The search of unconventional magnetic and nonmagnetic states is a major topic in the study of frustrated magnetism. Canonical examples of those states include various spin liquids and spin nematics. However, discerning their existence and the correct characterization is usually challenging. Here we introduce a machine-learning protocol that can identify general nematic order and their order parameter from seemingly featureless spin configurations, thus providing comprehensive insight on the presence or absence of hidden orders. We demonstrate the capabilities of our method by extracting the analytical form of nematic order parameter tensors up to rank 6. This may prove useful in the search for novel spin states and for ruling out spurious spin liquid candidates.The statistical learning of phases is nowadays an active field of research [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Despite the enormous recent progress, learning or classifying intricate phases in manybody systems remains a daunting task. Many recent algorithmic advances are tried and tested in only the simplest of models, and their applicability to more complex situations remains an open question. The ability to interpret results to gain physical insight has been identified as one of the key challenges in the application of machine learning techniques to the domain of physics. Still, recent approaches struggle and this is only exacerbated when going beyond those simple models. However, those situations can also be arenas for machine learning methods to demonstrate their features and prove their worth, in comparison to-or complementary to-traditional methods.One such arena may be found in frustrated spin and spin-orbital-coupled systems [18]. These systems have rich phase diagrams, supporting various spin nematic (multipolar ordered) [19][20][21][22][23][24][25][26][27][28] and spin liquid phases [29][30][31][32][33][34][35]. However, to distinguish these two types of phases is often tricky, since both of them are invisible to conventional magnetic measurements. Indeed, there have been steady reports of "hidden" multipolar orders from a magnetically disordering state [36][37][38][39][40][41][42][43][44][45][46][47][48]. Moreover, identifying the right characterization of a spin-nematic order can also be a nontrivial task. For instance, in the low temperature phase of the classical Heisenberg-Kagomé antiferromagnet, a hidden quadrupolar order was found first [36], followed by the realization of an additional octupolar order [37] and its optimal order parameter [39,40].The aforementioned multipolar orders are only the simplest ones admitted by the subgroup structure of O(3). There are indeed myriads of more complicated multipolar orders where even the abstract classification of their order parameters has only been accomplished two years ago [49][50][51]. Along with the diverse interactions and lattice geometries in frustrated systems, identifying or ruling out certain orders becomes a difficult task for traditional methods, as there is n...

The concept of symmetry breaking has been a propelling force in understanding phases of matter. While rotational-symmetry breaking is one of the most prevalent examples, the rich landscape of orientational orders breaking the rotational symmetries of isotropic space, i.e., O(3), to a three-dimensional point group remain largely unexplored, apart from simple examples such as ferromagnetic or uniaxial nematic ordering. Here we provide an explicit construction, utilizing a recently introduced gauge-theoretical framework, to address the three-dimensional point-group-symmetric orientational orders on a general footing. This unified approach allows us to enlist order parameter tensors for all three-dimensional point groups. By construction, these tensor order parameters are the minimal set of simplest tensors allowed by the symmetries that uniquely characterize the orientational order. We explicitly give these for the point groups {C n ,D n ,T ,O,I } ⊂ SO(3) andfor n,2n ∈ {1,2,3,4,6,∞}. This central result may be perceived as a road map for identifying exotic orientational orders that may become more and more in reach in view of rapid experimental progress in, e.g., nanocolloidal systems and novel magnets.

The physics of nematic liquid crystals has been the subject of intensive research since the late 19th century. However, the focus of this pursuit has been centered around uniaxial and biaxial nematics associated with constituents bearing a D ∞h or D 2h symmetry, respectively. In view of general symmetries, however, these are singularly special since nematic order can in principle involve any point-group symmetry. Given the progress in tailoring nanoparticles with particular shapes and interactions, this vast family of "generalized nematics" might become accessible in the laboratory. Little is known because the order parameter theories associated with the highly symmetric point groups are remarkably complicated, involving tensor order parameters of high rank. Here, we show that the generic features of the statistical physics of such systems can be studied in a highly flexible and efficient fashion using a mathematical tool borrowed from high-energy physics: discrete non-Abelian gauge theory. Explicitly, we construct a family of lattice gauge models encapsulating nematic ordering of general three-dimensional point-group symmetries. We find that the most symmetrical generalized nematics are subjected to thermal fluctuations of unprecedented severity. As a result, novel forms of fluctuation phenomena become possible. In particular, we demonstrate that a vestigial phase carrying no more than chiral order becomes ubiquitous departing from high point-group symmetry chiral building blocks, such as I, O, and T symmetric matter.

Machine-learning techniques are evolving into a subsidiary tool for studying phase transitions in many-body systems. However, most studies are tied to situations involving only one phase transition and one order parameter. Systems that accommodate multiple phases of coexisting and competing orders, which are common in condensed matter physics, remain largely unexplored from a machine-learning perspective. In this paper, we investigate multiclassification of phases using support vector machines (SVMs) and apply a recently introduced kernel method for detecting hidden spin and orbital orders to learn multiple phases and their analytical order parameters. Our focus is on multipolar orders and their tensorial order parameters whose identification is difficult with traditional methods. The importance of interpretability is emphasized for physical applications of multiclassification. Furthermore, we discuss an intrinsic parameter of SVM, the bias, which allows for a special interpretation in the classification of phases, and its utility in diagnosing the existence of phase transitions. We show that it can be exploited as an efficient way to explore the topology of unknown phase diagrams where the supervision is entirely delegated to the machine.

A novel gapped metallic state coined orthogonal Dirac semimetal is proposed in the honeycomb lattice in terms of Z2 slave-spin representation of Hubbard model. This state corresponds to the disordered phase of slave-spin and has the same thermaldynamical and transport properties as usual Dirac semimetal but its singe-particle excitation is gapped and has nontrivial topological order due to the Z2 gauge structure. The quantum phase transition from this orthogonal Dirac semimetal to usual Dirac semimetal is described by a mean-field decoupling with complementary fluctuation analysis and its criticality falls into the universality class of 2+1D Ising model while a large anomalous dimension for the physical electron is found at quantum critical point (QCP), which could be considered as a fingerprint of our fractionalized theory when compared to other non-fractionalized approaches. As byproducts, a path integral formalism for the Z2 slave-spin representation of Hubbard model is constructed and possible relations to other approaches and the sublattice pairing states, which has been argued to be a promising candidate for gapped spin liquid state found in the numerical simulation, are briefly discussed. Additionally, when spin-orbit coupling is considered, the instability of orthogonal Dirac semimetal to the fractionalized quantum spin Hall insulator (fractionalized topological insulator) is also expected. We hope the present work may be helpful for future studies in Z2 slave-spin theory and related non-Fermi liquid phases in honeycomb lattice.