Magic-angle twisted bilayer graphene (TBG), characterized by flat bands, provides a platform to study correlated electron phenomena. Doping TBG leads to insulating and superconducting states, with evidence of correlation-induced effects at the charge neutrality point (CNP) potentially stemming from spontaneous symmetry breaking. Previous studies have observed sample-dependent behavior influenced by alignment with the underlying Boron Nitride (h-BN), affecting the in-plane twofold rotational symmetry. Explaining the insulating states at integer fillings and the doping dependence of band widening requires incorporating correlation effects. The nature of these states and whether correlated states influence only flat bands or higher-energy bands remains a topic of debate. In the absence of symmetry breaking, the upper and lower flat bands are fourfold degenerate due to spin and valley degeneracy, exhibiting M_2y, C₃, and C_2T symmetry. Signatures of nematicity (C₃ symmetry breaking) have been observed in measurements of the superconducting upper critical field in doped samples and STM experiments, disappearing when flat bands are completely filled or empty, indicating that they are not simply a consequence of strain. Activated behavior at the CNP in some transport experiments suggests a gap opening at the Dirac points, but whether this is electronic or due to substrate coupling remains unclear. This study explores the impact of symmetry-breaking states on the optical response of two TBG models (fully relaxed (FR) and partially relaxed (PR)) with differing degrees of relaxation. The authors aim to demonstrate that optical conductivity can identify correlated states by detecting the gapping of Dirac points and the breaking of discrete rotational symmetries.

The introduction extensively reviews existing literature on magic-angle TBG. It cites numerous studies demonstrating the emergence of correlated insulator behavior, unconventional superconductivity, and the interplay between insulating and superconducting orders at various doping levels. It highlights experimental observations from STM (scanning tunneling microscopy) showing band widening and signatures of nematicity. The review also acknowledges the inconsistencies across different samples and the influence of substrate alignment on symmetry breaking. Furthermore, theoretical models proposing explanations for the observed phenomena, including Mott insulating behavior and the origin of superconductivity, are discussed, emphasizing the ongoing debate regarding the nature of the correlated states and their extent of influence on the electronic structure.

The authors employ two tight-binding models for TBG, one fully relaxed (FR) with θ_FR ≈ 0.9° and the other partially relaxed (PR) with θ_PR ≈ 1.05°. These models, based on effective moiré orbitals rather than carbon atomic orbitals, incorporate different degrees of lattice relaxation, and were previously proposed and validated against ab-initio calculations. The optical conductivity is calculated using the Kubo formula, adapted for multi-orbital systems, accounting for the coupling to the electromagnetic field via Peierls substitution. The calculations include ten effective moiré orbitals located at various lattice sites within the TBG structure (AA, AB, BA, and saddle points (SP)). The methodology carefully considers the impact of different atomic distances on the phase acquired in the Peierls substitution. The calculations focus on σ'(ω), the real part of the longitudinal electrical conductivity, considering different electronic fillings (CNP, empty flat bands, full flat bands). Symmetry breaking states (C_2T and nematic) are introduced phenomenologically at the mean-field level. The authors explicitly analyze the optical conductivity along different crystallographic directions to detect anisotropy arising from nematicity. The Drude weight, a measure of the dc conductivity, is calculated directly from the Hamiltonian and current matrix elements.

The optical conductivity calculations reveal several key findings: 1. **Non-correlated states:** In the absence of correlations, the optical conductivity exhibits no Drude peak at zero frequency at the CNP but shows inter-band transitions from zero energy due to Dirac points. The conductivity spectra show distinct peaks (γ_m) corresponding to different interband transitions, with variations in peak intensities related to matrix elements. Doping introduces new transitions between flat and higher-energy bands, leading to a Drude peak if the chemical potential remains within the flat bands. 2. **C_2T symmetry breaking:** A correlated order breaking C_2T symmetry gaps the Dirac points, resulting in activated behavior and a reorganization of spectral weight, observable in the optical conductivity. 3. **Nematic order:** Nematic states, breaking C₃ symmetry, exhibit directional dependence in optical conductivity. The optical conductivity along different directions becomes unequal, reflecting the reduced rotational symmetry. Surprisingly, a nematic order can induce Fermi pockets and a finite Drude weight at the CNP, indicating a semimetal-to-metal Lifshitz transition. The Drude weight anisotropy depends non-monotonically on the nematic order parameter and significantly varies with doping. 4. **Lifshitz Transition:** The emergence of Fermi pockets at the CNP due to nematicity results in a Lifshitz transition transforming the semimetallic TBG into a metallic state. This transition is observed in both PR and FR models but is more pronounced in the FR model due to the multiple maxima in the lower flat band. The sign of the Drude weight anisotropy is sensitive to both the model (PR vs. FR) and the type of nematic order. 5. **Doping dependence:** Analyzing the doping dependence of the optical conductivity helps distinguish between lattice and electronic origins of symmetry breaking. Shifts in peak frequencies with doping are expected only if the correlated states' signature (C₃ symmetry breaking or band structure modifications) depends on the flat band filling. Transitions involving higher energy bands (γ₃) provide further insights into the extent of correlation effects.

The findings demonstrate that optical conductivity measurements provide a powerful tool to characterize different symmetry-breaking states in TBG and reveal the correlated nature of these states. The observed anisotropy in optical conductivity, linked to nematicity, and the gap opening at the CNP (associated with C_2T symmetry breaking) can be directly observed in the spectra. The study also highlights the importance of considering higher energy bands, extending beyond just flat bands, in understanding correlated phenomena. The sensitivity of the Drude weight anisotropy to the TBG model and nematic order parameters suggests significant sample-to-sample variability in experimental observations. The reported Lifshitz transitions, driven by nematicity, affect various experimental measurements, including transport, quantum oscillations, and STM, emphasizing the significant influence of lattice relaxation on the system's response to nematic order.

This work reveals the potential of optical conductivity as a probe for distinguishing various symmetry-breaking states in magic-angle TBG and characterizing their correlated nature. The study's key contribution is the demonstration of how optical conductivity can identify both reduced rotational symmetry and gap openings at the CNP. The findings highlight the sensitivity of the system's response to the degree of lattice relaxation and the importance of considering the full electronic structure, including higher-energy bands, in understanding the complex correlated phenomena observed in TBG. Future research could focus on comparing these theoretical predictions with experimental optical conductivity measurements to validate the proposed models and further elucidate the nature of correlated states in TBG.

The study utilizes tight-binding models, incorporating simplifications and approximations. Excitonic effects are neglected in the optical conductivity calculations. The phenomenological introduction of symmetry-breaking orders at the mean-field level might not fully capture the complexity of interactions in the system. The sensitivity of the results to the details of the model parameters, particularly the degree of lattice relaxation, implies that the precise values of observables like the Drude weight anisotropy might show variations across different samples.