Non-Fermi liquid behaviour of CDW instabilities in fractionally-filled moiré flatbands

TL;DR

The paper addresses non-Fermi liquid behavior near a CDW quantum critical point in fractionally filled moiré flatbands. It develops a minimal patch theory with fermions near hot spots coupled to CDW bosons centered at the M points, analyzed via dimensional regularization around the upper critical dimension $d_c=5/2$ and an $\epsilon$-expansion. A controlled IR NFL fixed point emerges with $\hat g^* = \epsilon/\mathcal{U}_1$, $z^* = 1+\tfrac{2}{3}\epsilon$, and $\eta_\psi^* = \eta_\phi^* = \epsilon/2$; for $d=2$ ($\epsilon=1/2$) this yields $z^*=4/3$ and $\eta^*=1/4$, while vertex corrections remain finite. The results offer a coherent framework for NFL behavior at CDW-nesting instabilities in moiré materials and align with ED-based CDW tendencies at $\nu=1/4$; future work could incorporate the full cubic boson interaction via fRG and explore transport and collective modes near the QCP.

Abstract

Spin- and valley-polarized fractionally-filled moiré flatbands are known to host emergent Fermi-liquid phases, when analysed with the help of a dual description in terms of holes. The dominant Coulomb interactions in an almost flatband endow the fermions with a nontrivial dispersion, when the system is described in terms of the hole operators (rather than the particle operators). In particular, for one-fourth filling, the Fermi surface takes a quasi-triangular shape, which brings about the possibility of charge-density-wave (CDW) ordering in the ground state, characterised by the nesting vectors ($ \mathbf{Q}_n $). The $\mathbf{Q}_n$'s connect antipodal points of the Fermi surface (designated as hot-spots) and are found to belong to the space of reciprocal vectors of the underlying honeycomb structure. The resulting CDW order can be described in terms of instabilities caused by bosonic fields with momenta centred at $\lbrace \mathbf{Q}_n \rbrace $, coupling with the fermions residing in the vicinity of a pair of antipodal hot-spots. When there is a transition from a Fermi liquid to a CDW state, the bosons become massless (or critical), effectuating a non-Fermi liquid behaviour. We set out to identify such non-Fermi liquid phases after constructing a minimal effective action.

Figures (4)

• Figure 1: (a) Energy for electrons in the original filled valence band, $E_0(\mathbf{k})$, throughout the Brillouin zone of trilayer graphene stacked on hBN. (b) Renormalized energy of electrons in the empty valence band, $E_\text{ren}(\mathbf{k})$. The black closed curve in (b) represents the Fermi surface at a band-filling of $1/4$. The edges of the Fermi surface are connected by the vectors corresponding to $1.05 \,\mathbf{M}$. This figure is reproduced with the permission of Raul Perea Causin raul_unpub.
• Figure 2: Schematics of the hot-spots on the Fermi surface at $\nu = 1/4$ filling. We show the three pairs of hot-spots, with the $n^{\rm th}$ pair located at the ends of the wavevectors, $\lbrace \mathbf M_n \rbrace$. For a given value of $n$, the fermionic fields in the vicinity of the head (tail) of the $\mathbf M_n$-vector are designated as $\psi_+^{(n)}$ ($\psi_-^{(n)}$), which interact via the mediation of the order-parameter boson, $\phi_n$.
• Figure 3: The one-loop diagrams for (a) the boson self-energy, (b) the fermion self-energy, (c) $\Psi \Psi$-vertex correction, and (d) $\bar{\Psi} \bar{\Psi}$-vertex correction. Curves with arrows represent the bare fermion propagator, $G$, whereas the wiggly lines in (b), (c), and (d) represent the dressed bosonic propagator, $D_{(1)}$. The wiggly line in (a) represents the bare bosonic propagator, $D_{(0)}$.
• Figure 4: (a) Many-body energy spectrum at filling $\nu = 1/4$ for $N_\text{s}=32$ sites. The lowest 15 values are shown for each momentum sector and the energy is offset with respect to the lowest-energy state. (b) Pair-correlation function, $G(\mathbf{r})$, in the finite-sized system showing a CDW order in real space. This figure is reproduced with permission from Raul Perea Causin raul_unpub.