Daily Papers

Topological materials provide a platform that utilizes the geometric characteristics of structured materials to control the flow of waves, enabling unidirectional and protected transmission that is immune to defects or impurities. The topologically designed photonic materials can carry quantum states and electromagnetic energy, benefiting nanolasers or quantum photonic systems. This article reviews recent advances in the topological applications of photonic materials for radiative heat transfer, especially in the near field. When the separation distance between media is considerably smaller than the thermal wavelength, the heat transfer exhibits super-Planckian behavior that surpasses Planck's blackbody predictions. Near-field thermal radiation in subwavelength systems supporting surface modes has various applications, including nanoscale thermal management and energy conversion. Photonic materials and structures that support topological surface states show immense potential for enhancing or suppressing near-field thermal radiation. We present various topological effects, such as periodic and quasi-periodic nanoparticle arrays, Dirac and Weyl semimetal-based materials, structures with broken global symmetries, and other topological insulators, on near-field heat transfer. Also, the possibility of realizing near-field thermal radiation in such topological materials for alternative thermal management and heat flux guiding in nano-scale systems is discussed based on the existing technology.

The quantum Hall effect, topological insulators and quantum capacitance are usually treated as distinct topics, yet they share a common thread. Bulk topology determines a protected boundary state, and quantum capacitance is one of the few electrostatic methods able to detect the density of states. This paper examines that connection. The integer quantum Hall effect showed that quantised boundary transport has a topological origin given by the Chern number, later extended to zero field in Z2 topological insulators with helical edge and Dirac surface states. Quantum capacitance, developed for two-dimensional electron gases, has since become a key technique for mapping the Landau level density of states underlying the quantum Hall effect and the linear Dirac density of states of topological surface states. This picture is complicated by self-consistent screening theory, which shows that the quantum Hall bulk splits into compressible and incompressible regions that shift with field. Local quantum capacitance measurements confirm that the bulk is not uniformly insulating across a quantised plateau, even though the plateau stays quantised. The paper argues that quantum capacitance is not a secondary feature of these phenomena but one of the clearest windows into the boundary density of states that topology imposes, and that it reveals the local, field-dependent nature of the insulating bulk assumption underlying the analogy between the quantum Hall effect and topological insulators.

The discovery of topological quantum states marks a new chapter in both condensed matter physics and materials sciences. By analogy to spin electronic system, topological concepts have been extended into phonons, boosting the birth of topological phononics (TPs). Here, we present a high-throughput screening and data-driven approach to compute and evaluate TPs among over 10,000 materials. We have clarified 5014 TP materials and classified them into single Weyl, high degenerate Weyl, and nodal-line (ring) TPs. Among them, three representative cases of TPs have been discussed in detail. Furthermore, we suggest 322 TP materials with potential clean nontrivial surface states, which are favorable for experimental characterizations. This work significantly increases the current library of TP materials, which enables an in-depth investigation of their structure-property relations and opens new avenues for future device design related to TPs.

Quantum materials hosting Weyl fermions have opened a new era of research in condensed matter physics. First proposed in 1929 in particle physics, Weyl fermions have yet to be observed as elementary particles. In 2015, Weyl fermions were detected as collective electronic excitations in the strong spin-orbit coupled material tantalum arsenide, TaAs. This discovery was followed by a flurry of experimental and theoretical explorations of Weyl phenomena in materials. Weyl materials naturally lend themselves to the exploration of the topological index associated with Weyl fermions and their divergent Berry curvature field, as well as the topological bulk-boundary correspondence giving rise to protected conducting surface states. Here, we review the broader class of Weyl topological phenomena in materials, starting with the observation of emergent Weyl fermions in the bulk and of Fermi arc states on the surface of the TaAs family of crystals by photoemission spectroscopy. We then discuss some of the exotic optical and magnetic responses observed in these materials, as well as the progress in developing some of the related chiral materials. We discuss the conceptual development of high-fold chiral fermions, which generalize Weyl fermions, and we review the observation of high-fold chiral fermion phases by taking the rhodium silicide, RhSi, family of crystals as a prime example. Lastly, we discuss recent advances in Weyl-line phases in magnetic topological materials. With this Review, we aim to provide an introduction to the basic concepts underlying Weyl physics in condensed matter, and to representative materials and their electronic structures and topology as revealed by spectroscopic studies. We hope this work serves as a guide for future theoretical and experimental explorations of chiral fermions and related topological quantum systems with potentially enhanced functionalities.

We conducted a high-throughput search for topological magnetic materials on 522 new, experimentally reported commensurate magnetic structures from MAGNDATA, doubling the number of available materials on the Topological Magnetic Materials database. This brings up to date the previous studies which had become incomplete due to the discovery of new materials. For each material, we performed first-principle electronic calculations and diagnosed the topology as a function of the Hubbard U parameter. Our high-throughput calculation led us to the prediction of 250 experimentally relevant topologically non-trivial materials, which represent 47.89% of the newly analyzed materials. We present five remarkable examples of these materials, each showcasing a different topological phase: Mn{}_2AlB{}_2 (BCSID 1.508), which exhibits a nodal line semimetal to topological insulator transition as a function of SOC, CaMnSi (BCSID 0.599), a narrow gap axion insulator, UAsS (BCSID 0.594) a 5f-orbital Weyl semimetal, CsMnF{}_4 (BCSID 0.327), a material presenting a new type of quasi-symmetry protected closed nodal surface and FeCr{}_2S{}_4 (BCSID 0.613), a symmetry-enforced semimetal with double Weyls and spin-polarised surface states.

In fault-tolerant quantum computing with the surface code, non-Clifford gates are crucial for universal computation. However, implementing these gates using methods like magic state distillation and code switching requires significant resources. In this work, we propose a new protocol that combines magic state preparation and code switching to realize logical non-Clifford operations with the potential for fault tolerance. Our approach begins with a special logical state in the Z_4 surface code. By applying a sequence of transformations, the system goes through different topological codes, including the non-Abelian D_4 quantum double model. This process ultimately produces a magic state in a condensed Z_2 surface code, which enables the implementation of a logical T gate in the standard Z_2 surface code. In our analysis, we employ a framework where the topological codes are represented by their topological orders and all the transformations are considered as topological manipulations such as gauging symmetries and condensing anyons. This perspective is particularly useful for understanding code switching between topological codes.

Comprehensive study using DC transport, specific heat, magnetization, and two-coil mutual inductance measurements unveils an understanding of three temperature regimes in SmB_6: (i) T geq T^{*} (sim66K), (ii) T_g (sim40 K) leq T < T^{*}, and (iii) T < T_g. Onset of Kondo breakdown below T^{*} releases disorder-driven magnetic fluctuations, which splits the bulk (sim116K) and surface Kondo temperature (T_k^{s} approx 7 K). Below T_g, as magnetic fluctuations subside, surface Kondo screening revives, stabilizing the topological surface state and generating an in-gap feature (sim2.2 meV) across which Dirac-like carriers are excited. Nyquist impedance analysis reveals a crossover from purely capacitive to capacitive-inductive behavior, signalling a disorder-driven two-fluid phase of heavy quasiparticles and light, high-mobility carriers below T_g. We identify a characteristic length scale, L_{ν_0}(T), associated with the high-mobility phase, exhibiting an almost divergent trend below T_k^{s}. These findings underscore the complex nature of the surface conducting state in SmB_6.

Consider a topological surface Σ. We introduce the spectrum of a representation from the fundamental group of Σ to SL(2,R), which is a subset of projective measured lamination on the surface, which captures the directions along which the representation fails to be Fuchsian, and which characterizes the action of the mapping class group on this representation. In the case of the once-punctured torus, we show that the spectrum of a generic representation is a Cantor set, and that it completely describes the dynamics of the familly of locally constant cocycles above interval exchange transformations associated to the representation.

Sphere eversions have been described so far by either pictures with minimal topological complexity, numerical evolution or complex equations. We write down relatively simple explicit formulas for the whole eversion, both analytic and topologically simpler, including also Boy surface (real projective plane), using a family of ruled surfaces. We show their usefulness in visualizing the process using commonly available modeling software.

Characteristic classes, which are abstract topological invariants associated with vector bundles, have become an important notion in modern physics with surprising real-world consequences. As a representative example, the incredible properties of topological insulators, which are insulators in their bulk but conductors on their surface, can be completely characterized by a specific characteristic class associated with their electronic band structure, the first Chern class. Given their importance to next generation computing and the computational challenge of calculating them using first-principles approaches, there is a need to develop machine learning approaches to predict the characteristic classes associated with a material system. To aid in this program we introduce the {Haldane bundle dataset}, which consists of synthetically generated complex line bundles on the 2-torus. We envision this dataset, which is not as challenging as noisy and sparsely measured real-world datasets but (as we show) still difficult for off-the-shelf architectures, to be a testing ground for architectures that incorporate the rich topological and geometric priors underlying characteristic classes.

Topological flat bands (FBs) offer an ideal platform for realizing exotic topological phases, such as fractional Chern insulators, yet their realization with both exact flatness and stable topology in local lattice models has been long hindered by fundamental no-go theorems. Here, we overcome this barrier by demonstrating the existence of critical topological FBs (CTFBs) in finite-range hopping models. They saturate the no-go theorems via a unique structure of Bloch wavefunctions: While continuous over the whole Brillouin zone, the wavefunctions are non-analytic at isolated band touching points, thereby relaxing the inherent restrictions on the coexistence of exact flatness and stable topology. We establish a general principle to construct CTFBs, as well as their parent Hamiltonians, that carry desired topological invariants in given space groups. Explicit examples exhibiting Chern numbers, strong Z_2 index, and crystalline-symmetry-protected invariants in two and three dimensions are provided. Furthermore, an automated algorithm identifies more than 50,000 robust, symmetry-indicated CTFBs. Filling such CTFBs yields short-range entangled topological states that exhibit power-law correlations. Crucially, all filled CTFB states admit exact tensor-network representations with finite bond dimensions, providing a tractable starting point for exploring strongly correlated topological matter.

Topological materials present unconventional electronic properties that make them attractive for both basic science and next-generation technological applications. The majority of currently known topological materials have been discovered using methods that involve symmetry-based analysis of the quantum wavefunction. Here we use machine learning to develop a simple-to-use heuristic chemical rule that diagnoses with a high accuracy whether a material is topological using only its chemical formula. This heuristic rule is based on a notion that we term topogivity, a machine-learned numerical value for each element that loosely captures its tendency to form topological materials. We next implement a high-throughput procedure for discovering topological materials based on the heuristic topogivity-rule prediction followed by ab initio validation. This way, we discover new topological materials that are not diagnosable using symmetry indicators, including several that may be promising for experimental observation.

One of the main goals and challenges of materials discovery is to find the best candidates for each interest property or application. Machine learning rises in this context to efficiently optimize this search, exploring the immense materials space, consisting of simultaneously the atomic, compositional, and structural spaces. Topological insulators, presenting symmetry-protected metallic edge states, are a promising class of materials for different applications. However, further, development is limited by the scarcity of viable candidates. Here we present and discuss machine learning-accelerated strategies for searching the materials space for two-dimensional topological materials. We show the importance of detailed investigations of each machine learning component, leading to different results. Using recently created databases containing thousands of ab initio calculations of 2D materials, we train machine learning models capable of determining the electronic topology of materials, with an accuracy of over 90%. We can then generate and screen thousands of novel materials, efficiently predicting their topological character without the need for a priori structural knowledge. We discover 56 non-trivial materials, of which 17 novel insulating candidates for further investigation, for which we corroborate their topological properties with density functional theory calculations. This strategy is 10times more efficient than the trial-and-error approach while few orders of magnitude faster and is a proof of concept for guiding improved materials discovery search strategies.

The surface code family is a promising approach to implementing fault-tolerant quantum computations. Universal fault-tolerance requires error-corrected non-Clifford operations, in addition to Clifford gates, and for the former, it is imperative to experimentally demonstrate additional resources known as magic states. Another challenge is to efficiently embed surface codes into quantum hardware with connectivity constraints. This work simultaneously addresses both challenges by employing a qubit-efficient rotated heavy-hexagonal surface code for IBM quantum processors (ibm\_fez) and implementing the magic state injection protocol. Our work reports error thresholds for both logical bit- and phase-flip errors, of approx0.37% and approx0.31%, respectively, which are higher than the threshold values previously reported with traditional embedding. The post-selection-based preparation of logical magic states |H_Lrangle and |T_Lrangle achieve fidelities of 0.8806pm0.0002 and 0.8665pm0.0003, respectively, which are both above the magic state distillation threshold. Additionally, we report the minimum fidelity among injected arbitrary single logical qubit states as 0.8356pm0.0003. Our work demonstrates the potential for realising non-Clifford logical gates by producing high-fidelity logical magic states on IBM quantum devices.

We point out that there are two different chiral spin liquid states on the triangular lattice and discuss the conducting states that are expected on doping them. These states labeled CS1 and CS2 are associated with two distinct topological orders with different edge states, although they both spontaneously break time reversal symmetry and exhibit the same quantized spin Hall conductance. While CSL1 is related to the Kalmeyer-Laughlin state, CSL2 is the ν=4 member of Kitaev's 16 fold way classification. Both states are described within the Abrikosov fermion representation of spins, and the effect of doping can be accessed by introducing charged holons. On doping CSL2, condensation of charged holons leads to a topological d+id superconductor. However on doping CSL1 , in sharp contrast , two different scenarios can arise: first, if holons condense, a chiral metal with doubled unit cell and finite Hall conductivity is obtained. However, in a second novel scenario, the internal magnetic flux adjusts with doping and holons form a bosonic integer quantum Hall (BIQH) state. Remarkably, the latter phase is identical to a d+id superconductor. In this case the Mott insulator to superconductor transition is associated with a bosonic variant of the integer quantum Hall plateau transition for the holon. We connect the above two scenarios to two recent numerical studies of doped chiral spin liquids on triangular lattice. Our work clarifies the complex relation between topological superconductors, chiral spin liquids and quantum criticality .

In a novel application of the tools of topological data analysis (TDA) to nonperturbative quantum gravity, we introduce a new class of observables that allows us to assess whether quantum spacetime really resembles a ``quantum foam" near the Planck scale. The key idea is to investigate the Betti numbers of coarse-grained path integral histories, regularized in terms of dynamical triangulations, as a function of the coarse-graining scale. In two dimensions our analysis exhibits the well-known fractal structure of Euclidean quantum gravity.

The discoveries of intrinsically magnetic topological materials, including semimetals with a large anomalous Hall effect and axion insulators, have directed fundamental research in solid-state materials. Topological quantum chemistry has enabled the understanding of and the search for paramagnetic topological materials. Using magnetic topological indices obtained from magnetic topological quantum chemistry (MTQC), here we perform a high-throughput search for magnetic topological materials based on first-principles calculations. We use as our starting point the Magnetic Materials Database on the Bilbao Crystallographic Server, which contains more than 549 magnetic compounds with magnetic structures deduced from neutron-scattering experiments, and identify 130 enforced semimetals (for which the band crossings are implied by symmetry eigenvalues), and topological insulators. For each compound, we perform complete electronic structure calculations, which include complete topological phase diagrams using different values of the Hubbard potential. Using a custom code to find the magnetic co-representations of all bands in all magnetic space groups, we generate data to be fed into the algorithm of MTQC to determine the topology of each magnetic material. Several of these materials display previously unknown topological phases, including symmetry-indicated magnetic semimetals, three-dimensional anomalous Hall insulators and higher-order magnetic semimetals. We analyse topological trends in the materials under varying interactions: 60 per cent of the 130 topological materials have topologies sensitive to interactions, and the others have stable topologies under varying interactions. We provide a materials database for future experimental studies and open-source code for diagnosing topologies of magnetic materials.

We review theoretical and experimental highlights in transport in two-dimensional materials focussing on key developments over the last five years. Topological insulators are finding applications in magnetic devices, while Hall transport in doped samples and the general issue of topological protection remain controversial. In transition metal dichalcogenides valley-dependent electrical and optical phenomena continue to stimulate state-of-the-art experiments. In Weyl semimetals the properties of Fermi arcs are being actively investigated. A new field, expected to grow in the near future, focuses on the non-linear electrical and optical responses of topological materials, where fundamental questions are once more being asked about the intertwining roles of the Berry curvature and disorder scattering. In topological superconductors the quest for chiral superconductivity, Majorana fermions and topological quantum computing is continuing apace.

Berry curvature physics and quantum geometric effects have been instrumental in advancing topological condensed matter physics in recent decades. Although Landau level-based flat bands and conventional 3D solids have been pivotal in exploring rich topological phenomena, they are constrained by their limited ability to undergo dynamic tuning. In stark contrast, moiré systems have risen as a versatile platform for engineering bands and manipulating the distribution of Berry curvature in momentum space. These moiré systems not only harbor tunable topological bands, modifiable through a plethora of parameters, but also provide unprecedented access to large length scales and low energy scales. Furthermore, they offer unique opportunities stemming from the symmetry-breaking mechanisms and electron correlations associated with the underlying flat bands that are beyond the reach of conventional crystalline solids. A diverse array of tools, encompassing quantum electron transport in both linear and non-linear response regimes and optical excitation techniques, provide direct avenues for investigating Berry physics. This review navigates the evolving landscape of tunable moiré materials, highlighting recent experimental breakthroughs in the field of topological physics. Additionally, we delineate several challenges and offer insights into promising avenues for future research.

With their non-Abelian topological charges, real multi-bandgap systems challenge the conventional topological phase classifications. As the minimal sector of multi-bandgap systems, real triple degeneracies (RTPs), which serve as real 'Weyl points', lay the foundation for the research on real topological phases. However, experimental demonstration of physical systems with global band configurations consisting of multiple RTPs in crystals has not been reported. Here we present experimental evidence of RTPs in photonic meta-crystals, characterizing them using the Euler number, and establishing their connection with both Abelian and non-Abelian charges. By considering RTPs as the basic elements, we further propose the concept of a topological compound, akin to a chemical compound, where we find that certain phases are not topologically allowed. The topological classification of RTPs in crystals demonstrated in our work plays a similar role as the 'no-go' theorem in Weyl systems.

Topological magnons give rise to possibilities for engineering novel spintronics devices with critical applications in quantum information and computation, due to its symmetry-protected robustness and low dissipation. However, to make reliable and systematic predictions about material realization of topological magnons has been a major challenge, due to the lack of neutron scattering data for most materials. In this work, we significantly advance the symmetry-based approach for identifying topological magnons through developing a fully automated algorithm, utilizing the theory of symmetry indicators, that enables a highly efficient and large-scale search for candidate materials hosting field-induced topological magnons. This progress not only streamlines the discovery process but also expands the scope of materials exploration beyond previous manual or traditional methods, offering a powerful tool for uncovering novel topological phases in magnetic systems. Performing a large-scale search over all 1649 magnetic materials in Bilbao Crystallographic Server with a commensurate magnetic order, we discover 387 candidate materials for topological magnons, significantly expanding the pool of topological magnon materials. We further discuss examples and experimental accessibility of the candidate materials, shedding light on future experimental realizations of topological magnons in magnetic materials.

The study of twisted two-dimensional (2D) materials, where twisting layers create moiré superlattices, has opened new opportunities for investigating topological phases and strongly correlated physics. While systems such as twisted bilayer graphene (TBG) and twisted transition metal dichalcogenides (TMDs) have been extensively studied, the broader potential of a seemingly infinite set of other twistable 2D materials remains largely unexplored. In this paper, we define "theoretically twistable materials" as single- or multi-layer structures that allow for the construction of simple continuum models of their moiré structures. This excludes, for example, materials with a "spaghetti" of bands or those with numerous crossing points at the Fermi level, for which theoretical moiré modeling is unfeasible. We present a high-throughput algorithm that systematically searches for theoretically twistable semimetals and insulators based on the Topological 2D Materials Database. By analyzing key electronic properties, we identify thousands of new candidate materials that could host rich topological and strongly correlated phenomena when twisted. We propose representative twistable materials for realizing different types of moiré systems, including materials with different Bravais lattices, valleys, and strength of spin-orbital coupling. We provide examples of crystal growth for several of these materials and showcase twisted bilayer band structures along with simplified twisted continuum models. Our results significantly broaden the scope of moiré heterostructures and provide a valuable resource for future experimental and theoretical studies on novel moiré systems.

Motivated by topologically protected states in wave physics, we study localised eigenmodes in one-dimensional periodic media with defects. The Su-Schrieffer-Heeger model (the canonical example of a one-dimensional system with topologically protected localised defect states) is used to demonstrate the method. Our approach can be used to describe two broad classes of perturbations to periodic differential problems: those caused by inserting a finite-sized piece of arbitrary material and those caused by creating an interface between two different periodic media. The results presented here characterise the existence of localised eigenmodes in each case and, when they exist, determine their eigenfrequencies and provide concise analytic results that quantify the decay rate of these modes. These results are obtained using both high-frequency homogenisation and transfer matrix analysis, with good agreement between the two methods.

This article provides an introduction to surface code quantum computing. We first estimate the size and speed of a surface code quantum computer. We then introduce the concept of the stabilizer, using two qubits, and extend this concept to stabilizers acting on a two-dimensional array of physical qubits, on which we implement the surface code. We next describe how logical qubits are formed in the surface code array and give numerical estimates of their fault-tolerance. We outline how logical qubits are physically moved on the array, how qubit braid transformations are constructed, and how a braid between two logical qubits is equivalent to a controlled-NOT. We then describe the single-qubit Hadamard, S and T operators, completing the set of required gates for a universal quantum computer. We conclude by briefly discussing physical implementations of the surface code. We include a number of appendices in which we provide supplementary information to the main text.

Subwavelength atomic lattices have emerged as a promising platform for quantum applications, leveraging collective superradiant and subradiant effects to enhance light-matter interactions. Integrating atomic lattices into nanostructures is at the front of effort toward any application with atomic lattices, but is still a challenging theoretical task, as the dipole interactions are hard to model in arbitrary dielectric environment. Here, we develop a formalism that allows the embedding of emitters to a wide class of thin curved waveguides, by greatly simplifying the wave equation and consequentially the dipole interactions. We demonstrate the formalism for locally constant positive curved surfaces, where we derive a new surface Green function that allows exact computation of the collective superradiant and subradiant states. We also show how the curvature and the thickness of the waveguides affects the collective states through an effective surface wavelength.

Let L denote the Laplacian matrix of a graph G. We study continuous quantum walks on G defined by the transition matrix U(t)=expleft(itLright). The initial state is of the pair state form, e_a-e_b with a,b being any two vertices of G. We provide two ways to construct infinite families of graphs that have perfect pair transfer. We study a "transitivity" phenomenon which cannot occur in vertex state transfer. We characterize perfect pair state transfer on paths and cycles. We also study the case when quantum walks are generated by the unsigned Laplacians of underlying graphs and the initial states are of the plus state form, e_a+e_b. When the underlying graphs are bipartite, plus state transfer is equivalent to pair state transfer.

Tunneling spectroscopy is widely used to examine the subgap spectra in semiconductor-superconductor nanostructures when searching for Majorana zero modes (MZMs). Typically, semiconductor sections controlled by local gates at the ends of hybrids serve as tunnel barriers. Besides detecting states only at the hybrid ends, such gate-defined tunnel probes can cause the formation of non-topological subgap states that mimic MZMs. Here, we develop an alternative type of tunnel probes to overcome these limitations. After the growth of an InSb-Al hybrid nanowire, a precisely controlled in-situ oxidation of the Al shell is performed to yield a nm-thick Al oxide layer. In such thin isolating layer, tunnel probes can be arbitrarily defined at any position along the hybrid nanowire by shadow-wall angle-deposition of metallic leads. This allows us to make multiple tunnel probes along single nanowire hybrids and to successfully identify Andreev bound states (ABSs) of various spatial extension residing along the hybrids.

We present EuLearn, the first surface datasets equitably representing a diversity of topological types. We designed our embedded surfaces of uniformly varying genera relying on random knots, thus allowing our surfaces to knot with themselves. EuLearn contributes new topological datasets of meshes, point clouds, and scalar fields in 3D. We aim to facilitate the training of machine learning systems that can discern topological features. We experimented with specific emblematic 3D neural network architectures, finding that their vanilla implementations perform poorly on genus classification. To enhance performance, we developed a novel, non-Euclidean, statistical sampling method adapted to graph and manifold data. We also introduce adjacency-informed adaptations of PointNet and Transformer architectures that rely on our non-Euclidean sampling strategy. Our results demonstrate that incorporating topological information into deep learning workflows significantly improves performance on these otherwise challenging EuLearn datasets.

We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse rendering of geometry rely on silhouette gradients for topology changes, such signals are sparse. In contrast, our theory derives topological derivatives that relate the introduction of vanishing holes and phases to changes in image intensity. As a result, we enable differentiable shape perturbations in the form of hole or phase nucleation. We validate the proposed theory with optimization of closed curves in 2D and surfaces in 3D to lend insights into limitations of current methods and enable improved applications such as image vectorization, vector-graphics generation from text prompts, single-image reconstruction of shape ambigrams and multi-view 3D reconstruction.

Using an evolutionary algorithm in combination with first-principles density functional theory calculations, we identify two-dimensional (2D) CaP_3 monolayer as a new Dirac semimetal due to inversion and nonsymmorphic spatial symmetries of the structure. This new topological material, composed of light elements, exhibits high structural stability (higher than the phase known in the literature), which is confirmed by thermodynamic and kinetic stability analysis. Moreover, it satisfies the electron filling criteria, so that its Dirac state is located near the Fermi level. The existence of the Dirac state predicted by the theoretical symmetry analysis is also confirmed by first-principles electronic band structure calculations. We find that the energy position of the Dirac state can be tuned by strain, while the Dirac state is unstable against an external electric field since it breaks the spatial inversion symmetry. Our findings should be instrumental in the development of 2D Dirac fermions based on light elements for their application in nanoelectronic devices and topological electronics.

We build a team of specialized large language-model agents and present an agent-driven workflow for research-level formalization in theoretical physics, with the autoformalization of the fundamental theorem of matrix-product states as a demonstration. The agents, coordinated through a structured mathematical blueprint and periodic human review, orchestrated and executed the full formalization autonomously. For some statements, the agents were able to explore new proof routes that are not part of the standard literature. Along the way the agents produced extensive tensor-network and quantum-information libraries not previously available in Mathlib, Lean's mathematical library. As a physical application, the formalization also extends towards symmetry-protected topological phases in one dimension. We find that the main bottleneck in large-scale autoformalization is enforcing mathematical intent and we provide a detailed study of the full process and various subtleties involved. We release the codebase as the library https://github.com/LionSR/TNLean{TNLean}, together with a -chapter https://lionsr.github.io/TNLean/blueprint/{blueprint} of the formalization effort.

Quality of epitaxial films strongly depends on their structural and chemical match with the substrates: the more closely they match, the better the film quality is. Topological insulators (TI) such as Bi2Se3 thin films are of no exception. However, there do not exist commercial substrates that match with TI films both structurally and chemically, at the level commonly available for other electronic materials. Here, we introduce BiInSe3 bulk crystal as the best substrate for Bi2Se3 thin films. These films exhibit superior surface morphology, lower defect density and higher Hall mobility than those on other substrates, due to structural and chemical match provided by the BiInSe3 substrate. BiInSe3 substrate could accelerate the advance of TI research and applications.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

I review two classes of strong coupling problems in condensed matter physics, and describe insights gained by application of the AdS/CFT correspondence. The first class concerns non-zero temperature dynamics and transport in the vicinity of quantum critical points described by relativistic field theories. I describe how relativistic structures arise in models of physical interest, present results for their quantum critical crossover functions and magneto-thermoelectric hydrodynamics. The second class concerns symmetry breaking transitions of two-dimensional systems in the presence of gapless electronic excitations at isolated points or along lines (i.e. Fermi surfaces) in the Brillouin zone. I describe the scaling structure of a recent theory of the Ising-nematic transition in metals, and discuss its possible connection to theories of Fermi surfaces obtained from simple AdS duals.

We show that two properly embedded compact surfaces in an orientable 4-manifold are cobordant if and only if they are Z/2-homologous and either the 4-manifold has boundary or the surfaces have the same normal Euler number. If the 4-manifold is simply-connected and the surfaces are closed, non-orientable, and cobordant, we show that they are in fact concordant. This completes the classification of closed surfaces in simply-connected 4-manifolds up to concordance. Our methods give new constructions of cobordisms with prescribed boundaries, and completely determine when a given cobordism between the boundaries extends to a cobordism or concordance between the surfaces. We obtain our concordance results by extending Sunukjian's method of ambient surgery to the unoriented case using Pin^--structures. We also discuss conditions for an arbitrary codimension 2 properly embedded submanifold to admit an unoriented spanning manifold with prescribed boundary. All results hold in both the smooth and topological categories.

We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by sequences of elementary braids of quasiparticles with exotic fractional statistics in certain two-dimensional topological condensed matter systems, described by effective topological quantum field theories. We specifically focus on a non-semisimple version of topological field theory, which provides a foundation for an extended theory of Ising anyons and which has recently been shown by Iulianelli et al., Nature Communications {\bf 16}, 6408 (2025), to permit universal quantum computation. While the proofs of this pioneering result are existential in nature, the mixed integer programming provides an approach to explicitly construct quantum gates in topological systems. We demonstrate this by focusing specifically on the entangling controlled-NOT operation, and its local equivalence class, using braiding operations in the non-semisimple Ising system. This illustrates the utility of the Mixed-Integer Quadratically Constrained Quadratic Programming for topological quantum compilation.

We analyze the static and dynamical properties of a one-dimensional topological lattice, the fermionic Su-Schrieffer-Heeger model, in the presence of on-site interactions. Based on a study of charge and spin correlation functions, we elucidate the nature of the topological edge modes, which depending on the sign of the interactions, either display particles of opposite spin on opposite edges, or a pair and a holon. This study of correlation functions also highlights the strong entanglement that exists between the opposite edges of the system. This last feature has remarkable consequences upon subjecting the system to a quench, where an instantaneous edge-to-edge signal appears in the correlation functions characterizing the edge modes. Besides, other correlation functions are shown to propagate in the bulk according to the light-cone imposed by the Lieb-Robinson bound. Our study reveals how one-dimensional lattices exhibiting entangled topological edge modes allow for a non-trivial correlation spreading, while providing an accessible platform to detect spin-charge separation using state-of-the-art experimental techniques.

We study the transverse-field axial next-nearest-neighbor Ising (ANNNI) model on the square lattice and on a bilayer. At the classical multiphase point κ=J_2/J_0=1/2, every configuration of straight domain walls that separates domains of at least two sites is a ground state. The walls are rigid, so perturbation theory in the transverse field g=Γ/J_0 can be carried out for each configuration separately, and we take it to order g^8. Quantum fluctuations select langle 3rangle (uparrowuparrowuparrowdownarrowdownarrowdownarrow) at order g^2, because spins next to a wall gain the most energy from virtual flips. The structures langle 23rangle, langle 223rangle and langle 2223rangle then open between langle 3rangle and the antiphase langle 2rangle at orders g^4, g^6 and g^8, in the order of the Fisher-Selke sequence of the classical three-dimensional model. A second layer suppresses each successive phase more strongly, beyond the stiffening of the local fields. We test the two widest phases with stochastic series expansion quantum Monte Carlo, comparing the free energies of the competing ordered states by thermodynamic integration, since annealing freezes in the modulation that forms at the ordering transition. The simulations reproduce the langle 3rangle wedge to about 1%, and in a pre-registered test langle 23rangle is stable at six of seven points, with the seventh resolved by post-hoc runs.

Previous theoretical and experimental research has shown that current NISQ devices constitute powerful platforms for analogue quantum simulation. With the exquisite level of control offered by state-of-the-art quantum computers, we show that one can go further and implement a wide class of Floquet Hamiltonians, or timedependent Hamiltonians in general. We then implement a single-qubit version of these models in the IBM Quantum Experience and experimentally realize a temporal version of the Bernevig-Hughes-Zhang Chern insulator. From our data we can infer the presence of a topological transition, thus realizing an earlier proposal of topological frequency conversion by Martin, Refael, and Halperin. Our study highlights promises and limitations when studying many-body systems through multi-frequency driving of quantum computers.

Axion insulators possess a quantized axion field theta=pi protected by combined lattice and time-reversal symmetry, holding great potential for device applications in layertronics and quantum computing. Here, we propose a high-spin axion insulator (HSAI) defined in large spin-s representation, which maintains the same inherent symmetry but possesses a notable axion field theta=(s+1/2)^2pi. Such distinct axion field is confirmed independently by the direct calculation of the axion term using hybrid Wannier functions, layer-resolved Chern numbers, as well as the topological magneto-electric effect. We show that the guaranteed gapless quasi-particle excitation is absent at the boundary of the HSAI despite its integer surface Chern number, hinting an unusual quantum anomaly violating the conventional bulk-boundary correspondence. Furthermore, we ascertain that the axion field theta can be precisely tuned through an external magnetic field, enabling the manipulation of bonded transport properties. The HSAI proposed here can be experimentally verified in ultra-cold atoms by the quantized non-reciprocal conductance or topological magnetoelectric response. Our work enriches the understanding of axion insulators in condensed matter physics, paving the way for future device applications.

This short Perspective article presents an overview of the discovery of topological π modes as well as their physical significance in quantum computing and the understanding of an exotic phase of matter, i.e., the Floquet time crystal. The recent proposals of 2π/k modes as the generalizations of π modes are further elucidated.

We study theories of type D_4 in class-S, with nonabelian outer-automorphism twists around various cycles of the punctured Riemann surface C. We propose an extension of previous formulæ for the superconformal index to cover this case and classify the SCFTs corresponding to fixtures (3-punctured spheres). We then go on to study families of SCFTs corresponding to once-punctured tori and 4-punctured spheres. These exhibit new behaviours, not seen in previous investigations. In particular, the generic theory with 4 punctures on the sphere from non-commuting Z_2 twisted sectors has six distinct weakly-coupled descriptions.

We elaborate on the role of extremal surfaces probing the internal space in AdS/CFT. Extremal surfaces in AdS quantify the "geometric" entanglement between different regions in physical space for the dual CFT. This, however, is just one of many ways to split a given system into subsectors, and extremal surfaces in the internal space should similarly quantify entanglement between subsectors of the theory. For the case of AdS_5timesS^5, their area was interpreted as entanglement entropy between U(n) and U(m) subsectors of U(n+m) N=4 SYM. Making this proposal precise is subtle for a number of reasons, the most obvious being that from the bulk one usually has access to gauge-invariant quantities only, while a split into subgroups is inherently gauge variant. We study N=4 SYM on the Coulomb branch, where some of the issues can be mitigated and the proposal can be sharpened. Continuing back to the original AdS_5timesS^5 geometry, we obtain a modified proposal, based on the relation of the internal space to the R-symmetry group.

A general approach is formulated to describe spontaneous surface current distribution in a chiral p-wave superconductor. We use the quasiclassical Eilenberger formalism in the Ricatti parametrization to describe various types of the superconductor surface, including arbitrary roughness and metallic behaviour of the surface layer. We calculate angle resolved distributions of the spontaneous surface currents and formulate the conditions of their observability. We argue that local measurements of these currents by muSR technique may provide an information on the underlying pairing symmetry in the bulk superconductor.

Riemannian four-manifolds in which the triple contraction of the curvature tensor against itself yields a functional multiple of the metric are called weakly Einstein. We focus on weakly Einstein K\"ahler surfaces. We provide several conditions characterizing those K\"ahler surfaces which are weakly Einstein, classify weakly Einstein K\"ahler surfaces having some specific additional properties, and construct new examples.

This work investigates the q-deformation of (r,s)-Airy structures and their realization via q-difference operators, providing a bridge between quantum spectral curves and integrable systems. We construct an all-order q-WKB solution for the matrix systems associated with the q-quantized curve E_q(x,y)=0. We demonstrate that the resulting non-perturbative connected q-amplitudes satisfy a set of shifted q-loop equations, which can be interpreted as the Ward identities of a q-deformed W(gl_r) algebra. Our main result provides a rigorous classification of admissible (r,s,q) pairs and q-Casimir configurations that satisfy the q-topological type property. This ensures that the semi-classical expansion is uniquely governed by the q-topological recursion, offering new insights into the q-quantization of mirror curves and their underlying algebraic structures.

Topological kagome systems have been a topic of great interest in condensed matter physics due totheir unique electronic properties. The vanadium-based kagome materials are particularly intrigu-ing since they exhibit exotic phenomena such as charge density wave (CDW) and unconventionalsuperconductivity. The origin of these electronic instabilities is not fully understood, and the re-cent discovery of a charge density wave in ScV6Sn6provides a new avenue for investigation. In thiswork, we investigate the electronic structure of the novel kagome metal ScV6Sn6using angle resolvedphotoemission spectroscopy (ARPES), scanning tunneling microscopy (STM), and first-principlesdensity functional theory calculations. Our analysis reveals for the first time the temperature-dependent band changes of ScV6Sn6and identifies a new band that exhibits a strong signatureof a structure with CDW below the critical temperature. Further analysis revealed that this newband is due to the surface kagome layer of the CDW structure. In addition, a Lifshitz transition isidentified in the ARPES spectra that is related to the saddle point moving across the Fermi levelat the critical temperature for the CDW formation. This result shows the CDW behavior may alsobe related to nesting of the saddle point, similar to related materials. However, no energy gap is observed at the Fermi level and thus the CDW is not a typical Fermi surface nesting scenario. These results provide new insights into the underlying physics of the CDW in the kagome materials and could have implications for the development of materials with new functionality.

We identify the 2d topological theory underlying the N=2 4d superconformal index with an explicit model: q-deformed 2d Yang-Mills. By this route we are able to evaluate the index of some strongly-coupled 4d SCFTs, such as Gaiotto's T_N theories.

We study the superconformal index for the class of N=2 4d superconformal field theories recently introduced by Gaiotto. These theories are defined by compactifying the (2,0) 6d theory on a Riemann surface with punctures. We interpret the index of the 4d theory associated to an n-punctured Riemann surface as the n-point correlation function of a 2d topological QFT living on the surface. Invariance of the index under generalized S-duality transformations (the mapping class group of the Riemann surface) translates into associativity of the operator algebra of the 2d TQFT. In the A_1 case, for which the 4d SCFTs have a Lagrangian realization, the structure constants and metric of the 2d TQFT can be calculated explicitly in terms of elliptic gamma functions. Associativity then holds thanks to a remarkable symmetry of an elliptic hypergeometric beta integral, proved very recently by van de Bult.

We report the anomalous and topological Hall effect of the ferromagnetic Weyl semimetal Mn_5Ge_3. We observe a significant anisotropic anomalous Hall effect (AHE) due to nonzero Berry curvature in the momentum space, such that the anomalous Hall conductivity (AHC) is 965 S/cm for the xy-plane and 233 S/cm for the zx-plane of the single crystal. The band structure calculations predict several Weyl and nodal points span across the momentum space, gapped out under the spin-orbit coupling effect, leading to significant k-space Berry curvature and large AHC. Experimentally, we also demonstrate a sizeable topological Hall effect that is originated by the non-coplanar chiral spin structure due to the competition between the out-of-plane uniaxial magnetocrystalline anisotropy and the dipole-dipole interaction between two Mn sublattices. This study hints at the importance of dipole-dipole interactions in producing the skyrmion lattice in Mn_5Ge_3.

In 2021 Nature Physics published a paper by Vaitiekenas, Liu, Krogstrup and Marcus titled "Zero-bias peaks at zero magnetic field in ferromagnetic hybrid nanowires". The paper reports low temperature transport measurements on semiconductor InAs nanowires with two partly overlapping shells -- a shell of EuS, a magnetic insulator, and a shell of Al, a metal that becomes superconducting at temperatures below 1.2K. The paper claims that (1) the data are consistent with induced topological superconductivity and Majorana zero modes (MZMs), and (2) that this is facilitated by the breaking of the time reversal symmetry through a direct magnetic interaction with the EuS shell. In this Matters Arising, we present an alternative explanation which is based on trivial effects that are likely to appear in the reported geometry. Specifically, first, we find that data the authors present in support of the topological superconductivity claim can originate from unintended quantum dots in their devices, a widely known likely explanation that is not being discussed in the paper. Second, our analysis of the setup, supported by our numerical micromagnetic simulations, shows similar effects could be obtained due to stray magnetic fields from the region of the EuS shell damaged during Al etching. This basic picture should come before the exotic interpretation in terms of magnetic exchange interaction with a ferromagnetic insulator.

Transitions among quantum Hall plateaux share a suite of remarkable experimental features, such as semi-circle laws and duality relations, whose accuracy and robustness are difficult to explain directly in terms of the detailed dynamics of the microscopic electrons. They would naturally follow if the low-energy transport properties were governed by an emergent discrete duality group relating the different plateaux, but no explicit examples of interacting systems having such a group are known. Recent progress using the AdS/CFT correspondence has identified examples with similar duality groups, but without the DC ohmic conductivity characteristic of quantum Hall experiments. We use this to propose a simple holographic model for low-energy quantum Hall systems, with a nonzero DC conductivity that automatically exhibits all of the observed consequences of duality, including the existence of the plateaux and the semi-circle transitions between them. The model can be regarded as a strongly coupled analog of the old `composite boson' picture of quantum Hall systems. Non-universal features of the model can be used to test whether it describes actual materials, and we comment on some of these in our proposed model.

We evaluate electromagnetic-response observables in a half-filled Chern band, across a topological phase transition between a composite Fermi liquid (CFL) and a Fermi liquid (FL) phase. While a sharp gapped plasma mode exists deep in the CFL phase, we demonstrate that it is damped near the proposed continuous phase transition between CFL and FL. This plasmon-damping phenomenon originates from emergent gauge fields and a Dirac-fermion-like spectrum. Similar features also occur in other continuous deconfined topological phase transitions, such as the Laughlin to superfluid transition in a bosonic system. In particular, this damping behavior extends over a finite range across the phase boundary, and, hence, we expect it to persist even when the transition is weakly first-order. Furthermore, we analyze the behavior of the Drude weight, the wavevector-dependent conductivity, and the chiral mirror effect across these topological phase transitions.

Moiré semiconductors provide a physical realization of the Hubbard model. Bandwidth-tuned Mott transitions and complex phase diagrams mimicking those of high-temperature superconductors have been demonstrated near a weak Mott insulating state at half-band-filling. The platform also provides an opportunity to explore the physics of a partially filled Hubbard band at doping densities far away from the Mott insulator, a regime that has remained largely unexplored. Here we report the emergence of an extended strange metal (ESM), in both doping density and twist angle, in a partially filled Hubbard band of twisted bilayer WSe2. The ESM is in the clean limit, persists down to 100mK, and spans over multiple symmetry-breaking ground states; it also appears only at small twist angles, where electronic correlations are strong. These observations distinguish the state from the reported strange metals associated with quantum criticality and favor the viewpoint that it is an intrinsic state of matter. We also report the observation of a new superconducting dome being developed from the ESM but far away from both the Mott insulating state and the van Hove singularity of the material. Our work opens the door to exploring the new regime of partially filled Hubbard bands in the clean limit.

Cloth manipulation is a difficult problem mainly because of the non-rigid nature of cloth, which makes a good representation of deformation essential. We present a new representation for the deformation-state of clothes. First, we propose the dGLI disk representation, based on topological indices computed for segments on the edges of the cloth mesh border that are arranged on a circular grid. The heat-map of the dGLI disk uncovers patterns that correspond to features of the cloth state that are consistent for different shapes, sizes of positions of the cloth, like the corners and the fold locations. We then abstract these important features from the dGLI disk onto a circle, calling it the Cloth StatE representation (CloSE). This representation is compact, continuous, and general for different shapes. Finally, we show the strengths of this representation in two relevant applications: semantic labeling and high- and low-level planning. The code, the dataset and the video can be accessed from : https://jaykamat99.github.io/close-representation

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

It is well established that the spectral analysis of canonically quantized four-dimensional Seiberg-Witten curves can be systematically studied via the Nekrasov-Shatashvili functions. In this paper, we explore another aspect of the relation between N=2 supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlev\'e equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg-Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an O(2) matrix model. We then show that these eigenfunctions are computed by surface defects in SU(2) super Yang-Mills in the self-dual phase of the Omega-background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.

We explore the use of topological data analysis (TDA) combined with machine learning for discriminating standard model backgrounds from the invisible decay of the Z^prime boson associated with monophoton emission at a 3 TeV muon collider. Reconstructed events are mapped into a six-dimensional kinematic space and aggregated into bags of events, from which persistent homology is used to extract Betti number distributions. Within the Multiple Instance Learning paradigm, classifiers trained on these topological descriptors demonstrate significantly improved classification accuracy compared to the conventional ML approaches based on event-wise kinematic inputs. We also draw exclusion contours at 95\% CL in the (m_{Z^prime}, m_chi) parameter space, highlighting the potential of topological features to extend the discovery reach of future collider experiments.

Ising spin-orbit coupling (SOC) is widely known for locking electronic spins and protecting superconductivity against in-plane paramagnetic pair breaking. Here we show that, in a correlated multiorbital system, Ising SOC can play a qualitatively different role: it reconstructs the hierarchy of spin-fluctuation-mediated pairing channels by reshaping the low-energy Fermi surfaces, orbital-spin coherence factors, and momentum-resolved scattering phase space. We study a noncentrosymmetric two-orbital triangular-lattice model with Rashba SOC, Ising SOC, and onsite SOC. We solve the pairing problem within a multiorbital random-phase-approximation framework. In the single-orbital limit, an E-representation state with a dominant S_z=0 component emerges only at relatively strong Ising SOC. Orbital degrees of freedom facilitate the stabilization of this pairing state at a smaller nominal Ising SOC strength and generate a broad competition between E and A_1 pairing channels. The E state can maintain the same S_z=0 triplet character, whereas the A_1 state becomes a fully gapped sign-changing splusminus superconductor. The latter also realizes a topologically nontrivial phase in the time-reversal-invariant weak-pairing regime, with a nonzero Z_2 index, Wilson-loop spectral flow, and boundary modes. Its topology is not carried by a single orbital pairing component, but emerges cooperatively from the dominant intra- and inter-orbital pairing components. These results establish a microscopic route by which Ising SOC and orbital hybridization can reconstruct interaction-driven superconductivity, producing both Ising-triplet and topological singlet-dominated phases in correlated non-centrosymmetric systems.

The spin liquid phase is one of the prominent strongly interacting topological phases of matter whose unambiguous confirmation is yet to be reached despite intensive experimental efforts on numerous candidate materials. Recently, a new family of correlated honeycomb materials, in which strong spin-orbit coupling allows for various bond-dependent spin interactions, have been promising candidates to realize the Kitaev spin liquid. Here we study a model with bond-dependent spin interactions and show numerical evidence for the existence of an extended quantum spin liquid region, which is possibly connected to the Kitaev spin liquid state. These results are used to provide an explanation of the scattering continuum seen in neutron scattering on α-RuCl_3.

Entanglement entropy plays a central role in holography. The quantum extremal surface (QES) formula is the standard tool in settings where both dynamics and bulk quantum corrections are important, such as in black hole evaporation. We derive and explore QES-equivalent prescriptions --- quantum covariant bit threads --- that formulate holographic entanglement entropy in terms of minimal or maximal bulk flows rather than extremal surfaces. Like classical covariant bit threads, our prescriptions come in two types: (typically) spacelike V-flows, and timelike U-flows. To account for bulk quantum corrections, the flow lines can start and end at points in the bulk, with the distribution of endpoints controlled by bulk entanglement entropies. In our bit thread language, the QES is a quantum bottleneck to the flow, with some threads passing through it (accounting for the area term in the generalized entropy) and others jumping across it (accounting for the entropy term). Our prescriptions provide a complementary perspective to the QES formula and offer new insights on important phenomena such as black hole evaporation, islands, and baby universes.

Among a huge variety of known two-dimensional materials, some of them have anisotropic crystal structures; examples include so different systems as a few-layer black phoshphorus (phosphorene), beryllium nitride BeN_4, van der Waals magnet CrSBr, rhenium dichalgogenides ReX_2. As a consequence, their optical and electronic properties turn out to be highly anisotropic as well. In some cases, the anisotropy results not just in a smooth renormalization of observable properties in comparison with the isotropic case but in the appearance of dramatically new physics. The examples are hyperbolic plasmons and excitons, strongly anisotropic ordering of adatoms at the surface of two-dimensional or van der Waals materials, essential change of transport and superconducting properties. Here, we present a systematic review of electronic structure, transport and optical properties of several representative groups of anisotropic two-dimensional materials including semiconductors, anisotropic Dirac and semi-Dirac materials, as well as superconductors.

We investigate modified gravity with quadratic curvature terms (R^2, R_{AB}R^{AB}, and R_{ABCD}R^{ABCD}). In this framework, the geometry of the extra dimensions is determined by the higher-dimensional field equations together with auxiliary conditions. For inhomogeneous metrics, we derive new stability conditions under radion excitations. The central result is that stable solutions satisfying these conditions are not isolated: a typical stable regular metric belongs to a continuous family of stable metrics in its immediate neighbourhood. Since the effective four-dimensional physical parameters depend on the extra-dimensional geometry, this geometrical landscape of stable metrics induces a continuous landscape of effective four-dimensional parameters. Transitions between distinct static solutions within the landscape cannot be induced by spatially localized perturbations, whose asymptotic decay (|x|toinfty) preserves the boundary data distinguishing the states. We also note a possible topological transition from a two-brane to a one-brane configuration, associated with a change in the number of regular zeros of the metric function r(u).

I introduce a novel mathematical framework integrating topological dynamics, operator algebras, and ergodic geometry to study lattices of asynchronous metric dynamical systems. Each node in the lattice carries an internal flow represented by a one-parameter family of operators, evolving on its own time scale. I formalize stratified state spaces capturing multiple levels of synchronized behavior, define an asynchronous evolution metric that quantifies phase-offset distances between subsystems, and characterize emergent coherent topologies arising when subsystems synchronize. Within this framework, I develop formal operators for the evolution of each subsystem and give precise conditions under which phase-aligned synchronization occurs across the lattice. The main results include: (1) the existence and uniqueness of coherent (synchronized) states under a contractive coupling condition, (2) stability of these coherent states and criteria for their emergence as a collective phase transition in a continuous operator topology, and (3) the influence of symmetries, with group-invariant coupling leading to flow-invariant synchrony subspaces and structured cluster dynamics. Proofs are given for each theorem, demonstrating full mathematical rigor. In a final section, I discuss hypothetical applications of this framework to symbolic lattice systems (e.g. subshifts), to invariant group actions on dynamical lattices, and to operator fields over stratified manifolds in the spirit of noncommutative geometry. Throughout, I write in the first person to emphasize the exploratory nature of this work. The paper avoids any reference to cosmology or observers, focusing instead on clean, formal mathematics suitable for a broad array of dynamical systems.

We present a topology-informed inverse rendering approach for reconstructing high-genus surface meshes from multi-view images. Compared to 3D representations like voxels and point clouds, mesh-based representations are preferred as they enable the application of differential geometry theory and are optimized for modern graphics pipelines. However, existing inverse rendering methods often fail catastrophically on high-genus surfaces, leading to the loss of key topological features, and tend to oversmooth low-genus surfaces, resulting in the loss of surface details. This failure stems from their overreliance on Adam-based optimizers, which can lead to vanishing and exploding gradients. To overcome these challenges, we introduce an adaptive V-cycle remeshing scheme in conjunction with a re-parametrized Adam optimizer to enhance topological and geometric awareness. By periodically coarsening and refining the deforming mesh, our method informs mesh vertices of their current topology and geometry before optimization, mitigating gradient issues while preserving essential topological features. Additionally, we enforce topological consistency by constructing topological primitives with genus numbers that match those of ground truth using Gauss-Bonnet theorem. Experimental results demonstrate that our inverse rendering approach outperforms the current state-of-the-art method, achieving significant improvements in Chamfer Distance and Volume IoU, particularly for high-genus surfaces, while also enhancing surface details for low-genus surfaces.

Topology is, by its nature and design, suited to structure that is nonlinear, multiscale, and nonstationary - however, within machine learning, its use remains largely confined to topological data analysis. We advocate that tools from low-dimensional topology which have remained almost exclusively contained within the domain of pure mathematics (such as Morse theory) offer a strong, complementary, and yet virtually unexplored perspective on the hidden structure of data-generating processes and learning tasks built upon them. Here we introduce concepts from cobordism theory and harness tools from discrete Morse theory to improve the performance of graph diffusion models through our pipeline MG-Diff. Further, we derive theoretical guarantees and sufficient conditions so that under a positive decision-gap, the Morse-theoretic tools and their application for induced diffusion guidance are stable under small perturbations. Finally, we illustrate the utility of discrete Morse theory in application to graph diffusion models for spatio-temporal graph forecasting and graph regeneration, and argue that these applications are only a small window into the part of what low-dimensional topology can offer to the field of machine learning.

Two-dimensional (2D) materials have emerged as a versatile and powerful platform for quantum technologies, offering atomic-scale control, strong quantum confinement, and seamless integration into heterogeneous device architectures. Their reduced dimensionality enables unique quantum phenomena, including optically addressable spin defects, tunable single-photon emitters, low-dimensional magnetism, gate-controlled superconductivity, and correlated states in Moiré superlattices. This Roadmap provides a comprehensive overview of recent progress and future directions in exploiting 2D materials for quantum sensing, computation, communication, and simulation. We survey advances spanning spin defects and quantum sensing, quantum emitters and nonlinear photonics, computational theory and data-driven discovery of quantum defects, spintronic and magnonic devices, cavity-engineered quantum materials, superconducting and hybrid quantum circuits, quantum dots, Moiré quantum simulators, and quantum communication platforms. Across these themes, we identify common challenges in defect control, coherence preservation, interfacial engineering, and scalable integration, alongside emerging opportunities driven by machine-learning-assisted design and integrated experiment-theory feedback loops. By connecting microscopic quantum states to mesoscopic excitations and macroscopic device architectures, this Roadmap outlines a materials-centric framework for integrating coherent quantum functionalities and positions 2D materials as foundational building blocks for next-generation quantum technologies.

The Kohn-Luttinger mechanism for unconventional superconductivity (SC) driven by weak repulsive electron-electron interactions on a periodic lattice is generalized to the quasicrystal (QC) via a real-space perturbative approach. The repulsive Hubbard model on the Penrose lattice is studied as an example, on which a classification of the pairing symmetries is performed and a pairing phase diagram is obtained. Two remarkable properties of these pairing states are revealed, due to the combination of the presence of the point-group symmetry and the lack of translation symmetry on this lattice. Firstly, the spin and spacial angular momenta of a Cooper pair is de-correlated: for each pairing symmetry, both spin-singlet and spin-triplet pairings are possible even in the weak-pairing limit. Secondly, the pairing states belonging to the 2D irreducible representations of the D_5 point group can be time-reversal-symmetry-breaking topological SCs carrying spontaneous bulk super current and spontaneous vortices. These two remarkable properties are general for the SCs on all QCs, and are rare on periodic lattices. Our work starts the new area of unconventional SCs driven by repulsive interactions on the QC.

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

To each four dimensional Ngeq 2 supersymmetric quantum field theory, one can associate an algebraic completely integrable (ACI) system that encodes the low energy dynamics of theory. In this paper we explicitly derive the appropriate ACI systems for the global forms of N=4 super Yang-Mills (sYM) using isogenies of polarised abelian varieties. In doing so, we relate the complex moduli of the resulting varieties to the exactly marginal coupling of the theory, thus allowing us to probe the S-duality groups of the global forms. Finally, we comment on whether the resulting varieties are the Jacobians of a minimal genus Riemann surface, coming to the conclusion that many global forms of N=4 sYM do not admit a minimal genus Seiberg-Witten curve that correctly reproduces the global form.

Almost one century ago, string states - complex bound states (Wellenkomplexe) of magnetic excitations - have been predicted to exist in one-dimensional quantum magnets and since then become a subject of intensive theoretical study. However, experimental realization and identification of string states in condensed-matter systems remains an unsolved challenge up to date. Here we use high-resolution terahertz spectroscopy to identify string states in the antiferromagnetic Heisenberg-Ising chain SrCo2V2O8 in strong longitudinal magnetic fields. We observe complex bound states (strings) and fractional magnetic excitations (psinons and antipsinons) in the field-induced critical regime, which are precisely described by the Bethe ansatz. Our study reveals that two-string and three-string states govern the quantum spin dynamics close to the quantum criticality, while the fractional excitations are dominant at low energies, reflecting the antiferromagnetic quantum fluctuations.

The mobility edge (ME) is a critical energy delineates the boundary between extended and localized states within the energy spectrum, and it plays a crucial role in understanding the metal-insulator transition in disordered or quasiperiodic systems. While there have been extensive studies on MEs in one-dimensional non-Hermitian (NH) quasiperiodic lattices recently, the investigation of exact NH MEs in two-dimensional (2D) cases remains rare. In the present study, we introduce a 2D dissipative Lieb lattice (DLL) model with imaginary quasiperiodic potentials applied solely to the vertices of the Lieb lattice. By mapping this DLL model to the 2D NH Aubry-Andr{é}-Harper (AAH) model, we analytically derive the exact ME and find it associated with the absolute eigenenergies. We find that the eigenvalues of extended states are purely imaginary when the quasiperiodic potential is strong enough. Additionally, we demonstrate that the introduction of imaginary quasiperiodic potentials does not disrupt the flat bands inherent in the system. Finally, we propose a theoretical framework for realizing our model using the Lindblad master equation. Our results pave the way for further investigation of exact NH MEs and flat bands in 2D dissipative quasiperiodic systems.

We have investigated the effects of strain on two-dimensional square lattices and examined the methods for inducing pseudo-magnetic fields. In both the columnar and staggered pi-flux square lattices, we have found that strain only modulates Fermi velocities rather than inducing pseudo-magnetic fields. However, spatially non-uniform on-site potentials (anisotropic hoppings) can create pseudo-magnetic fields in columnar (staggered) pi-flux square lattices. On the other hand, we demonstrate that strain does induce pseudo-magnetic fields in staggered zero-flux square lattices. By breaking a quarter of the bonds, we clarify that a staggered zero-flux square lattice is topologically equivalent to a honeycomb lattice and displays pseudo-vector potentials and pseudo-Landau levels at the Dirac points.

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

The paper is a continuation of the study started in Yorzh1. Schrodinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of delta type. Either an infinite series of trace formulae (provided that edge potentials are infinitely smooth) or a finite number of such formulae (in the cases of L_1 and C^M edge potentials) are obtained which link together two different quantum graphs under the assumption that their spectra coincide. Applications are given to the problem of recovering matching conditions for a quantum graph based on its spectrum.

We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These Bonnet pair tori are the first examples of compact Bonnet pairs. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique smooth compact immersion. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. Our construction uses the relationship between Bonnet pairs and isothermic surfaces. The Bonnet pair tori arise as conformal transformations of an isothermic torus with one family of planar curvature lines. We classify such isothermic tori in our companion paper (arXiv:2312.14956). The above approach stems from computational investigations of a 5x7 quad decomposition of a torus using a discrete differential geometric analog of isothermic surfaces and Bonnet pairs.

Motivated by the recent experimental observation of the topological Anderson insulator in disordered atomic wires based on the Su-Schrieffer-Heeger (SSH) model, we study disorder effects on a dimerized Kitaev superconductor chain which is regarded as the superconductor version of the SSH model. By computing the real-space winding number and the zero-bias differential conductance, we analyze the topological phase transitions occurring in a dimerized Kitaev superconductor chain with disorder. It is found that disorder can induce a topologically nontrivial superconductor phase hosting Majorana zero modes (MZMs). We can regulate the appearance of disorder-induced MZMs by adjusting the dimerization parameter. Finally, we use the self-consistent Born approximation method to verify the numerical results.

Classifying the topology of closed curves is a central problem in low dimensional topology with applications beyond mathematics spanning protein folding, polymer physics and even magnetohydrodynamics. The central problem is how to determine whether two embeddings of a closed arc are equivalent under ambient isotopy. Given the striking ability of neural networks to solve complex classification tasks, it is therefore natural to ask if the knot classification problem can be tackled using Machine Learning (ML). In this paper, we investigate generic shortcut methods employed by ML to solve the knot classification challenge and specifically discover hidden non-topological features in training data generated through Molecular Dynamics simulations of polygonal knots that are used by ML to arrive to positive classifications results. We then provide a rigorous foundation for future attempts to tackle the knot classification challenge using ML by developing a publicly-available (i) dataset, that aims to remove the potential of non-topological feature classification and (ii) code, that can generate knot embeddings that faithfully explore chosen geometric state space with fixed knot topology. We expect that our work will accelerate the development of ML models that can solve complex geometric knot classification challenges.

Adversarially perturbed images of text can cause sophisticated OCR systems to produce misleading or incorrect transcriptions from seemingly invisible changes to humans. Some of these perturbations even survive physical capture, posing security risks to high-stakes applications such as document processing, license plate recognition, and automated compliance systems. Existing defenses, such as adversarial training, input preprocessing, or post-recognition correction, are often model-specific, computationally expensive, and affect performance on unperturbed inputs while remaining vulnerable to unseen or adaptive attacks. To address these challenges, TopoReformer is introduced, a model-agnostic reformation pipeline that mitigates adversarial perturbations while preserving the structural integrity of text images. Topology studies properties of shapes and spaces that remain unchanged under continuous deformations, focusing on global structures such as connectivity, holes, and loops rather than exact distance. Leveraging these topological features, TopoReformer employs a topological autoencoder to enforce manifold-level consistency in latent space and improve robustness without explicit gradient regularization. The proposed method is benchmarked on EMNIST, MNIST, against standard adversarial attacks (FGSM, PGD, Carlini-Wagner), adaptive attacks (EOT, BDPA), and an OCR-specific watermark attack (FAWA).

Structural analyses are an integral part of computational research on nucleation and supercooled water, whose accuracy and efficiency can impact the validity and feasibility of such studies. The underlying molecular mechanisms of these often elusive and computationally expensive processes can be inferred from the evolution of ice-like structures, determined using appropriate structural analysis techniques. We present d-SEAMS, a free and open-source post-processing engine for the analysis of molecular dynamics trajectories, which is specifically able to qualitatively classify ice structures, in both strong confinement and bulk systems. For the first time, recent algorithms for confined ice structure determination have been implemented, along with topological network criteria for bulk ice structure determination. Recognizing the need for customization in structural analysis, d-SEAMS has a unique code architecture, built with `nix`, employing a `YAML`-`Lua` scripting pipeline. The software has been designed to be user-friendly and easy to extend. The engine outputs are compatible with popular graphics software suites, allowing for immediate visual insights into the systems studied. We demonstrate the features of d-SEAMS by using it to analyze nucleation in the bulk regime and for quasi-one and quasi-two-dimensional systems. Structural time evolution and quantitative metrics are determined for heterogenous ice nucleation on a silver-exposed beta-AgI surface, homogenous ice nucleation, flat monolayer square ice formation and freezing of an ice nanotube.

We attempt to reveal the geometry, emerged from the entanglement structure of any general N-party pure quantum many-body state by representing entanglement entropies corresponding to all 2^N bipartitions of the state by means of a generalized adjacency matrix. We show this representation is often exact and may lead to a geometry very different than suggested by the Hamiltonian. Moreover, in all the cases, it yields a natural entanglement contour, similar to previous proposals. The formalism is extended for conformal invariant systems, and a more insightful interpretation of entanglement is presented as a flow among different parts of the system.

Along the way initiated by Carleo and Troyer [1], we construct the neural-network quantum state of transverse-field Ising model(TFIM) by an unsupervised machine learning method. Such a wave function is a map from the spin-configuration space to the complex number field determined by an array of network parameters. To get the ground state of the system, values of the network parameters are calculated by a Stochastic Reconfiguration(SR) method. We provide for this SR method an understanding from action principle and information geometry aspects. With this quantum state, we calculate key observables of the system, the energy, correlation function, correlation length, magnetic moment and susceptibility. As innovations, we provide a high efficiency method and use it to calculate entanglement entropy (EE) of the system and get results consistent with previous work very well.

Classical simulation of quantum physics is a central approach to investigating physical phenomena. Quantum computers enhance computational capabilities beyond those of classical resources, but it remains unclear to what extent existing limited quantum computers can contribute to this enhancement. In this work, we explore a new hybrid, efficient quantum-classical representation of quantum states, the multiple-basis representation. This representation consists of a linear combination of states that are sparse in some given and different bases, specified by quantum circuits. Such representation is particularly appealing when considering depth-limited quantum circuits within reach of current hardware. We analyze the expressivity of multiple-basis representation states depending on the classical simulability of their quantum circuits. In particular, we show that multiple-basis representation states include, but are not restricted to, both matrix-product states and stabilizer states. Furthermore, we find cases in which this representation can be used, namely approximation of ground states, simulation of deeper computations by specifying bases with shallow circuits, and a tomographical protocol to describe states as multiple-basis representations. We envision this work to open the path of simultaneous use of several hardware-friendly bases, a natural description of hybrid computational methods accessible for near-term hardware.

We consider the superconformal index of class S theories of type D, which arise by compactification of the (2,0) D_n theories on a punctured Riemann surface C. We also allow for the presence of twist lines on C associated to the Z_2 outer automorphism of D_n. For a two-parameter slice (p=0, q, t) in the space of superconformal fugacities, we determine the 2d TQFT that computes the index.

Topological data analysis (TDA) is a powerful technique for extracting complex and valuable shape-related summaries of high-dimensional data. However, the computational demands of classical algorithms for computing TDA are exorbitant, and quickly become impractical for high-order characteristics. Quantum computers offer the potential of achieving significant speedup for certain computational problems. Indeed, TDA has been purported to be one such problem, yet, quantum computing algorithms proposed for the problem, such as the original Quantum TDA (QTDA) formulation by Lloyd, Garnerone and Zanardi, require fault-tolerance qualifications that are currently unavailable. In this study, we present NISQ-TDA, a fully implemented end-to-end quantum machine learning algorithm needing only a short circuit-depth, that is applicable to high-dimensional classical data, and with provable asymptotic speedup for certain classes of problems. The algorithm neither suffers from the data-loading problem nor does it need to store the input data on the quantum computer explicitly. The algorithm was successfully executed on quantum computing devices, as well as on noisy quantum simulators, applied to small datasets. Preliminary empirical results suggest that the algorithm is robust to noise.

We present deep observations in targeted regions of the string landscape through a combination of analytic and dedicated numerical methods. Specifically, we devise an algorithm designed for the systematic construction of Type IIB flux vacua in finite regions of moduli space. Our algorithm is universally applicable across Calabi-Yau orientifold compactifications and can be used to enumerate flux vacua in a region given sufficient computational efforts. As a concrete example, we apply our methods to a two-modulus Calabi-Yau threefold, demonstrating that systematic enumeration is feasible and revealing intricate structures in vacuum distributions. Our results highlight local deviations from statistical expectations, providing insights into vacuum densities, superpotential distributions, and moduli mass hierarchies. This approach opens pathways for precise, data-driven mappings of the string landscape, complementing analytic studies and advancing the understanding of the distribution of flux vacua. This allows us to obtain different types of solutions with hierarchical suppressions, e.g.~vacua with small values of the Gukov-Vafa-Witten superpotential |W_0|. We find an example with |W_0| = 5.547 times 10^{-5} at large complex structure, without light directions and the use of non-perturbative effects.

We construct new classical global solutions of the one-phase Bernoulli free boundary problem in R^3 whose positive phases are periodic perturbations of the product of the Hauswirth--Hélein--Pacard double hairpin domain with a line. In contrast with previous constructions of undulating cylinders, the free boundary of the base solution is not compact modulo translations. More generally, we show that the same scheme yields periodic solutions in R^{n+1}, n geq 3, bifurcating from product solutions U(x,t)=u(x), when the base u:R^nto [0,infty) has nontrivial finite Morse index, is nondegenerate within a symmetry class, and satisfies certain natural geometric conditions at infinity. In dimensions nin{3,4}, where the stable Bernstein theorem is available, the recent classification of finite Morse index solutions by Fernández-Real, Florit-Simon and Serra shows that every such solution satisfies these geometric conditions. Thus, the existence of undulating solutions in R^{4} and R^{5} bifurcating from their respective products reduces to a nondegeneracy property of the base solution. In order to confirm that the perturbed solutions are positive, we prove that finite Morse index solutions with noncompact free boundary satisfy a strong distance nondegeneracy estimate.

Spintronics aims to utilize the spin degree of freedom for information storage and computing applications. One major issue is the generation and detection of spins via spin and charge conversion. Quantum materials have recently exhibited many unique spin-dependent properties, which can be used as promising material candidates for efficient spin and charge conversion. Here, we review recent findings concerning spin and charge conversion in quantum materials, including Rashba interfaces, topological insulators, two-dimensional materials, superconductors, and non-collinear antiferromagnets. Important progress in using quantum materials for spin and charge conversion could pave the way for developing future spintronics devices.

A fundamental problem in quantum many-body physics is that of finding ground states of local Hamiltonians. A number of recent works gave provably efficient machine learning (ML) algorithms for learning ground states. Specifically, [Huang et al. Science 2022], introduced an approach for learning properties of the ground state of an n-qubit gapped local Hamiltonian H from only n^{O(1)} data points sampled from Hamiltonians in the same phase of matter. This was subsequently improved by [Lewis et al. Nature Communications 2024], to O(log n) samples when the geometry of the n-qubit system is known. In this work, we introduce two approaches that achieve a constant sample complexity, independent of system size n, for learning ground state properties. Our first algorithm consists of a simple modification of the ML model used by Lewis et al. and applies to a property of interest known beforehand. Our second algorithm, which applies even if a description of the property is not known, is a deep neural network model. While empirical results showing the performance of neural networks have been demonstrated, to our knowledge, this is the first rigorous sample complexity bound on a neural network model for predicting ground state properties. We also perform numerical experiments that confirm the improved scaling of our approach compared to earlier results.

We design a motion planning algorithm to coordinate the movements of two robots along a figure eight track, in such a way that no collisions occur. We use a topological approach to robot motion planning that relates instabilities in motion planning algorithms to topological features of configuration spaces. The topological complexity of a configuration space is an invariant that measures the complexity of motion planning algorithms. We show that the topological complexity of our problem is 3 and construct an explicit algorithm with three continuous instructions.

In this long paper we give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this 2-dimensional non-compact system has a 3-dimensional compactification, a certain polyhedron exchange map, and that this compactification has a renormalization scheme. These two features allow us to make some sharp statements concerning the distribution, large-scale geometry, fine-scale geometry, and hidden algebraic symmetries of the orbits. For instance, one of our results is that the union of the unbounded orbits has Hausdorff dimension 1. We give a computer-aided proof of the results concerning the compactification and the renormalization. This proof involves finitely many calculations done with exact integer arithmetic.

There is extensive current interest about electronic topology in correlated settings. In strongly correlated systems, contours of Green's function zeros may develop in frequency-momentum space, and their role in correlated topology has increasingly been recognized. However, whether and how the zeros contribute to electronic properties is a matter of uncertainty. Here we address the issue in an exactly solvable model for Mott insulator. We show that the Green's function zeros contribute to several physically measurable correlation functions, in a way that does not run into inconsistencies. In particular, the physical properties remain robust to chemical potential variations up to the Mott gap as it should be based on general considerations. Our work sets the stage for further understandings on the rich interplay among topology, symmetry and strong correlations.

Representing 3D surfaces as level sets of continuous functions over R^3 is the common denominator of neural implicit representations, which recently enabled remarkable progress in geometric deep learning and computer vision tasks. In order to represent 3D motion within this framework, it is often assumed (either explicitly or implicitly) that the transformations which a surface may undergo are homeomorphic: this is not necessarily true, for instance, in the case of fluid dynamics. In order to represent more general classes of deformations, we propose to apply this theoretical framework as regularizers for the optimization of simple 4D implicit functions (such as signed distance fields). We show that our representation is capable of capturing both homeomorphic and topology-changing deformations, while also defining correspondences over the continuously-reconstructed surfaces.

Symmetries such as gauge invariance and anyonic symmetry play a crucial role in quantum many-body physics. We develop a general approach to constructing gauge invariant or anyonic symmetric autoregressive neural network quantum states, including a wide range of architectures such as Transformer and recurrent neural network (RNN), for quantum lattice models. These networks can be efficiently sampled and explicitly obey gauge symmetries or anyonic constraint. We prove that our methods can provide exact representation for the ground and excited states of the 2D and 3D toric codes, and the X-cube fracton model. We variationally optimize our symmetry incorporated autoregressive neural networks for ground states as well as real-time dynamics for a variety of models. We simulate the dynamics and the ground states of the quantum link model of U(1) lattice gauge theory, obtain the phase diagram for the 2D Z_2 gauge theory, determine the phase transition and the central charge of the SU(2)_3 anyonic chain, and also compute the ground state energy of the SU(2) invariant Heisenberg spin chain. Our approach provides powerful tools for exploring condensed matter physics, high energy physics and quantum information science.

We propose a general theory of the Open Gromov-Witten invariant on Calabi-Yau three-folds. We introduce the moduli space of multi-curves and show how it leads to invariants. Our construction is based on an idea of Witten. In the special case that each connected component of the Lagrangian submanifold has the rational homology of a sphere we define rational numbers F_{g,h} for each genus g and h boundary components.

In this paper we study holomorphic properties of infinite dimensional spin factors. Among the infinite dimensional Banach spaces with homogeneous open unit balls, we show that the spin factors are natural outlier spaces in which to ask the question (as was proved in the early 1970s for Hilbert spaces): Do biholomorphic automorphisms g of the open unit ball B have fixed points in overline B? In this paper, for infinite dimensional spin factors, we provide reasonable conditions on g that allow us to explicitly construct fixed points of g lying on partial B. En route, we also prove that every spin factor has the density property. In another direction, we focus on (compact) holomorphic maps f:Brightarrow B, having no fixed point in B and examine the sequence of iterates (f^n). As (f^n) does not generally converge, we instead trace the target set T(f) of f, that is, the images of all accumulation points of (f^n)_n, for any topology finer than the topology of pointwise convergence on B. We prove for a spin factor that T(f) lies on the boundary of a single bidisc unique to f.

A key step in any scanning-based asset creation workflow is to convert unordered point clouds to a surface. Classical methods (e.g., Poisson reconstruction) start to degrade in the presence of noisy and partial scans. Hence, deep learning based methods have recently been proposed to produce complete surfaces, even from partial scans. However, such data-driven methods struggle to generalize to new shapes with large geometric and topological variations. We present Points2Surf, a novel patch-based learning framework that produces accurate surfaces directly from raw scans without normals. Learning a prior over a combination of detailed local patches and coarse global information improves generalization performance and reconstruction accuracy. Our extensive comparison on both synthetic and real data demonstrates a clear advantage of our method over state-of-the-art alternatives on previously unseen classes (on average, Points2Surf brings down reconstruction error by 30\% over SPR and by 270\%+ over deep learning based SotA methods) at the cost of longer computation times and a slight increase in small-scale topological noise in some cases. Our source code, pre-trained model, and dataset are available on: https://github.com/ErlerPhilipp/points2surf

Unconventional superconductivity can emerge in various forms in strongly correlated systems. Historically, the most extensively studied examples include nodal and triplet pairing, which were experimentally detected in cuprates and superfluid helium-3, respectively. Beyond these canonical cases, there are also nematic superconductors, in which the rotational symmetry of the lattice is broken in the superconducting state. While there are several examples of nematic superconductors whose pairing emerges inside a preexisting nematic normal state, experimental evidence for superconducting states that intrinsically break rotational symmetry on their own has been exceedingly scarce. Here, we report the experimental realization and identification of an intrinsic multi-component nodal nematic superconducting state in magic-angle twisted trilayer graphene (MATTG). By integrating tunneling spectroscopy with quantum transport measurements, we directly probe the symmetry of low-energy superconducting quasiparticles. Deep inside the superconducting state, we observe a pronounced two-fold anisotropy in the zero-bias tunneling conductance under an in-plane magnetic field, which is strongly suppressed together with superconductivity as the temperature or perpendicular magnetic field is increased. Remarkably, the major axis of anisotropy rotates by up to 90^{circ} across the superconducting dome, indicating an intrinsic, field- and density-tunable symmetry of the low-energy excitations. These behaviors imply a nodal superconducting state in MATTG that spontaneously breaks the six-fold rotational symmetry of the moiré superlattice. Our results establish moiré graphene as a versatile platform for realizing and interrogating nodal nematic superconductivity, opening a new avenue for exploring unconventional symmetry-breaking superconducting states in two-dimensional materials.

Rydberg atom arrays have emerged as a powerful platform to simulate a number of exotic quantum ground states and phase transitions. To verify these capabilities numerically, we develop a versatile quantum Monte Carlo sampling technique which operates in the reduced Hilbert space generated by enforcing the constraint of a Rydberg blockade. We use the framework of stochastic series expansion and show that in the restricted space, the configuration space of operator strings can be understood as a hard rod gas in d+1 dimensions. We use this mapping to develop cluster algorithms which can be visualized as various non-local movements of rods. We study the efficiency of each of our updates individually and collectively. To elucidate the utility of the algorithm, we show that it can efficiently generate the phase diagram of a Rydberg atom array, to temperatures much smaller than all energy scales involved, on a Kagom\'e link lattice. This is of broad interest as the presence of a Z_2 spin liquid has been hypothesized recently.

We propose a Transformer-based Reinforcement Learning architecture, "LB-Explorer", to search for heterotic line bundle standard models arising from compactifications on smooth Calabi-Yau (CY) threefolds. We construct E_8times E_8 vacua with SU(5) symmetry, where the SU(5) can be further broken to the Standard Model gauge group via discrete Wilson lines. We test the LB-Explorer environment on complete intersection Calabi-Yau (CICY) manifolds, though the neural network architecture naturally generalizes to any CY admitting a smooth, simplicial Mori cone and a freely-acting discrete symmetry. The LB-Explorer efficiently learns constraints on the line bundle sums, guaranteeing the E_8 gauge embedding, anomaly cancellation, poly-stability (supersymmetry), chirality of the spectrum, and the absence of exotic matter. Valid configurations can be subsequently filtered by imposing the missing constraints, such as the equivariant structure of the line bundle sum and further requirements on the particle spectrum. In this direction, we introduce a hybrid architecture incorporating CP-SAT solvers that aims to impose some of the conditions exactly by perturbing solutions found by the LB-Explorer. The versatility and scalability of the LB-Explorer make it a powerful tool for navigating the string landscape with a large number of moduli. The code and tools necessary to reproduce our findings are available at https://github.com/alexmininno/LB-Explorer

We review recent efforts to machine learn relations between knot invariants. Because these knot invariants have meaning in physics, we explore aspects of Chern-Simons theory and higher dimensional gauge theories. The goal of this work is to translate numerical experiments with Big Data to new analytic results.

The natural world is full of complex systems characterized by intricate relations between their components: from social interactions between individuals in a social network to electrostatic interactions between atoms in a protein. Topological Deep Learning (TDL) provides a comprehensive framework to process and extract knowledge from data associated with these systems, such as predicting the social community to which an individual belongs or predicting whether a protein can be a reasonable target for drug development. TDL has demonstrated theoretical and practical advantages that hold the promise of breaking ground in the applied sciences and beyond. However, the rapid growth of the TDL literature has also led to a lack of unification in notation and language across Topological Neural Network (TNN) architectures. This presents a real obstacle for building upon existing works and for deploying TNNs to new real-world problems. To address this issue, we provide an accessible introduction to TDL, and compare the recently published TNNs using a unified mathematical and graphical notation. Through an intuitive and critical review of the emerging field of TDL, we extract valuable insights into current challenges and exciting opportunities for future development.

A bilayer formed by stacking two distinct materials creates a moiré lattice, which can serve as a platform for novel electronic phases. In this work we study a unique example of such a system: the graphene-black phosphorus heterostructure (G/BP), which has been suggested to have an intricate band structure. Most notably, the valence band hosts a quasi-one-dimensional region in the Brillouin zone of high density of states, suggesting that various many-body electronic phases are likely to emerge. We derive an effective tight-binding model that reproduces this band structure, and explore the emergent broken-symmetry phases when interactions are introduced. Employing a mean-field analysis, we find that the favored ground-state exhibits a striped spin density wave (SDW) order, characterized by either one of two-fold degenerate wave-vectors that are tunable by gating. Further exploring the phase-diagram controlled by gate voltage and the interaction strength, we find that the SDW-ordered state undergoes a metal to insulator transition via an intermediate metallic phase which supports striped SDW correlations. Possible experimental signatures are discussed, in particular a highly anisotropic dispersion of the collective excitations which should be manifested in electric and thermal transport.

We present an algorithm that constructs Calabi-Yau threefold orientifolds and their F-theory uplifts to elliptically-fibered Calabi-Yau fourfolds, embedded in toric varieties at codimension one and two respectively. The resulting Calabi-Yau fourfolds arise from triangulations of 6d reflexive polytopes -- which our method constructs from orientifold data -- and are smooth away from isolated terminal singularities. For many of our fourfolds, the construction of the mirror manifold is immediate, enabling the computation of fourfold periods, and thus the seven-brane superpotential. We present multiple examples that demonstrate these capabilities. Our algorithms work with CYTools and are available through a GitHub repository.

We develop a differential-geometric framework for variational quantum circuits in which noisy single- and multi-qubit parameter spaces are modeled by weighted projective lines (WPLs). Starting from the pure-state Bloch sphere CP1, we show that realistic hardware noise induces anisotropic contractions of the Bloch ball that can be represented by a pair of physically interpretable parameters (lambda_perp, lambda_parallel). These parameters determine a unique WPL metric g_WPL(a_over_b, b) whose scalar curvature is R = 2 / b^2, yielding a compact and channel-resolved geometric surrogate for the intrinsic information structure of noisy quantum circuits. We develop a tomography-to-geometry pipeline that extracts (lambda_perp, lambda_parallel) from hardware data and maps them to the WPL parameters (a_over_b, b, R). Experiments on IBM Quantum backends show that the resulting WPL geometries accurately capture anisotropic curvature deformation across calibration periods. Finally, we demonstrate that WPL-informed quantum natural gradients (WPL-QNG) provide stable optimization dynamics for noisy variational quantum eigensolvers and enable curvature-aware mitigation of barren plateaus.

It was previously believed that the Bloch electronic states of non-magnetic materials with inversion symmetry cannot have finite spin polarizations. However, since the seminal work by Zhang et al. [Nat. Phys. 10, 387-393 (2014)] on local spin polarizations of Bloch states in non-magnetic, centrosymmetric materials, the scope of spintronics has been significantly broadened. Here, we show, using a framework that is universally applicable independent of whether hidden spin polarizations are small (e.g., diamond, Si, Ge, and GaAs) or large (e.g., MoS2 and WSe2), that the corresponding quantity arising from orbital - instead of spin - degrees of freedom, the hidden orbital polarization, is (i) much more abundant in nature since it exists even without spin-orbit coupling and (ii) more fundamental since the interband matrix elements of the site-dependent orbital angular momentum operator determines the hidden spin polarization. We predict that the hidden spin polarization of transition metal dichalcogenides is reduced significantly upon compression. We suggest experimental signatures of hidden orbital polarization from photoemission spectroscopies and demonstrate that the current-induced hidden orbital polarization may play a far more important role than its spin counterpart in antiferromagnetic information technology by calculating the current-driven antiferromagnetism in compressed silicon.

Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a many-body quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, 2D random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.

We study complex saddle points of the worldline path integral for massive charged particles in two-dimensional maximally symmetric backgrounds threaded by homogeneous electric fields. Within the worldline formalism, we show that the semiclassical saddles generically require complexifying both the target spacetime and the proper-time parameter. The resulting solutions comprise two slices of a single complex family, distinguished by the direction of proper-time evolution. The one evolving in imaginary proper time carries an action that typically controls the exponential suppression of non-perturbative effects, while that evolving in real proper time may encode additional complex phases responsible for quantum interference between saddles. Treating this family as a whole, we give a prescription for constructing and classifying all topologically inequivalent instanton sectors, and we find that the worldline action depends only on their topological data, namely winding numbers around the compact directions and certain isolated punctures. Moreover, in Anti-de Sitter spacetime, the qualitative structure of the solutions is governed by the dimensionless ratio R |eE|/m (R being the curvature radius), which distinguishes sub-extremal, extremal, and super-extremal regimes. By contrast, no analogous distinction occurs in de Sitter---where the instantons are insensitive to this ratio---or in Minkowski spacetime, which lacks a curvature scale altogether. We also recast the action as a sum over residues of the relevant (non-conserved) conjugate momentum, making transparent which analytic data control each contribution. Finally, we discuss the subtleties that arise when solutions built in various local patches are lifted to global (Anti-)de Sitter space. These impose further necessary conditions on which saddles can in principle contribute to the one-loop path integral.

Complex prediction models such as deep learning are the output from fitting machine learning, neural networks, or AI models to a set of training data. These are now standard tools in science. A key challenge with the current generation of models is that they are highly parameterized, which makes describing and interpreting the prediction strategies difficult. We use topological data analysis to transform these complex prediction models into pictures representing a topological view. The result is a map of the predictions that enables inspection. The methods scale up to large datasets across different domains and enable us to detect labeling errors in training data, understand generalization in image classification, and inspect predictions of likely pathogenic mutations in the BRCA1 gene.

We investigate the stability of superconducting strings as bound states of strings and fermion zero modes at both the classical and quantum levels. The dynamics of these superconducting strings can result in a stable configuration, known as a vorton. We mainly focus on global strings, but the majority of the discussion can be applied to local strings. Using lattice simulations, we study the classical dynamics of superconducting strings and confirm that they relax to the vorton configuration through Nambu-Goldstone boson radiation, with no evidence of over-shooting that would destabilize the vorton. We explore the tunneling of fermion zero modes out of the strings. Both our classical analysis and quantum calculations yield consistent results: the maximum energy of the zero mode significantly exceeds the fermion mass, in contrast to previous literature. Additionally, we introduce a world-sheet formalism to evaluate the decay rate of zero modes into other particles, which constitute the dominant decay channel. We also identify additional processes that trigger zero-mode decay due to non-adiabatic changes of the string configuration. In these decay processes, the rates are suppressed by the curvature of string loops, with exponential suppression for large masses of the final states. We further study the scattering with light charged particles surrounding the string core produced by the zero-mode current and find that a wide zero-mode wavefunction can enhance vorton stability.

Preparing high-fidelity logical magic states is crucial for fault-tolerant quantum computation. Among prior attempts to reduce the substantial cost of magic state preparation, magic state cultivation (MSC), a recently proposed protocol for preparing T states without magic state distillation, achieves state-of-the-art efficiency. Inspired by this work, we propose a new MSC procedure that would produce a logical T state on a rotated surface code at a further reduced cost. For our MSC protocol, we define a new code family, the RP^2 code, by putting the rotated surface code on RP^2 (a two-dimensional manifold), as well as two self-dual CSS codes named SRP-3 and SRP-5 respectively. Small RP^2 codes are used to hold logical information and checked by syndrome extraction (SE) circuits. We design fast morphing circuits that enable switching between a distance 3 (5) RP^2 code and an SRP-3 (SRP-5) code on which we can efficiently check the correctness of the logical state. To preserve the high accuracy of the cultivated logical T state, we design an efficient and easy-to-decode expansion stage that grows a small RP^2 code to a large rotated surface code in one round. Our MSC protocol utilizes non-local connectivity, available on both neutral atom array and ion trap platforms. According to our Monte Carlo sampling results, our MSC protocol requires about an order of magnitude smaller space-time volume to reach a target logical error rate around 10^{-9} compared to the original MSC protocol.

Topological Neural Networks (TNNs) incorporate higher-order relational information beyond pairwise interactions, enabling richer representations than Graph Neural Networks (GNNs). Concurrently, topological descriptors based on persistent homology (PH) are being increasingly employed to augment the GNNs. We investigate the benefits of integrating these two paradigms. Specifically, we introduce TopNets as a broad framework that subsumes and unifies various methods in the intersection of GNNs/TNNs and PH such as (generalizations of) RePHINE and TOGL. TopNets can also be readily adapted to handle (symmetries in) geometric complexes, extending the scope of TNNs and PH to spatial settings. Theoretically, we show that PH descriptors can provably enhance the expressivity of simplicial message-passing networks. Empirically, (continuous and E(n)-equivariant extensions of) TopNets achieve strong performance across diverse tasks, including antibody design, molecular dynamics simulation, and drug property prediction.

Understanding the topology of decision regions is central to explaining the inner workings of deep neural networks. Prior empirical work has provided evidence that these regions are path connected. We study a stronger topological question: whether closed loops inside a decision region can be contracted without leaving that region. To this end, we propose an iterative quad-mesh filling procedure that constructs a finite-resolution label-preserving surface bounded by a given loop and lying entirely within the same decision region. We further connect this construction to natural Coons patches in order to quantify its deviation from a canonical geometric interpolation of the loop. By evaluating our method across several modern image-classification models, we provide empirical evidence supporting the hypothesis that decision regions in deep neural networks are not only path connected, but also simply connected.

We investigate the entanglement properties in the symmetric subspace of N-partite d-dimensional systems (qudits). For diagonal symmetric states, we show that there is no bound entanglement for d = 3,4 and N = 3. Further, we present a constructive algorithm to map multipartite diagonal symmetric states of qudits onto bipartite symmetric states of larger local dimension. This technique greatly simplifies the analysis of multipartite states and allows to infer entanglement properties for any even N geq 4 due to the fact that the PPT conditions that arise from the bipartite symmetric state correspond to the same PPT conditions that appear in the multipartite diagonal symmetric state.

We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of both the input and latent space to derive a topological loss term. Under weak theoretical assumptions, we construct this loss in a differentiable manner, such that the encoding learns to retain multi-scale connectivity information. We show that our approach is theoretically well-founded and that it exhibits favourable latent representations on a synthetic manifold as well as on real-world image data sets, while preserving low reconstruction errors.

In 1983, Gromov introduced the notion of distortion of a knot, and asked if there are knots with arbitrarily large distortion. In 2011, Pardon proved that the distortion of T_{p,q} is at least min{p,q} up to a constant factor. We prove that the distortion of T_{p, p+1}# K is at least p up to a constant, independent of K. We also prove that any embedding of a minimal genus Seifert surface for T_{p,p+1}# K in R^3 has small extrinsic systole, in the sense that it contains a non-contractible loop with small R^3-diameter relative to the length of the knot. These results are related to combinatorial properties of the monodromy map associated to torus knots.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

η pairing describes a staggered superconducting condensate whose order parameter changes sign between neighboring unit cells while maintaining a uniform magnitude. Nearly four decades after its theoretical proposal by C. N. Yang, a material platform for stabilizing η pairing has remained elusive. Here, we reveal that altermagnetism, despite being pair breaking for conventional BCS pairing, naturally favors η-like pairing in V_2Se_2O-family altermagnets. Their valley-contrasting spin structure gives rise to oppositely spin-polarized, symmetry-related valleys centered at X=(π,0) and Y=(0,π), which form spin-singlet pairing partners separated by the commensurate momentum mathbf Q=(π,π), thereby favoring an η-like condensate. Self-consistent calculations demonstrate that the resulting η-like state remains robust over broad variations of chemical potential, hopping parameters, and Fermi-surface topology. To detect this hidden staggered order, we propose a phase-sensitive Josephson measurement with a uniform superconductor, where destructive interference suppresses single-Cooper-pair tunneling while preserving two-Cooper-pair processes. This constitutes a charge-4e-like Josephson response despite a bulk condensate composed of charge-2e Cooper pairs, providing a direct phase-sensitive signature of the otherwise elusive η-like condensate in V_2Se_2O-family altermagnets.

We consider fermions in a zero-temperature superconducting anti-de Sitter domain wall solution and find continuous bands of normal modes. These bands can be either partially filled or totally empty and gapped. We present a semi-classical argument which approximately captures the main features of the normal mode spectrum.

We construct modular-invariant torus partition functions with a unique vacuum, discrete energy levels and positive integer degeneracies by recursively repairing the Maloney--Witten--Keller modular completion of the Virasoro vacuum. Exact modular repairs and finite moment matching discretize successive spectral bands while preserving earlier levels and controlling convergence. For c_L=c_R=c and a=(c-1)/12, the construction realizes primary dimension gaps Δ_1=(1+κ)a for sufficiently small fixed κ>0, and Δ_1=a+δ for any fixed δge0, at every sufficiently large real c. Every nonvacuum primary satisfies h,bar hge(c-1)/24. In the fixed-δ family, spectra can be chosen whose densities of states, smoothed with a fixed nonnegative normalized smooth kernel of compact support, match the correspondingly smoothed prediction of a single perturbative BTZ saddle through every fixed finite order in 1/c. The count includes all spins and Virasoro descendants. Its logarithm reproduces the Bekenstein--Hawking entropy and its corrections at every fixed positive E/c, where E=Δ-c/12. This includes 0<E<c/12, where thermal AdS dominates the canonical ensemble.