Chern insulators are distinct from standard topological insulators
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The retrieved evidence mentions Chern insulators and standard topological insulators in separate contexts or references Chern numbers in connection to quantum Hall and topological states, but lacks an explicit, full comparison establishing their distinctness as stated in the claim.
Magnetic topological insulators (MTIs) are a group of materials that feature topological band structures with concurrent magnetism, which can offer new opportunities for technological advancements in various applications, such as spintronics and quantum computing. The combination of topology and magnetism introduces a rich spectrum of topological phases in MTIs, which can be controllably manipulated by tuning material parameters such as doping profiles, interfacial proximity effect, or external conditions such as pressure and electric field. In this paper, we first review the mainstream MTI material platforms where the quantum anomalous Hall effect can be achieved, along with other exotic topological phases in MTIs. We then focus on highlighting recent developments in modulating topological properties in MTI with finite-size limit, pressure, electric field, and magnetic proximity effect. The manipulation of topological phases in MTIs provides an exciting avenue for advancing both fundamental research and practical applications. As this field continues to develop, further investigations into the interplay between topology and magnetism in MTIs will undoubtedly pave the way for innovative breakthroughs in the fundamental understanding of topological physics as well as practical applications.
Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( https://creativecommons.org/licenses/by/4.0/ ). Abstract Magnetic topological insulators (MTIs)
Shortly after, theorists worked out that when two-dimensional electron gases are subject to magnetic fields and periodic potentials, the filling factor υ of the QHE states is equivalent to the topological invariant [ 2 , 3 ], now known as the Thouless–Kohmoto–Nightingale–Nijs (TKNN) number or the Chern number. These topological invariants remain unchanged under smooth deformation, which gives rise to the robustness of quantization in the QHE. Since then, the concept of topological classification has quickly expanded to other gapped electronic systems, leading to the discovery of topological insulators.
Depending on the number of layers and spin configurations, MnBi 2 Te 4 can also host a variety of topological phases including QAH, Chern insulators, Weyl semimetals, and axion insulator states, which will be further elaborated below. 1.1.3. Twisted Moiré Materials By twisting 2D materials (either homogenously or heterogeneously), Moiré superlattice can be created with a superlattice periodicity varying in a wide range, typically much larger than the lattice constant. This allows a rich phase diagram in these twistronics with strong correlations and/or breaking symmetries. In order to observe QAH states in twisted Moiré materials, breaking time-reversal symmetry is required.
Topological States in Magnetic Topological Insulators Beyond QAH states, MTIs can also host or are closely related to other topological phases of matter through topological phase transitions, including: axion insulators, Weyl semimetals, high-Chern-number insulators, and higher-order topological states, just to name a few. Here, we provide an overview of a few representative examples of topological states that are accessible by tuning material parameters of MTI within an experimentally feasible range. 1.2.1.
As a result, there exists a range of magnetic field where the top and bottom magnetizations point in opposite directions, leading to an axion insulator, as shown in Figure 1 d. Such a material platform has been investigated by combining Cr-doped and V-doped MTIs, and signatures of axion insulators such as zero-Hall plateaus distinct from those in a trivial insulating transition state in CBST have been observed [ 46 , 47 , 48 ]. It remains an active research area to directly observe quantized topological magnetoelectric effect in axion insulators. 1.2.2.
In particular, because even- and odd-layered MBT films have different compensated/uncompensated net magnetizations, their ground states’ evolution under pressure also differs drastically. In an even-layered MBT with fully compensated magnetization, Chern insulating states can be achieved even in spin-flop states at a moderate magnetic field of 2.5 T, which is significantly lower compared to the onset of FM states (~5 T) under ambient pressure, whereas an odd-layered MBT at a zero field will undergo a phase transition from a trivial insulator to a QAH insulator by tuning the pressure [ 80 ]. 4. Electric Field Tuning of Topological Phases 4.1.
Theoretical Models In thin films of MTIs, the interaction between the surface hybridization gap ( m 0 ) and the magnetic exchange gap (Δ) gives rise to distinct topological phases. Applying an electric field is equivalent to changing the potential difference ( V ) between the two surfaces. Based on the effective Hamiltonian model for 2D surface states [ 81 , 82 ], the surface wavefunctions can be modulated by V , leading to changes in the global band gap and the magnetic exchange gap. The band gap can be expressed as E g = 2 ∆ − 2 m 0 2 + V 2 . The gap closing point occurs at V c = ∆ 2 − m 0 2 , indicating a topological phase transition between the QAH insulator and normal insulator states.
The effective distinction between carrier density and electric field excludes the possibility of a metal-to-insulator transition resulting from the localization-induced phase transition [ 92 ], which typically arises from disorder-induced charge states localized at the hole band edge. The mechanism of the topological phase transition can be understood as the band closing and reopening modulated by an electric field, as illustrated by the surface band structures shown in Figure 5 c. The band structure of the CBST lattice exhibits an inverted surface band gap, and the calculated in-gap Hall conductivity indicates a QAH phase with a Chern number of 1 in the absence of an electric field.
We investigate the localization and topological properties of the Haldane model under the influence of random flux and Anderson disorder. Our localization analysis reveals that random flux induces a transition from insulating to metallic states, while Anderson localization only arises under the modulation of Anderson disorder. By employing real-space topological invariant methods, we demonstrates that the system undergoes topological phase transitions under different disorder manipulations, whereas random flux modulation uniquely induces topological Anderson insulator phases, with the potential to generate states with opposite Chern numbers. These findings highlight the distinct roles of disorder in shaping the interplay between topology and localization, providing insights into stabilizing topological states and designing robust topological quantum materials.
Fractional Chern insulators (FCIs) generalize the fractional quantum Hall effects of Landau levels to lattice systems in the absence of an external magnetic field, which arise from the interplay between strong electron-electron interactions and non-trivial topology. Despite extensive theoretical proposals over more than a decade, realizing intrinsic FCIs in realistic materials has remained a long-standing challenge, owing to stringent requirements on band flatness, topology and quantum geometry. Advances in two-dimensional moiré superlattices have overcome these obstacles, enabling observations of the fractional quantum anomalous Hall effect. In this Perspective, we review the theoretical foundations of FCIs and discuss their experimental realizations in moiré materials, with a focus on two complementary platforms: twisted MoTe<sub>2</sub> and rhombohedral multilayer graphene/hexagonal boron nitride moiré superlattices. We highlight the distinct physical mechanisms underlying FCIs in these systems, outline open questions concerning their microscopic origin and stability and discuss future opportunities towards new FCIs and non-Abelian topological order in quantum materials.
3: FCIs in tMoTe 2 studied by multiple probes. Fig. 4: Experimental observations of FCIs and FQAHE in moiré RMG. Fig. 5: Multidimensional tunability of FCIs in moiré RMG. Similar content being viewed by others Tunable fractional Chern insulators in rhombohedral graphene superlattices Article 22 April 2025 Fractional high-Chern insulator in twisted rhombohedral graphene Article 15 July 2026 Fractional quantization in insulators from Hall to Chern Article 07 November 2025 Subjects Electronics, photonics and device physics Topological matter References Regnault, N. & Bernevig, B. A. Fractional Chern insulator. Phys. Rev. X 1 , 021014 (2011). Google Scholar Sheng, D. N., Gu, Z. C., Sun, K.
W. & Wen, X. G. High-temperature fractional quantum Hall states. Phys. Rev. Lett. 106 , 236802 (2011). Article PubMed Google Scholar Sun, K., Gu, Z., Katsura, H. & Das Sarma, S. Nearly flatbands with nontrivial topology. Phys. Rev. Lett. 106 , 236803 (2011). Article PubMed Google Scholar Bernevig, B. A. & Regnault, N. Emergent many-body translational symmetries of Abelian and non-Abelian fractionally filled topological insulators. Phys. Rev. B 85 , 075128 (2012). Article Google Scholar Liu, T., Repellin, C., Bernevig, B. A. & Regnault, N. Fractional Chern insulators beyond Laughlin states. Phys. Rev. B 87 , 205136 (2013). Article Google Scholar Wang, Y.-F., Yao, H., Gu, Z.-C., Gong, C.-D.
& Sheng, D. N. Non-Abelian quantum Hall effect in topological flat bands. Phys. Rev. Lett. 108 , 126805 (2012). Article PubMed Google Scholar Wu, Y.-L., Bernevig, B. A. & Regnault, N. Zoology of fractional Chern insulators. Phys. Rev. B 85 , 075116 (2012). Article Google Scholar Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556 , 80–84 (2018). Article CAS PubMed Google Scholar Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556 , 43–50 (2018). Article CAS PubMed Google Scholar Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure.
Article CAS PubMed Google Scholar Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. H. Topological insulators in twisted transition metal
Phys. Rev. Res. 3 , L032070 (2021). Article CAS Google Scholar Morales-Durán, N., Wei, N., Shi, J. & MacDonald, A. H. Magic angles and fractional Chern insulators in twisted homobilayer transition metal dichalcogenides. Phys. Rev. Lett. 132 , 096602 (2024). Article PubMed Google Scholar Zhang, Y.-H., Mao, D. & Senthil, T. Twisted bilayer graphene aligned with hexagonal boron nitride: anomalous Hall effect and a lattice model. Phys. Rev. Res. 1 , 033126 (2019). Article CAS Google Scholar Chittari, B. L., Chen, G., Zhang, Y., Wang, F. & Jung, J. Gate-tunable topological flat bands in trilayer graphene boron-nitride moiré superlattices. Phys. Rev. Lett. 122 , 016401 (2019).
Article CAS PubMed Google Scholar Chang, X. et al. Evidence of competing ground states between fractional Chern insulator and antiferromagnetism in moiré MoTe 2 . Nat. Commun. 17 , 4874 (2026). Wang, Y. et al. Hidden states and dynamics of fractional fillings in twisted MoTe 2 bilayers. Nature 641 , 1149–1155 (2025). Article CAS PubMed Google Scholar Anderson, E. et al. Magnetoelectric control of helical light emission in a moiré Chern magnet. Phys. Rev. X 15 , 031057 (2025). CAS Google Scholar Huber, O. et al. Optical control over topological Chern number in moiré materials. Nature 649 , 1153–1158 (2026). Article CAS PubMed Google Scholar Holtzmann, W. et al.
Exotic non-Abelian anyons from conventional fractional quantum Hall states. Nat. Commun. 4 , 1348 (2013). Article PubMed Google Scholar Wu, Y.-L., Regnault, N. & Bernevig, B. A. Bloch model wave functions and pseudopotentials for all fractional Chern insulators. Phys. Rev. Lett. 110 , 106802 (2013). Article PubMed Google Scholar Barkeshli, M. & Qi, X. Topological nematic states and non-Abelian lattice dislocations. Phys. Rev. X 2 , 031013 (2012). Google Scholar Sterdyniak, A. et al. Series of Abelian and non-Abelian states in C > 1 fractional Chern insulators. Phys. Rev. B 87 , 205137 (2013). Article Google Scholar Möller, G. & Cooper, N.
Fractional Chern insulators in Harper–Hofstadter bands with higher Chern number. Phys. Rev. Lett. 115 , 126401 (2015). Article PubMed Google Scholar Jaworowski, B., Regnault, N. & Liu, Z. Characterization of quasiholes in two-component fractional quantum Hall states and fractional Chern insulators in | C | = 2 flat bands. Phys. Rev. B 99 , 045136 (2019). Article Google Scholar Wang, Y.-F. et al. Fractional quantum Hall effect in topological flat bands with Chern number two. Phys. Rev. B 86 , 201101 (2012). Article Google Scholar Andrews, B., Neupert, T. & Möller, G.
expected that the existence of a topological Dirac surface state in this material would lead to a topological insulator with strong electronic correlations
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases, that arise from electromagnetic forces between atoms and electrons. More generally, the subject deals with condensed phases of matter: systems of many constituents with strong interactions among them. More exotic condensed phases i
The study of phase transitions and the critical behavior of observables, termed critical phenomena, was a major field of interest in the 1960s. Leo Kadanoff, Benjamin Widom and Michael Fisher developed the ideas of critical exponents and widom scaling. These ideas were unified by Kenneth G. Wilson in 1972, under the formalism of the renormalization group in the context of quantum field theory.
The quantum Hall effect was discovered by Klaus von Klitzing, Dorda and Pepper in 1980 when they observed the Hall conductance to be integer multiples of a fundamental constant
e
2
/
h
{\displaystyle e^{2}/h}
.(see figure) The effect was observed to be independent of parameters such as system size and impurities. In 1981, theorist Robert Laughlin proposed a theory explaining the unanticipated precision of the integral plateau. It also implied that the Hall conductance is proportional to a topological invariant, called Chern number, whose relevance for the band structure of solids was formulated by David J. Thouless and collaborators. Shortly after, in 1982, Horst Störmer and Daniel Tsui observed the fractional quantum Hall effect where the conductance was now a rational multiple of the constant
e
2
/
The study of phase transitions and the critical behavior of observables, termed critical phenomena, was a major field of interest in the 1960s. Leo Kadanoff, Benjamin Widom and Michael Fisher developed the ideas of critical exponents and widom scaling. These ideas were unified by Kenneth G. Wilson in 1972, under the formalism of the renormalization group in the context of quantum field theory.
The quantum Hall effect was discovered by Klaus von Klitzing, Dorda and Pepper in 1980 when they observed the Hall conductance to be integer multiples of a fundamental constant
e
2
/
h
{\displaystyle e^{2}/h}
.(see figure) The effect was observed to be independent of parameters such as system size and impurities. In 1981, theorist Robert Laughlin proposed a theory explaining the unanticipated precision of the integral plateau. It also implied that the Hall conductance is proportional to a topological invariant, called Chern number, whose relevance for the band structure of solids was formulated by David J. Thouless and collaborators. Shortly after, in 1982, Horst Störmer and Daniel Tsui observed the fractional quantum Hall effect where the conductance was now a rational multiple of the constant
e
2
/
h
{\displaystyle e^{2}/h}
. Laughlin, in 1983, realized that this was a consequence of quasiparticle interaction in the Hall states and formulated a variational method solution, named the Laughlin wavefunction. The study of topological properties of the fractional Hall effect remains an active field of research. Decades later, the aforementioned topological band theory advanced by David J. Thouless and collaborators was further expanded leading to the discovery of topological insulators.
In 1986, Karl Müller and Johannes Bednorz discovered the first high temperature superconductor, La2-xBaxCuO4, which is superconducting at temperatures as high as 39 K. It was realized that the high temperature superconductors are examples of strongly correlated materials where the electron–electron interactions play an important role. A satisfactory theoretical description of high-temperature superconductors is still not known and the field of strongly correlated materials continues to be an active research topic.
In 2012, several groups released preprints which suggest that samarium hexaboride has the properties of a topological insulator in accord with the earlier theoretical predictions. Since samarium hexaboride is an established Kondo insulator, i.e. a strongly correlated electron material, it is expected that the existence of a topological Dirac surface state in this material would lead to a topological insulator with strong electronic correlations.
Phase transition refers to the change of phase of a system, which is brought about by change in an external parameter such as temperature, pressure, or molar composition. In a single-component system, a classical phase transition occurs at a temperature (at a specific pressure) where there is an abrupt change in the order of the system. For example, when ice melts and becomes water, the ordered hexagonal crystal structure of ice is modified to a hydrogen bonded, mobile arrangement of water molecules.
In quantum phase transitions, the temperature is set to absolute zero, and the non-thermal control parameter, such as pressure or magnetic field, causes the phase transitions when order is destroyed by quantum fluctuations originating from the Heisenberg uncertainty principle. Here, the different quantum phases of the system refer to distinct ground states of the Hamiltonian matrix. Understanding the behavior of quantum phase transition is important in the difficult tasks of explaining the properties of rare-earth magnetic insulators, high-temperature superconductors, and other substances.
Two classes of phase transitions occur: first-order transitions and second-order or continuous transitions. For the latter, the two phases involved do not
This work focuses on the thermodynamic behavior of electronic systems (either topological or conventional) in their ground state, when some external parameters are varied, i.e. thickness of heterostructures or external electromagnetic fields. This behavior, which completely ignores interactions, is a promise that one-electron physics is still of great importance (quite successful in early experiments in various distinct areas) and quite often leads to topological properties. Starting my work, I review the field of topological insulators which is a milestone in physics today, and remind the reader that a topological insulator may be found from two distinct cases: the first one is that it is an emergent phenomenon resulting from conventional materials when placed in a magnetic field, or it is a pure phenomenon resulting from compounds with large spin-orbit coupling. These cases involve non-zero Chern number, which is the topological index analogous to well-known Euler characteristics in geometry. In Chapter 2, Berry physics (which directly leads to the Chern number) is thoroughly discussed, highlighting some new consequences on Hellmann – Feynman theorem, and the resulting quantization of real magnetic charge in 3-D space, through a fictitious magnetic charge that appears in the parameter space. In addition, we study some non-Hermitian influences on Hellmann – Feynman and Ehrenfest theorems that come out of boundary contributions, resulting in the correction of some paradoxes a
Topological insulators feature a number of topologically protected boundary modes linked to the value of their bulk invariant. While in one-dimensional systems the boundary modes are zero dimensional and localized, in two-dimensional topological insulators the boundary modes are chiral, one-dimensional propagating modes along the edges of the system. Thus, topological photonic insulators with large Chern numbers naturally display a topologically protected multimode waveguide at their edges. Here, we show how to take advantage of these topologically protected propagating modes by interfacing them with quantum emitters. In particular, using a Harper-Hofstadter lattice, we find situations in which the emitters feature quasiquantized decay rates due to the increasing number of edge modes, and where their spontaneous emission spatially separates in different modes. We also show how using a single $π$-pulse the combination of such spatial separation and the interacting character of the emitters leads to the formation of a single-photon time-bin entangled state with no classical analog, which we characterize computing its entanglement entropy. Finally, we also show how the emitters can selectively interact with the different channels using nonlocal light-matter couplings such as the ones that can be obtained with giant atoms. Such capabilities pave the way for generating quantum gates among topologically protected photons as well as generating more complex entangled states of light
Topological photonics explores photonic systems that exhibit robustness against defects and disorder, enabled by protection from underlying topological phases. These phases are typically realized in linear optical systems and characterized by their intrinsic photonic band structures. Here we experimentally study Floquet Chern insulators in periodically driven nonlinear photonic crystals, where the topological phase is controlled by the polarization and the frequency of the driving field. Our transient sum-frequency generation measurements reveal strong hybridization of the Floquet photonic bands. The measured spectrum remains gapless under a linearly polarized drive but becomes gapped under a circularly polarized drive. Theoretical analysis confirms that the Floquet gap is topological, characterized by a non-zero Chern number—a consequence of time-reversal symmetry breaking induced by the circularly polarized driving field. Furthermore, this work offers opportunities to explore the role of classical optical nonlinearity in topological phases and their applications in nonlinear optoelectronics.
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