Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.12258

Stable Topology in Exactly Flat Bands

Yan-Qi Li, Yi-Jie Wang, Pei-han Lin, Bin Wang, Zhi-Da Song

Published Mar 12, 2026• 0 citations
arXivHTMLPDF

Abstract

Topological flat bands (FBs) offer an ideal platform for realizing exotic topological phases and exploring quantum geometric effects, yet their realization with both exact flatness and stable topology in local lattice models has been hindered by fundamental no-go theorems. The obstruction is also manifested as the absence of exact Gaussian TNS representations for topological insulators and superconductors. Here, we overcome this barrier by demonstrating the existence of critical topological FBs (CTFBs) in finite-range hopping models. They saturate the no-go theorems via a unique structure of Bloch wavefunctions: While continuous over the whole Brillouin zone, the projector $P(k)$ onto FBs is non-analytic at isolated band touching points. Filling such CTFBs yields short-range entangled topological states with power-law correlations due to the non-analyticity. We then establish a general symmetry-based principle to systematically construct CTFBs that carry desired topological invariants in given space groups. It utilizes the bipartite structure and requires no further fine-tuning, and the topology is robust against arbitrary symmetry-preserving gap-opening perturbations. Remarkably, independent tuning of the quantum geometry is allowed. Examples exhibiting Chern numbers 1, 2, 3, 6, in 2D, and strong $\mathbb{Z}_2$ indices in 2D and 3D are demonstrated for concreteness. An automated algorithm further identifies more than 50,000 symmetry-indicated CTFBs, including crystalline and higher-order topological ones. In the end, we show that the bipartite structure naturally endows the filled CTFB states with exact TNS representations with finite bond dimensions. By bridging the topological band theory to TNS methods, CTFB provides a novel tractable starting point for exploring strongly correlated topological matter, with potential relevance to realistic material realizations.

DOI
10.48550/arxiv.2603.12258

Related papers

Band flatness optimization through complex analysis

Ching Hua Lee, Daniel P. Arovas, Ronny Thomale

Narrow-band electron systems are particularly likely to exhibit correlated many-body phases driven by interaction effects. Examples include magnetic materials, heavy-fermion systems, and topological phases such as fractional quantum Hall states and their lattice-based cousins, the fractional Chern insulators (FCIs). Here we discuss the problem of designing models with optimal band flatness, subject to constraints on the range of electron hopping. In particular, we show how the imaginary gap, which serves as a proxy for band flatness, can be optimized by appealing to Rouch\'e's theorem, a familiar result from complex analysis. This leads to an explicit construction which we illustrate through its application to two-band FCI models with nontrivial topology (i.e., nonzero Chern numbers). We show how the imaginary-gap perspective leads to an elegant geometric picture of how topological properties can obstruct band flatness in systems with finite-range hopping.

arXiv:1511.03295•Physical review. B./Physical review. B•Apr 26, 2016•48 citations

New class of exactly flat topological bands - compact localised states protected by local graph topology

Tamaghna Hazra

Strongly correlated quantum matter is fundamentally defined by the tension between non-commuting quantum operators. Hamiltonians exhibiting macroscopic degeneracies are of general interest in this field because they imply an infinite susceptibility to any non-commuting perturbation. In moiré heterostructures, engineering such extensive degeneracies in the kinetic Hamiltonian creates a fertile garden for exotic strongly correlated phases of matter to emerge from the resulting flat bands. Here, we introduce a prescription to construct an infinite family of exact flat band Hamiltonians supported on the faces of arbitrary graphs. We demonstrate this algorithm on the faces of four Bravais lattices. Using a discrete graph generalization of the Atiyah-Singer index theorem, we prove that the extensive degeneracies of such face-graph Hamiltonians are protected by the local topology of the face-graph connectivity. The resulting macroscopic null spaces yield compact localised states that remain localised over time due to frustration in hopping pathways. We discuss the broad implications of such non-dispersing quantum modes in diverse settings, from arrested dynamics in quantum networks and quantum machine learning algorithms to Majorana-free topological quantum computation.

cond-mat.str-elCross-listedarXiv:2607.22831•Jul 28, 2026•0 citations

Interacting topological quantum chemistry in 2D with many-body real space invariants

Jonah Herzog-Arbeitman, Bogdan Andrei Bernevig, Zhi-Da Song

The topological phases of non-interacting fermions have been classified by their symmetries, culminating in a modern electronic band theory where wavefunction topology can be obtained from momentum space. Recently, Real Space Invariants (RSIs) have provided a spatially local description of the global momentum space indices. The present work generalizes this real space classification to interacting 2D states. We construct many-body local RSIs as the quantum numbers of a set of symmetry operators on open boundaries, but which are independent of the choice of boundary. Using the U(1) particle number, they yield many-body fragile topological indices, which we use to identify which single-particle fragile states are many-body topological or trivial at weak coupling. To this end, we construct an exactly solvable Hamiltonian with single-particle fragile topology that is adiabatically connected to a trivial state through strong coupling. We then define global many-body RSIs on periodic boundary conditions. They reduce to Chern numbers in the band theory limit, but also identify strongly correlated stable topological phases with no single-particle counterpart. Finally, we show that the many-body local RSIs appear as quantized coefficients of Wen-Zee terms in the topological quantum field theory describing the phase.

arXiv:2212.00030•Nature Communications•Feb 8, 2024•25 citations

Universal Topological Signatures of Non-Abelian Anyons in Engineered Flat Bands

Revista, Zen, PHYSICS, 10

The quest for non-Abelian anyons, exotic quasiparticles fundamental to topological quantum computing, remains a forefront challenge in condensed matter physics. This paper explores the universal topological signatures that characterize non-Abelian anyons when hosted within engineered flat band systems. Flat bands, distinguished by their quenched kinetic energy, dramatically enhance interaction effects, providing a unique platform for stabilizing exotic many-body phases and fostering the emergence of non-Abelian statistics. We theoretically investigate how specific lattice geometries and tuning parameters in various engineered platforms, such as Moiré superlattices, photonic crystals, and ultracold atomic gases, can lead to the formation of topological flat bands conducive to non-Abelian anyonic excitations. Our analysis identifies several universal signatures, including robust ground state degeneracy sensitive to system topology, characteristic patterns in topological entanglement entropy, and distinct braiding statistics manifested through non-Abelian holonomies and fractional quantum numbers. These signatures are shown to be robust against local perturbations and largely independent of microscopic details, providing reliable indicators for experimental detection. We discuss the implications for developing fault-tolerant quantum computation and outline potential experimental methodologies for probing these predicted signatures, emphasizing the transformative potential of engineered flat bands in advancing topological quantum science.

•Nov 28, 2025•0 citations

Many-body superconductivity in topological flat bands

Jonah Herzog-Arbeitman, Aaron Chew, Kukka-Emilia Huhtinen +2 more

In a flat band superconductor, bosonic excitations disperse while unpaired electrons are immobile. To study this strongly interacting system, we construct a family of multi-band Hubbard models with exact eta-pairing ground states in all space groups. We analytically compute their many-body excitations and find that the Cooper pair bound states obey an effective single-boson hopping Hamiltonian written in terms of the non-interacting wavefunctions. These bound states possess a unique zero-energy excitation whose quadratic dispersion is determined by the minimal quantum metric. The rest of the bound state spectrum is classified by topological quantum chemistry, which we use to identify Cooper pairs with Weyl nodes, higher angular momentum pairing, and fragile topology. Broadening the band dispersion, we find the strongest pairing occurs at half filling and not at the highest density of states. Our analytical results prove that the elementary fermionic and bosonic excitations, corresponding respectively to pairing and coherence, are governed by distinct properties of the flat band, rigorously establishing its unique phenomenology. Superconductivity in flat bands is a central feature of quantum geometric effects, supported by mean-field theory and numerics. Here, the authors exhibit a large family of flat band Hubbard models whose ground states, Bogoliubov-de-Gennes particles, and Cooper pairs are exactly solvable with analytically calculable dispersions.

arXiv:2209.00007•Communications Physics•Jul 4, 2026•9 citations

Localization and Criticality in Non-crystalline Quantum Matter

Moein N. Ivaki

The concept of non-trivial electronic topology in condensed matter physics is crucial for understanding the electronic spectrum of solids and for tailoring various thermodynamic and transport properties, useful for real-world applications. Research on topological insulators and superconductors primarily utilizes the electronic band theory and symmetry classes of gapped quantum systems, with the existence of spectral or mobility gaps ensured even in the presence of impurities and defects. However, structural disorder, naturally manifested in materials such as glasses and amorphous solids, can significantly impact the critical properties of phase transitions in topological quantum matters. This dissertation systematically studies topological phase transitions in non-crystalline free-fermion systems via tight-binding quantum Hamiltonians on 2D random geometries. Leveraging various theoretical and experimentally accessible measures, we demonstrate the resilience of topological phases in non-crystalline systems against structural randomness, described by classical percolation theory, and characterize their unique critical behavior through finite-size scaling. We also extend the notion of amorphous topological states to non-integer dimensional random systems. We reveal that these 2D-embedded random fractals can admit well-defined thermodynamic topological phases with unconventional critical properties. We further study quantum dynamics in these random topological systems. By employing a family of discrete-time topological quantum walks on random percolation lattices, we address wavefunction propagation, superdiffusive quantum speedup, and localization transition by identifying the critical probabilities and timescales. Lastly, we introduce topological lattices with adjustable spatial correlation to investigate Landau levels and quantum oscillations in a low magnetic field regime.

•Jan 1, 2023•0 citations
Topological flat bands (FBs) offer an ideal platform for realizing exotic topological phases and exploring quantum geometric effects, yet their realization with both exact flatness and stable topology in local lattice models has been hindered by fundamental no-go theorems. The obstruction is also manifested as the absence of exact Gaussian TNS representations for topological insulators and superconductors. Here, we overcome this barrier by demonstrating the existence of critical topological FBs (CTFBs) in finite-range hopping models. They saturate the no-go theorems via a unique structure of Bloch wavefunctions: While continuous over the whole Brillouin zone, the projector P(k)P(k)P(k) onto FBs is non-analytic at isolated band touching points. Filling such CTFBs yields short-range entangled topological states with power-law correlations due to the non-analyticity. We then establish a general symmetry-based principle to systematically construct CTFBs that carry desired topological invariants in given space groups. It utilizes the bipartite structure and requires no further fine-tuning, and the topology is robust against arbitrary symmetry-preserving gap-opening perturbations. Remarkably, independent tuning of the quantum geometry is allowed. Examples exhibiting Chern numbers 1, 2, 3, 6, in 2D, and strong Z2\mathbb{Z}_2Z2​ indices in 2D and 3D are demonstrated for concreteness. An automated algorithm further identifies more than 50,000 symmetry-indicated CTFBs, including crystalline and higher-order topological ones. In the end, we show that the bipartite structure naturally endows the filled CTFB states with exact TNS representations with finite bond dimensions. By bridging the topological band theory to TNS methods, CTFB provides a novel tractable starting point for exploring strongly correlated topological matter, with potential relevance to realistic material realizations.
Narrow-band electron systems are particularly likely to exhibit correlated many-body phases driven by interaction effects. Examples include magnetic materials, heavy-fermion systems, and topological phases such as fractional quantum Hall states and their lattice-based cousins, the fractional Chern insulators (FCIs). Here we discuss the problem of designing models with optimal band flatness, subject to constraints on the range of electron hopping. In particular, we show how the imaginary gap, which serves as a proxy for band flatness, can be optimized by appealing to Rouch\'e's theorem, a familiar result from complex analysis. This leads to an explicit construction which we illustrate through its application to two-band FCI models with nontrivial topology (i.e., nonzero Chern numbers). We show how the imaginary-gap perspective leads to an elegant geometric picture of how topological properties can obstruct band flatness in systems with finite-range hopping.
Strongly correlated quantum matter is fundamentally defined by the tension between non-commuting quantum operators. Hamiltonians exhibiting macroscopic degeneracies are of general interest in this field because they imply an infinite susceptibility to any non-commuting perturbation. In moiré heterostructures, engineering such extensive degeneracies in the kinetic Hamiltonian creates a fertile garden for exotic strongly correlated phases of matter to emerge from the resulting flat bands. Here, we introduce a prescription to construct an infinite family of exact flat band Hamiltonians supported on the faces of arbitrary graphs. We demonstrate this algorithm on the faces of four Bravais lattices. Using a discrete graph generalization of the Atiyah-Singer index theorem, we prove that the extensive degeneracies of such face-graph Hamiltonians are protected by the local topology of the face-graph connectivity. The resulting macroscopic null spaces yield compact localised states that remain localised over time due to frustration in hopping pathways. We discuss the broad implications of such non-dispersing quantum modes in diverse settings, from arrested dynamics in quantum networks and quantum machine learning algorithms to Majorana-free topological quantum computation.
The topological phases of non-interacting fermions have been classified by their symmetries, culminating in a modern electronic band theory where wavefunction topology can be obtained from momentum space. Recently, Real Space Invariants (RSIs) have provided a spatially local description of the global momentum space indices. The present work generalizes this real space classification to interacting 2D states. We construct many-body local RSIs as the quantum numbers of a set of symmetry operators on open boundaries, but which are independent of the choice of boundary. Using the U(1) particle number, they yield many-body fragile topological indices, which we use to identify which single-particle fragile states are many-body topological or trivial at weak coupling. To this end, we construct an exactly solvable Hamiltonian with single-particle fragile topology that is adiabatically connected to a trivial state through strong coupling. We then define global many-body RSIs on periodic boundary conditions. They reduce to Chern numbers in the band theory limit, but also identify strongly correlated stable topological phases with no single-particle counterpart. Finally, we show that the many-body local RSIs appear as quantized coefficients of Wen-Zee terms in the topological quantum field theory describing the phase.
The quest for non-Abelian anyons, exotic quasiparticles fundamental to topological quantum computing, remains a forefront challenge in condensed matter physics. This paper explores the universal topological signatures that characterize non-Abelian anyons when hosted within engineered flat band systems. Flat bands, distinguished by their quenched kinetic energy, dramatically enhance interaction effects, providing a unique platform for stabilizing exotic many-body phases and fostering the emergence of non-Abelian statistics. We theoretically investigate how specific lattice geometries and tuning parameters in various engineered platforms, such as Moiré superlattices, photonic crystals, and ultracold atomic gases, can lead to the formation of topological flat bands conducive to non-Abelian anyonic excitations. Our analysis identifies several universal signatures, including robust ground state degeneracy sensitive to system topology, characteristic patterns in topological entanglement entropy, and distinct braiding statistics manifested through non-Abelian holonomies and fractional quantum numbers. These signatures are shown to be robust against local perturbations and largely independent of microscopic details, providing reliable indicators for experimental detection. We discuss the implications for developing fault-tolerant quantum computation and outline potential experimental methodologies for probing these predicted signatures, emphasizing the transformative potential of engineered flat bands in advancing topological quantum science.
In a flat band superconductor, bosonic excitations disperse while unpaired electrons are immobile. To study this strongly interacting system, we construct a family of multi-band Hubbard models with exact eta-pairing ground states in all space groups. We analytically compute their many-body excitations and find that the Cooper pair bound states obey an effective single-boson hopping Hamiltonian written in terms of the non-interacting wavefunctions. These bound states possess a unique zero-energy excitation whose quadratic dispersion is determined by the minimal quantum metric. The rest of the bound state spectrum is classified by topological quantum chemistry, which we use to identify Cooper pairs with Weyl nodes, higher angular momentum pairing, and fragile topology. Broadening the band dispersion, we find the strongest pairing occurs at half filling and not at the highest density of states. Our analytical results prove that the elementary fermionic and bosonic excitations, corresponding respectively to pairing and coherence, are governed by distinct properties of the flat band, rigorously establishing its unique phenomenology. Superconductivity in flat bands is a central feature of quantum geometric effects, supported by mean-field theory and numerics. Here, the authors exhibit a large family of flat band Hubbard models whose ground states, Bogoliubov-de-Gennes particles, and Cooper pairs are exactly solvable with analytically calculable dispersions.
The concept of non-trivial electronic topology in condensed matter physics is crucial for understanding the electronic spectrum of solids and for tailoring various thermodynamic and transport properties, useful for real-world applications. Research on topological insulators and superconductors primarily utilizes the electronic band theory and symmetry classes of gapped quantum systems, with the existence of spectral or mobility gaps ensured even in the presence of impurities and defects. However, structural disorder, naturally manifested in materials such as glasses and amorphous solids, can significantly impact the critical properties of phase transitions in topological quantum matters. This dissertation systematically studies topological phase transitions in non-crystalline free-fermion systems via tight-binding quantum Hamiltonians on 2D random geometries. Leveraging various theoretical and experimentally accessible measures, we demonstrate the resilience of topological phases in non-crystalline systems against structural randomness, described by classical percolation theory, and characterize their unique critical behavior through finite-size scaling. We also extend the notion of amorphous topological states to non-integer dimensional random systems. We reveal that these 2D-embedded random fractals can admit well-defined thermodynamic topological phases with unconventional critical properties. We further study quantum dynamics in these random topological systems. By employing a family of discrete-time topological quantum walks on random percolation lattices, we address wavefunction propagation, superdiffusive quantum speedup, and localization transition by identifying the critical probabilities and timescales. Lastly, we introduce topological lattices with adjustable spatial correlation to investigate Landau levels and quantum oscillations in a low magnetic field regime.