The package addresses a class of problems that sits at the heart of both physics and computer science: finding the lowest-energy configuration of spins on a lattice, the classic task of computing a ground state. These problems are computationally brutal in general, and they matter well beyond condensed matter theory. Many scheduling, routing and circuit-design challenges can be mapped onto Ising models, which is why quantum annealers and specialized Ising machines have attracted so much attention. Tensor networks offer a classical alternative, compressing the exponentially large space of spin configurations into structured objects that can be manipulated with controlled approximations. SpinGlassPEPS.jl implements this approach using projected entangled-pair-style constructions adapted to quasi-two-dimensional connectivity, including the Chimera graphs used by D-Wave quantum hardware.

The consolidation alone changes the user experience substantially. The original release consisted of four packages, SpinGlassTensors, SpinGlassNetworks, SpinGlassExhaustive and SpinGlassEngine, whose version numbers were kept in sync but whose interfaces gradually drifted apart. Renamed symbols lacked deprecation paths, leaving users with breakages and no guidance. Version 2.0.1 folds all four into internal modules of one package, so a single command, adding SpinGlassPEPS, installs the complete solver. The public surface shrinks from roughly 240 re-exported symbols to about 80 documented ones, while lower-level kernels remain accessible through the submodules for those who need them. It is a classic engineering trade: less clutter at the top, the same power underneath.

Two correctness fixes deserve particular attention because they affected results. The corner_matrix function, a core building block of the contraction scheme, failed on every SiteTensor input and permuted the trailing dimensions of VirtualTensor outputs. Meanwhile, the exhaustive GPU search launched too few execution blocks and returned incomplete spectra for problems with ten or more spins. Both defects are now covered by regression tests against independent dense references on CPU and GPU, including a deterministic ten-spin test that compares all 1024 GPU state codes and energies, produced by two 512-thread blocks, against complete CPU enumeration. The release also bundles the benchmark instances behind the original publication’s figures, which had previously been referenced but not distributed, closing a reproducibility gap that plagues much of computational science.

The most scientifically interesting addition is a discarded-weight diagnostic. Every truncating factorization in the contraction now records the relative weight it discards, accumulated in a task-local accumulator so that concurrent solves cannot mix statistics. A solve reports the sum and maximum of these discarded weights, along with how many truncations were forced by the bond-dimension cap rather than the singular-value tolerance, and how many of the offered singular values were retained. On a 128-spin instance, bond dimension 4 yields a total discarded weight of 3.1 times ten to the minus four, with all 18 truncations limited by the bond bound; at bond dimension 32 the figure drops to 5.6 times ten to the minus fourteen, with none of its four truncations bond-limited. In effect, users gain an internal warning light that was previously available only to those with an exact solution to compare against.

The diagnostics come with honest caveats, which is refreshing in a field where optimism often outruns rigor. Discarded weight measures truncation loss in cold contractions, not warm-start error: a warm start optimizes within a fixed bond dimension without a truncating factorization, so it can report essentially zero discarded weight despite a nonzero variational gap. The package warns when the two are combined. Nor does discarded weight reliably rank solution quality. In benchmarks on ten 2500-spin square-lattice instances, the median accumulated discarded weight peaked at inverse temperature 4, where the median energy error was smaller, while at inverse temperature 2 the discarded weight was near its minimum but the energy error was largest. The documentation now positions the diagnostic as a contraction indicator and optional preference filter, not a selection criterion.

Performance work runs throughout the release. A new beta_ladder feature evaluates an increasing schedule of the inverse temperature, which controls how strongly the solver’s branch probabilities favor low-energy states. When a retained boundary matrix product state has the dimensions required by the next rung, it warm-starts variational compression instead of building the target state exactly and truncating it. On a 2048-spin instance where boundary construction dominates, the two warmed rungs saw wall times fall by about 24 and 25 percent, with identical energies, translating to roughly 16 percent over the full ladder. Meanwhile, a sweep over eight lattice transformations now runs concurrently, with a calibration solve estimating peak memory and admitting tasks within a byte budget. On an Intel Xeon Platinum 8462Y+ processor, CPU speed-ups reached 3.1 times for a 36-spin case at eight concurrent solves, and identical energies were returned in every serial and concurrent configuration tested.

Perhaps the most surprising finding concerns hardware. The original publication recommended GPU execution for larger examples, but on the tested Xeon and NVIDIA H100 system, the GPU won in only one matched configuration: a 2048-spin sparse instance at bond dimension 32, where the CPU-to-GPU wall-clock ratio was 1.45. At 36 spins, CPU wall time was roughly 2 percent of GPU time. Profiling explains why: on a 128-spin solve, host-side CUDA API calls occupied 27 percent of the profiled interval while GPU activities occupied just 6.6 percent, meaning at least two-thirds of the time lay outside both categories. Even eliminating the entire measured CUDA API overhead would cap speed-up at about 1.4 times by Amdahl’s law. Faster tensor kernels alone cannot fix a problem that lives on the host side.

So the developers attacked host allocation instead. In the previous implementation, a function called branch_states accounted for 52.7 percent of the bytes allocated by a 128-spin solve, spawning tens of thousands of small vectors per call; a single matrix now stores branched configurations instead. Contraction temporaries were moved off the garbage-collected heap, cutting allocated bytes on a corresponding CPU solve from 90.9 to 32 gibibytes, a reduction of about two thirds. The post-change totals were 32.3 gibibytes on CPU and 24.1 gibibytes on GPU for a 2048-spin, bond-32 solve. These are the unglamorous numbers that determine whether a scientific tool feels responsive or sluggish, and they illustrate how modern performance engineering often means fighting the runtime environment rather than the mathematics.

Version 2.0.1, archived on Zenodo and released under the Apache License 2.0, arrives as debate intensifies over whether classical algorithms can match specialized quantum and analog Ising machines. A companion study comparing tensor-network approaches to quantum and classical Ising machines suggests the answer is nuanced, and honest tooling like this strengthens the comparison. With one public API, deprecation shims for renamed entry points, distributed benchmark instances and raw per-run data with runnable drivers, the release models what reproducible computational research should look like. The work was supported by the National Science Centre, Poland, and the package’s developers, Łukasz Pawela and Bartłomiej Gardas, have made clear-eyed limitations part of the documentation itself. For anyone using tensor networks to hunt ground states, the message is simple: the toolkit got faster, safer and considerably more transparent, and it now tells you when it might be wrong.

Subject of Research: A tensor-network software package for solving Ising-like optimization problems on quasi-two-dimensional graphs

Article Title: Version 2.0.1 – SpinGlassPEPS.jl: Tensor-network package for Ising-like optimization on quasi-two-dimensional graphs

Article References: Pawela, Ł., & Gardas, B. (2026). Version 2.0.1 – SpinGlassPEPS.jl: Tensor-network package for Ising-like optimization on quasi-two-dimensional graphs. SoftwareX, 36, Article 103027. https://doi.org/10.1016/j.softx.2026.103027

Image Credits: AI Generated

DOI: 10.1016/j.softx.2026.103027

Keywords: tensor networks, Ising optimization, SpinGlassPEPS.jl, Julia language, ground state computation, GPU computing, high-performance computing, combinatorial optimization, reproducibility, software engineering, spin glasses, quantum annealing

The work, published in Nature Physics by Kaixin Ji, Yu Zhao, Ce Shen, Yidun Wan and Ling-Yan Hung, draws on some of the deepest ideas in modern condensed matter theory. The authors’ strategy does not start from spins or magnets at all. Instead, they engineer the boundary conditions of three-dimensional topological orders, exotic phases of matter whose excitations, called anyons, can carry quantum statistics that are neither bosonic nor fermionic. These topological orders are described concretely by string-net models, exactly soluble constructions introduced by Michael Levin and Xiao-Gang Wen in 2005, in which the vacuum is pictured as a tangle of fluctuating strings whose allowed patterns are dictated by algebraic data known as a fusion category.

The key innovation lies in how the critical points are created. In a topological phase, certain anyon types can undergo condensation, a process analogous to the condensation of a Bose-Einstein condensate, in which the anyon becomes part of the vacuum and other excitations are reorganized accordingly. When a single set of anyons condenses, the system typically flows from one gapped topological phase to another. The researchers instead arranged for non-commuting anyons to condense in a carefully balanced, commensurate fashion, meaning that two or more condensation channels that cannot coexist in an ordinary gapped phase are forced into competition. The tug-of-war between these incompatible orders prevents the system from settling into any gapped phase, and the resulting critical points flow in the infrared limit to conformal field theories. By tuning the relative weights of the competing condensates, the algorithm generates a lattice Hamiltonian whose low-energy behavior is precisely the desired conformal theory.

The machinery relies on a holographic device known as the strange correlator, a quantity computed as a three-dimensional path integral that maps the boundary lattice model onto the bulk topological order. In this picture, the two-dimensional critical model lives on the boundary of the three-dimensional string-net system, and the algebraic rules governing anyon fusion in the bulk translate directly into the interaction terms of the boundary model. The critical couplings, the parameter values at which the phase transitions occur, are encoded exactly in algebraic data associated with the string-net construction, specifically in the Frobenius algebras that specify which anyons condense. This means that instead of laboriously scanning parameter space numerically to hunt for critical points, physicists can read off where the transitions happen from the underlying category theory, a level of precision control that is rare in the study of strongly correlated systems.

The practical payoff is an infinite family of critical lattice models. The authors demonstrate that their procedure recovers known conformal field theories that preserve the so-called Haagerup symmetries, exotic non-invertible symmetries named after the mathematician Uffe Haagerup, whose fusion categories have intrigued both mathematicians and physicists since the 1990s. Haagerup-symmetric theories have become a testing ground for the emerging theory of categorical symmetry, in which ordinary symmetry groups are replaced by richer algebraic structures. Critical lattice models realizing these symmetries had been proposed only recently, and the new algorithm reproduces them as a special case of a much more general construction, providing independent confirmation of earlier numerical work that had reported evidence for Haagerup conformal field theories in tensor network calculations.

More strikingly, the factory does not merely recycle known results. Among the models it generates, the researchers identified three previously unknown candidate conformal field theories, critical points that had never been observed or catalogued before. These discoveries suggest that the space of two-dimensional conformal field theories is far more densely populated by accessible lattice realizations than the traditional, largely ad hoc methods of statistical mechanics had revealed. Historically, finding a lattice model for a given critical phenomenon was a matter of insight and luck, from Onsager’s solution of the Ising model to the Ashkin-Teller models studied in the early 1980s. The new algorithm replaces that serendipity with a recipe: choose a fusion category, select competing condensable algebras, and compute the resulting boundary model and its phase diagram.

The numerical verification of the construction is itself technically notable. The team developed symmetry-preserving tensor network algorithms to map out the phase diagrams of their models, coloring the parameter space by the numerically determined central charge, a fundamental invariant of a conformal field theory that measures the number of its degrees of freedom. In the phase diagrams, regions corresponding to different condensed anyon orders meet along critical lines and surfaces, and the interpolation between multiple competing condensates can be visualized in ternary diagrams representing three-condensate mixtures. The agreement between the predicted critical couplings extracted from the algebraic data and the numerical scans provides a stringent consistency check of the entire framework, and the MATLAB code and source data used to regenerate the phase diagrams have been made available with the paper.

The broader implications extend beyond two-dimensional statistical mechanics. Conformal field theories occupy a central role in high-energy theoretical physics as well, appearing as limits of quantum field theories, as building blocks of string theory, and through the AdS/CFT correspondence as dual descriptions of quantum gravity. A systematic method for discretizing conformal field theories onto lattices offers a potential route to studying them with the numerical tools of condensed matter, including tensor networks and quantum simulation. The authors and other researchers in the field have also drawn connections to topological holography and the idea that symmetries themselves can be understood as shadows of topological order, suggesting that the factory could illuminate how generalized, non-invertible symmetries emerge at quantum critical points.

The work also raises tantalizing prospects for classification. One of the great unsolved problems in theoretical physics is to classify all possible conformal field theories, a task that has proved formidable even in two dimensions where the machinery is most powerful. By establishing a structured scheme in which critical theories arise from combinatorial algebraic data, the conformal field theory factory provides a framework for discovering and potentially organizing these theories in families. If every entry in a suitable catalogue of fusion categories and condensable algebras yields a critical model, physicists may be able to enumerate, or at least systematically sample, far more of the landscape of critical behavior than ever before. For a field that has spent half a century stitching together critical phenomena one painstaking example at a time, the prospect of a factory that produces them by the dozen marks a genuine shift in method, and the three brand-new candidate theories that emerged from its first run hint at how much of that landscape still lies unexplored.

Subject of Research: An algorithm generating two-dimensional critical lattice models from competing anyon condensation in three-dimensional topological orders

Article Title: An algorithm to generate two-dimensional critical lattice models using competing anyon condensation

Article References: Ji, K., Zhao, Y., Shen, C., Wan, Y., & Hung, L.-Y. (2026). An algorithm to generate two-dimensional critical lattice models using competing anyon condensation. Nature Physics. https://doi.org/10.1038/s41567-026-03438-6

Image Credits: AI Generated

DOI: 10.1038/s41567-026-03438-6

Keywords: conformal field theory, anyon condensation, topological order, string-net models, critical phenomena, lattice models, Haagerup symmetry, phase transitions, fusion categories, tensor networks, categorical symmetry, theoretical physics

CuGeO3 has occupied a special place in quantum magnetism since 1993, when Masashi Hase, Isao Terasaki and Kunimitsu Uchinokura first reported that chains of spin-1/2 copper ions running through this inorganic compound undergo a spin-Peierls transition. Below a characteristic temperature, the crystal lattice itself distorts in three dimensions, and the magnetic ions pair up into spin singlets, forming a nonmagnetic ground state built from dimers. This is the magnetic analogue of the Peierls instability familiar in conducting polymers, and CuGeO3 remains one of the rare inorganic materials in which it occurs. The transition is driven by an interplay of one-dimensional magnetic frustration, in which next-nearest-neighbour antiferromagnetic exchanges compete with the dominant nearest-neighbour coupling, and weak explicit dimerization imposed by the three-dimensional crystal structure. For three decades, physicists have debated how precisely these ingredients combine and what they imply for the excitation spectrum of the dimerized phase.

The theoretical backdrop is the frustrated spin-1/2 Heisenberg chain, a model in which nearest-neighbour interactions J1 compete with next-nearest-neighbour interactions J2. In the unfrustrated chain, the ground state is a quantum critical spin liquid whose excitations are deconfined spinons, a fact established by Hans Bethe in 1931 and elaborated by Faddeev and Takhtajan half a century later. When frustration is strong enough, the chain spontaneously dimerizes into one of two degenerate patterns of singlet pairs, as shown by Haldane in 1982 and by the exactly solvable Majumdar-Ghosh point. In such a dimerized state, the elementary excitations are no longer free spinons but triplons, triplet bound states localized on dimers that hop through the lattice. Whether a real material sits close to the boundary between these regimes, and how explicit dimerization from lattice distortions tips the balance, determines the entire character of its magnetic spectrum.

To resolve these questions, the team grew high-quality single crystals of CuGeO3 and measured their full excitation spectrum below the spin-Peierls transition temperature using time-of-flight inelastic neutron scattering. The experiments were performed at the SEQUOIA spectrometer at the Spallation Neutron Source at Oak Ridge National Laboratory and at the 4SEASONS spectrometer at the Japan Proton Accelerator Research Complex. Neutron scattering is uniquely suited to this task because neutrons couple directly to the spin fluctuations of the material, allowing researchers to record the dynamical structure factor, a comprehensive map of magnetic excitations as a function of energy and momentum in all three crystallographic directions. By combining data from multiple incident neutron energies, the team captured both the low-energy triplon modes and the high-energy continuum with exceptional coverage and resolution.

The resulting spectra revealed a striking energy-dependent transformation of quasiparticle character. At high energies, the excitations form a broad, diffuse continuum, the unmistakable fingerprint of weakly interacting, deconfined spinons propagating through the chain. At lower energies, in contrast, the spectrum resolves into sharp, highly coherent dispersive modes, the signature of tightly bound triplons, each a composite of two spinons locked together by the dimerization. The researchers traced this confinement-deconfinement crossover across both energy and temperature scales, demonstrating that a single quantum magnet can exhibit fractionalized behaviour in one part of its spectrum and conventional bound-state behaviour in another. This observation provides direct experimental confirmation of theoretical scenarios, proposed in the 1990s by Uhrig, Schulz, Singh and Weihong and others, in which the crossover from triplons to spinons occurs dynamically as a function of energy in dimerized and frustrated chains.

To interpret the data quantitatively, Bo Xiao performed extensive tensor network simulations of the frustrated, dimerized spin-1/2 chain, building on the density matrix renormalization group methods pioneered by Steven White and extended to dynamical response functions by Vidal and collaborators. By comparing the simulated dynamical structure factor with the neutron data across the full energy range, the team extracted the microscopic spin Hamiltonian of CuGeO3 with unprecedented precision. The analysis revealed substantial next-nearest-neighbour frustration, confirming that the material lies deep in the regime where spontaneous dimerization would occur even in a purely one-dimensional chain. At the same time, the three-dimensional lattice structure contributes only a weak explicit dimerization. CuGeO3 therefore occupies a delicate regime dominated by spontaneous dimerization, gently biased by the lattice, a conclusion that reconciles decades of seemingly contradictory parameter estimates in the literature.

One of the most visually compelling results concerns the two-particle regime. The triplon character of the low-energy quasiparticles persists when pairs of triplons are excited, producing a structured two-triplon continuum rather than a featureless background. Within this continuum, the team identified a pronounced spectral feature at its lower boundary associated with a van Hove singularity, a logarithmic enhancement of the spectral weight that arises where the triplon dispersion becomes flat at extrema of the band. Van Hove singularities, familiar from the electronic density of states of solids and recently observed in magnon spectra of two-dimensional quantum magnets, had not been resolved so clearly at the boundary of a two-triplon continuum in a spin-Peierls system. Their observation underscores the remarkable coherence of the triplon excitations even in the multiparticle sector, and it demonstrates that the confinement picture remains valid well beyond the one-particle regime.

The findings carry broader implications for the study of fractionalization and confinement in quantum matter. Fractionalized quasiparticles are a defining feature of quantum spin liquids and appear in contexts ranging from the fractional quantum Hall effect to Kitaev materials, and understanding how they confine into bound states is a central theme of modern condensed matter physics. The CuGeO3 results show that the interplay between magnetic frustration and dimerization, whether spontaneous or explicitly imposed by the lattice, can reshape fractionalization and confinement within a single material, tuning the quasiparticle character continuously from deconfined spinons at high energy to tightly bound triplons at low energy. Because the understanding of this crossover requires accounting for both spontaneous and explicit dimerization simultaneously, the work establishes a quantitative framework that can be applied to other quasi-one-dimensional frustrated magnets, including spin ladders and chain compounds under chemical substitution or pressure.

The study also exemplifies the power of pairing modern neutron spectroscopy with modern computational many-body methods. Tensor network techniques, which compress the exponentially complex quantum wavefunction into an efficient matrix product form, have matured to the point where they can reproduce entire measured spectra of realistic spin Hamiltonians, allowing experimental data to be translated directly into microscopic coupling constants. The raw neutron scattering data from the SEQUOIA measurements have been made openly available through the Oak Ridge Neutron Catalog, ensuring that the community can reanalyse and build upon the results. As researchers continue to hunt for fractionalized excitations and emergent bound states in quantum magnets, CuGeO3 now stands as a benchmark system in which the full life cycle of a spinon, from free fractional particle to confined triplon and structured two-particle continuum, has been observed and understood within a single, quantitatively validated theoretical picture.

Subject of Research: Fractionalized spinon and bound triplon excitations in the frustrated quantum spin-Peierls chain compound CuGeO3

Article Title: Weakly interacting spinons and tightly bound triplons in the frustrated quantum spin-Peierls chain

Article References: Park, P., Xiao, B., Górnicka, K., May, A. F., Yan, J., Kajimoto, R., Nakamura, M., Stone, M. B., Halász, G. B., & Christianson, A. D. (2026). Weakly interacting spinons and tightly bound triplons in the frustrated quantum spin-Peierls chain. Nature Physics. https://doi.org/10.1038/s41567-026-03447-5

Image Credits: AI Generated

DOI: 10.1038/s41567-026-03447-5

Keywords: CuGeO3, spinons, triplons, spin-Peierls transition, quantum magnetism, magnetic frustration, dimerization, neutron scattering, tensor networks, quasiparticles, van Hove singularity, one-dimensional spin chains