Topological & fault-tolerant approaches

We consider the global aspects of the 6-dimensional N=(1, 0) theory arising from the coupling of the vector multiplet to the tensor multiplet. We show that the Yang-Mills field and its dual, when both are abelianized, combine to define a class in twisted cohomology with the twist arising from the class of the $B$-field, in a duality-symmetric manner. We then show that this lifts naturally to a class in twisted (differential) K-theory. Alternatively, viewing the B-field in both N=(1,0) and N=(2,0) theories, not as a twist but as an invertible element, leads to a description within untwisted chromatic level two generalized cohomology theories, including forms of elliptic cohomology and Morava K-theory.

Moiré materials host fractional Chern insulators and electron crystals in close proximity, but the mechanism selecting between them remains an open question. We address this competition in Chern bands with ideal but momentum-dependent quantum geometry -- Aharonov-Casher bands. We present an ansatz wave function for generalized Wigner crystals and, by comparing its energy to that of the competing Laughlin-like state, map out the phase diagram at filling fraction $ν=1/m$ as a function of the degree of geometric non-uniformity. Our work identifies quantum geometry-controlled zero point fluctuations of the charge density of the generalized Wigner crystal as the mechanism controlling its relative stability, implying a kind of quantum Lindemann criterion for the crystal-liquid phase boundary.

Abstract The idea that topologically protected quantum states, such as anyons, may be attached to super/semiconductor heterostructures has received enormous attention, but experimental signatures in 1D systems remain elusive. Here we revisit theoretical underpinnings of anyons in 2D fractional quantum Hall (FQH) systems, whose signatures have been experimentally observed by independent groups. We highlight novel theorems about the Hopfion or CP 1-model, understood as quantization of FQH surplus flux in 2-Cohomotopy, which retrodicts abelian FQH anyon properties in fine detail. Examining this theoretical model further, we find that islands of expelled flux enlarge the ground-state monodromy from an abelian group to a (framed spherical) braid group, nonabelian for two or more islands, so that nonabelian anyonic states become topologically admissible; superconducting islands are one candidate realization of such flux expulsion.

Topological Anderson insulators provide a paradigm for disorder-induced topology, in which the underlying system turns from a trivial to a topological phase. It is widely recognized that the latter disappears at large disorder amplitude. Here, and contrary to the general belief, we provide evidence for successive disorder-driven topological transitions ending up in an unconventional topological Anderson phase that remains unexpectedly robust at strong disorder. The corresponding topological invariant increases stepwise with disorder, and then saturates in the strong-disorder regime, without reverting to a trivial Anderson insulator. We experimentally realize these topological Anderson staircase transitions in a one-dimensional topolectrical circuit that emulates single-wall nanotubes. Our work opens the road to topological disordertronics, in which robust topological phases can be tuned by disorder. Using a topolectrical circuit, the authors realise an unconventional disorder-tuned cascade of topological Anderson phase transitions ending up, contrary to the general belief, in an extremely robust topological phase. This work opens the gate for topological disordertronics, where topological channels can be tuned by disorder.

This chapter establishes a geometric homotopy model for twisted orbifold K-theory, formulated via mapping stacks from an orbifold domain to a universal bundle of Fredholm operators over the moduli stack of quantum symmetries. By extending the projective unitary group with PCT symmetries — time reversal, charge conjugation, and parity — the authors demonstrate that the fixed loci of the resulting quantum symmetry group on the space of Fredholm operators classify the ten flavors of topological K-theory known as the 10-fold way. The text defines twisted orbifold K-theory as the connected components of slice maps into this universal stack, a definition that unifies and reduces to standard equivariant K-theory and KR-theory under suitable conditions. Finally, the chapter constructs a low-dimensional equivariant orientation by analyzing the quaternionic Hopf fibration; this involves identifying the four-sphere with the quaternionic projective line and proving that the pullback of the tautological line bundle (realized via parameterized Fredholm operators) is equivariantly trivializable along the fibration.

This chapter surveys the approach of the book, a rigorous mathematical frame- work for extraordinary nonabelian twisted geometric orbifold cohomology, motivated by its applications to topological phases of quantum materials and brane charges in quantum gravity. Identifying the limitations of traditional approaches — such as Chen-Ruan cohomology, which effectively acts as Borel cohomology and consequently misses proper equivariant structures — the authors lay out a unifying theory based on higher topos theory. By utilizing systems of adjoint modalities to axiomatize singular-cohesive homotopy theory, the text creates a bridge between the differential geometric aspects of orbifolds and their singular, equivariant nature. The introduction previews key results, specifically that this new intrinsic cohomology subsumes proper equivariant theories like Bredon cohomology and equivariant K-theory, and demonstrates that tangentially twisted orbifold cohomology reduces to https://www.w3.org/1998/Math/MathML" display="inline"> RO (G) -graded equivariant cohomology near singularities.

This chapter establishes a foundation for geometric homotopy theory by constructing a hierarchy of convenient categories of generalized spaces — specifically higher topoi — designed to resolve the limitations of traditional topology and manifold theory regarding universal constructions like mapping spaces. The text adopts a probe based methodology, where spaces are defined by how they are charted by test objects: starting with smooth sets (sheaves on Cartesian spaces), the framework expands to haloed and super smooth sets by incorporating infinitesimal and super-geometric probes (dual numbers and Grassmann algebras) to capture synthetic differential geometry and fermionic fields. Finally, the authors generalize this to smooth ∞-groupoids by adding cellular simplices as probes, allowing the theory to describe higher gauge transformations and homotopy types within a cohesive ∞-topos.

This chapter explores the parallel between topological phases in crystalline materials and brane charges in supergravity, proposing that both are best described by specific generalized cohomology theories. It posits that while traditional stable cohomology theories (like K-theory or elliptic cohomology) offer valid approximations, the most precise microscopic description for systems like 5D/11D supergravity with probe L1/M5-branes is given by unstable non-abelian cohomology, specifically twisted relative Cohomotopy. This proper flux quantization reflects the non-linear Gauss laws of flux densities and, when the probe brane wraps a spacetime M-fiber, completely determines the topological quantum observables. The text further demonstrates that measuring these microscopic charges in a coarser stable cohomology theory is mathematically equivalent to that theory possessing a specific geometric orientation (complex or quaternionic). Finally, it notes that these charge theories must be twisted by the tangential structure of spacetime (e.g., via U(2) or Sp(2) actions), which naturally reduces them to equivariant cohomology near orbifold singularities. Finally, the authors provide a rigorous construction of the generator of KU 1 https://www.w3.org/1998/Math/MathML" display="inline"> (S 3) using the Atiyah-Singer exponential map and self-adjoint Fredholm operators, thereby completing and validating a prior proposal by Hořava and Witten regarding the K-theoretic description of D-brane charges.

This chapter establishes synthetic axiomatic foundations for a geometric homotopy theory of orbifolds by unifying cohesive geometric homotopy theory with global-equivariant homotopy theory. It systematically introduces cohesive, elastic, and solid ∞-toposes to formulate differential topology, differential geometry (via infinitesimal neighborhoods), and super-geometry through adjoint systems of modal operators. Building upon a smooth cohesive base, the text constructs a singular-cohesive ∞-topos defined as ∞-sheaves over a site of singularities, which consists of delooping groupoids of finite groups (a global orbit category). This framework establishes an adjoint triple of modalities — singular, smooth, and orbi-singular — to characterize generic orbi-singularities and elucidates the dichotomy between geometric and homotopy fixed points, thereby formalizing the proper equivariant structure inherent in singular cohesion.

This chapter explores the application of generalized cohomology theories to the description of topological phases and charges in quantum physics, extending the classical insight that magnetic flux is classified by ordinary cohomology. It defines these charges via homotopy theory as classes of classifying maps, arguing that while stable theories like K-theory are traditional candidates, the most fine-grained microscopic description—particularly for probe branes in supergravity—is provided by unstable non-abelian cohomology, specifically twisted relative Cohomotopy. The text demonstrates that mapping these microscopic charges to a stable cohomology theory acts as a coarse-graining “measurement” via cohomology operations. Furthermore, it establishes that the ability to measure these charges in a stable theory is equivalent to that theory possessing a complex or quaternionic orientation. Finally, the chapter generalizes these concepts to include charges twisted by the tangential structure of spacetime, such as U(2) or Sp(2) structures.

This chapter establishes a framework for topological stacks defined as the geometric homotopy theory of topological groupoids subject to Morita equivalences. Utilizing D-topological spaces as a convenient category for these constructions, the text details the structure of groupoids—specifically delooping groupoids and action groupoids (homotopy quotients) — to demonstrate that nonabelian cohomology is isomorphic to the connected components of the mapping stack from the given domain to a delooping groupoid. This approach subsumes equivariant cohomology via maps out of equivariant Čech groupoids and introduces slice mapping stacks to define twisted cohomology as maps into a fiber bundle over a base stack. The chapter concludes by recalling the definition of orbifolds as proper étale groupoids and applies the stacky machinery to provide a streamlined definition of twisted orbifold cohomology.

This chapter formulates the theory of orbifolds within singular-cohesive homotopy theory by combining two distinct geometric aspects: treating them as cohesive spaces with orbi-singularities and as spaces locally equivalent to a model space. The section on “Higher Orbispaces” reconciles topological definitions with global equivariant homotopy theory, establishing that the shape of an orbi-singularized topological groupoid is equivalent to an orbispace in the global equivariant sense. This section further analyzes cohesive orbispaces, relating their shape to the fixed-point loci of their universal covering spaces. The subsequent section introduces V-folds — generalizing manifolds as objects locally diffeomorphic to a group object V (such as Cartesian or super-Cartesian space) — and develops their internal geometry, including frame bundles, structural groups, and https://www.w3.org/1998/Math/MathML" display="inline"> G-structures. The text establishes that V-fold structures and their integrability are classified by Haefliger stacks. Finally, the chapter synthesizes these concepts to define Orbi-V-folds, which are argued to constitute the proper ∞-category of higher orbifolds.

This chapter establishes the foundations of higher geometry through the lens of geometric homotopy theory and ∞-topos theory, aiming to provide an internal formulation for group actions and fiber bundles. It begins by outlining the necessary category theory and topology, introducing D-topological spaces as a convenient category that relates to diffeological spaces via an idempotent adjunction. The text defines smooth ∞-groupoids as sheaves on a site of Cartesian spaces and reviews equivariant homotopy theory, including Elmendorf’s theorem which relates https://www.w3.org/1998/Math/MathML" display="inline"> G-spaces to systems of fixed-point spaces. Additionally, the chapter details universal constructions within ∞-toposes, such as limits, colimits, and n-truncation modalities. Finally, it presents an internal Galois theory where groups are characterized as loop space objects, group actions are defined via homotopy fiber sequences, and principal bundles are classified by maps into delooping stacks.

This chapter presents a general construction of orbifold étale cohomology within the framework of a singular-cohesive ∞-topos, designed to unify the tangentially twisted cohomology of smooth curved spaces with the proper equivariant cohomology of flat singular spaces. The text defines proper equivariant cohomology through orbi-singularization, demonstrating that it subsumes traditional theories such as Bredon cohomology and equivariant K-theory. A central result is the formulation of tangentially twisted proper orbifold cohomology, which is proven to reduce to proper equivariant cohomology on flat orbifolds and to tangentially twisted V-fold cohomology on smooth orbifolds. This framework is subsequently applied to define J-twisted proper orbifold Cohomotopy using Tate spheres and the Tate J-homomorphism, a theory that mathematically unifies the physical models required for Hypothesis H in M-theory by connecting twisted Cohomotopy outside singularities with RO-graded equivariant Cohomotopy at singularities.

Topological phases of quantum materials and brane charges in M-theory are measured by extraordinary cohomology theories defined on orbifold spacetimes. Geometric Orbifold Cohomology presents a modernized and enhanced formulation of these theories, establishing a rigorous framework for nonabelian and differential cohomology in the setting of higher geometry. Motivated by cutting-edge problems in mathematical physics—specifically the analysis of M-brane charges and topological insulators—this monograph bridges the gap between classical equivariant topology and modern cohesive homotopy theory. It fills a critical gap in the literature by offering a unified perspective where orbifold geometry is treated synthetically via modal operators in higher topos theory, allowing for a precise treatment of geometric singularities and differential structures. • Accessible Foundations: Begins with a streamlined reconstruction of twisted nonabelian orbifold cohomology and orbifold K-theory using the language of topological groupoids and stacks, offering a pedagogical entry point for new- comers. • Higher Topos Theory: Provides a detailed exposition of the transition from classical perspectives to the modern context of cohesive higher topos theory and global equivariant unstable homotopy theory. • Synthetic Orbifold Geometry: Lays out a powerful synthetic differential theory of higher orbifold Cartan geometry, utilizing systems of modal operators to rigorously capture orbi-singularities and proper equivariant homotopy types. • Unification of Methods: Unifies orbifold geometry with classical differential geometry and cohomology, compatibly enhancing the picture with modalities that reflect proper equivariant structures. • Concrete Applications: Showcases the theory’s utility through the analysis of tangentially twisted nonabelian orbifold cohomology, specifically J-twisted orbifold Cohomotopy, applied to "Hypothesis H" and the classification of topo- logical phases. This volume is designed to serve a dual purpose: the first part acts as an invitation for advanced graduate students and beyond, in mathematics and theoretical physics, leading them from basic topological charges to modern research applications. The subsequent parts provide a comprehensive resource for academic researchers in al- gebraic topology, differential geometry, and string theory interested in a cutting edge formulation of geometric cohomology and its applications to quantum systems.

This chapter extends the previous framework to equivariant homotopy theory by introducing orbit spaces as the necessary probes for detecting group actions and orbi-singularities. Leveraging Elmendorf’s theorem, the text characterizes equivariant ∞-groupoids as ∞-presheaves on an orbit category, effectively treating fixed-point loci as plots. This approach is further generalized to a global equivariant setting via a category of abstract singularities (a global orbit category), which consists of delooping groupoids of finite groups, leading to the definition of singular-cohesive ∞-topoi. The authors argue this specific formulation is required to define geometric orbifold cohomology and differential equivariant cohomology as intrinsic notions within the topos, rather than as extrinsic constructions.

It is an old idea that higher-dimensional super-gravity (SuGra) is put on-shell just by imposing Bianchi identities on super-field strengths over super-spacetimes subject to super-torsion constraints. We give a modernized, rigorous account and review recent developments, pointing out how this perspective lends itself to the construction of infrared completions of SuGra by electromagnetic flux quantization in differential nonabelian cohomology theories Adff lifting the coefficient L∞-algebra lA of the super-Bianchi identities. After surveying necessary background, we highlight: (1.) our recent proof that solutions of 11D SuGra are equivalent to the lS⁴-Bianchi identities on super $C$-field flux over super-torsion-free 11D super-spacetimes, (2.) how from this the nonlinearly self-dual gauge sector on 6D M5-brane worldvolumes is equivalent to the lS⁴S⁷-Bianchi identity on super $B$-field flux, (3.) and the recent complete discussion of superspace dimensional reduction of this situation to 10D IIA SuGra via lCyc(S⁴)-Bianchi identities on super NS/RR-flux, together with its extension to further reduction to 9 SuGra via lTor(S⁴)-Bianchis on the corresponding super-fluxes. We close by indicating how this implies consistent infrared completions of 11D SuGra by $C$-field flux quantization in 4-Cohomotopy, of 10D IIA SuGra by NS/RR-field flux-quantization in a form of twisted unstable K-theory, and of M5-brane worldvolumes by $B$-field flux-quantization in twisted (twistorial) relative Cohomotopy. The latter admits geometric engineering of experimentally relevant topological quantum orders.

In this study, we establish the topological adaptability and structural resilience of periodic soft matter entanglements using the DNA tensegrity triangle, a foundational motif in structural DNA nanotechnology, as a model system. Motivated by the Reidemeister moves, which consist of three local moves on knot or link diagrams, we systematically slide the strand termini relative to junction crossings to yield 47 unique biomolecular structures. With this method, we identify three major results: (1) a “poke crossover” between unlinked helices with inter- locking nicks that drastically increases lattice rigidity, alters lattice geometry, and provides a convenient metal cation binding site; (2) a new class of four-arm junction, the semi-junction, which is held together by a single rotatable bond and features globally relaxed torsional strain and novel crystal packing; and (3) a facile topological inversion between left- and right-handed tertiary structures enabled by the rotational freedom of the semi-junction, demonstrating how local flexibility at a single connection can reorganize global molecular chirality without disrupting the underlying architecture. Together, these findings reveal how controlled topological operations at the molecular level can tune local flexibility and stiffness at key sites to direct long-range lattice geometry, crystal packing, and emergent chiral organization. This work positions DNA-based architectures as an eminently programmable platform for the design of nanoscale devices with tunable structural and functional properties.

Random circuits giving rise to unitary designs are key tools in quantum information science and many-body physics. In this work, we investigate a class of random quantum circuits with a specific gate structure. Within this framework, we prove that one-dimensional structured random circuits with non-Haar-random local gates can exhibit substantially more global randomness compared to Haar-random circuits with the same underlying circuit architecture. In particular, under a solvability condition, we exactly diagonalize the second-moment operators for these structured random circuits by establishing a link to free-fermionic models, and we show that their spectral gaps can exceed those of Haar-random circuits. While solvable structured random circuits exhibit free-fermionic behavior, the underlying fixed gate need not be a matchgate. Our findings have applications in improving circuit depth bounds for randomized benchmarking, in generating approximate unitary 2-designs from shallow random circuits, and in enabling exact analyses of quantum many-body dynamics.

Abstract The electric Gauss law in 11D SuGra is famously non-linear, whence its flux quantization must be in nonabelian cohomology. We have previously shown that the minimal admissible choice is 4-Cohomotopy, which in the presence of magnetized M5-probes takes its relative twistorial form. Here we discuss how this situation is further refined in the presence of M-string probes on the M5-worldvolume. Based on the superspace formulation of 11D SuGra, we find the nested Bianchi identities by iterating the superembedding construction for super p-branes. The resulting probe brane hierarchy (M1 on magnetized M5 in 11D bulk) turns out to admit flux quantization in a doubly-relative form of twisted Cohomotopy, classified by the factorization of the quaternionic Hopf fibration through the twistor fibration. The further equivariant refinement of this cohomology theory reduces on A-type singularities to a form of relative Cohomotopy which geometrically engineers Chern-insulator phases on M 5 ∩ A n, with the M-string playing the role of gapped nodal lines.

We produce the dimensional reduction to 10D IIA supergravity (SuGra), via cyclification, of the remarkable result that full 11D SuGra is put on shell just by imposing the duality-symmetric Bianchi identities on C-field super-flux densities over supertorsion-free superspace. Generally, we highlight that when duality-symmetric superspace Bianchi identities are characterized by Whitehead bracket L∞-algebras lA of a classifying space A, their dimensional reduction is characterized by the cyclic loop space Cyc(A). We promote this to a general mechanism of dimensional reduction on super-spacetime, compatible with the global (infrared) completion of supergravity theories by flux quantization in non-abelian cohomology with coefficients in A and Cyc(A), respectively. In the case of 11D SuGra, the characteristic L∞-algebra is lS⁴ and hence we obtain that full on-shell 10D IIA SuGra is equivalent to lCyc(S⁴)-Bianchi imposed identities on NS/RR super-flux densities over supertorsion-free 10D super-spacetime. This implies that any space which is R-rationally equivalent to Cyc(S⁴) classifies an admissible flux quantization law, which provides a global completion of 10D IIA SuGra that admits oxidation to 11D.

After global completion of higher gauge fields (as appearing in higher-dimensional supergravity) by proper flux quantization in extraordinary nonabelian cohomology, the (non-perturbative, renormalized) topological quantum observables and quantum states of solitonic field histories are completely determined through a topological form of light-front quantization. We survey the logic of this construction and expand on aspects of the quantization argument. In the instructive example of 5D Maxwell-Chern-Simons theory (the gauge sector of 5D SuGra) dimensionally reduced to 3D, a suitable choice of flux quantization in Cohomotopy (“Hypothesis h”) recovers this way the fine detail of the traditionally renormalized (Wilson loop) quantum observables of abelian Chern-Simons theory and makes novel predictions about anyons in fractional quantum (anomalous) Hall systems. An analogous choice (“Hypothesis H”) of global completion of 11D higher Maxwell-Chern-Simons theory (the higher gauge sector of 11D SuGra) realizes various aspects of the topological sector of the conjectural “M-theory” and its M5-branes.

More profound than bulk topological order of quantum materials is only its unwinding via gapless excitations along boundaries of the sample. We recast this bulk-edge correspondence -- for the experimentally relevant case of fractional quantum Hall (FQH) systems -- in terms of effective relative higher gauge theory, controlled by choices of classifying fibrations. For FQH systems, we identify the complex Hopf fibration as classifying the bulk/boundary topological effects, and find that it yields a non-Lagrangian reconstruction of Floreanini-Jackiw/Wess-Zumino-Witten chiral edge currents. Remarkably, the resulting effective FQH higher gauge theory turns out to be "geometrically engineered" on M2/M5-branes probing A-type orbi-singularities in 11D supergravity, globally completed by flux-quantization in twisted equivariant differential (TED) Cohomotopy: Here the M-string ends of M2-branes on M5-branes engineer the FQH liquid's boundary. This geometric engineering on M-branes might naturally elucidate the curious combination of W∞-symmetry and of super-symmetry that is known to govern the collective excitations of FQH liquids at long wavelengths.

A question raised by Freedman & Hastings [16] still stands: To produce a mathematical theory that would unify quantum entanglement/tensor-structure with parameterized/bundle-structure via their amalgamation (a hypothetical pushout) along bare quantum (information) theory—a question motivated by the role that vector bundles of spaces of quantum states play in the K-theoretic classification of topological phases of matter. Here, we produce a possible answer to this question. To that end, first, we make precise a form of the relevant pushout diagram in monoidal category theory. With the question thus formalized, we proceed to compute this pushout and prove that it gives what is known as the external tensor product on vector bundles/K-classes, or rather on flat such bundles (flat K-theory), i.e., those equipped with monodromy encoding topological Berry phases. The external tensor product was recently highlighted in the context of topological phases of matter in [36] and through our work in quantum programming theory [49] but has not otherwise found due attention in quantum theory yet.

Parameterized stable homotopy theory organizes local systems of spectra over homotopy types, governed by a "yoga" of six functors. To provide semantics for the recently developed Linear Homotopy Type Theory (LHoTT), good model categories of these spectra are required, preferably monoidal with respect to the external smash product. We focus on the case of parameterized HK-module spectra (∞-local systems), motivated by recent applications of parameterized homotopy to topological quantum computing. While traditionally treated via dg-categories, we leverage combinatorial model structures on simplicial chain complexes to construct the first dedicated global model structure for K-linear ∞-local systems, which offers better control than existing models for general parameterized spectra. In particular, when restricted to base 1-types, our model structure is monoidal with respect to the external tensor product, making it a candidate target semantics for the multiplicative fragment of LHoTT.

We provide a systematic approach to twisting differential KO-theory leading to a construction of the corresponding twisted differential Atiyah-Hirzebruch spectral sequence (AHSS). We relate and contrast the degree two and the degree one twists, whose description involves appropriate local systems. Along the way, we provide a complete and explicit identification of the differentials at the E2 and E3 pages in the topological case, which has been missing in the literature and which is needed for the general case. The corresponding differentials in the refined theory reveal an intricate interplay between topological and geometric data, the former involving the flat part and the latter requiring the construction of the twisted differential Pontrjagin character. We illustrate with examples and applications from geometry, topology and physics. For instance, quantization conditions show how to lift differential $4k$-forms to twisted differential KO-theory leading to integrality results, while considerations of anomalies in type I string theory allow for characterization of twisted differential Spin structures.

Formalizations of quantum information theory in category theory and type theory, for the design of verifiable quantum programming languages, need to express its two fundamental characteristics: (1) parameterized linearity and (2) metricity, namely Hermiticity. The first is naturally addressed by dependent-linearly typed languages such as Proto- Quipper or, following our recent observations (Sati and Schreiber in Quantum Stud: Math Found 12:25, 2025; Sati and Schreiber in Quantum Stud: Math Found 2026): Linear Homotopy Type Theory (LHoTT). The second point has received substantial attention (only) in the form of semantics in “dagger-categories”, where operator adjoints are axiomatized, but their specification to Hermitian adjoints still needs to be imposed by hand. In this brief note, we describe a natural emergence of Hermiticity which is rooted in principles of equivariant homotopy theory, lends itself to homotopically-typed languages, and naturally connects to topological quantum states classified by twisted equivariant Real K-theory (with capital “R”: KR-theory). Namely, we observe that when the complex numbers are considered as a monoid internal to Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}Z2\end{document}-equivariant real linear types, via complex conjugation (the “Real numbers”, with capital “R”), then (finite-dimensional) Hilbert spaces do become self-dual objects among internally complex Real modules. This move absorbs the dagger-structure into the type structure; for instance, a complex linear map is unitary iff seen internally to Real modules it is orthogonal. The point is that this construction of Hermitian forms requires of the ambient linear type theory nothing further than a negative unit term of tensor unit type. We observe that just such a term is constructible in plain LHoTT, where it interprets as the non-trivial degree=0 element of the ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}∞\end{document}-group of units of the sphere spectrum, interestingly tying the foundations of quantum theory to homotopy theory. We close by indicating how this observation allows for encoding (and verifying) the unitarity of quantum gates and of quantum channels in quantum languages embedded into LHoTT, as described in Sati and Schreiber (Quantum Stud: Math Found 12:25, 2025).

Abstract Collective excitations of Fractional Quantum Hall (FQH) liquids at long wavelengths are thought to be of a generally covariant geometric nature, governed by area-preserving diffeomorphisms (SDiff). But current analyses rely solely on the corresponding perturbative w ∞ Lie algebra. We argue this is insufficient: We identify a non-perturbative construction of the effective Maxwell-Chern-Simons quantum field theory which carries unitary SDiff equivariance. But this turns out to be non-differentiable, suggesting underappreciated subtleties when the usual Hilbert space truncation is removed.

The Drinfeld center fusion category Z(VecG) famously models anyons in certain lattice models. Here we demonstrate how its fusion rules may also describe topological order in fractional topological insulator materials, in the vicinity of point defects in the Brillouin zone. Concretely, we prove that Z(VecG) reflects, locally over a punctured disk in the Brillouin zone, the monodromy (topological order) of gapped quantum states over the parameter space of Bloch Hamiltonians whose classifying space has fundamental group $G$.

The Drinfeld center fusion category Z(VecG) famously models anyons in certain lattice models. Here we demonstrate how its fusion rules may also describe topological order in fractional topological insulator materials, in the vicinity of point defects in the Brillouin zone. Concretely, we prove that Z(VecG) reflects, locally over a punctured disk in the Brillouin zone, the monodromy (topological order) of gapped quantum states over the parameter space of Bloch Hamiltonians whose classifying space has fundamental group $G$.

The idea that topologically protected quantum states, such as anyons, may be attached to super/semiconductor heterostructures has received enormous attention, but experimental signatures in 1D systems remain elusive. Here we revisit theoretical underpinnings of anyons in 2D fractional quantum Hall (FQH) systems, whose signatures have been experimentally observed by independent groups. Invoking novel theorems about the Hopfion or CP¹-model understood as flux quantization in 2-Cohomotopy, we demonstrate a robust prediction for possibly nonabelian anyonic states induced by superconducting islands.

We have previously observed that the theory of solutions of partial differential equations, regarded as diffieties inside jet bundles, acquires a powerful comonadic formulation after passage from the category of Fréchet smooth manifolds to the Cahiers topos of formal smooth sets (a well-adapted model for Synthetic Differential Geometry). However, the tacit assumption that this passage preserves the projective limits that define infinite jet bundles had remained unproven. Here we provide a detailed proof.

One-dimensional topological Anderson insulators provide a paradigm for disorder-induced topological phases in which the underlying system turns from a trivial to a topological phase. It is widely recognized that the latter vanishes at large disorder amplitude. Here, and contrary to the general belief, we provide evidence for a successive disorder-driven topological transitions in a single-wall nanotube, culminating in a topological Anderson phase that remains unexpectedly robust at strong disorder. This phenomenon is confirmed by analysis of the corresponding topological invariant, which increases stepwise as disorder increases, giving evidence for the emergence of edge states. We experimentally implement these topological Anderson staircase phase transitions in a one-dimensional topolectrical circuit, where the persistence of edge states is revealed by node-voltage measurements. The robustness of the edge states is corroborated by numerical calculations of their localization properties. Our work opens the road to topological disordertronics, where topological phases can be tuned by disorder.

Recently observed fractional quantum anomalous Hall (FQAH) materials are candidates for topological quantum hardware, but their required anyon states are elusive. We point out dependence on monodromy of the fragile band topology in 2-cohomotopy. An algebro-topological theorem of L. Larmore and E. Thomas, [Math. Scand. 47, 232 (1980)], identifies FQAH anyons over momentum space. Admissible braiding phases involve 2C-th roots of unity, with C the Chern number. This lays the foundation for understanding symmetry-protected topological order in FQAH systems, reducing the problem to computations in equivariant cohomotopy.

We highlight that integer Heisenberg groups at level 2 underlie topological quantum phenomena: their group algebras coincide with the algebras of quantum observables of abelian anyons in fractional quantum Hall (FQH) systems on closed surfaces. Decades ago, these groups were shown to arise as the fundamental groups of the space of maps from the surface to the 2-sphere -- which has recently been understood as reflecting an effective FQH flux quantization in 2-Cohomotopy. Here we streamline and generalize this theorem using the homotopy theory of H-groups, showing that for k ∈ 1,2,4, the non-torsion part of π1 Map((S2k-1)², S2k) is an integer Heisenberg group of level 2, where we identify this level with 2 divided by the Hopf invariant of the generator of π4k−1(S2k). This result implies the existence of higher-dimensional analogs of FQH anyons in the cohomotopical completion of 11D supergravity ("Hypothesis H").

Topological superconductivity could host quasi-particles that are key candidates for fault-tolerant quantum computation due to their immunity to noise as they obey non-Abelian exchange statistics. For example, in the case of Majorana zero modes (MZM), braiding enables two topologically protected quantum gates. While their direct manipulation in solid-state systems remains experimentally challenging, digital emulation of MZM behavior has provided insight as well as a deeper understanding of controlling these topological quantum systems. This emulation is typically accomplished by mapping the topological and trivial phases of a Majorana system to ferromagnetic and paramagnetic Hamiltonians of a spin-glass model. This approach usually relies on adiabatic evolution of superconducting Hamiltonians, which require circuits with very large depths. In this work, we present a resource-efficient method to emulate MZM braiding in a trijunction geometry using a quantum processor. We introduce direct braiding operators which simulate the evolution more efficiently, reducing the quantum gate overhead. We then further generalize this method to emulate braiding operations in extended trijunction architectures based on Kitaev chains.

We apply methods of equivariant homotopy theory, which may not previously have found due attention in condensed matter physics, to classify first the fragile/unstable topological phases of 2D crystalline Chern insulator materials, and second the possible topological order of their fractional cousins. We highlight that the phases are given by the equivariant 2-Cohomotopy of the Brillouin torus of crystal momenta (with respect to wallpaper point group actions) -- which, despite the attention devoted to crystalline Chern insulators, seems not to have been considered before. Arguing then that any topological order must be reflected in the adiabatic monodromy of gapped quantum ground states over the covariantized space of these band topologies, we compute the latter in examples where this group is non-abelian, showing that any potential FQAH anyons must be localized in momentum space. We close with an outlook on the relevance for the search for topological quantum computing hardware. Mathematical details are spelled out in a supplement.

Investigating the interplay of dualities, generalized symmetries, and topological defects beyond theoretical models is an important challenge in condensed matter physics and quantum materials. A simple model exhibiting this physics is the transverse-field Ising model, which can host a topological defect that performs the Kramers-Wannier duality transformation. When acting on one point in space, this duality defect imposes the duality twisted boundary condition and binds a single zero mode. This zero mode is unusual as it lacks a localized partner in the same Z 2 sector and has an infinite lifetime, even in finite systems. Using Floquet driving of a closed Ising chain with a duality defect, we generate this zero mode in a digital quantum computer. We detect the mode by measuring its associated persistent autocorrelation function using an efficient sampling protocol and a compound strategy for error mitigation. We also show that the zero mode resides at the domain wall between two regions related by a Kramers-Wannier duality transformation. Finally, we highlight the robustness of the isolated zero mode to integrability- and symmetry-breaking perturbations. Our findings provide a method for exploring exotic topological defects, associated with noninvertible generalized symmetries, in digitized quantum devices.

Following arguments that the (hidden) M-algebra serves as the maximal super-exceptional tangent space for 11D supergravity, we make explicit here its integration to a (super-Lie) group. This is equipped with a left-invariant extension of the''decomposed''M-theory 3-form, such that it constitutes the Kleinian space on which super-exceptional spacetimes are to be locally modeled as Cartan geometries. As a simple but consequential application, we highlight how to describe lattice subgroups Zk ≤ 528 of the hidden M-group that allow to toroidially compactify also the''hidden''dimensions of a super-exceptional spacetime, akin to the familiar situation in topological T-duality. In order to deal with subtleties in these constructions, we (i) provide a computer-checked re-derivation of the''decompose''M-theory 3-form, and (ii) present a streamlined conception of super-Lie groups, that is both rigorous while still close to physics intuition and practice. Thereby this article highlights modernized super-Lie theory along the example of the hidden M-algebra, with an eye towards laying foundations for super-exceptional geometry. Among new observations is the dimensional reduction of the hidden M-algebra to a''hidden IIA-algebra''which in a companion article we explain as the exceptional extension of the T-duality doubled super-spacetime.

This is the second in a series of papers that aim to develop rigorous and most encompassing foundations for field theory, where in the first installment we laid out the natural formulation of bosonic variational field theory via the ``functorial geometry'' of smooth sets, namely in the topos over the site of spaces with smooth maps between them. Here, we extend this to the category ThickenedSmoothSets of {\it infinitesimally thickened smooth sets}. We first describe the Cahiers topos in a simplified, but still fully rigorous, R-algebraic setting -- which should serve as a more accessible introduction to the theory of Synthetic Differential Geometry to both physicists and mathematicians. Then, we formulate local Lagrangian field theory in this setting in which infinitesimal spaces exist and interact correctly with the field-theoretic spaces of infinite jet bundles, off-shell and on-shell spaces of fields etc. This discussion subsumes previous constructions and further recovers all the relevant tangent bundles of traditional (off-shell and on-shell) field theory considerations via the synthetic tangent bundle construction, i.e., as ``infinitesimal curves'' in those spaces, which were previously defined only in an indirect manner. Beyond finally putting such aspects of the theory on a firm foundation, this approach recognizes the variational principle of local Lagrangian field theory, equivalently, as an intersection of thickened smooth sets. Lastly, as we will show in detail, it permits a mathematical formalization (and recovery) of perturbative field theory as the actual restriction to the infinitesimal neighborhood around a field configuration -- a statement which thus far had remained only in the realm of intuition. Crucially, all of these results are obtained with no use of infinite dimensional manifold theory, topology or functional analysis of mapping spaces.

In the practice of physics model building, the process of renormalization, resummation, and anomaly cancellation is to incrementally repair initially ill-defined Lagrangian quantum field theories by a successive choice of partial fixes. Impressive as this is, one would rather have concisely defined complete theories to begin with, and understand these choices as emergent from fundamental principles. As an instructive example, we recall renormalization choices for Wilson loop observables in abelian Chern–Simons theory. Then we show that these emerge in a novel non-Lagrangian topological completion of 5D Maxwell–Chern–Simons QFT, by means of proper flux quantization in 2-Cohomotopy. This result is a modest cousin, with applications to topologically ordered quantum materials, of the more ambitious completion of 11D supergravity by flux quantization in 4-Cohomotopy (“Hypothesis H”).

Topological Chern phases of quantum materials, as well as brane charges on M-theory orbifolds, have famously been argued to be classified by (orbi) topological K-theory, or possibly by other stable and, notably, complex-oriented cohomology theories, such as elliptic cohomology or Morava K-theory. However, closer inspection reveals that the most fine-grained "fragile" microscopic classification in both cases is in (orbi) Cohomotopy, which is the primordial "unstable" or nonabelian generalized cohomology. Coarsening takes the latter (fragile) to the former (stable) cohomology along an nonabelian cohomology operations. But what then is the role of complex orientation on the stable side? We observe here (i) that over gapped nodal lines in the 2D Brillouin torus and on probe M5-branes in 11D spacetime, the cohomotopical phases/charges lift through the complex/quaternionic Hopf fibration, and (ii) that measuring this fragile situation in stable cohomology means equivalently to ask for universal complex/quaternionic orientation on stable cohomology in four/ten dimensions! Then we give an explicit realization of such unstable four/ten-dimensional complex/quaternionic orientation in U(2)/Sp(2)-equivariant K-theory, using real division-algebraic tools within a new model of twisted orbifold K-theory in cohesive homotopy theory; and we explain this as an extraordinary character map from orbi Cohomotopy-twisted Cohomotopy to relative orbi K-theory. Finally, we discuss an application to the classification of 2-band crystalline topological insulator phases sensitive to the topology in the gapping process of their nodal line semimetal parent phase, and to the measurement of M-string charges inside M5-brane probes.

These extended lecture notes survey a novel derivation of anyonic topological order (as seen in fractional quantum Hall systems) on single magnetized M5-branes probing Seifert orbi-singularities (“geometric engineering” of anyons), which we motivate from fundamental open problems in the field of quantum computing. The rigorous construction is non-Lagrangian and non-perturbative, based on previously neglected global completion of the M5-brane’s tensor field by flux-quantization consistent with its non-linear self-duality and its twisting by the bulk C-field. This exists only in little-studied non-abelian generalized cohomology theories, notably in a twisted equivariant (and “twistorial”) form of unstable Cohomotopy (“Hypothesis H”). As a result, topological quantum observables form Pontrjagin homology algebras of mapping spaces from the orbi-fixed worldvolume into a classifying 2-sphere. Remarkably, results from algebraic topology imply from this the quantum observables and modular functor of abelian Chern-Simons theory, as well as braid group actions on defect anyons of the kind envisioned as hardware for topologically protected quantum gates.

We first give a brief exposition of our recent realization of anyonic quantum states on single M5-brane probes in 11D super-gravity backgrounds, by non-perturbative quantization of the topological sector of the self-dual tensor field on the 6Dworldvolume, after its proper flux-quantization. This opens the prospect of holographic models for topological qbits away from the usual but unrealistic limit of large numbers of branes. At the same time, the elementary homotopy-theoretic nature of the construction yields a slick expression of topological quantum gates in homotopically-typed programming languages, opening the prospect of topological-hardware aware quantum programming. In view of these results, we end with some more meta-physical remarks on (cohesive) homotopy (type) theory in view of emergent fundamental physics and, possibly, M-theory.

We propose a novel floating-point encoding scheme that builds on prior work involving fixed-point encodings. We encode floating-point numbers using Two's Complement fixed-point mantissas and Two's Complement integral exponents. We used our proposed approach to develop quantum algorithms for fundamental arithmetic operations, such as bit-shifting, reciprocation, multiplication, and addition. We prototyped and investigated the performance of the floating-point encoding scheme on quantum computer simulations by performing reciprocation on randomly drawn inputs and by solving first-order ordinary differential equations, while varying the number of qubits in the encoding. We observed rapid convergence to the exact solutions as we increased the number of qubits and a significant reduction in the number of ancilla qubits required for reciprocation when compared with similar approaches.

The jellium model is a paradigmatic problem in condensed matter physics, exhibiting a phase transition between metallic and Wigner crystal phases. However, its vanishing Berry curvature makes it ill suited for studying recent experimental platforms that combine strong interactions with nontrivial quantum geometry. These experiments inspired the anomalous Hall crystal (AHC)-a topological variant of the Wigner crystal. The AHC spontaneously breaks continuous translation symmetry but has a nonzero Chern number. In this Letter, we introduce λ-jellium, a minimal extension of the two-dimensional jellium model. Its Berry curvature distribution is controlled by a single parameter λ, where λ=0 corresponds to the standard jellium model. This setup facilitates the systematic exploration of Berry curvature's impact on electron crystallization. The phase diagram of this model, established using self-consistent Hartree Fock calculations, reveals several interesting features: (i) The AHC phase occupies a large region of the phase diagram. (ii) Two distinct Wigner crystal phases, the latter enabled by quantum geometry, and two distinct Fermi liquid phases are present. (iii) A continuous phase transition separates the AHC and one of the Wigner crystal phases. (iv) In some parts of the AHC phase, the lattice geometry is nontriangular, unlike in the classical Wigner crystal. In addition to elucidating the physics of correlated electrons with nonzero Berry curvature, we expect that the simplicity of the model makes it an excellent starting point for more advanced numerical methods.

In previous work (Sati and Voronov in Commun Math Phys 400:1915–1960, 2023. https://doi.org/10.1007/s00220-023-04643-7, in Adv Theor Math Phys 28(8):2491–2601, 2024. https://doi.org/10.4310/atmp.241119034750), we introduced Mysterious Triality, extending the Mysterious Duality (Iqbal et al. in Adv Theor Math Phys 5:769–808, 2002. https://doi.org/10.4310/ATMP.2001.v5.n4.a5) between physics and algebraic geometry to include algebraic topology in the form of rational homotopy theory. Starting with the rational Sullivan minimal model of the 4-sphere S4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}S⁴\end{document}, capturing the dynamics of M-theory via Hypothesis H, this progresses to the dimensional reduction of M-theory on torus Tk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}Tk\end{document}, k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}k ≥ 1\end{document}, with its dynamics described via the iterated cyclic loop space LckS4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}Lck S⁴\end{document} of the 4-sphere. From this, we also extracted data corresponding to the maximal torus/Cartan subalgebra and the Weyl group of the exceptional Lie group/algebra of type Ek\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}Ek\end{document}. In this paper, we discover much richer symmetry by extending the action of the Cartan subalgebra by symmetries of the equations of motion of (11-k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(11-k)$$\end{document}d supergravity to a maximal parabolic subalgebra pkk(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}pkk(k)\end{document} of the Lie algebra ek(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}ek(k)\end{document} of the U-duality group. We do this by constructing the action on the rational homotopy model of the slightly more symmetric than LckS4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}Lck S⁴\end{document} toroidification TkS4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}Tk S⁴\end{document}, which is another bookkeeping device for the equations of motion. To justify these results, we identify the minimal model of the toroidification TkS4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}Tk S⁴\end{document}, generalizing the results of Vigué-Poirrier, Sullivan, and Burghelea, and establish an algebraic toroidification/totalization adjunction.

Demonstrating superconductivity in purely repulsive Hubbard models is a compelling goal which underscores the counter-intuitive ability of Coulomb interactions to mediate superconductivity. Here, we present numerical evidence for robust superconductivity in a triangular Hubbard-Hofstadter model at $π/2$ flux per plaquette. Employing infinite density matrix renormalization group calculations on infinite cylinders of finite circumference, we observe superconducting ground states for a wide range of dopings, whose pair-correlations strengthen as the 2D limit is approached. At a density of one electron per site, Hubbard interactions have been reported to drive the insulating parent state of the superconductor from an integer quantum Hall (IQH) state to a chiral spin liquid (CSL). Our findings give credence to a recent proposal that proximity to the IQH-CSL transition serves to make chiral superconductivity energetically favorable on doping, and also correctly predicts the nature of the edge modes in the superconductor. On the CSL side, this suggests the superconductor can be thought of as arising from Laughlin's `anyon superconductivity' mechanism. Thus the Hubbard-Hofstadter model studied here offers a clean and experimentally accessible setup, potentially realizable in moiré heterostructures, for exploring the properties of anyonic matter at finite density and the interplay of topological order, quantum criticality and superconductivity.

Abstract Representations are a foundational component of any modeling protocol, including on molecules and molecular solids. For tasks that depend on knowledge of both molecular conformation and 3D orientation, such as the modeling of molecular dimers, clusters, or condensed phases, we desire a rotatable representation that is provably complete in the types and positions of atomic nuclei and roto-inversion equivariant with respect to the input point cloud. In this paper, we develop, train, and evaluate a new type of autoencoder, molecular O(3) encoding net (Mo3ENet), for multi-type point clouds, for which we propose a new reconstruction loss, capitalizing on a Gaussian mixture representation of the input and output point clouds. Mo3ENet is end-to-end equivariant, meaning the learned representation can be manipulated on O(3), a practical bonus. An appropriately trained Mo3ENet latent space comprises a universal embedding for scalar, vector, and tensorial molecule property prediction tasks, as well as other downstream tasks incorporating the 3D molecular pose, and we demonstrate its fitness on several such tasks.

We provide a brief invitation to the novel understanding of anyonic topological order in fractional quantum (anomalous) Hall systems, via "extraordinary" quantization of effective magnetic flux in Cohomotopy -- following our presentation at ISQS29.

Over a decade before the modern formulation of AdS/CFT duality, Duff et al. had observed a candidate microscopic explanation by identifying the CFT fields with fluctuations of probe p-branes stretched out in parallel near the horizon of their own black brane incarnation. A profound way to characterize these and more general probe p-brane configurations, especially for M5-branes, is expected to be as “super-embeddings” of their super-worldvolumes into target super-spacetime — but no concrete example of these had appeared in the literature. Here we fill this gap by constructing the explicit holographic [Formula: see text]-BPS super-embeddings of probe M5-branes and M2-branes into their corresponding super-AdS backgrounds.

We argue that the combination of strong repulsive interactions and high magnetic fields can generate electron pairing and superconductivity. Inspired by the large lattice constants of moiré materials, which make large flux per unit cell accessible at laboratory fields, we study the triangular lattice Hofstadter-Hubbard model at one-quarter flux quantum per plaquette, where previous literature has argued that a chiral spin liquid separates a weak-coupling integer quantum Hall phase and a strong-coupling topologically trivial antiferromagnetic insulator at a density of one electron per site. We argue that topological superconductivity emerges upon doping in the vicinity of the integer quantum Hall to chiral spin liquid transition. We employ exact diagonalization and density matrix renormalization group methods to examine this theoretical scenario and find that electronic pairing indeed occurs on both sides of criticality over a remarkably broad range of interaction strengths. On the chiral spin liquid side, our results provide a concrete model realization of the long-hypothesized mechanism of anyon superconductivity. Our study thus establishes a beyond-Bardeen-Cooper-Schrieffer route to electron pairing in a well-controlled limit, relying crucially on the interplay between electron correlations and band topology.

Physical quantities with long lifetimes have both theoretical significance in the study of quantum many-body systems and practical implications for quantum technologies. In this manuscript, we investigate the roles played by topological defects in the construction of quasiconserved quantities, using as a prototypical example the Kramers-Wannier duality defect in a deformed one-dimensional quantum transverse-field Ising model. We construct the duality defect Hamiltonian in three different ways: half-chain Kramers-Wannier transformation, utilization of techniques in the Ising fusion category, and defect-modified weak integrability breaking deformation. The third method is also applicable for the study of generic integrable defects under weak integrability breaking deformations. We also work out the deformation of defect-modified higher charges in the model and study their slower decay behavior. Furthermore, we consider the corresponding duality defect twisted deformed Floquet transverse-field Ising model and investigate the stability of the isolated zero mode associated with the duality defect in the integrable Floquet Ising model, under such weak integrability breaking deformation.

The modern theory of functional programming languages uses monads for encoding computational side-effects and side-contexts, beyond bare-bone program logic. Even though quantum computing is intrinsically side-effectful (as in quantum measurement) and context-dependent (as on mixed ancillary states), little of this monadic paradigm has previously been brought to bear on quantum programming languages. Here, we systematically analyze the (co)monads on categories of parameterized module spectra which are induced by Grothendieck’s “motivic yoga of operations”—for the present purpose specialized to HC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}HC\end{document}-modules and further to set-indexed complex vector spaces, as discussed in a companion article (Sati and Schreiber in Entanglement of sections. arXiv:2309.07245). Interpreting an indexed vector space as a collection of alternative possible quantum state spaces parameterized by quantum measurement results, as familiar from Proto- -semantics, we find that these (co)monads provide a comprehensive natural language for functional quantum programming with classical control and with “dynamic lifting” of quantum measurement results back into classical contexts. We close by indicating a domain-specific quantum programming language () expressing these monadic quantum effects in transparent do-notation, embeddable into the recently constructed Linear Homotopy Type Theory () which interprets into parameterized module spectra. Once embedded into, this should make for formally verifiable universal quantum programming with linear quantum types, classical control, dynamic lifting, and notably also with topological effects (as discussed in the companion article Myers et al. in Commun Math Phys 405:172, 2024).

Planar Josephson junctions (JJs) have emerged as a promising platform for the realization of topological superconductivity and Majorana zero modes. To obtain robust quasi one-dimensional (1D) topological superconducting states using planar JJs, limiting the number of 1D Andreev bound states'subbands that can be present, and increasing the size of the topological superconducting gap, are two fundamental challenges. It has been suggested that both problems can be addressed by properly designing the interfaces between the JJ's normal region and the superconducting leads. We fabricated Josephson junctions with periodic hole structures on the superconducting contact leads on InAs heterostructures with epitaxial superconducting Al. By depleting the chemical potential inside the hole region with a top gate, we observed an enhancement of the supercurrent across the junction. Such an enhancement is reproduced in theoretical simulations. The theoretical analysis shows that the enhancement of the JJ's critical current is achieved when the hole depletion is such to optimize the matching of quasiparticles'wave-function at the normal/superconductor interface. These results show how the combination of carefully designed patterns for the Al coverage, and external gates, can be successfully used to tune the density and wave functions'profiles in the normal region of the JJ, and therefore open a new avenue to tune some of the critical properties, such as number of subbands and size of the topological gap, that must be optimized to obtain robust quasi 1D superconducting states supporting Majorana bound states.

Chiral magnetic textures are of fundamental interest for studying exotic spin behavior, but also promising for the development of energy-efficient memory and logic devices. Recent computational work has suggested that in contrast to well-studied static chiral textures like skyrmions, there exists dynamic fluctuation-driven magnetic chirality even in the absence of long-range magnetic order. Here, we present definitive magnetotransport signatures of fluctuation-driven chirality in a model material system of Pt/Li_{0.5}Al_{1.0}Fe_{1.5}O_{4} (LAFO) bilayers, where LAFO is a ferromagnetic insulator. We establish a fluctuation-driven origin for the magnetic chirality by observing a strong correlation between the onset of the topological Hall effect and the destruction of magnetic order. Corroborated by Monte Carlo simulations, we develop a rigorous analytical framework to gain a first-principles understanding of scaling behaviors in the transport data. Our results bring novel insights to fluctuation-driven chirality and analysis of magnetotransport data.

The discovery of fractional quantum anomalous Hall states in moiré systems has raised the possibility of realizing phases of itinerant anyons, whose mobility requires the absence of continuous magnetic translation symmetry (CMTS). Motivated by this, we consider anyons on the sphere in the presence of a nonuniform magnetic field that breaks the SU(2) rotation symmetry, the analog of CMTS on the sphere, down to a U(1). This allows us to study the energy dispersion of the anyons as a function of the angular momentum L_{z}, while maintaining the perfect flatness of the single-particle dispersion. We parametrize the nonuniform field by a real parameter R that concentrates the field at the north (south) pole for R>1 (R<1). For this choice of field, any p-body correlation function evaluated in the space of Laughlin quasiholes maps exactly onto the corresponding p-body correlation function in uniform field. In the thermodynamic limit, this exact mapping lets us analytically compute the interaction-generated, spatially varying potential felt by the anyons. Remarkably, this static potential alone is sufficient to generate anyon dispersion: because each anyon experiences an emergent background magnetic field, the two components of its position do not commute, so the potential is converted into a finite dispersion, which we compute exactly up to an overall scaling constant. The resulting anyon dispersion describes the azimuthal motion around the sphere at constant height, the analog of spin precession. Our Letter thus provides an exactly solvable realization of a general mechanism for interaction-induced anyon dispersion-a symmetry-breaking potential converted into kinetic energy by the anyon's many-body Berry phase-with no single-particle dispersion or hopping of any kind.

Super [Formula: see text]-algebras unify extended super-symmetry with rational classifying spaces for higher flux densities: The super-invariant super-fluxes which control super p-branes and their supergravity target super-spaces are, together with their (nonlinear) Bianchi identities, neatly encoded in (non-abelian) super-[Formula: see text] cocycles. These are the rational shadows of flux-quantization laws (in ordinary cohomology, K-theory, Cohomotopy, iterated K-theory, etc.). We first review, in streamlined form while filling some previous gaps, double-dimensional reduction/oxidation and 10D superspace T-duality along higher-dimensional super-tori. We do so tangent super-space wise, by viewing it as an instance of adjunctions (dualities) between super-[Formula: see text]-extensions and-cyclifications, applied to the avatar super-flux densities of 10D supergravity. In particular, this yields a derivation, at the rational level, of the traditional laws of “topological T-duality” from the super-[Formula: see text] structure of type II superspace. At this level, we also discuss a higher categorical analog of T-duality involving M-branes. Then, by considering super-space T-duality along all 1+9 spacetime dimensions while retaining the 11th dimension as in F-theory, we find the M-algebra appearing as the complete brane-charge extension of the fully T-doubled/correspondence super-spacetime. On this backdrop, we recognize the “decomposed” M-theory 3-form on the “hidden M-algebra” as an M-theoretic lift of the Poincaré super 2-form that controls superspace T-duality as the integral kernel of the super Fourier–Mukai transform. This provides the super-space structure of an M-theory lift of the doubled/correspondence space geometry, which controls T-duality.

Interacting fermionic chains exhibit extended regions of topological degeneracy of their ground states as a result of the presence of Majorana or parafermionic zero modes localized at the edges. In the opposite limit of infinite temperature, the corresponding nonintegrable spin chains, obtained via generalized Jordan-Wigner mapping, are known to host so-called almost strong zero modes, which are long-lived with respect to any bulk excitations. Here we study the fairly unexplored territory that bridges these two extreme cases of zero and infinite temperature. We blend two established techniques for states, the Lanczos series expansion and a tensor network ansatz, uplifting them to the level of operator algebra. This allows us to efficiently simulate large system sizes for arbitrarily long timescales and to extract the temperature-dependent decay rates. We observe that for the Kitaev-Hubbard model, the decay rate of the edge mode depends exponentially on the inverse temperature β, and on an effective energy scale Δ eff that is greater than the thermodynamic gap of the system Δ.

In this study, we investigate the topological adaptability and structural resilience of periodic soft matter entanglements using the DNA tensegrity triangle, a foundational motif in structural DNA nanotechnology, as a model system. By simulating the Reidemeister moves from knot theory, which describe a series of “moves” by which the knot equivalence is preserved, we demonstrate that many variants of the tensegrity triangle maintain their lattice geometry, underscoring the motif’s inherent topological robustness. Using granular deformations in a series of closely related motifs, we systematically twist the helices and slide their ends relative to junction crossings to yield 48 distinct crystal structures. Notably, we Identify a novel poke-DX feature (PDX), which introduces rigid crossover configurations with enhanced crystallographic resolution and site-specific metal ion coordination. Further exploration reveals the formation of semi-junctions – a new class of four-arm junctions held together by a single rotatable bond, which feature relaxed torsional strain and altered crossover geometries. These configurations support lattice transformations into tetragonal and distorted rhombohedral forms as well as facilitate topological inversion between left- and right- handed triangles. Altogether, these findings illustrate how controlled topological operations at the molecular level can tune local flexibility and stiffness at key sites to affect long-range lattice geometry. This work positions DNA-based frameworks as a programmable platform for the design of architected materials, topological metamaterials, and nanoscale devices with tunable structural and functional properties.

Topology has emerged as a field for describing and controlling order and matter, and thereby the physical properties of materials. There are several largely disparate fields focused on examining and manipulating topology. One of these arenas is in the realm of real space, manipulating systems in terms of their spatial properties, to control the corresponding structural, mechanical, and self- assembling responses. Much of the work in soft matter topology falls within this domain. A second arena is in the domain of momentum or k-space wherein topology controls the features of the electronic band structure of materials, and topologically non-trivial features result in the development of materials with truly unique properties. This work focuses squarely on the realm of condensed matter physics. Here, we review concepts of real- and k-space topology and propose areas for convergence between these two disparate fields.

DNA nanotechnology leverages the molecular design resolution of the DNA double helix to fold and tile matter into designer architectures. Recent advances in bioinorganic chemistry have exploited the affinity of soft nucleobase functional groups for silver ions in order to template the growth of silver nanoclusters by templated reduction. The coupling of the spatial resolution of DNA nanotechnology and the atomic precision of DNA- based nanocluster synthesis has not been realized. Here we develop a method using 3D DNA crystals to employ silver-ion-mediated base pairs as nucleation sites for atomically- precise nanocluster growth. By leveraging the topology of DNA tensegrity triangles, we provide a mesoporous 3D lattice that is robust to reducing conditions, enabling precise spatial templating. Use of in situ confocal fluorescence microscopy allows for the direct observation of reaction kinetics and reconstruction of the optical bandgap. Control over reaction time and stoichiometry, base pair identity, and buffer composition enable precise tuning of the atomic composition and optical properties of the ensuing nanoclusters. The resulting crystals are of diffraction quality, yielding molecular structures of Ag4 and Ag6 in 3D. Inter-cluster distances of less than 2 nm show strong plasmonic coupling, with red shifting observed relative to literature standards. A thorough computational investigation establishes a theoretical basis for our observed behavior, and establishes the interplay between cluster size, charge, geometry, and resonance. We anticipate that these results will yield advances in materials synthesis, DNA-based plasmonic crystals, and optically-active nanoelectronics

A (2+1)D topological ordered phase with U(1) symmetry may or may not have a symmetric gapped edge state, even if both thermal and electric Hall conductivity are vanishing. It is recently discovered that there are "higher" versions of Hall conductivity valid for fermionic fractional quantum Hall (FQH) states, which obstructs symmetry-preserving gapped edge state beyond thermal and electric Hall conductivity. In this paper, we show that one can extract higher Hall conductivity from a single wave function of an FQH state, by evaluating the expectation value of the "partial rotation" unitary which is a combination of partial spatial rotation and a U(1) phase rotation. This result is verified numerically with the fermionic Laughlin state with $ν=1/3$, $1/5$, as well as the non-Abelian Moore-Read state. Together with topological entanglement entropy, we prove that the expectation values of the partial rotation completely determines if a bosonic/fermionic Abelian topological order with U(1) symmetry has a symmetry-preserving gappable edge state or not. We also show that thermal and electric Hall conductivity of Abelian topological order can be extracted by partial rotations. Even in non-Abelian FQH states, partial rotation provides the Lieb-Schultz-Mattis type theorem constraining the low-energy spectrum of the bulk-boundary system. The generalization of higher Hall conductivity to the case with Lie group symmetry is also presented.

Fractional quantum Hall systems (FQH), due to their experimentally observed anyonic topological order, are a main contender for future hardware-implementation of error-protected quantum registers ("topological qbits") subject to error-protected quantum operations ("topological quantum gates"), both plausibly necessary for future quantum computing at useful scale, but both remaining insufficiently understood. Here we present a novel non-Lagrangian effective description of FQH anyons, based on previously elusive proper global quantization of effective topological flux in extraordinary non-abelian cohomology theories. This directly translates the system's quantum-observables, -states, -symmetries, and-measurement channels into purely algebro-topological analysis of local systems of Hilbert spaces over the quantized flux moduli spaces. Under the hypothesis -- for which we provide a fair bit of evidence -- that the appropriate effective flux quantization of FQH systems is in 2-Cohomotopy theory (a cousin of Hypothesis H in high-energy physics), the results here are rigorously derived and as such might usefully inform laboratory searches for novel anyonic phenomena in FQH systems and hence for topological quantum hardware.

Symmetry-protected topological (SPT) phases in insulators and superconductors are known for their robust edge modes, linked to bulk invariants through the bulk-boundary correspondence. While this principle traditionally applies to gapped phases, recent advances have extended it to gapless systems, where topological edge-states persist even in the absence of a bulk gap. We extend this framework to periodically driven chains with chiral symmetry, revealing the existence of topological edge zero and π modes despite the lack of bulk gaps in the quasienergy spectrum. By examining the half-period decomposition of chiral evolutions, we construct topological invariants that circumvent the need to define the Floquet Hamiltonian, making them more suitable for generalization to the gapless regime. We provide explicit examples, including generalizations of the Kitaev chain and related spin models, where localized π modes emerge even when the bulk is gapless at the same quasienergy as the edge modes. We numerically study the effect of interactions, which give a finite lifetime to the edge modes in the thermodynamic limit with the decay rate consistent with Fermi's golden rule.

We observe that there is a rigorous derivation of (abelian) anyonic quantum states, hence of “topological order”, on the 1+2-dimensional fixed locus of M5-probes wrapped over a trivially Seifert-fibered 3-orbifold singularity. Similar statements have previously been conjectured by appeal to the unknown dynamics of “coincident” M5-branes, but neglecting effects of flux-quantization that, as we highlight, entail anyonic solitons already in the rigorously tractable case of single M5-brane probes. This is possible after globally completing the “self-dual” tensor field on probe M5-branes by flux-quantization in the non-abelian cohomology theory called equivariant twistorial Cohomotopy, which is admissible by recent results.

Recursion methods such as Krylov techniques map complex dynamics to an effective noninteracting problem in one dimension. For example, the operator Krylov space for Floquet dynamics can be mapped to the dynamics of an edge operator of the one-dimensional Floquet inhomogeneous transverse field Ising model (ITFIM), where the latter, after a Jordan-Wigner transformation, is a Floquet model of noninteracting Majorana fermions and the couplings correspond to Krylov angles. We present an application of this showing that a moment method exists where given an autocorrelation function, one can construct the corresponding Krylov angles and from that the corresponding Floquet ITFIM. Consequently, when no solutions for the Krylov angles are obtained, it indicates that the autocorrelation is not generated by unitary dynamics. We highlight this by studying certain special cases: stable $m$-period dynamics derived using the method of continued fractions, exponentially decaying, and power-law decaying stroboscopic dynamics. Remarkably, our examples of stable $m$-period dynamics correspond to $m$-period edge modes for the Floquet ITFIM where, deep in the chain, the couplings correspond to a critical phase. Our results pave the way to engineer Floquet systems with desired properties of edge modes and also provide examples of persistent edge modes in gapless Floquet systems.

Most modern theoretical considerations of the physical world suggest that nature is: (1) field-theoretic, (2) smooth, (3) local, (4) gauged, (5) containing fermions, and (6) non-perturbative. Tautologous as this may sound to experts, it is remarkable that the mathematical notion of geometry which reflects all of these aspects - namely, ``supergeometric homotopy theory''- has received little attention. Elaborate algebraic machinery is known for perturbative field theories both at the classical and quantum level, but to tackle the deep open questions of the subject, these will need to be lifted to a global geometry of physics. Our aim in this series is to introduce inclined physicists to this theory, to fill mathematical gaps in the existing literature, and to rigorously develop the full power of supergeometric homotopy theory and apply it to the analysis of fermionic (not necessarily super-symmetric) field theories. Secondarily, this will also lead to a streamlined and rigorous perspective we hope would also be desirable to mathematicians. In this first part, we explain how classical bosonic Lagrangian field theory (variational Euler-Lagrange theory) finds a natural home in the ``topos of smooth sets'', thereby neatly setting the scene for the higher supergeometry discussed in later parts of the series. This introductory material will be largely known to a few experts but has never been comprehensively laid out before. A key technical point we make is to regard jet bundle geometry systematically in smooth sets instead of just its subcategories of diffeological spaces or even Fr{\'e}chet manifolds -- or worse simply as a formal object. Besides being more transparent and powerful, it is only on this backdrop that a reasonable supergeometric jet geometry exists, needed for satisfactory discussion of any field theory with fermions.

A comprehensive investigation of the entanglement characteristics is carried out on tripartite spin-1/2 systems, examining prototypical tripartite states, the thermal Heisenberg model, and the transverse field Ising model. The entanglement is computed using the Rényi relative entropy. In the traditional Rényi relative entropy, the generalization parameter α can take values only in the range [Formula: see text] due to the requirements of joint convexity of the measure. To use the Rényi relative entropy over a wider range of α, we use the sandwiched form which is jointly convex in the regime [Formula: see text]. In prototypical tripartite states, we find that GHZ states are monogamous, but surprisingly so are W states. On the other hand, star states exhibit polygamy, due to the higher level of purity of the bipartite subsystems. For spin models, we study the dependence of entanglement on various parameters such as temperature, spin-spin interaction, and anisotropy, and identify regions where entanglement is the largest. The Rényi parameter α scales the amount of entanglement in the system. The entanglement measure based on the traditional and the sandwiched Rényi relative entropies obey the Araki-Lieb-Thirring inequality. In the Heisenberg models, namely the XYZ, XXZ, and XY models, the system is always monogamous. However, in the transverse field Ising model, the state is initially polygamous and becomes monogamous with temperature and coupling.

The fundamental notion of non-abelian generalized cohomology gained recognition in algebraic topology as the non-abelian Poincaré-dual to "factorization homology", and in theoretical physics as providing flux-quantization for non-linear Gauß laws.However, already the archetypical example-unstable Cohomotopy, first studied almost a century ago by Pontrjagin-has remained underappreciated as a cohomology theory and has only recently received attention as a flux-quantizaton law ("Hypothesis H").Here we lay out a general construction of the analogue of the Chern character map on twisted equivariant non-abelian cohomology theories (with equivariantly simply-connected classifying spaces) and illustrate the construction by spelling out a twisted equivariant form of Cohomotopy as an archetypical and intriguing running example, essentially by computing its equivariant Sullivan model.We close with an outlook on the application of this result to the rigorous deduction of anyonic quantum states on M5-branes wrapped over Seifert 3-orbifolds.

The conjectured symmetries of M-theory famously involve (1.) brane-extended super-symmetry (the M-algebra) and (2.) exceptional duality symmetry (the e 11-algebra); but little attention has been given to their inevitable combination. In this short note, we highlight (by combining results available in the literature) that the local exceptional duality symmetry (the hyperbolic involutory k 1, 10 ⊂ e 11) acts on the M-algebra, through the “brane-rotating symmetry” sl 32, in a way which extends the known hierarchy of finite-dimensional n-exceptional tangent spaces compatibly beyond the traditional bound of n ≤ 7 all the way to n = 11.

While it has become widely appreciated that defining (higher) gauge theories requires, in addition to ordinary phase space data, also “flux quantization” laws in generalized differential cohomology, there has been little discussion of the general rules, if any, for lifting Poisson brackets of (flux-)observables and their quantization from traditional phase spaces to the resulting higher moduli stacks of flux-quantized gauge fields. In this short note, we present a systematic analysis of (i) the canonical quantization of flux observables in Yang–Mills theory and (ii) of valid flux quantization laws in abelian Yang–Mills, observing (iii) that the resulting topological quantum observables form the homology Pontrjagin algebra of the loop space of the moduli space of flux-quantized gauge fields. This is remarkable because the homology Ponrjagin algebra on loops of moduli makes immediate sense in broad generality for higher and non-abelian (nonlinearly coupled) gauge fields, such as for the C field in 11d supergravity, where it recovers the quantum effects previously discussed in the context of “Hypothesis H.”

We observe that there is a rigorous derivation of (abelian) anyonic quantum states, hence of "topological order", on the 1+2-dimensional fixed locus of M5-probes wrapped over a trivially Seifert-fibered 3-orbifold singularity. Similar statements have previously been conjectured by appeal to the unknown dynamics of "coincident" M5-branes, but neglecting effects of flux-quantization that, as we highlight, entail anyonic solitons already in the rigorously tractable case of single M5-brane probes. This is possible after globally completing the "self-dual" tensor field on probe M5-branes by flux quantization in the non-abelian cohomology theory called equivariant twistorial Cohomotopy, which is admissible by recent results.

Can superconductivity arise from strong repulsive interactions between electrons in a lattice model with topological bands? Here, the authors answer this question in the affirmative via DMRG numerical simulations of an interacting spinful bilayer Hofstadter model featuring low-lying bands of opposite Chern number. Furthermore, by developing a new technique to probe the pairing symmetry in number-conserving DMRG, the authors demonstrate that superconductivity is $p$-wave, tantalizingly reminiscent of nodal pairing recently observed in twisted bilayer graphene, which shares many structural similarities with their model. This work thus showcases the possibility of an all-electronic pairing mechanism in the strong coupling limit in two dimensions, which can be tested directly with cold fermions in an optical lattice.

Recent experiments on rhombohedral pentalayer graphene with a substrate-induced moiré potential have identified both Chern insulators and fractional quantum Hall states at zero magnetic field. Surprisingly, these states are observed in strong displacement fields where the effects of the moiré lattice are weak, and seem to be readily accessed without fine-tuning. To address these experimental puzzles, we study a model of interacting electrons in this geometry. Within self-consistent Hartree-Fock (SCHF) calculations, we find an isolated Chern band with small bandwidth and good quantum geometry. Exact diagonalization and density-matrix renormalization group calculations both confirm the band hosts fractional quantum Hall states without a magnetic field. Remarkably, the Chern band is stable at a wide range of angles, at four through six rhombohedral layers, at varying rhombohedral hopping parameters, and-most strikingly-survives in SCHF calculations when the moiré potential vanishes. In this limit, the state spontaneously breaks time-reversal and translation symmetry simultaneously, giving a topological crystalline state that we term the "anomalous Hall crystal." We argue this is a general mechanism to create stable Chern bands in rhombohedral multilayer graphene, opening the door to studying the interplay between electronic topology, fractionalization, and spontaneous translation symmetry breaking.

Organic molecular crystals constitute a class of materials of critical importance in numerous industries. Despite the ubiquity of these systems, our ability to predict molecular crystal structures starting only from a two-dimensional diagram of the constituent compound(s) remains a significant challenge. Most structure-prediction protocols require a customized interatomic interaction model on which the quality of the results can depend sensitively. To overcome this problem, we introduce a new topological approach to molecular crystal structure prediction. The approach posits that in a stable structure, molecules are oriented such that principal axes and normal ring plane vectors are aligned with specific crystallographic directions and that heavy atoms occupy positions that correspond to minima of a set of geometric order parameters. By minimizing an objective function that encodes these orientations and atomic positions, and filtering based on the vdW free volume and intermolecular close contact distributions derived from the Cambridge Structural Database, stable structures and polymorphs for a given crystal can be predicted entirely mathematically without reliance on an interaction model.

A bstract We highlight the need for global completion of the field content in the M5-brane sigma-model analogous to Dirac’s charge/flux quantization, and we point out that the superspace Bianchi identities on the worldvolume and on its ambient supergravity background constrain the M5’s flux-quantization law to be in a non-abelian cohomology theory rationally equivalent to a twisted form of co-Homotopy. In order to clearly bring out this subtle point we give a streamlined re-derivation of the worldvolume 3-flux via M5 “super-embeddings”. Finally, assuming the flux-quantization law to actually be in co-Homotopy (“Hypothesis H”) we show how this implies Skyrmion-like solitons on general M5-worldvolumes and (abelian) anyonic solitons on the boundaries of “open M5-branes” in heterotic M-theory.

It is shown that the stroboscopic time evolution under a Floquet unitary, in any spatial dimension, and of any Hermitian operator, can be mapped to an operator Krylov space, which is identical to that generated by the edge operator of the noninteracting Floquet transverse-field Ising model (TFIM) in one-spatial dimension, and with inhomogeneous Ising and transverse field couplings. The latter has four topological phases reflected by the absence (topologically trivial) or presence (topologically nontrivial) of edge modes at 0 and/or π quasienergies. It is shown that the Floquet dynamics share certain universal features characterized by how the Krylov parameters vary in the topological phase diagram of the Floquet TFIM with homogeneous couplings. These results are highlighted through examples, all chosen for numerical convenience to be in one spatial dimension: nonintegrable Floquet spin $1/2$ chains and Floquet Z3 clock model where the latter hosts period-tripled edge modes.

Majorana bound states (MBSs) are quasiparticles that are their own antiparticles. They are predicted to emerge as zero-energy modes localized at the boundary of a topological superconductor. No intrinsic topological superconductor is known to date. However, by interfacing conventional superconductors and semiconductors with strong spin-orbit coupling, it is possible to create a system hosting topological states. Hence, epitaxial superconductors and semiconductors have emerged as an attractive material system with atomically sharp interfaces and broad flexibility in device fabrications incorporating Josephson junctions. We discuss the basics of topological superconductivity and provide insight into how to go beyond current state-of-the-art experiments. We argue that the ultimate success in realizing MBS physics requires the observation of non-Abelian braiding and fusion experiments. Published by the American Physical Society 2024

We establish a natural identification of cobordism classes of framed links with the fundamental group of the group-completed configuration space of points in the plane, by appeal to Okuyama's previously underappreciated interval configuration model for the latter. Under Segal's theorem, these classes are integers generated by the Hopf generator in the 2-cohomotopy of the 3-sphere, and we identify these knot-theoretically with the writhe sum of the linking and framing number of links. We observe that this link invariant is the properly regularized Wilson line observable of abelian Chern-Simons theory, and show, in consequence of the main theorem, that this arises equivalently as the expectation value of pure quantum states on the group algebra, under link sum, of cobordism classes of framed links. Observing that these quantum states regard framed links as worldlines of anyons in that they assign a fixed complex phase factor to each crossing (braiding) of strands, we close with an outlook on implications for the identification of anyonic solitons in 2D electron gases as exotic flux quantized in Cohomotopy instead of in ordinary cohomology.

Planar Josephson junctions provide a platform to host topological superconductivity, which, through manipulating Majorana bound states (MBS), could enable fault-tolerant quantum computing. However, what constitutes experimental signatures of topological superconductivity and how MBS can be detected remains strongly debated. In addition to spurious effects that mimic MBS, there is a challenge to discern the inherent topological signals in realistic systems with many topologically trivial Andreev bound states, determining the transport properties of Josephson junctions. Guided by the advances in microwave spectroscopy, we theoretically study Al/InAs-based planar Josephson junction embedded into a radio-frequency superconducting quantum interference device to identify microwave signatures of topological superconductivity. Remarkably, by exploring the closing and reopening of a topological gap, we show that even in a wide planar Josephson junction with many Andreev bound states, such a topological signature is distinguishable in the resonance frequency shift of a microwave drive and the half-slope feature of the microwave absorption spectrum. Our findings provide an important step towards experimental detection of non-Abelian statistics and implementing scalable topological quantum computing.

Results are presented for the dynamics of edge modes in interacting Floquet Ising chains. It is shown that in addition to the quasistable 0 and π edge modes, a third long lived edge mode arising from the operator product of the 0 and π edge modes exists. Depending on the microscopic parameters, this Floquet product mode is shown to have a substantially longer lifetime than the individual 0 and π modes. This is triggered by a scattering process which converts a 0 mode into a π mode while scattering two bulk excitations. This process can lead to a rapid decay of both 0 and π mode without affecting the product mode.

It is a classic result that certain interacting integrable spin chains host robust edge modes known as strong zero modes (SZMs). In this Letter, we extend this result to the Floquet setting of local quantum circuits, focusing on a prototypical model providing an integrable Trotterization for the evolution of the XXZ Heisenberg spin chain. By exploiting the algebraic structures of integrability, we show that an exact SZM operator can be constructed for these integrable quantum circuits in certain regions of parameter space. Our construction, which recovers a well-known result by Paul Fendley in the continuous-time limit, relies on a set of commuting transfer matrices known from integrability, and allows us to easily prove important properties of the SZM, including normalizabilty. Our approach is different from previous methods and could be of independent interest even in the Hamiltonian setting. Our predictions, which are corroborated by numerical simulations of infinite-temperature autocorrelation functions, are potentially interesting for implementations of the XXZ quantum circuit on available quantum platforms.

A bstract Flux quantization of the C-field in 11d supergravity is arguably necessary for the (UV-)completion of the theory, in that it determines the torsion charges carried by small numbers N ≪ ∞ of M-branes. However, hypotheses about C-field flux-quantization (“models of the C-field”) have previously been discussed only in the bosonic sector of 11d supergravity and ignoring the supergravity equations of motion. Here we highlight a duality-symmetric formulation of on-shell 11d supergravity on superspace, observe that this naturally lends itself to completion of the theory by flux quantization, and indeed that 11d super-spacetimes are put on-shell by carrying quantizable duality-symmetric super-C-field flux; the proof of which we present in detail.

Despite the plausible necessity of topological protection for realizing scalable quantum computers, the conceptual underpinnings of topological quantum logic gates had arguably remained shaky, both regarding their physical realization as well as their information-theoretic nature. Building on recent results on defect branes in string/M-theory (Sati and Schreiber in Rev Math Phys, 2023. https://doi.org/10.1142/S0129055X23500095. [arXiv:2203.11838]) and on their holographically dual anyonic defects in condensed matter theory (Sati and Schreiber in Rev Math Phys 35(03):2350001, 2023. https://doi.org/10.1142/S0129055X23500010. [arXiv:2206.13563]), here we explain [as announced in Sati and Schreiber (PlanQC 2022:33, 2022, [arXiv:2209.08331], [ncatlab.org/schreiber/show/Topological+Quantum+Programming+in+TED-K])] how the specification of realistic topological quantum gates, operating by anyon defect braiding in topologically ordered quantum materials, has a surprisingly slick formulation in parameterized point-set topology, which is so fundamental that it lends itself to certification in modern homotopically typed programming languages, such as cubical Agda. We propose that this remarkable confluence of concepts may jointly kickstart the development of topological quantum programming proper as well as of real-world application of homotopy type theory, both of which have arguably been falling behind their high expectations; in any case, it provides a powerful paradigm for simulating and verifying topological quantum computing architectures with high-level certification languages aware of the actual physical principles of realistic topological quantum hardware. In companion articles (Sati and Schreiber in The Quantum Monadology, [arXiv:2310.15735], Sati and Schreiber in Entanglement of Sections: The pushout of entangled and parameterized quantum information [arXiv:2309.07245]) [announced in Schreiber (Quantum types via Linear Homotopy Type Theory, talk at Workshop on Quantum Software @ QTML2022, Naples, 2022, [ncatlab.org/schreiber/files/QuantumDataInLHoTT-221117.pdf])], we explain how further passage to “dependent linear” homotopy types naturally extends this scheme to a full-blown quantum programming/certification language in which our topological quantum gates may be compiled to verified quantum circuits, complete with quantum measurement gates and classical control.

A set of high dimensional points X=x1, x2,…, xn ⊂ Rd in isotropic position is said to be $δ$-anti concentrated if for every direction $v$, the fraction of points in $X$ satisfying |⟨ xi,v ⟩ |≤ δ is at most $O(δ)$. Motivated by applications to list-decodable learning and clustering, recent works have considered the problem of constructing efficient certificates of anti-concentration in the average case, when the set of points $X$ corresponds to samples from a Gaussian distribution. Their certificates played a crucial role in several subsequent works in algorithmic robust statistics on list-decodable learning and settling the robust learnability of arbitrary Gaussian mixtures, yet remain limited to rotationally invariant distributions. This work presents a new (and arguably the most natural) formulation for anti-concentration. Using this formulation, we give quasi-polynomial time verifiable sum-of-squares certificates of anti-concentration that hold for a wide class of non-Gaussian distributions including anti-concentrated bounded product distributions and uniform distributions over Lp balls (and their affine transformations). Consequently, our method upgrades and extends results in algorithmic robust statistics e.g., list-decodable learning and clustering, to such distributions. Our approach constructs a canonical integer program for anti-concentration and analysis a sum-of-squares relaxation of it, independent of the intended application. We rely on duality and analyze a pseudo-expectation on large subsets of the input points that take a small value in some direction. Our analysis uses the method of polynomial reweightings to reduce the problem to analyzing only analytically dense or sparse directions.

While it has become widely appreciated that (higher) gauge theories need, besides their variational phase space data, to be equipped with “flux quantization laws” in generalized differential cohomology, there used to be no general prescription for how to define and construct the resulting flux-quantized phase space stacks. In this short note, we observe that all higher Maxwell-type equations have solution spaces given by flux densities on a Cauchy surface subject to a higher Gauß law and no further constraint: The metric duality-constraint is all absorbed into the evolution equation away from the Cauchy surface. Moreover, we observe that the higher Gauß law characterizes the Cauchy data as flat differential forms valued in a characteristic L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L∞\end{document}-algebra. Using the recent construction of the non-abelian Chern–Dold character map, this implies that compatible flux quantization laws on phase space have classifying spaces whose rational Whitehead L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L∞\end{document}-algebra is this characteristic one. The flux-quantized higher phase space stack of the theory is then simply the corresponding (generally non-abelian) differential cohomology moduli stack on the Cauchy surface. We show how this systematic prescription reproduces existing proposals for flux-quantized phase spaces of vacuum Maxwell theory and of the chiral boson and its higher siblings, but reveals that there are other choices of (non-abelian) flux quantization laws even in these basic cases, further discussed in a companion article (Sati and Schreiber in Quantum observables on quantized fluxes. arXiv:2312.13037). Moreover, for the case of NS/RR-fields in type II supergravity/string theory, the traditional “Hypothesis K” of flux quantization in topological K-theory is naturally implied, without the need, on phase space, of the notorious further duality constraint. Finally, as a genuinely non-abelian example we consider flux quantization of the C-field in 11d supergravity/M-theory given by unstable differential 4-Cohomotopy (“Hypothesis H”) and emphasize again that, implemented on Cauchy data, this qualifies as the full phase space without the need for a further duality constraint.

We study a periodically driven one-dimensional Kitaev model in the presence of disorder. In the clean limit our model exhibits four topological phases corresponding to the existence or nonexistence of edge modes at zero and π quasienergy. When potential disorder is added, the system parameters get renormalized and the system may exhibit a topological phase transition. When starting from the Majorana π mode (MPM) phase, which hosts only edge Majoranas with quasienergy π, disorder induces a transition into a neighboring phase with both π and zero modes on the edges. We characterize the disordered system using (i) exact diagonalization, (ii) Arnoldi mapping onto an effective tight-binding chain, and (iii) topological entanglement entropy.

In superconducting systems in which inversion and time-reversal symmetry are simultaneously broken the critical current for positive and negative current bias can be different. For superconducting systems formed by Josephson junctions (JJs) this effect is termed Josephson diode effect. In this paper, we study the Josephson diode effect for a superconducting quantum interference device (SQUID) formed by a topological JJ with a 4π-periodic current-phase relationship and a topologically trivial JJ. We show how the fractional Josephson effect manifests in the Josephson diode effect with the application of a magnetic field and how tuning properties of the trivial SQUID arm can lead to diode polarity switching. We then investigate the ac response and show that the polarity of the diode effect can be tuned by varying the ac power and discuss differences between the ac diode effect of asymmetric SQUIDs with no topological JJ and SQUIDs in which one JJ is topological.

We introduce a Floquet circuit describing the driven Ising chain with topological defects. The corresponding gates include a defect that flips spins as well as the duality defect that explicitly implements the Kramers-Wannier duality transformation. The Floquet unitary evolution operator commutes with such defects, but the duality defect is not unitary, as it projects out half the states. We give two applications of these defects. One is to analyze the return amplitudes in the presence of “space-like” defects stretching around the system. We verify explicitly that the return amplitudes are in agreement with the fusion rules of the defects. The second application is to study unitary evolution in the presence of “time-like” defects that implement anti-periodic and duality-twisted boundary conditions. We show that a single unpaired localized Majorana zero mode appears in the latter case. We explicitly construct this operator, which acts as a symmetry of this Floquet circuit. We also present analytic expressions for the entanglement entropy after a single time step for a system of a few sites, for all of the above defect configurations.

The use of electric fields to modify chemical reactions is a promising, emerging technique in catalysis. However, there exist few guiding principles, and rational design requires assumptions about the transition state or explicit atomistic calculations. Here, we present a linear free energy relationship, familiar in other areas of physical organic chemistry and catalysis, that microscopically relates field-induced changes in the activation energy to those in the reaction energy, connecting kinetic and thermodynamic behaviors. We verify our theory using first-principles electronic structure calculations of a symmetric SN2 reaction and the dehalogenation of an aryl halide on gold surfaces and observe hallmarks of linear free energy relationships, such as the shifting to early and late transition states. We also report and explain a counterintuitive case, where the constant of proportionality relating the activation and reaction energies is negative.

Flux- and charge-quantization laws for higher gauge fields of Maxwell type -- e.g. the common electromagnetic field (the "A-field") but also the B-, RR-, and C-fields considered in string/M-theory -- specify non-perturbative completions of these fields by encoding their solitonic behaviour and hence by specifying the discrete charges carried by the individual branes (higher-dimensional monopoles or solitons) that source the field fluxes. This article surveys the general (rational-)homotopy theoretic understanding of flux- and charge-quantization via the Chern-Dold character map generalized to the non-linear (self-sourcing) Bianchi identities that appear in higher-dimensional supergravity theories, notably for B&RR-fields in D=10, for the C-field in D=11 supergravity, and for the B-field on fivebrane worldvolumes.

Inertia orbifolds homotopy-quotiented by rotation of geometric loops play a fundamental role not only in ordinary cyclic cohomology, but more recently in constructions of equivariant Tate-elliptic cohomology and generally of transchromatic characters on generalized cohomology theories. Nevertheless, existing discussion of such cyclified stacks has been relying on ad-hoc component presentations with intransparent and unverified stacky homotopy type. Following our previous formulation of transgression of cohomological charges (“double-dimensional reduction”), we explain how cyclification of ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}∞\end{document}-stacks is a fundamental and elementary base-change construction over moduli stacks in cohesive higher topos theory (cohesive homotopy-type theory). We prove that Ganter/Huan’s extended inertia groupoid used to define equivariant quasi-elliptic cohomology is indeed a model for this intrinsically defined cyclification of orbifolds, and we show that cyclification implements transgression in group cohomology in general, and hence in particular the transgression of degree-4 twists of equivariant Tate-elliptic cohomology to degree-3 twists of orbifold K-theory on the cyclified orbifold. As an application, we show that the universal shifted integral 4-class of equivariant 4-Cohomotopy theory on ADE-orbifolds induces the Platonic 4-twist of ADE-equivariant Tate-elliptic cohomology, and we close by explaining how this should relate to elliptic M5-brane genera, under our previously formulated Hypothesis H.

There has been significant interest in understanding how practical constraints on contemporary quantum devices impact the complexity of quantum learning. For the classic question of tomography, recent work tightly characterized the copy complexity for any protocol that can only measure one copy of the unknown state at a time, showing it is polynomially worse than if one can make fully-entangled measurements. While we now have a fairly complete picture of the rates for such tasks in the near-term and fault-tolerant regimes, it remains poorly understood what the landscape in between looks like. In this work, we study tomography in the natural setting where one can make measurements of $t$ copies at a time. For sufficiently small $ε$, we show that for any t ≤ d², Θ(d³tε²) copies are necessary and sufficient to learn an unknown $d$-dimensional state $ρ$ to trace distance $ε$. This gives a smooth and optimal interpolation between the known rates for single-copy and fully-entangled measurements. To our knowledge, this is the first smooth entanglement-copy tradeoff known for any quantum learning task, and for tomography, no intermediate point on this curve was known, even at $t = 2$. An important obstacle is that unlike the optimal single-copy protocol, the optimal fully-entangled protocol is inherently biased and thus precludes naive batching approaches. Instead, we devise a novel two-stage procedure that uses Keyl's algorithm to refine a crude estimate for $ρ$ based on single-copy measurements. A key insight is to use Schur-Weyl sampling not to estimate the spectrum of $ρ$, but to estimate the deviation of $ρ$ from the maximally mixed state. When $ρ$ is far from the maximally mixed state, we devise a novel quantum splitting procedure that reduces to the case where $ρ$ is close to maximally mixed.

A (2+1)D topologically ordered phase may or may not have a gappable edge, even if its chiral central charge c− is vanishing. Recently, it was discovered that a quantity regarded as a ``higher'' version of chiral central charge gives a further obstruction beyond c− to gapping out the edge. In this Letter, we show that the higher central charges can be characterized by the expectation value of the partial rotation operator acting on the wave function of the topologically ordered state. This allows us to extract the higher central charge from a single wave function, which can be evaluated on a quantum computer. Our characterization of the higher central charge is analytically derived from the modular properties of edge conformal field theory, as well as the numerical results with the ν=1/2 bosonic Laughlin state and the non-Abelian gapped phase of the Kitaev honeycomb model, which corresponds to U(1)2 and Ising topological order, respectively. The Letter establishes a numerical method to obtain a set of obstructions to the gappable edge of (2+1)D bosonic topological order beyond c−, which enables us to completely determine if a (2+1)D bosonic Abelian topological order has a gappable edge or not. We also point out that the expectation values of the partial rotation on a single wave function put a constraint on the low-energy spectrum of the bulk-boundary system of (2+1)D bosonic topological order, reminiscent of the Lieb-Schultz-Mattis-type theorems.

It is shown that the stroboscopic time-evolution under a Floquet unitary, in any spatial dimension, and of any Hermitian operator, can be mapped to an operator Krylov space which is identical to that generated by the edge operator of the non-interacting Floquet transverse-field Ising model (TFIM) in one-spatial dimension, and with inhomogeneous Ising and transverse field couplings. The latter has four topological phases reflected by the absence (topologically trivial) or presence (topologically non-trivial) of edge modes at $0$ and/or $π$ quasi-energies. It is shown that the Floquet dynamics share certain universal features characterized by how the Krylov parameters vary in the topological phase diagram of the Floquet TFIM with homogeneous couplings. These results are highlighted through examples, all chosen for numerical convenience to be in one spatial dimension: non-integrable Floquet spin $1/2$ chains and Floquet Z3 clock model where the latter hosts period-tripled edge modes.

Weight systems on chord diagrams play a central role in knot theory and Chern–Simons theory; and more recently in stringy quantum gravity. We highlight that the noncommutative algebra of horizontal chord diagrams is canonically a star-algebra and ask which weight systems are positive with respect to this structure; hence, we ask: Which weight systems are quantum states, if horizontal chord diagrams are quantum observables? We observe that the fundamental gl(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}gl(n)\end{document}-weight systems on horizontal chord diagrams with N strands may be identified with the Cayley distance kernel at inverse temperature β=ln(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}β = ln(n)\end{document} on the symmetric group on N elements. In contrast to related kernels like the Mallows kernel, the positivity of the Cayley distance kernel had remained open. We characterize its phases of indefinite, semi-definite and definite positivity, in dependence of the inverse temperature β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}β\end{document}; and we prove that the Cayley distance kernel is positive (semi-)definite at β=ln(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}β = ln(n)\end{document} for all n=1,2,3,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}n = 1,2,3, …\end{document}. In particular, this proves that all fundamental gl(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}gl(n)\end{document}-weight systems are quantum states, and hence, so are all their convex combinations. We close with briefly recalling how, under our “Hypothesis H”, this result impacts on the identification of bound states of multiple M5-branes.

The transverse field Ising model (TFIM) on the half-infinite chain possesses an edge zero mode. This work considers an impurity model: TFIM perturbed by a boundary integrability-breaking interaction. For sufficiently large transverse field, but in the ordered phase of the TFIM, the zero mode is observed to decay. The decay is qualitatively different from zero modes where the integrability-breaking interactions are nonzero all along the chain. It is shown that for the impurity model, the zero mode decays by relaxing to a nonlocal quasiconserved operator, the latter being exactly conserved when the opposite edge of the chain has no noncommuting perturbations so as to ensure perfect degeneracy of the spectrum. In the thermodynamic limit, the quasiconserved operator vanishes, and a regime is identified where the decay of the zero mode obeys Fermi's golden rule. A toy model for the decay is constructed in Krylov space and it is highlighted how Fermi's golden rule may be recovered from this toy model.