Coupling ultracold matter to dynamical gauge fields in optical lattices: From flux attachment to ?2 lattice gauge theories
Luca Barbiero, Christian Schweizer, Monika Aidelsburger, Eugene Demler, Nathan Goldman and Fabian Grusdt.
Science Advances 5, (2019).
From the standard model of particle physics to strongly correlated electrons, various physical settings are formulated in terms of matter coupled to gauge fields. Quantum simulations based on ultracold atoms in optical lattices provide a promising avenue to study these complex systems and unravel the underlying many-body physics. Here, we demonstrate how quantized dynamical gauge fields can be created in mixtures of ultracold atoms in optical lattices, using a combination of coherent lattice modulation with strong interactions. Specifically, we propose implementation of ?2 lattice gauge theories coupled to matter, reminiscent of theories previously introduced in high-temperature superconductivity. We discuss a range of settings from zero-dimensional toy models to ladders featuring transitions in the gauge sector to extended two-dimensional systems. Mastering lattice gauge theories in optical lattices constitutes a new route toward the realization of strongly correlated systems, with properties dictated by an interplay of dynamical matter and gauge fields.
Analogue quantum chemistry simulation
Javier Argüello-Luengo, Alejandro González-Tudela, Tao Shi, Peter Zoller & J. Ignacio Cirac.
Nature 574, 215-218 (2019).
Computing the electronic structure of molecules with high precision is a central challenge in the field of quantum chemistry. Despite the success of approximate methods, tackling this problem exactly with conventional computers remains a formidable task. Several theoretical and experimental attempts have been made to use quantum computers to solve chemistry problems, with early proof-of-principle realizations done digitally. An appealing alternative to the digital approach is analogue quantum simulation, which does not require a scalable quantum computer and has already been successfully applied to solve condensed matter physics problems. However, not all available or planned setups can be used for quantum chemistry problems, because it is not known how to engineer the required Coulomb interactions between them. Here we present an analogue approach to the simulation of quantum chemistry problems that relies on the careful combination of two technologies: ultracold atoms in optical lattices and cavity quantum electrodynamics. In the proposed simulator, fermionic atoms hopping in an optical potential play the role of electrons, additional optical potentials provide the nuclear attraction, and a single-spin excitation in a Mott insulator mediates the electronic Coulomb repulsion with the help of a cavity mode. We determine the operational conditions of the simulator and test it using a simple molecule. Our work opens up the possibility of efficiently computing the electronic structures of molecules with analogue quantum simulation.
Floquet approach to Z2 lattice gauge theories with ultracold atoms in optical lattices
Christian Schweizer, Fabian Grusdt, Moritz Berngruber, Luca Barbiero, Eugene Demler, Nathan Goldman, Immanuel Bloch, Monika Aidelsburger.
Nature Physics (2019).
Quantum simulation has the potential to investigate gauge theories in strongly-interacting regimes, which are up to now inaccessible through conventional numerical techniques. Here, we take a first step in this direction by implementing a Floquet-based method for studying Z2 lattice gauge theories using two-component ultracold atoms in a double-well potential. For resonant periodic driving at the on-site interaction strength and an appropriate choice of the modulation parameters, the effective Floquet Hamiltonian exhibits Z2 symmetry. We study the dynamics of the system for different initial states and critically contrast the observed evolution with a theoretical analysis of the full time-dependent Hamiltonian of the periodically-driven lattice model. We reveal challenges that arise due to symmetry-breaking terms and outline potential pathways to overcome these limitations. Our results provide important insights for future studies of lattice gauge theories based on Floquet techniques.
Confined phases of one-dimensional spinless fermions coupled to Z2 gauge theory
Umberto Borla, Ruben Verresen, Fabian Grusdt, Sergej Moroz
We investigate a quantum many-body lattice system of one-dimensional spinless fermions interacting with a dynamical Z2 gauge field. The gauge field mediates long-range attraction between fermions resulting in their confinement into bosonic dimers. At strong coupling we develop an exactly solvable effective theory of such dimers with emergent constraints. Even at generic coupling and fermion density, the model can be rewritten as a local spin chain. Using the Density Matrix Renormalization Group the system is shown to form a Luttinger liquid, indicating the emergence of fermionic fractionalized excitations despite the confinement of lattice fermions. In a finite chain we observe the doubling of the period of Friedel oscillations which paves the way towards experimental detection of confinement in this system. We discuss the possibility of a Mott phase at the commensurate filling 2/3.
Photon Correlation Spectroscopy of Luminescent Quantum Defects in Carbon Nanotubes
Manuel Nutz, Jiaxiang Zhang, Mijin Kim, Hyejin Kwon, Xiaojian Wu, YuHuang Wang, Alexander Högele.
Nano Letters (2019).
Defect-decorated single-wall carbon nanotubes have shown rapid growing potential for imaging, sensing, and the development of room-temperature single-photon sources. The key to the highly nonclassical emission statistics is the discrete energy spectrum of defect-localized excitons. However, variations in defect configurations give rise to distinct spectral bands that may compromise single-photon efficiency and purity in practical devices, and experimentally it has been challenging to study the exciton population distribution among the various defect-specific states. Here, we performed photon correlation spectroscopy on hexyl-decorated single-wall carbon nanotubes to unravel the dynamics and competition between neutral and charged exciton populations. With autocorrelation measurements at the single-tube level, we prove the nonclassical photon emission statistics of defect-specific exciton and trion photoluminescence and identify their mutual exclusiveness in photoemissive events with cross-correlation spectroscopy. Moreover, our study reveals the presence of a dark state with population-shelving time scales between 10 and 100 ns. These new insights will guide further development of chemically tailored carbon nanotube states for quantum photonics applications.
Many-body chaos near a thermal phase transition
Alexander Schuckert, Michael Knap.
SciPost Physics 7, 022 (2019).
We study many-body chaos in a (2+1)D relativistic scalar field theory at high temperatures in the classical statistical approximation, which captures the quantum critical regime and the thermal phase transition from an ordered to a disordered phase. We evaluate out-of-time ordered correlation functions (OTOCs) and find that the associated Lyapunov exponent increases linearly with temperature in the quantum critical regime, and approaches the non-interacting limit algebraically in terms of a fluctuation parameter. OTOCs spread ballistically in all regimes, also at the thermal phase transition, where the butterfly velocity is maximal. Our work contributes to the understanding of the relation between quantum and classical many-body chaos and our method can be applied to other field theories dominated by classical modes at long wavelengths.
Topological proximity effects in a Haldane graphene bilayer system
Peng Cheng, Philipp W. Klein, Kirill Plekhanov, Klaus Sengstock, Monika Aidelsburger, Christof Weitenberg, and Karyn Le Hur.
Physical Review B 100, 081107(R) (2019).
We reveal a proximity effect between a topological band (Chern) insulator described by a Haldane model and spin-polarized Dirac particles of a graphene layer. Coupling weakly the two systems through a tunneling term in the bulk, the topological Chern insulator induces a gap and an opposite Chern number on the Dirac particles at half filling, resulting in a sign flip of the Berry curvature at one Dirac point. We study different aspects of the bulk-edge correspondence and present protocols to observe the evolution of the Berry curvature as well as two counterpropagating (protected) edge modes with different velocities. In the strong-coupling limit, the energy spectrum shows flat bands. Therefore we build a perturbation theory and address further the bulk-edge correspondence. We also show the occurrence of a topological insulating phase with Chern number one when only the lowest band is filled. We generalize the effect to Haldane bilayer systems with asymmetric Semenoff masses. Moreover, we propose an alternative definition of the topological invariant on the Bloch sphere.
Emergent Glassy Dynamics in a Quantum Dimer Model
Johannes Feldmeier, Frank Pollmann, and Michael Knap.
Physical Review Letters 123, 040601 (2019).
We consider the quench dynamics of a two-dimensional quantum dimer model and determine the role of its kinematic constraints. We interpret the nonequilibrium dynamics in terms of the underlying equilibrium phase transitions consisting of a Berezinskii-Kosterlitz-Thouless (BKT) transition between a columnar ordered valence bond solid (VBS) and a valence bond liquid (VBL), as well as a first-order transition between a staggered VBS and the VBL. We find that quenches from a columnar VBS are ergodic and both order parameters and spatial correlations quickly relax to their thermal equilibrium. By contrast, the staggered side of the first-order transition does not display thermalization on numerically accessible timescales. Based on the model’s kinematic constraints, we uncover a mechanism of relaxation that rests on emergent, highly detuned multidefect processes in a staggered background, which gives rise to slow, glassy dynamics at low temperatures even in the thermodynamic limit.
String patterns in the doped Hubbard model
Christie S. Chiu, Geoffrey Ji, Annabelle Bohrdt, Muqing Xu, Michael Knap, Eugene Demler, Fabian Grusdt, Markus Greiner, Daniel Greif.
Science 365, 251-256 (2019).
Understanding strongly correlated quantum many-body states is one of the most difficult challenges in modern physics. For example, there remain fundamental open questions on the phase diagram of the Hubbard model, which describes strongly correlated electrons in solids. In this work, we realize the Hubbard Hamiltonian and search for specific patterns within the individual images of many realizations of strongly correlated ultracold fermions in an optical lattice. Upon doping a cold-atom antiferromagnet, we find consistency with geometric strings, entities that may explain the relationship between hole motion and spin order, in both pattern-based and conventional observables. Our results demonstrate the potential for pattern recognition to provide key insights into cold-atom quantum many-body systems.
Classifying snapshots of the doped Hubbard model with machine learning
Annabelle Bohrdt, Christie S. Chiu, Geoffrey Ji, Muqing Xu, Daniel Greif, Markus Greiner, Eugene Demler, Fabian Grusdt und Michael Knap.
Nature Physics (2019).
Quantum gas microscopes for ultracold atoms can provide high-resolution real-space snapshots of complex many-body systems. We implement machine learning to analyse and classify such snapshots of ultracold atoms. Specifically, we compare the data from an experimental realization of the two-dimensional Fermi–Hubbard model to two theoretical approaches: a doped quantum spin liquid state of resonating valence bond type (1,2), and the geometric string theory (3,4), describing a state with hidden spin order. This technique considers all available information without a potential bias towards one particular theory by the choice of an observable and can therefore select the theory that is more predictive in general. Up to intermediate doping values, our algorithm tends to classify experimental snapshots as geometric-string-like, as compared to the doped spin liquid. Our results demonstrate the potential for machine learning in processing the wealth of data obtained through quantum gas microscopy for new physical insights.
Site-selectively generated photon emitters in monolayer MoS2 via local helium ion irradiation
Julian Klein, Michael Lorke, Matthias Florian, Florian Sigger, Jakob Wierzbowski, John Cerne, Kai Müller, Takashi Taniguchi, Kenji Watanabe, Ursula Wurstbauer, Michael Kaniber, Michael Knap, Richard Schmidt, Jonathan J. Finley, Alexander W. Holleitner.
Nature Communications 10, Article number: 2755 (2019).
Quantum light sources in solid-state systems are of major interest as a basic ingredient for integrated quantum photonic technologies. The ability to tailor quantum emitters via site-selective defect engineering is essential for realizing scalable architectures. However, a major difficulty is that defects need to be controllably positioned within the material. Here, we overcome this challenge by controllably irradiating monolayer MoS2 using a sub-nm focused helium ion beam to deterministically create defects. Subsequent encapsulation of the ion exposed MoS2 flake with high-quality hBN reveals spectrally narrow emission lines that produce photons in the visible spectral range. Based on ab-initio calculations we interpret these emission lines as stemming from the recombination of highly localized electron–hole complexes at defect states generated by the local helium ion exposure. Our approach to deterministically write optically active defect states in a single transition metal dichalcogenide layer provides a platform for realizing exotic many-body systems, including coupled single-photon sources and interacting exciton lattices that may allow the exploration of Hubbard physics.
Atomtronics with a spin: Statistics of spin transport and nonequilibrium orthogonality catastrophe in cold quantum gases
Jhih-Shih You, Richard Schmidt, Dmitri A. Ivanov, Michael Knap, and Eugene Demler.
Physical Review B 99, 214505 (2019).
We propose to investigate the full counting statistics of nonequilibrium spin transport with an ultracold atomic quantum gas. The setup makes use of the spin control available in atomic systems to generate spin transport induced by an impurity atom immersed in a spin-imbalanced two-component Fermi gas. In contrast to solid-state realizations, in ultracold atoms spin relaxation and the decoherence from external sources is largely suppressed. As a consequence, once the spin current is turned off by manipulating the internal spin degrees of freedom of the Fermi system, the nonequilibrium spin population remains constant. Thus one can directly count the number of spins in each reservoir to investigate the full counting statistics of spin flips, which is notoriously challenging in solid-state devices. Moreover, using Ramsey interferometry, the dynamical impurity response can be measured. Since the impurity interacts with a many-body environment that is out of equilibrium, our setup provides a way to realize the nonequilibrium orthogonality catastrophe. Here, even for spin reservoirs initially prepared in a zero-temperature state, the Ramsey response exhibits an exponential decay, which is in contrast to the conventional power-law decay of Anderson's orthogonality catastrophe. By mapping our system to a multistep Fermi sea, we are able to derive analytical expressions for the impurity response at late times. This allows us to reveal an intimate connection of the decay rate of the Ramsey contrast and the full counting statistics of spin flips.
Secure quantum remote state preparation of squeezed microwave states
S. Pogorzalek, K. G. Fedorov, M. Xu, A. Parra-Rodriguez, M. Sanz, M. Fischer, E. Xie, K. Inomata, Y. Nakamura, E. Solano, A. Marx, F. Deppe, R. Gross.
Nature Communicationsvolume 10, 2604 (2019).
Quantum communication protocols based on nonclassical correlations can be more efficient than known classical methods and offer intrinsic security over direct state transfer. In particular, remote state preparation aims at the creation of a desired and known quantum state at a remote location using classical communication and quantum entanglement. We present an experimental realization of deterministic continuous-variable remote state preparation in the microwave regime over a distance of 35 cm. By employing propagating two-mode squeezed microwave states and feedforward, we achieve the remote preparation of squeezed states with up to 1.6 dB of squeezing below the vacuum level. Finally, security of remote state preparation is investigated by using the concept of the one-time pad and measuring the von Neumann entropies. We find nearly identical values for the entropy of the remotely prepared state and the respective conditional entropy given the classically communicated information and, thus, demonstrate close-to-perfect security.
Bosonic superfluid on lowest Landau level
Sergej Moroz, Dam Thanh Son
Physic Review Letters 122, 235301 (2019).
We develop a low-energy effective field theory of a two-dimensional bosonic superfluid on the lowest Landau level at zero temperature and identify a Berry term that governs the dynamics of coarse-grained superfluid degrees of freedom. For an infinite vortex crystal we compute how the Berry term affects the low-energy spectrum of soft collective Tkachenko oscillations and non-dissipative Hall responses of the particle number current and stress tensor. This term gives rise to a quadratic in momentum term in the Hall conductivity, but does not generate a non-dissipative Hall viscosity.
Avoided quasiparticle decay from strong quantum interactions
Ruben Verresen, Roderich Moessner & Frank Pollmann.
Nature Physics 15, 750-753 (2019).
Quantum states of matter—such as solids, magnets and topological phases—typically exhibit collective excitations (for example, phonons, magnons and anyons)1. These involve the motion of many particles in the system, yet, remarkably, act like a single emergent entity—a quasiparticle. Known to be long lived at the lowest energies, quasiparticles are expected to become unstable when encountering the inevitable continuum of many-particle excited states at high energies, where decay is kinematically allowed. Although this is correct for weak interactions, we show that strong interactions generically stabilize quasiparticles by pushing them out of the continuum. [...]
Quantum phases and topological properties of interacting fermions in one-dimensional superlattices
Leo Stenzel, Andrew L. C. Hayward, Claudius Hubig, Ulrich Schollwöck, Fabian Heidrich-Meisner
Physical Review A 99, 053614 (2019).
The realization of artificial gauge fields in ultracold atomic gases has opened up a path towards experimental studies of topological insulators and, as an ultimate goal, topological quantum matter in many-body systems. As an alternative to the direct implementation of two-dimensional lattice Hamiltonians that host the quantum Hall effect and its variants, topological charge-pumping experiments provide an additional avenue towards studying many-body systems. Here, we consider an interacting two-component gas of fermions realizing a family of one-dimensional superlattice Hamiltonians with onsite interactions and a unit cell of three sites, the ground states of which would be visited in an appropriately defined charge pump. First, we investigate the grand canonical quantum phase diagram of individual Hamiltonians, focusing on insulating phases. For a certain commensurate filling, there is a sequence of phase transitions from a band insulator to other insulating phases (related to the physics of ionic Hubbard models) for some members of the manifold of Hamiltonians. Second, we compute the Chern numbers for the whole manifold in a many-body formulation and show that, related to the aforementioned quantum phase transitions, a topological transition results in a change of the value and sign of the Chern number. We provide both an intuitive and a conceptual explanation and argue that these properties could be observed in quantum-gas experiments.
Quantum Zeno effect generalized
Tim Möbus, Michael M. Wolf
Journal of Mathematical Physics 60, 052201 (2019).
The quantum Zeno effect, in its original form, uses frequent projective measurements to freeze the evolution of a quantum system that is initially governed by a fixed Hamiltonian. We generalize this effect simultaneously in three directions by allowing open system dynamics, time-dependent evolution equations and general quantum operations in place of projective measurements. More precisely, we study Markovian master equations with bounded generators whose time dependence is Lipschitz continuous. Under a spectral gap condition on the quantum operation, we show how frequent measurements again freeze the evolution outside an invariant subspace. Inside this space, the evolution is described by a modified master equation.
Bounds on the bipartite entanglement entropy for oscillator systems with or without disorder
Vincent Beaud, Julian Sieber and Simone Warzel.
Journal of Physics A: Mathematical and Theoretical 52, 235202 (2019).
We give a direct alternative proof of an area law for the entanglement entropy of the ground state of disordered oscillator systems—a result due to Nachtergaele et al (2013 J. Math. Phys. 54 042110). Instead of studying the logarithmic negativity, we invoke the explicit formula for the entanglement entropy of Gaussian states to derive the upper bound. We also contrast this area law in the disordered case with divergent lower bounds on the entanglement entropy of the ground state of one-dimensional ordered oscillator chains.
Observation of many-body localization in an one-dimensional system with a single-particle mobility edge
Thomas Kohlert, Sebastian Scherg, Xiao Li, Henrik P. Lüschen, Sankar Das Sarma, Immanuel Bloch, Monika Aidelsburger.
Physical Review Letters 122, 170403 (2019).
We experimentally study many-body localization (MBL) with ultracold atoms in a weak one-dimensional quasiperiodic potential, which in the noninteracting limit exhibits an intermediate phase that is characterized by a mobility edge. We measure the time evolution of an initial charge density wave after a quench and analyze the corresponding relaxation exponents. We find clear signatures of MBL when the corresponding noninteracting model is deep in the localized phase. We also critically compare and contrast our results with those from a tight-binding Aubry-André model, which does not exhibit a single-particle intermediate phase, in order to identify signatures of a potential many-body intermediate phase.
Finite-temperature properties of interacting bosons on a two-leg flux ladder
Maximilian Buser, Fabian Heidrich-Meisner, Ulrich Schollwöck
Physical Review A 99, 053601 (2019).
Quasi-one-dimensional lattice systems such as flux ladders with artificial gauge fields host rich quantum-phase diagrams that have attracted great interest. However, so far, most of the work on these systems has concentrated on zero-temperature phases while the corresponding finite-temperature regime remains largely unexplored. The question if and up to which temperature characteristic features of the zero-temperature phases persist is relevant in experimental realizations. We investigate a two-leg ladder lattice in a uniform magnetic field and concentrate our study on chiral edge currents and momentum-distribution functions, which are key observables in ultracold quantum-gas experiments. These quantities are computed for hard-core bosons as well as noninteracting bosons and spinless fermions at zero and finite temperatures. We employ a matrix-product-state based purification approach for the simulation of strongly interacting bosons at finite temperatures and analyze finite-size effects. Our main results concern the vortex-fluid-to-Meissner crossover of strongly interacting bosons. We demonstrate that signatures of the vortex-fluid phase can still be detected at elevated temperatures from characteristic finite-momentum maxima in the momentum-distribution functions, while the vortex-fluid phase leaves weaker fingerprints in the local rung currents and the chiral edge current. In order to determine the range of temperatures over which these signatures can be observed, we introduce a suitable measure for the contrast of these maxima. The results are condensed into a finite-temperature crossover diagram for hard-core bosons.
Eigenstate thermalization and quantum chaos in the Holstein polaron model
David Jansen, Jan Stolpp, Lev Vidmar, and Fabian Heidrich-Meisner
Physical Review B 99, 155130 (2019).
The eigenstate thermalization hypothesis (ETH) is a successful theory that provides sufficient criteria for ergodicity in quantum many-body systems. Most studies were carried out for Hamiltonians relevant for ultracold quantum gases and single-component systems of spins, fermions, or bosons. The paradigmatic example for thermalization in solid-state physics are phonons serving as a bath for electrons. This situation is often viewed from an open-quantum-system perspective. Here, we ask whether a minimal microscopic model for electron-phonon coupling is quantum chaotic and whether it obeys ETH, if viewed as a closed quantum system. Using exact diagonalization, we address this question in the framework of the Holstein polaron model. Even though the model describes only a single itinerant electron, whose coupling to dispersionless phonons is the only integrability-breaking term, we find that the spectral statistics and the structure of Hamiltonian eigenstates exhibit essential properties of the corresponding random-matrix ensemble. Moreover, we verify the ETH ansatz both for diagonal and off-diagonal matrix elements of typical phonon and electron observables, and show that the ratio of their variances equals the value predicted from random-matrix theory.
Time-dependent study of disordered models with infinite projected entangled pair states
Claudius Hubig, J. Ignacio Cirac.
SciPost Physics 6, 031 (2019).
Infinite projected entangled pair states (iPEPS), the tensor network ansatz for two-dimensional systems in the thermodynamic limit, already provide excellent results on ground-state quantities using either imaginary-time evolution or variational optimisation. Here, we show (i) the feasibility of real-time evolution in iPEPS to simulate the dynamics of an infinite system after a global quench and (ii) the application of disorder-averaging to obtain translationally invariant systems in the presence of disorder. To illustrate the approach, we study the short-time dynamics of the square lattice Heisenberg model in the presence of a bi-valued disorder field.
Learning multiple order parameters with interpretable machines
Ke Liu, Jonas Greitemann, and Lode Pollet
Physical Review B 99, 104410 (2019).
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.
Interaction quench and thermalization in a one-dimensional topological Kondo insulator
I. Hagymási, C. Hubig, and U. Schollwöck
Physical Review B 99, 075145 (2019).
We study the nonequilibrium dynamics of a one-dimensional topological Kondo insulator, modelled by a p-wave Anderson lattice model, following a quantum quench of the on-site interaction strength. Our goal is to examine how the quench influences the topological properties of the system, and therefore our main focus is the time evolution of the string order parameter, entanglement spectrum, and the topologically protected edge states. We point out that postquench local observables can be well captured by a thermal ensemble up to a certain interaction strength. Our results demonstrate that the topological properties after the interaction quench are preserved. Though the absolute value of the string order parameter decays in time, the analysis of the entanglement spectrum, Loschmidt echo and the edge states indicates the robustness of the topological properties in the time-evolved state. These predictions could be directly tested in state-of-the-art cold-atom experiments.
Probing hidden spin order with interpretable machine learning
Jonas Greitemann, Ke Liu, and Lode Pollet
Phyical Review B 99, 060404(R) (2019).
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.