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In strongly correlated materials, interacting electrons are entangled and form collective quantum states, resulting in rich low-temperature phase diagrams. Notable examples include cuprate superconductors, in which superconductivity emerges at low doping out of an unusual ``pseudogap'' metallic state above the critical temperature. The Fermi-Hubbard model, describing a wide range of phenomena associated with strong electron correlations, still offers major computational challenges despite its simple formulation. In this context, ultracold atoms quantum simulators have provided invaluable insights into the microscopic nature of correlated quantum states. Here, we use a quantum gas microscope Fermi-Hubbard simulator to explore a wide range of doping levels and temperatures in a regime where a pseudogap is known to develop. By measuring multi-point correlation functions up to fifth order, we uncover a novel universal behaviour in magnetic and higher-order spin-charge correlations. This behaviour is characterized by a doping-dependent energy scale that governs the exponential growth of the magnetic correlation length upon cooling. Accurate comparisons with determinant Quantum Monte Carlo and Minimally Entangled Typical Thermal States simulations confirm that this energy scale agrees well with the pseudogap temperature $T^{*}$. Our findings establish a qualitative and quantitative understanding of the magnetic origin and physical nature of the pseudogap and pave the way towards the exploration of pairing and collective phenomena among charge carriers expected to emerge at lower temperatures.

Classical machine learning has proven remarkably useful in post-processing quantum data, yet typical learning algorithms often require prior training to be effective. In this work, we employ a tensorial kernel support vector machine (TK-SVM) to analyze experimental data produced by trapped-ion quantum computers. This unsupervised method benefits from directly interpretable training parameters, allowing it to identify the non-trivial string-order characterizing symmetry-protected topological (SPT) phases. We apply our technique to two examples: a spin-1/2 model and a spin-1 model, featuring the cluster state and the AKLT state as paradigmatic instances of SPT order, respectively. Using matrix product states, we generate a family of quantum circuits that host a trivial phase and an SPT phase, with a sharp phase transition between them. For the spin-1 case, we implement these circuits on two distinct trapped-ion machines based on qubits and qutrits. Our results demonstrate that the TK-SVM method successfully distinguishes the two phases across all noisy experimental datasets, highlighting its robustness and effectiveness in quantum data interpretation., Comment: 16 pages, 11 figures

Rydberg tweezer arrays provide a versatile platform to explore quantum magnets with dipolar XY or van-der-Waals Ising ZZ interactions. Here, we propose a scheme combining dipolar and van-der-Waals interactions between two Rydberg states, where the amplitude of the latter can be greater than that of the former, realizing an extended Hubbard model with long-range tunnelings in optical tweezer arrays. On the triangular lattice with repulsive interactions, we predict the existence of a robust supersolid phase with a critical entropy per particle $S/N \approx 0.19$ accessible in current Rydberg tweezer experiments supported by large-scale quantum Monte Carlo simulations. We further demonstrate the experimental feasibility by identifying pairs of Rydberg states in ${}^{87}$Rb realizing the required interactions. Such a lattice supersolid is long-lived, found over a wide parameter range in an isotropic and flat two-dimensional geometry, and can be realized for 100s of particles allowing one to directly probe the defect-induced picture of supersolids. Its thermodynamical and dynamical properties can hence be studied at a far larger scale than hitherto possible., Comment: 5+2 pages, 3+3 figures

Fracton models host unconventional topological orders in three and higher dimensions and provide promising candidates for quantum memory platforms. Understanding their robustness against quantum fluctuations is an important task but also poses great challenges due to the lack of efficient numerical tools. In this work, we establish neural quantum states (NQS) as new tools to study phase transitions in these models. Exact and efficient parametrizations are derived for three prototypical fracton codes - the checkerboard and X-cube model, as well as Haah's code - both in terms of a restricted Boltzmann machine (RBM) and a correlation-enhanced RBM. We then adapt the correlation-enhanced RBM architecture to a perturbed checkerboard model and reveal its strong first-order phase transition between the fracton phase and a trivial field-polarizing phase. To this end, we simulate this highly entangled system on lattices of up to 512 qubits with high accuracy, representing a cutting-edge application of variational neural-network methods. Our work demonstrates the remarkable potential of NQS in studying complicated three-dimensional problems and highlights physics-oriented constructions of NQS architectures., Comment: 36 pages, 19 figures, Submission to SciPost

The recent experimental detection of the onset of a dynamically prepared, gapped $Z_2$ quantum spin liquid on the ruby lattice brought the physics of frustrated magnetism and lattice gauge theory to Rydberg tweezer arrays (Semeghini et al, Science 374, 1242 (2021)). The thermodynamic properties of such models remain inadequately addressed, yet knowledge thereof is indispensable if one wants to prepare large, robust, and long-lived quantum spin liquids. Using large scale quantum Monte Carlo simulations we find in the PXP model a renormalized classical spin liquid with constant entropy density $S/N$ approaching $\ln(2)/6$ in the thermodynamic limit for all moderate and large values of the detuning $\delta$ and starting from $T/\Omega \sim 0.5$ (in units of the Rabi frequency $\Omega$) down to the lowest temperatures we could simulate, $T/\Omega \sim 0.01$. With Van der Waals interactions, constant entropy plateaus are still found but its value shifts with $\delta$. We comment the adiabatic approximation to the dynamical ramps for the electric degrees of freedom., Comment: 4 pages, 6 figures + supplemental

Even when ideal solids are insulating, their states with crystallographic defects may have superfluid properties. It became clear recently that edge dislocations in $^4$He featuring a combination of microscopic quantum roughness and superfluidity of their cores may represent a new paradigmatic class of quasi-one-dimensional superfluids. The new state of matter, termed transverse quantum fluid (TQF), is found in a variety of physical setups. The key ingredient defining the class of TQF systems is infinite compressibility, which is responsible for all other unusual properties such as the quadratic spectrum (or even the absence) of normal modes, irrelevance of the Landau criterion, off-diagonal long-range order at $T = 0$, and the exponential dependence of the phase slip probability on the inverse flow velocity. From a conceptual point of view, the TQF state is a striking demonstration of the conditional character of many dogmas associated with superfluidity, including the necessity of elementary excitations, in general, and the ones obeying Landau criterion in particular., Comment: Review for Annu. Rev. Condens. Matter Phys

Monochromatic gamma-ray signals constitute a potential smoking gun signature for annihilating or decaying dark matter particles that could relatively easily be distinguished from astrophysical or instrumental backgrounds. We provide an updated assessment of the sensitivity of the Cherenkov Telescope Array (CTA) to such signals, based on observations of the Galactic centre region as well as of selected dwarf spheroidal galaxies. We find that current limits and detection prospects for dark matter masses above 300 GeV will be significantly improved, by up to an order of magnitude in the multi-TeV range. This demonstrates that CTA will set a new standard for gamma-ray astronomy also in this respect, as the world's largest and most sensitive high-energy gamma-ray observatory, in particular due to its exquisite energy resolution at TeV energies and the adopted observational strategy focussing on regions with large dark matter densities. Throughout our analysis, we use up-to-date instrument response functions, and we thoroughly model the effect of instrumental systematic uncertainties in our statistical treatment. We further present results for other potential signatures with sharp spectral features, e.g.~box-shaped spectra, that would likewise very clearly point to a particle dark matter origin., Comment: 44 pages JCAP style (excluding author list and references), 19 figures; minor changes to match published version

Machine learning algorithms thrive on large data sets of good quality. Here we show that they can also excel in a typical research setting with little data of limited quality, through an interplay of insights coming from machine, and human researchers. The question we address is the unsolved problem of ordering out of a spin-liquid phase described by an emergent rank-2 $U(1)$ gauge theory, as described by [H. Yan it et al., Phys. Rev. Lett. 124, 127203 (2020)]. Published Monte Carlo simulations for this problem are consistent with a strong first-order phase transition, out of the R2-U1 spin liquid, but were too noisy for the form of low-temperature order to be identified. Using a highly-interpretable machine learning approach based on a support vector machine with a tensorial kernel (TKSVM), we re-analyze this Monte Carlo data, gaining new information about the form of order that could in turn be interpreted by traditionally-trained physicists. We find that the low-temperature ordered phase is a form of hybrid nematic order with emergent $Z_2$ symmetry, which allows for a sub-extensive set of domain walls at zero energy. This complex form of order arises due to a subtle thermal order-by-disorder mechanism, that can be understood from the fluctuations of the tensor electric field of the parent rank-2 gauge theory. These results were obtained by a back-and-forth process which closely resembles a collaboration between human researchers and machines. We argue that this "collaborative" approach may provide a blueprint for solving other problems that have not yielded to human insights alone.

Lattice gauge theories (LGTs) were introduced in 1974 by Wilson to study quark confinement. These models have been shown to exhibit (de-)confined phases, yet it remains challenging to define experimentally accessible order parameters. Here we propose percolation-inspired order parameters (POPs) to probe confinement of dynamical matter in $\mathbb{Z}_2$ LGTs using electric field basis snapshots accessible to quantum simulators. We apply the POPs to study a classical $\mathbb{Z}_2$ LGT and find a confining phase up to temperature $T=\infty$ in 2D (critical $T_c$, i.e. finite-$T$ phase transition, in 3D) for any non-zero density of $\mathbb{Z}_2$ charges. Further, using quantum Monte Carlo we demonstrate that the POPs reproduce the square lattice Fradkin-Shenker phase diagram at $T=0$ and explore the phase diagram at $T>0$. The correlation length exponent coincides with the one of the 3D Ising universality class and we determine the POP critical exponent characterizing percolation. Our proposed POPs provide a geometric perspective of confinement and are directly accessible to snapshots obtained in quantum simulators, making them suitable as a probe for quantum spin liquids., Comment: 5+9 pages, 4+6 figures

Fracton codes host unconventional topological states of matter and are promising for fault-tolerant quantum computation due to their large coding space and strong resilience against decoherence and noise. In this work, we investigate the ground-state properties and phase transitions of two prototypical self-dual fracton spin models -- the tetrahedral Ising model and the fractal Ising model -- which correspond to error-correction procedures for the representative fracton codes of type-I and type-II, the checkerboard code and the Haah's code, respectively, in the error-free limit. They are endowed with exotic symmetry-breaking properties that contrast sharply with the spontaneous breaking of global symmetries and deconfinement transition of gauge theories. To show these unconventional behaviors, which are associated with sub-dimensional symmetries, we construct and analyze the order parameters, correlators, and symmetry generators for both models. Notably, the tetrahedral Ising model acquires an extended semi-local ordering moment, while the fractal Ising model fits into a polynomial ring representation and leads to a fractal order parameter. Numerical studies combined with analytical tools show that both models experience a strong first-order phase transition with an anomalous $L^{-(D-1)}$ scaling, despite the fractal symmetry of the latter. Our work provides new understanding of sub-dimensional symmetry breaking and makes an important step for studying quantum-error-correction properties of the checkerboard and Haah's codes., Comment: 13 pages, 6 figures. v2: published version

We study universal off-diagonal correlations in transverse quantum fluids (TQF) -- a new class of quasi-one-dimensional superfluids featuring long-range-ordered ground states. These exhibit unique self-similar space-time relations scaling with $x^2/D\tau$ that serve as fingerprints of the specific states. The results obtained with the effective field theory are found to be in perfect agreement with {\it ab initio} simulations of hard-core bosons on a lattice -- a simple microscopic realization of TQF. This allows an accurate determination -- at nonzero temperature and finite system size -- of such key ground-state properties as the condensate and superfluid densities, and characteristic parameter $D$.

The Large-Sized Telescopes (LSTs) of the Cherenkov Telescope Array Observatory (CTAO) will play a crucial role in the study of transient gamma-ray sources, such as gamma-ray bursts and flaring active galactic nuclei. The low energy threshold of LSTs makes them particularly well suited for the detection of these phenomena. The ability to detect and analyze gamma-ray transients in real-time is essential for quickly identifying and studying these rare and fleeting events. In this conference, we will present recent advances in the real-time analysis of data from the LST-1, the first prototype of LST located in the Canary island of La Palma. We will discuss in particular the development of new algorithms for event reconstruction and background rejection. These advances will enable rapid identification and follow-up observation of transient gamma-ray sources, making the LST-1 a powerful tool for the study of the dynamic universe. The implementation of this framework in the future Array Control and Data Acquisition System (ACADA) of CTAO will be discussed as well, based on the experience with LST., Comment: Presented at the 38th International Cosmic Ray Conference (ICRC 2023), 2023 (arXiv:2309.08219)

Speech generation for machine dubbing adds complexity to conventional Text-To-Speech solutions as the generated output is required to match the expressiveness, emotion and speaking rate of the source content. Capturing and transferring details and variations in prosody is a challenge. We introduce phrase-level cross-lingual prosody transfer for expressive multi-lingual machine dubbing. The proposed phrase-level prosody transfer delivers a significant 6.2% MUSHRA score increase over a baseline with utterance-level global prosody transfer, thereby closing the gap between the baseline and expressive human dubbing by 23.2%, while preserving intelligibility of the synthesised speech., Comment: Accepted to INTERSPEECH 2023

Prosody transfer is well-studied in the context of expressive speech synthesis. Cross-lingual prosody transfer, however, is challenging and has been under-explored to date. In this paper, we present a novel solution to learn prosody representations that are transferable across languages and speakers for machine dubbing of expressive multimedia contents. Multimedia contents often contain field recordings. To enable prosody transfer from noisy audios, we introduce a novel noise modelling module that disentangles noise conditioning from prosody conditioning, and thereby gains independent control of noise levels in the synthesised speech. We augment noisy training data with clean data to improve the ability of the model to map the denoised reference audio to clean speech. Our proposed system can generate speech with context-matching prosody and closes the gap between a strong baseline and human expressive dialogs by 11.2%., Comment: Submitted to INTERSPEECH

Motivated by a recent experiment on a square-lattice Rydberg atom array realizing a long-range dipolar XY model [Chen et al., Nature (2023)], we numerically study the model's equilibrium properties. We obtain the phase diagram, critical properties, entropies, variance of the magnetization, and site-resolved correlation functions. We consider both ferromagnetic and antiferromagnetic interactions and apply quantum Monte Carlo and pseudo-Majorana functional renormalization group techniques, generalizing the latter to a U(1) symmetric setting. Our simulations perform extensive thermometry for the first time in dipolar Rydberg atom arrays and establish conditions for adiabaticity and thermodynamic equilibrium. On the ferromagnetic side of the experiment, we determine the entropy per particle S/N~0.5, close to the one at the critical temperature, S_c/N = 0.585(15). The simulations suggest the presence of an out-of-equilibrium plateau at large distances in the correlation function, thus motivating future studies on the non-equilibrium dynamics of the system., Comment: 17 pages

We contend that the arguments provided in Phys. Rev. Lett. 130, 016001 (2023), purporting to show the absence of off-diagonal long-range order in hcp $^4$He dislocation cores are misleading and incorrect. In particular, the one-body density matrix averaged over the whole crystalline sample provides no useful information on the possible superfluid behavior inside the quasi-one-dimensional core of a dislocation in the crystal.

The centred pyrochlore lattice is a novel geometrically frustrated lattice, realized in the metal-organic framework Mn(ta)$_2$ (arXiv:2203.08780) where the basic unit of spins is a five site centred tetrahedron. Here, we present an in-depth theoretical study of the $J_1-J_2$ classical Heisenberg model on this lattice, using a combination of mean-field analytical methods and Monte Carlo simulations. We find a rich phase diagram with low temperature states exhibiting ferrimagnetic order, partial ordering, and a highly degenerate spin liquid with distinct regimes of low temperature correlations. We discuss in detail how the regime displaying broadened pinch points in its spin structure factor is consistent with an effective description in terms of a fluid of interacting charges. We also show how this picture holds in two dimensions on the analogous centred kagome lattice and elucidate the connection to the physics of thin films in ($d+1$) dimensions. Furthermore, we show that a Coulomb phase can be stabilized on the centred pyrochlore lattice by the addition of further neighbour couplings. This demonstrates the centred pyrochlore lattice is an experimentally relevant geometry which naturally hosts emergent gauge fields in the presence of charges at low energies., Comment: 29 pages, 9 figures, resubmission to SciPost with minor revisions

We study the phase diagram of the one-dimensional Bose-Fermi-Hubbard model at unit filling for the scalar bosons and half filling for the $S=1/2$ fermions using quantum Monte Carlo simulations. The bare interaction between the fermions is set to zero. A central question of our study is what type of interactions can be induced between the fermions by the bosons, for both weak and strong interspecies coupling. We find that the induced interactions can lead to competing instabilities favoring phase separation, superconducting phases, and density wave structures, in many cases at work on length scales of more than 100 sites. Marginal bosonic superfluids with a density matrix decaying faster than what is allowed for pure bosonic systems with on-site interactions, are also found., Comment: 19 pages, 37 figures

We study the phase diagram of two different mixed-dimensional $t-J_z-J_{\perp}$-models on the square lattice, in which the hopping amplitude $t$ is only nonzero along the $x$-direction. In the first, bosonic, model, the spin exchange amplitude $J_{\perp}$ is negative and isotropic along the $x$ and $y$ directions of the lattice, and $J_z$ is isotropic and positive. The low-energy physics is characterized by spin-charge separation: the holes hop as free fermions in an easy-plane ferromagnetic background. In the second model, $J_{\perp}$ is restricted to the $x$-axis while $J_z$ remains isotropic and positive. The model is agnostic to particle statistics, and shows stripe patterns with anti-ferromagnetic N{\'e}el order at low temperature and high hole concentrations, in resemblance of the mixed-dimensional $t-J_z$ and $t-J$ models. At lower hole concentration, a very strong first order transition and hysteresis loop is seen extending to a remarkably high 14(1)% hole doping., Comment: 8 pages, 9 figures

We investigate two concrete cases of phase transitions breaking a subsystem symmetry. The models are two classical compass models featuring line-flip and plane-flip symmetries and correspond to special limits of a Heisenberg-Kitaev Hamiltonian on a cubic lattice. We show that these models experience a hybrid symmetry breaking by which the system display distinct symmetry broken patterns in different submanifolds. For instance, the system may look magnetic within a chain or plane but nematic-like when observing from one dimensionality higher. We fully characterize the symmetry-broken phases by a set of subdimensional order parameters and confirm numerically both cases undergo a non-standard first-order phase transition. Our results provide new insights into phase transitions involving subsystem symmetries and generalize the notion of conventional spontaneous symmetry breaking., Comment: 8 pages, 6 figures

Experimental progress in qubit manufacturing calls for the development of new theoretical tools to analyze quantum data. We show how an unsupervised machine-learning technique can be used to understand short-range entangled many-qubit systems using data of local measurements. The method successfully constructs the phase diagram of a cluster-state model and detects the respective order parameters of its phases, including string order parameters. For the toric code subject to external magnetic fields, the machine identifies the explicit forms of its two stabilizers. Prior information of the underlying Hamiltonian or the quantum states is not needed; instead, the machine outputs their characteristic observables. Our work opens the door for a first-principles application of hybrid algorithms that aim at strong interpretability without supervision., Comment: 9 pages, 7 figures

Microscopically understanding competing orders in strongly correlated systems is a key challenge in modern quantum many-body physics. For example, the origin of stripe order and its relation to pairing in the Fermi-Hubbard model remains one of the central questions, and may help to understand the origin of high-temperature superconductivity in cuprates. Here, we analyze stripe formation in the doped mixed-dimensional (mixD) variant of the $t-J$ model, where charge carriers are restricted to move only in one direction, whereas magnetic SU(2) interactions are two-dimensional. Using the density matrix renormalization group at finite temperature, we find a stable vertical stripe phase in the absence of pairing, featuring incommensurate magnetic order and long-range charge density wave profiles over a wide range of dopings. We find high critical temperatures on the order of the magnetic coupling $\sim J/2$, hence being within reach of current quantum simulators. Snapshots of the many-body state, accessible to quantum simulators, reveal hidden spin correlations in the mixD setting, whereby antiferromagnetic correlations are enhanced when considering purely the magnetic background. The proposed model can be viewed as realizing a parent Hamiltonian of the stripe phase, whose hidden spin correlations lead to the predicted resilience against quantum and thermal fluctuations., Comment: 5 + 6 pages

We investigate the ground-state nature of the honeycomb Kitaev-$\Gamma$ model in the material-relevant parameter regime through a combination of classical and quantum simulations. The classical model is imprinted with a tangle of highly structured spin double helices. This helix tangle exhibits $18$ inequivalent helical axes and features a spontaneous periodicity anisotropy and a ${\rm sgn}(\Gamma)$-determined chirality pattern. Infinite PEPS simulations with clusters up to $36$ sites identify hallmarks of this many-body order in the quantum spin-$1/2$ model. Our findings provide a fresh perspective of the Kitaev-$\Gamma$ model and enrich the physics of Kitaev magnetism., Comment: 4+ $\epsilon$ pages, 5 figures plus supplemental materials; v2: expanded discussions of the modulated spin helicity, additional simulations in SM

We present a novel and open-source implementation of the worm algorithm, which is an algorithm to simulate Bose-Hubbard and sign-positive spin models using a path integral representation of the partition function. The code can deal with arbitrary lattice structures and assumes spin-exchange terms, or bosonic hopping amplitudes, between nearest-neighbor sites, and local or nearest-neighbor interactions of the density-density type. We explicitly demonstrate the near-linear scaling of the algorithm with respect to the system volume and the inverse temperature and analyze the autocorrelation times in the vicinity of a U(1) second order phase transition. The code is written in such a way that extensions to other lattice models as well as closely-related sign-positive models can be done straightforwardly on top of the provided framework., Comment: 21 pages, 8 figures; open source code available under https://github.com/LodePollet/worm

Geometric frustration inhibits magnetic systems from ordering, opening a window to unconventional phases of matter. The paradigmatic frustrated lattice in three dimensions to host a spin liquid is the pyrochlore, although there remain few experimental compounds thought to realize such a state. Here we go beyond the pyrochlore via molecular design in the metal-azolate framework [Mn(II)(ta)$_2$], which realizes a closely related centred pyrochlore lattice of Mn-spins with $S=5/2$. Despite a Curie-Weiss temperature of $-21$ K indicating the energy scale of magnetic interactions, [Mn(II)(ta)$_2$] orders at only 430 mK, putting it firmly in the category of highly frustrated magnets. Comparing magnetization and specific heat measurements to numerical results for a minimal Heisenberg model, we predict that this material displays distinct features of a classical spin liquid with a structure factor reflecting Coulomb physics in the presence of charges., Comment: Main: 6 pages, 3 figures. Supplementary: 15 pages, 12 figures. Updated Version

The unique superflow-through-solid effect observed in solid Helium-4 and attributed to the quasi-one-dimensional superfluidity along the dislocation cores exhibits two extraordinary features: (i) an exponentially strong suppression of the flow by a moderate increase in pressure, and (ii) an unusual temperature dependence of the flow rate with no analogy to any known system and in contradiction with the standard Luttinger liquid paradigm. Based on ab initio and model simulations, we argue that the two features are closely related: Thermal fluctuations of the shape of a superclimbing edge dislocation induce large, correlated, and asymmetric stress fields acting on the superfluid core. The critical flux is most sensitive to strong rare fluctuations and hereby acquires a sharp temperature dependence observed in experiments., Comment: replaced by updated version accepted by PRL

Fracton models provide examples of novel gapped quantum phases of matter that host intrinsically immobile excitations and therefore lie beyond the conventional notion of topological order. Here, we calculate optimal error thresholds for quantum error correcting codes based on fracton models. By mapping the error-correction process for bit-flip and phase-flip noises into novel statistical models with Ising variables and random multi-body couplings, we obtain models that exhibit an unconventional subsystem symmetry instead of a more usual global symmetry. We perform large-scale parallel tempering Monte Carlo simulations to obtain disorder-temperature phase diagrams, which are then used to predict optimal error thresholds for the corresponding fracton code. Remarkably, we found that the X-cube fracton code displays a minimum error threshold ($7.5\%$) that is much higher than 3D topological codes such as the toric code ($3.3\%$), or the color code ($1.9\%$). This result, together with the predicted absence of glass order at the Nishimori line, shows great potential for fracton phases to be used as quantum memory platforms., Comment: 6+10 pages, 4+4 figures, Fig 1 selected for PRL cover. v2: minor changes to introduction, more quantum error correction details added to supplementary materials, typos corrected, published version

Deep learning models are becoming predominant in many fields of machine learning. Text-to-Speech (TTS), the process of synthesizing artificial speech from text, is no exception. To this end, a deep neural network is usually trained using a corpus of several hours of recorded speech from a single speaker. Trying to produce the voice of a speaker other than the one learned is expensive and requires large effort since it is necessary to record a new dataset and retrain the model. This is the main reason why the TTS models are usually single speaker. The proposed approach has the goal to overcome these limitations trying to obtain a system which is able to model a multi-speaker acoustic space. This allows the generation of speech audio similar to the voice of different target speakers, even if they were not observed during the training phase.

We use a recently developed interpretable and unsupervised machine-learning method, the tensorial kernel support vector machine (TK-SVM), to investigate the low-temperature classical phase diagram of a generalized Heisenberg-Kitaev-$\Gamma$ ($J$-$K$-$\Gamma$) model on a honeycomb lattice. Aside from reproducing phases reported by previous quantum and classical studies, our machine finds a hitherto missed nested zigzag-stripy order and establishes the robustness of a recently identified modulated $S_3 \times Z_3$ phase, which emerges through the competition between the Kitaev and $\Gamma$ spin liquids, against Heisenberg interactions. The results imply that, in the restricted parameter space spanned by the three primary exchange interactions -- $J$, $K$, and $\Gamma$, the representative Kitaev material $\alpha$-${\rm RuCl}_3$ lies close to the boundaries of several phases, including a simple ferromagnet, the unconventional $S_3 \times Z_3$ and nested zigzag-stripy magnets. A zigzag order is stabilized by a finite $\Gamma^{\prime}$ and/or $J_3$ term, whereas the four magnetic orders may compete in particular if $\Gamma^{\prime}$ is anti-ferromagnetic., Comment: 13 pages, 12 figures, 1 table; expanded discussions, added references; as published