Coupled-wire descriptions of unconventional quantum states in twisted nanostructures

TL;DR

The paper surveys the coupled-wire approach for realizing unconventional quantum states in twisted nanostructures, highlighting moiré domain-wall networks as a tunable platform of interacting one-dimensional channels embedded in two dimensions. By formulating a quadratic fixed point for the network and analyzing forward-scattering couplings, it derives electrically tunable scaling exponents and spectroscopic signatures, and shows how non-quadratic perturbations—backscattering, pairing, and moiré umklapp—generate a rich phase diagram including density waves, superconductivity, Anderson localization, and topological edge states such as fractional quantum anomalous Hall states. The framework provides a unified, experimentally accessible route to explore the interplay of topology and strong correlations in nanoscale moiré materials, with concrete predictions for STM spectra and edge transport that distinguish bulk domain-wall physics from topological edge modes. It further extends to non-electronic platforms and magnetic phenomena, predicting 2D spin helices and magnon-induced singularities arising from the coupling between itinerant electrons and localized moments, and outlines future directions involving superconducting proximity and Majorana physics. Overall, the work positions coupled-wire networks in moiré systems as a versatile venue for engineering and probing interaction-driven topological and correlated states beyond conventional band theory.

Abstract

Coupled-wire description has been developed as a powerful framework for providing bosonic descriptions of strongly correlated quantum matter, with early applications to systems such as the cuprates and the integer and fractional quantum Hall states. In this topical review, we discuss recent developments of coupled-wire description in nanoscale systems, where it emerges not only as a theoretical tool but also as a highly tunable physical platform. In these nanoscale realizations, coupled-wire networks are formed by one-dimensional channels embedded in two-dimensional materials, most prominently in moiré and twisted structures. Such networks host a broad range of unconventional states of matter, including superconductivity, charge density waves, spin density waves, Mott insulating phases, Anderson insulating phases, quantum spin Hall states, quantum anomalous Hall states, and their fractionalized counterparts. The ability to electrically control interaction strength, confinement, and coupling between wires makes these systems qualitatively different from earlier realizations and allows continuous tuning between competing phases. Notably, recent work has demonstrated that the coupled-wire framework in moiré networks completes the trio of quantum Hall phenomena, encompassing quantum Hall, quantum spin Hall, and quantum anomalous Hall states, together with their fractional analogues. This development highlights coupled-wire networks in nanoscale materials as a versatile and experimentally relevant setting for exploring the interplay of topology, strong correlations, and low-dimensional physics.

Figures (11)

• Figure 1: Left: Schematic illustration of the moiré pattern in twisted bilayer graphene (purple). Spatially varying stacking configurations generate AB- and BA-stacking domains, with the domain walls (green, red, and orange) forming a triangular network. Right: Illustration of the device nanostructure, consisting of twisted graphene layers (purple), a dielectric spacer (blue), and a metallic gate (light blue).
• Figure 2: Left: Schematic illustration of the domain wall network and its low-energy spectrum. In the enlarged spectrum of a domain wall labeled by $m$, there are two energy band branches $\delta=1$ (solid) and $\delta=2$ (dashed) with spin up (red) and spin down (blue) states. The illustrated scattering and tunneling processes include intrawire backscattering (green), interwire tunneling (light blue), and interarray tunneling (orange). Middle: Scaling exponent ($\beta_{9}$) of the local density of states defined in Eq. \ref{['Eq:DOS']}. This plot is reproduced from Ref. Wang:2024. Right: Universal collapse of the density-of-states curves measured at different temperatures.
• Figure 3: Phase diagram of correlated coupled wires in twisted bilayer graphene as a function of interlayer bias ($V_d$) and electron-phonon coupling. The phases include a gapless correlated network phase (purple), a density-wave-dominated regime (orange), a phonon-assisted superconducting regime (blue), and an electron-phonon-coupled liquid (white). The figure is reproduced from Ref. Wang:2024.
• Figure 4: Localization length ($\xi_{\rm loc}$) and temperature ($T_{\rm loc}$) of the domain wall network as a function of interlayer bias ($V_d$) and screening length ($d$). The figures are reproduced from Ref. Wang:2024.
• Figure 5: Schematic illustration of electrons, $\psi_{\ell m \sigma}$ (with $\ell \in \{R,L\}$ and $\sigma \in \{\uparrow, \downarrow \}$), propagating along the domain walls and experiencing a periodic potential, $V(x)$, generated by the moiré pattern. The colors of the domain walls correspond to those used in Fig. \ref{['Fig:TBG_dw']}.