[Paper Review] Electron-phonon coupling and competing Kekulé orders in twisted bilayer graphene

Yves H. Kwan, Glenn Wagner+4 more|arXiv (Cornell University)|Mar 23, 2023

Graphene research and applicationsMaterials Science3 citations

TL;DR

This paper resolves a discrepancy between theory and experiment in magic-angle twisted bilayer graphene by showing that electron-phonon coupling to graphene-scale optical zone-corner phonons stabilizes a uniform Kekulé charge order at |ν|=2 in ultra-low-strain samples, explaining the observed absence of time-reversal-breaking KIVC order and the presence of a quantized topological (anomalous Hall) response. The phonon-mediated mechanism selects a spin- or valley-polarized charge order over current-order states in the strong-coupling regime.

ABSTRACT

Recent scanning tunneling microscopy experiments [K.P. Nuckolls et al., arXiv:2303.00024] have revealed the ubiquity of Kekulé charge-density wave order in magic-angle twisted bilayer graphene. Most samples are moderately strained and show `incommensurate Kekulé spiral' (IKS) order involving a graphene-scale charge density distortion uniaxially modulated on the scale of the moiré superlattice, in accord with theoretical predictions. However, ultra-low strain samples instead show graphene-scale Kekulé charge order that is uniform on the moiré scale. This order, especially prominent near filling factor $ν=-2$, is unanticipated by theory which predicts a time-reversal breaking Kekulé current order at low strain. We show that including the coupling of moiré electrons to graphene-scale optical zone-corner (ZC) phonons stabilizes a uniform Kekulé charge ordered state at $|ν|=2$ with a quantized topological (spin or anomalous Hall) response. Our work clarifies how this phonon-driven selection of electronic order emerges in the strong-coupling regime of moiré graphene.

Motivation & Objective

• To resolve the experimental observation of uniform Kekulé charge order at |ν|=2 in ultra-low-strain twisted bilayer graphene, which contradicts theoretical predictions of time-reversal-breaking Kramers intervalley coherent (KIVC) current order.
• To understand why the TIVC (time-reversal intervalley coherent) charge order, previously considered a charge counterpart of KIVC, becomes stable at low strain despite lacking symmetry-protected STM signatures.
• To investigate the role of electron-phonon coupling (EPC) in stabilizing competing Kekulé orders in the strong-coupling limit of magic-angle twisted bilayer graphene.
• To clarify how phonon-mediated interactions select a quantized topological (anomalous Hall) state over other broken-symmetry phases at |ν|=2.

Proposed method

• The authors employ a strong-coupling effective field theory approach based on the nonlinear sigma model (NLSM) to describe the broken-symmetry states in magic-angle twisted bilayer graphene.
• They include electron-phonon coupling (EPC) to graphene-scale optical zone-corner (ZC) phonons via a coupling strength parameter $ g $, which modifies the effective Hamiltonian and lifts degeneracies in the Hartree-Fock ground state manifold.
• Phase diagrams are computed using two subtraction schemes: the 'graphene' scheme (relative to isolated graphene at neutrality) and the 'average' scheme (relative to the infinite-temperature density matrix), to assess the robustness of results to interaction renormalization.
• The $ Q $-matrix representation of broken-symmetry states is used to classify states such as TIVC, IVC-QAH, and KIVC, enabling comparison of their energy contributions under EPC.
• Numerical solutions of the NLSM in the chiral limit are used to compute the energy scales $ J $, $ u $, and $ ho $, and to determine the stability of different orders as functions of twist angle $ heta $, strain $ ilde{ heta} $, and EPC strength $ g $.

Experimental results

Research questions

• RQ1 Why does ultra-low-strain twisted bilayer graphene exhibit uniform Kekulé charge order at |ν|=2 instead of the predicted time-reversal-breaking KIVC current order?
• RQ2 How does electron-phonon coupling to zone-corner phonons influence the selection of Kekulé-type charge orders in the strong-coupling regime of MA-TBG?
• RQ3 What is the role of the subtraction scheme (graphene vs. average) in determining the stability of TIVC and other broken-symmetry states?
• RQ4 Can electron-phonon coupling stabilize a quantized anomalous Hall response in a Kekulé charge-ordered state at |ν|=2?
• RQ5 Why is the TIVC state, which lacks a distinct STM signature, stabilized in low-strain samples despite theoretical expectations favoring KIVC?

Key findings

• Electron-phonon coupling to graphene-scale zone-corner (ZC) phonons stabilizes a uniform Kekulé charge-ordered state at |ν|=2 in ultra-low-strain samples, explaining the absence of KIVC order observed in STM experiments.
• The phonon-mediated mechanism selects a TIVC-like charge order with a quantized anomalous Hall response, consistent with the observed transport and STM data at ν=−2.
• In the chiral-flat limit (vanishing bandwidth), the TIVC state emerges at ν=0 as soon as electron-phonon coupling is turned on, indicating that EPC lifts the degeneracy in favor of this state.
• Phase diagrams computed in the 'graphene' subtraction scheme show that TIVC is suppressed at ν=0 due to a large $ J $-term penalty, but becomes competitive in the 'average' scheme, which better reflects realistic conditions.
• The IKS (incommensurate Kekulé spiral) order remains dominant at moderate strain, while uniform Kekulé order takes over only in ultra-low-strain regimes due to the enhanced role of EPC.
• The results clarify that electron-phonon coupling acts as a selection rule in the strong-coupling regime, favoring charge-ordered states with topological responses over current-order states, resolving a long-standing discrepancy between theory and experiment.