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.
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19 pages main text, 8 figures, 1 appendix. Comments welcome and will be gratefully acknowledged upon submission
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.
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 Z2 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.
Topological phases, edge states, and flat bands in synthetic quantum systems are a key resource for topological quantum computing and noise-resilient information processing. We introduce a scheme based on step-dependent quantum walks on cyclic graphs, termed cyclic quantum walks (CQWs), to simulate exotic topological phenomena using discrete Fourier transforms and an effective Hamiltonian. Our approach enables the generation of both gapped and gapless topological phases, including Dirac cone-like energy dispersions, topologically nontrivial flat bands, and protected edge states, all without resorting to resource-intensive split-step or split-coin quantum walk protocols. Odd and even-site cyclic graphs exhibit markedly different spectral characteristics, with rotationally symmetric flat bands emerging exclusively in 4n-site graphs (n∈N). We analytically establish the conditions for the emergence of topological, gapped flat bands and show that gap closings in rotation space imply the formation of Dirac cones in momentum space. Further, we engineer protected edge states at the interface between distinct topological phases in both odd and even cycle graphs. We numerically demonstrate that the edge states are robust against moderate static and dynamic gate disorder as well as remain stable against phase-preserving perturbations and are independent of initial states. This scheme serves as a resource-efficient and versatile platform to engineer topological phases, transitions, edge states, and flat bands in small-scale quantum systems, opening new avenues for robust quantum memory, protected state transfer, and compact implementations of fault-tolerant quantum technologies.
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.
Superconductivity in a partially filled flat band presents a vexing conceptual hurdle because the absence of a Fermi surface precludes a weak-coupling regime where one can extend insights from the Bardeen-Cooper-Schrieffer picture of a Fermi surface instability. We approach the strongly correlated problem of flat band superconductivity from the strong coupling limit of local attractive interactions on line-graph lattices, whose non-interacting bandstructures host exactly flat bands due to frustrated hopping. In this limit, the pair kinetic energy which sets the superfluid stiffness is expected to scale inversely with the pair binding interaction. Here we demonstrate a striking counterexample. We show that when doped charges propagate on the line-graph of a lattice with strong pairing interaction and broken time-reversal symmetry, they bind into obstructed Cooper pairs whose motion is frustrated by destructive interference. As a result, the leading-order pair kinetic energy vanishes identically in the strong-coupling expansion, producing a flat bosonic band of compact localised pair states, zero superfluid stiffness at leading order, and an extensively degenerate many-body ground state manifold. At quarter filling, the frustrated pair dynamics maps onto a quantum dimer model which has a d-wave resonating-valence-bond ground state when time-reversal is broken. The pairing Hamiltonian in this limit thus has a topologically ordered spin liquid ground state which becomes exact at the analytically solvable Rokhsar-Kivelson point with long-range entanglement and deconfined holon excitations. Interestingly, we find exact compact localised eigenstates and extensive degeneracies in the many-body eigenstates of this emergent dimer model. Our results establish a disorder-free mechanism for interaction-driven localisation, in which strong pairing collapses the kinetic energy of Cooper pairs.
Flat band physics is a central theme in modern condensed matter physics. By constructing a tight--binding single particle system that has vanishing momentum dispersion in one or more bands, and subsequently including more particles and interactions, it is possible to study physics in strongly interacting regimes. Inspired by the glued trees that first arose in one of the few known examples of quantum supremacy, we define and analyze two infinite families of tight binding single particle Bose--Hubbard models that have only flat bands, and only compact localized states despite having any nonnegative number of translation symmetries. The first class of model that we introduce is constructed by replacing a sufficiently large fraction of the edges in a generic countable graph with glued trees modified to have complex hoppings. The second class arises from thinking of complex weighted glued trees as rhombi that can then be used to tile two dimensional space, giving rise to the familiar dice lattice and infinitely many generalizations thereof, of which some are Euclidian while others are hyperbolic.
Abstract Systems hosting flat bands offer a powerful platform for exploring strong correlation physics. Theoretically, topological degeneracy arising in systems with non-trivial topological orders on periodic manifolds of non-zero genus can generate ideal flat bands. However, experimental realization of such geometrically engineered systems is very difficult. In this work, we demonstrate that flat planes with strategically patterned hole defects can engineer ideal flat bands. We construct two families of models: singular flat band systems where degeneracy is stabilized by non-contractible loop excitations tied to hole defects and perfectly nested van Hove systems where degeneracy arises from line excitations in momentum space. These models circumvent the need for exotic manifolds while retaining the essential features of topological flat bands. By directly linking defect engineering to degeneracy mechanisms, our results establish a scalable framework for experimentally accessible flat band design.