, a threshold never reached in the rigid point-dipole model. Moreover, the honeycomb hole lattice supports paramagnetic, antiferromagnetic, and stripe-ordered spin states. Complex polarization indicates a metal-insulator transition, while the states on the honeycomb hole lattice are consistent with a stripe-ordered Dirac semimetal and paramagnetic or antiferromagnetic Mott insulators via mapping to the half-filled Hubbard model. We further propose a BLG/hBN/WS $_2$ -WSe $_2$ heterostructure as a platform for the predicted honeycomb crystal, at an experimentally feasible density $n\sim10^{11}~\text{cm}^{-2}$ .

Figures & tables

Figure 1: Fixed-hole model for a charge-neutral, mass-asymmetric electron–hole bilayer with interlayer separation $d$ : $N$ mobile quantum electrons (blue) and $N$ classical holes fixed as point charges in the Born–Oppenheimer limit (orange).

Figure 2: Phase diagram of the ground-state hole lattice and electronic character for the

N=36

spin-unpolarized (

N_{\uparrow}=N_{\downarrow}=18

) system. Marker shapes give the ground-state hole lattice (hexagons, HC; triangles, T), and colors give the electronic character (red, itinerant; blue, localized). Dashed lines show the interpolated boundaries from the energy comparison between hole lattices. Finite-size checks at

N=48

near the phase boundaries support the phase diagram, with the inferred boundaries remaining nearly unchanged.

Figure 3: Charge and spin phases of the optimized $N=36$ states. The upper panels show the real-space charge $n(\mathbf{r})$ (left) and spin $m_{z}(\mathbf{r})$ (right) for each state, with each panel normalized independently. The lower panels plot the normalized radial-shell maxima $\widetilde{S}_{c}^{\max}(q)$ (purple solid), $\widetilde{S}_{s}^{\max}(q)$ (teal dashed) and $\widetilde{S}_{s,\rm coh}^{\max}(q)$ (orange solid) against $q/q_{\min}$ [Eqs. ( 4 )–( 6 )]. Each panel title gives the hole lattice [triangular (T) or honeycomb (HC)], the electronic character (itinerant or localized), and the spin state.

Figure 4: Mechanism and electronic properties of the optimized

N=36

states. (a) Scaled electronic energy difference

r_{s}\Delta E_{\mathrm{el}}/N

between honeycomb and triangular states. Solid and dotted curves show the NN-VMC and rigid point-dipole results, respectively. The red dashed line marks the gain required to overcome the hole–hole penalty,

-r_{s}\Delta V_{hh}/N=-37.7

mHa. Marker shapes give the ground-state hole lattice, and line colors give

r_{s}

. (b) Nearest-neighbor hopping

t

versus on-site repulsion

U

for the ground states on the honeycomb hole lattice, with a logarithmic vertical axis. Colors distinguish the AFM, stripe, and PM states, and point labels give

(r_{s},d)

. The dashed line is

U/t=3.87

[ 84 , 62 ] . (c) Complex polarization

z

of the electrons [ 76 , 75 ] . Marker shapes give the hole lattice and marker colors the electronic character (red, itinerant; blue, localized); line colors give

d

(color bar).

Quantum Many-Body Physics

Quantum many-body physics studies how interacting quantum particles produce collective behavior, phases of matter, and complex dynamics.

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QERNEL: a Scalable Large Electron Model

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Department of Physics, Massachusetts Institute of Technology, Cambridge, MA-02139,USA

Neural Quantum States Materials Science Quantum Many-Body Physics Quantum Neural Networks

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We introduce QERNEL, a foundational neural wavefunction that variationally solves families of parameterized many-electron Hamiltonians and captures their ground states throughout parameter space within a single model. QERNEL combines FiLM-based parameter conditioning with scale-efficient architectural elements -- mixture of experts and grouped-query attention, substantially improving expressivity at low computational cost. We apply QERNEL to interacting electrons in semiconductor moiré heterobilayers, training a single weight-shared model for systems of up to 150 electrons. By solving the many-electron Schrödinger equation conditioned on moiré potential depth, QERNEL captures both quantum liquid and crystal states and discovers the sharp phase transition between them, marked by abrupt changes in interaction energy and charge density. Our work establishes a foundation model for moiré quantum materials and a scalable architecture toward a Large Electron Model for solids.

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