Electronic crystals and quasicrystals in semiconductor quantum wells: an AI-powered discovery

TL;DR

This work tackles strongly correlated electrons in semiconductor quantum wells with finite thickness, revealing a rich phase diagram that includes monolayer and bilayer metallic and crystalline states, and identifying a novel bilayer electronic quasicrystal stabilized by quantum fluctuations. An unbiased attention-based NN-VMC method is developed to solve the 3D many-body Hamiltonian from first principles, constructing a flexible variational wavefunction as a sum of Slater determinants augmented by a Jastrow factor. The results show a sequence of phases with density and well thickness: bilayer Fermi liquid → bilayer crystal → monolayer Fermi liquid → monolayer Wigner crystal, with a 30° twisted bilayer quasicrystal emerging in the bilayer regime. These findings demonstrate AI-powered, first-principles discovery of new quantum phases in realistic semiconductor platforms and point to experimental tests in quantum wells and heterostructures, with implications for quantum-device design and digital-twin modeling.

Abstract

The homogeneous electron gas is a cornerstone of quantum condensed matter physics, providing the foundation for developing density functional theory and understanding electronic phases in semiconductors. However, theoretical understanding of strongly-correlated electrons in realistic semiconductor systems remains limited. In this work, we develop a neural network based variational approach to study quantum wells in three dimensional geometry for a variety of electron densities and well thicknesses. Starting from first principles, our unbiased AI-powered method reveals metallic and crystalline phases with both monolayer and bilayer charge distributions. In the emergent bilayer, we discover a new quantum phase of matter: the electronic quasicrystal.

Figures (4)

• Figure 1: Monolayer and bilayer metals: Out-of-plane charge density distribution at $(r_{2D}/a_{ B}, d/a_{ B}) = (10, 10)$$(a)$ and $(10, 25)$$(b)$. In the mononolayer metal all the electrons live in the lowest subband $n_z = 1$, giving rise to a single Fermi surface in the $2$D momentum space occupations obtained from the $1$-RDM $(c)$. In the bilayer metal, electrons are distributed between multiple subbands (for this specific example, $n_z=1,2$) and their momentum-space occupations form distinct Fermi surfaces $(d)$.
• Figure 2: Phase diagram: $(a)$ Out-of-plane charge distribution as a function of the electron density and the quantum well thickness. Blue and red dots correspond to monolayer (one peak at $z= 0$, e.g., Fig. \ref{['fig:mono_to_bilayer']}$\,(a)$) and bilayer states (two peaks spaced by a finite $\Delta$, e.g., Fig. \ref{['fig:mono_to_bilayer']}$\,(b)$), respectively. $(b)$ In-plane charge distribution, as characterized by the average height of the structure factor $S({\bm k})$ (Bragg) peaks. In the Fermi liquid phase (cool colors), the charge distribution is homogeneous and the structure factor does not have sharp peaks. In the crystalline phase (warm colors), the charge arrangement gives rise to strong peaks in $S({\bm k})$ (cf insets in Fig. \ref{['fig:bilayer_WC']}). Dashed lines qualitatively separate the mono- ($1$-WC) and bilayer ($2$-WC) crystals from the Fermi liquid regions ($1$-FL and $2$-FL).
• Figure 3: Bilayer (quasi-)crystal: Square $(a)$, honeycomb $(b)$ and quasicrystal $(c)$ bilayer, corresponding to the ground state at $(r_{2D}/a_{ B}, d/a_{ B}) = (25, 70), (25, 75)\text{ and }(r_{2D}/a_{ B}, d/a_{ B}) = (20, 80)$, respectively. The commensurate stackings $(a)$ and $(b)$ can be understood from minimizing the intra-layer and inter-layer Coulomb repulsion. The electronic quasicrystal, on the other hand, does not have a classical analogue and is stabilized purely by quantum fluctuations, via the zero-point energy contribution of its extensive low-lying excitations. The colored insets show the Bragg peaks of the in-plane structure factor $S({\bm k})$. This highlights the twelve-fold rotational symmetry of the quantum quasicrystal.
• Figure S1: Comparison of the energy per particle between the honeycomb lattice $E_{\rm hc}$ and (a) the square-stacked lattice $E_{\rm sq}$, (b) the quasicrystal $E_{\rm qc}$. The unit cell area for these calculations is set to $1$. (c,d) Configurations of the two lowest-energy states for $30$ electrons at $d/r_{2D} = 3$. Red and black points represent particles in different layers. Insets show the corresponding in-plane structure factor $S({\bm k})$. Notice the presence of additional Bragg peaks at higher momenta, which cannot be resolved in the quantum mechanical case \ref{['fig:bilayer_WC']} due to the quantum "smearing" of the electrons.