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Introduction
The Hofstadter butterfly, a fractal energy spectrum of electrons in a magnetic field, arises from magnetic translational symmetry in crystals. While typically requiring strong magnetic fields, recent advancements in two-dimensional van der Waals materials have enabled its observation in systems like graphene superlattices, magic-angle TBG, and twisted double-bilayer graphene. The connection between magnetic translational symmetry and symmetry-protected topological phases has been recently established. In systems with multiple sites per unit cell, the Hofstadter spectrum exhibits approximate replicative behavior under flux periodicity, described by a unitary transformation of the Hamiltonian. At half-flux periodicity, effective time-reversal symmetry is restored, leading to various symmetry-protected topological states. This research focuses on higher-order topological insulator (HOTI) phases within the Hofstadter butterflies of TBG. Unlike archetypal HOTIs with explicit symmetry protection, the replica HOTIs found here recur quasiperiodically without such protection, relying instead on the self-similarity of the Hofstadter butterflies. The study proves that the full lattice model of TBG possesses exact flux periodicity at all commensurate angles, rigorously characterizing band topological protection in a magnetic field. Two original HOTIs exist at time-reversal invariant fluxes (TRIFs), and numerous replica HOTIs appear at specific fluxes. Real-space topological markers are extended to diagnose HOTIs, revealing that replica HOTIs share features with original HOTIs, with the origin attributed to reduced Peierls path area due to interlayer hopping in TBG.
Literature Review
Previous studies have explored the Hofstadter butterfly effect in various 2D materials and its relation to topological phases. Works on graphene superlattices, magic-angle twisted bilayer graphene, and twisted double-bilayer graphene have experimentally observed the Hofstadter butterfly. Research has linked the magnetic translational symmetry to symmetry-protected topological phases, particularly focusing on the restoration of time-reversal symmetry at half-flux periodicity. Archetypal HOTIs have been extensively studied in terms of their symmetry protection. However, the existence and characteristics of replica HOTIs in systems like TBG, where self-similarity plays a crucial role, have not been fully explored until this study. The concept of real-space topological markers has been used to diagnose topological phases, but extending this to HOTIs within a self-similar fractal spectrum is a novel contribution of this work.
Methodology
The research utilizes the Moon-Koshino tight-binding model for TBG, incorporating magnetic flux via the Peierls substitution. The model considers the atomic structure of TBG, preserving specific rotational symmetries. The twist angle reduces translational symmetry, leading to a moiré lattice with an enlarged unit cell. A magnetic field introduces flux translational symmetry, restoring local crystalline symmetries at specific fluxes. The nearest-neighbor tight-binding model of graphene displays a flux periodicity determined by the magnetic field strength and unit cell area. The inclusion of next-nearest neighbor hoppings modifies the minimal Peierls path area, altering the flux periodicity. For TBG, the exact flux periodicity at a given twist angle is determined by a formula involving the greatest common divisor of specific integer combinations related to the twist angle parameters. The Hofstadter butterflies for graphene and TBG are calculated using the kernel polynomial method. The tight-binding model includes electron hopping beyond nearest neighbors, resulting in quasi-periodicity in the spectrum. Symmetries of the Hofstadter butterflies, including translational flux symmetry and C2x symmetry (broken under flux but restored at TRIFs), are analyzed. The model accurately reproduces the HOTI phase at zero flux, characterized by localized corner states within the bulk spectral gap. Two topological invariants, the second Stiefel-Whitney number and Z2 rotation-winding number, protect this phase. A new HOTI marker, χ(r), is introduced to diagnose HOTIs in real space, which is applicable even under symmetry-breaking perturbations. The spectral flow of corner states as a function of flux reveals topological changes and quantum Hall chiral edge states at discontinuity transitions. The re-entrant HOTI phase at half-flux periodicity is characterized using the HOTI marker, demonstrating localization along the edge. Replica HOTIs are identified at specific fluxes using the highest occupied and lowest unoccupied states as indicators. The HOTI marker successfully characterizes these replica states, despite the breaking of symmetries at these fluxes. The study verifies the presence of replica HOTIs at other large twist angles, showing consistent behavior of corner states and HOTI markers. Finally, the importance of out-of-plane rotational symmetry (C2x) for the re-entrant and replica HOTI phases is discussed. The calculations involved use both open boundary conditions (OBC) and periodic boundary conditions (PBC).
Key Findings
This research makes several key findings: 1. **Exact Flux Periodicity in TBG:** The study rigorously proves the existence of exact flux periodicity in TBG at all commensurate angles, providing a fundamental understanding of topological protection in the presence of magnetic fields. 2. **Replica HOTIs:** The research identifies numerous replica HOTI phases in TBG that occur at specific fluxes as quasiperiodic counterparts of TRIFs. These replica HOTIs exhibit localized corner states and edge-localized real-space topological markers, similar to the original HOTIs. 3. **HOTI Marker:** A novel real-space HOTI marker, χ(r), is introduced and successfully used to diagnose HOTI phases, even in the absence of protecting symmetries. This marker quantifies the localization of corner states and is shown to be applicable even at small magnetic fields and under OBC conditions. This offers significant advantages over the traditional momentum-space methods that are computationally costly at low magnetic fields. 4. **Re-entrant HOTI Phase:** A re-entrant HOTI phase is discovered at half-flux periodicity, where the effective twofold rotation symmetry is preserved. This re-entrance is shown to rely on a composite symmetry (UC2x). 5. **Symmetry Dependence:** The research highlights the importance of the out-of-plane rotational symmetry C2x in realizing the re-entrant and replica HOTI phases. The absence of this symmetry in some models prevents the re-entrant behavior. 6. **Quantitative Analysis:** The exponential decay of the HOTI marker provides a quantitative measure of localization strength, enabling detailed analysis of the topological states, even under symmetry-breaking conditions. 7. **Generalizability:** The findings are extended to other twist angles, demonstrating the generality of replica HOTIs in TBG. The localization strength of the markers is shown to weaken with decreasing twist angle due to reduced bulk gap.
Discussion
The findings address the research question by demonstrating the existence and characteristics of replica HOTIs in TBG, arising from the self-similar nature of the Hofstadter spectrum. The significance of these results lies in expanding the understanding of HOTI phases beyond conventional symmetry-protected scenarios. The development of the real-space HOTI marker provides a powerful tool for studying topological phases in systems where traditional momentum-space methods are computationally challenging. The discovery of the re-entrant HOTI phase highlights the role of specific symmetries in stabilizing these topological states. The results are relevant to the broader field of topological materials and quantum fractals, opening new avenues for exploring the interplay between symmetry, topology, and self-similarity in moiré systems. The observed replica topology may have implications for experiments at low magnetic fields, although realizing the observation requires high magnetic fields currently. The study suggests future exploration of the Coulomb repulsion effect on replica phases at smaller angles.
Conclusion
This work demonstrates the existence of replica HOTI phases in TBG, driven by the self-similarity of the Hofstadter butterfly spectrum. The introduction of the real-space HOTI marker is a key contribution, enabling efficient characterization of topological states even under symmetry-breaking conditions. The findings emphasize the critical role of symmetry in stabilizing HOTI phases and suggest future directions exploring the interplay between topology and many-body interactions in moiré systems. The study lays the groundwork for observing replica topology at potentially lower magnetic fields.
Limitations
The study primarily focuses on theoretical analysis using a tight-binding model. Experimental verification of the predicted replica HOTI phases requires high magnetic fields, which may present challenges in achieving experimental stability. While the model accurately reproduces the behavior of TBG, it does not explicitly include all possible interactions, and further study considering the effects of disorder and electron-electron interactions is warranted. The precise relationship between discrete scale invariance and band topology in this quantum fractal requires further investigation.
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