Moiré Collapse and Luttinger Liquids In Twisted Anisotropic Homobilayers

TL;DR

This work identifies moiré collapse as a distinctive, large-angle phenomenon in twisted anisotropic homobilayers, where the moiré unit cell and Brillouin zone collapse to quasi-1D, enabling quasi-one-dimensional electronic behavior and potential sliding Luttinger liquids. Using ab initio-informed continuum modeling, the authors derive the collapsed 1D Hamiltonian and its fully interacting bosonized form, and compute Luttinger parameters $K$, plasmon velocity $u$, and transport exponents, predicting observable power-law conductance near the collapse. They show that approaching the collapse angle $θ_M$ can stabilize strongly correlated multichannel Luttinger liquids, with a collapse-driven suppression of inter-wire coupling $t_{2M}/t_{1M} \to 0$ and a tunable critical temperature $T_c$. Ab initio results for additional anisotropic bilayers (e.g., SnSe) indicate that moiré collapse is a generic feature, suggesting broad experimental relevance for realizing quasi-1D correlated electronic states in twisted anisotropic moiré systems.

Abstract

We introduce twisted anisotropic homobilayers as a distinct class of moiré systems, characterized by a distinctive ``magic angle", $θ_M$, where both the moiré unit cell and Brillouin zone collapse. Unlike conventional studies of moiré materials, which primarily focus on small lattice misalignments, we demonstrate that this moiré collapse occurs at large twist angles in generic twisted anisotropic homobilayers. The collapse angle, $θ_M$, is likely to give rise quasi-crystal behavior as well as to the formation of strongly correlated states, that arise not from flat bands, but from the presence of ultra-anisotropic electronic states, where non-Fermi liquid phases can be stabilized. In this work, we develop a continuum model for electrons based on extensive \textit{ab initio} calculations for twisted bilayer black phosphorus, enabling a detailed study of the emerging moiré collapse features in this archetypal system. We show that the (temperature) stability criterion for the emergence of (sliding) Luttinger liquids is generally met as the twist angle approaches $θ_M$. Furthermore, we explicitly formulate the collapsed single-particle one-dimensional (1D) continuum Hamiltonian and provide the \textit{fully interacting}, bosonized Hamiltonian applicable at low doping levels. Our analysis reveals a rich landscape of multichannel Luttinger liquids, potentially enhanced by valley degrees of freedom at large twist angles.

Figures (3)

• Figure 1: Moiré collapse in twisted anisotropic homobilayers. (a) 90$^\circ$ twisted black phosphorus homobilayer. (b) Unit cell (left) and Brillouin zone (right) of monolayer black phosphorus. The anisotropic monolayer unit cell, with dimensions $l_x$ and $l_y$, contains four atomic sites denoted as $A$, $B$, $C$ and $D$. The monolayer Brillouin zone inherits the real space anisotropy. The shaded ellipse represents the Fermi surface centered at the $\Gamma$ point. The distribution of the smallest transfer momenta, $\textbf{q}_{1,2,3,4}$, for the twisted black phosphorus homobilayer is represented in panels (c) and (d). While the transfer momenta span a two-dimensional (2D) space in the $90^{\circ}$ twisted configuration in (c), panel (d) depicts a situation where all transfer momenta align along a particular direction at a finite twist angle, $\theta_M$, configuring a collapse into a one-dimensional (1D) transfer momenta space. Panels (e) and (f) display the moiré pattern of a representative twisted anisotropic homobilayer at $90^{\circ}$ and near $\theta_M$. The real space axis for the zigzag and armchair orientations are $x'$ and $y'$. We reserve $x$ and $y$ axis to describe the real space orientation of the moiré pattern, as defined above.
• Figure 2: The evolution of the electronic structure in twisted black phosphorus homobilayers towards the collapse limit. (a)-(d) The progression of the moiré Brillouin zone in twisted bilayer black phosphorus, starting from the $90^{\circ}$ ($\theta = 0^{\circ}$ in our convention) configuration. The red and blue rectangular Brillouin zones, with reciprocal lattice vectors $\textbf{b}_{1,2}$ and $\textbf{b}'_{1,2}$, respectively, represent the k-space of the bottom and top monolayers of black phosphorus. The moiré Brillouin zone, shown in green, is confined within the k-space region between the corners of the top and bottom Brillouin zones, denoted as tBZ and bBZ. As the twist angle $\theta$ increases, the initially isotropic mBZ becomes increasingly compressed until it collapses at a finite angle $\theta_M$. (e)-(h) The evolution of the electronic structure and density of states in twisted bilayer black phosphorus as it approaches the collapse limit. As the twist angle $\theta$ deviates from $0^{\circ}$, the initially isotropic conduction and valence bands begin to develop anisotropy. This leads to dramatic changes in the electronic behavior as $\theta$ approaches $\theta_M$, where the bands are described by a single crystal momentum over the collapsed moiré Brillouin zone, and 1D van Hove singularities emerge.
• Figure 3: The moiré collapse through the lens of ab initio calculations. (a) and (b) show the possible choices of commensurate supercells for twisted homobilayer black phosphorus (BP) and SnSe, respectively, highlighting the number of atoms at a given twist angle. The vertical lines at $90^{\circ} \pm \theta_M$ in both plots are the predicted collapse magic angle, closest to the $90^{\circ}$ configuration. The divergent number of atoms per supercell around these lines reflect collapse tendency. (c) and (d) show the ab initio electronic structure of homobilayer BP twisted at $90^\circ$ and $69.11^\circ$, respectively. Panels (d) and (f) show the same, but for twisted SnSe homobilayers at $90^\circ$ and $87.92^\circ$. The insets sketch the moiré Brillouin zone for each case. The $X'$ and $Y'$ high-symmetry points in panel (c) refer to the zigzag and armchair crystallographic orientations of the individual BP monolayers. Near the collapse, the real space moiré unit cell becomes elongated along the $\textbf{a}_{2M}$ direction, with the associated reciprocal space direction considerably shortened ($\Gamma - X$).The $\Gamma$ point charge densities, at the lowest conduction state, for the two nearly collapsed systems are shown in panels (e) and (d), where the quasi-1D feature is evident. The moiré unit cells are highlighted.