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In this paper, we propose a novel method of model-based time series clustering with mixtures of general state space models (MSSMs). Each component of MSSMs is associated with each cluster. An advantage of the proposed method is that it enables the use of time series models appropriate to the specific time series. This not only improves clustering and prediction accuracy but also enhances the interpretability of the estimated parameters. The parameters of the MSSMs are estimated using stochastic variational inference, a subtype of variational inference. The proposed method estimates the latent variables of an arbitrary state space model by using neural networks with a normalizing flow as a variational estimator. The number of clusters can be estimated using the Bayesian information criterion. In addition, to prevent MSSMs from converging to the local optimum, we propose several optimization tricks, including an additional penalty term called entropy annealing. To our best knowledge, the proposed method is the first computationally feasible one for time series clustering based on general (possibly nonlinear, non-Gaussian) state space models. Experiments on simulated datasets show that the proposed method is effective for clustering, parameter estimation, and estimating the number of clusters., Comment: 23 pages, 4 figures

We report $^{79}$Br and $^{63}$Cu nuclear quadrupole resonance (NQR) in the paramagnetic state above $T_\text{N} = 15$ K of the antiferromagnetic orthorhombic phase of barlowite Cu$_4$(OD)$_6$FBr consisting of a layered kagome structure. The divergent behavior of the longitudinal $^{79}(1/T_{1})$ and transverse $^{79}(1/T_{2})$ relaxation rates observed at $^{79}$Br sites evidences that critical slowing down of Cu spin fluctuations sets in below $\sim20$ K. This means that one or more Cu sites, most likely at the interlayer Cu(3,4,5) sites between the kagome planes, undergo the antiferromagnetic phase transition in a fairly conventional way. On the other hand, the $^{63}$Cu NQR signal intensity is gradually wiped out below $\sim30$ K, pointing toward gradual spin freezing of the kagome layers instead. These contrasting findings suggest significant roles played by magnetic frustration effects within the kagome layers., Comment: Accepted for publication in Phys. Rev. Mater. 5 figures

Kagome lattice Heisenberg antiferromagnets are known to be highly sensitive to perturbations caused by structural disorder. NMR is a local probe ideally suited for investigating such disorder-induced effects, but in practice large distributions in the conventional one-dimensional NMR data make it difficult to distinguish the intrinsic behavior expected for pristine kagome quantum spin liquids from disorder induced effects. Here we report the development of a two-dimensional NMR data acquisition scheme applied to Zn-barlowite (Zn$_{0.95}$Cu$_{0.05}$)Cu$_{3}$(OD)$_{6}$FBr kagome lattice, and successfully correlate the distribution of the low energy spin excitations with that of the local spin susceptibility. We present evidence for the gradual growth of domains with a local spin polarization induced by 5\% Cu$^{2+}$ defect spins occupying the interlayer non-magnetic Zn$^{2+}$ sites. These spin polarized domains account for $\sim60$\% of the sample volume at 2~K, where gapless excitations induced by interlayer defects dominate the low energy sector of spin excitations within the kagome planes., Comment: Revised manuscript accepted for publication in npj Quantum Materials

In this paper, we consider the task of clustering a set of individual time series while modeling each cluster, that is, model-based time series clustering. The task requires a parametric model with sufficient flexibility to describe the dynamics in various time series. To address this problem, we propose a novel model-based time series clustering method with mixtures of linear Gaussian state space models, which have high flexibility. The proposed method uses a new expectation-maximization algorithm for the mixture model to estimate the model parameters, and determines the number of clusters using the Bayesian information criterion. Experiments on a simulated dataset demonstrate the effectiveness of the method in clustering, parameter estimation, and model selection. The method is applied to real datasets commonly used to evaluate time series clustering methods. Results showed that the proposed method produces clustering results that are as accurate or more accurate than those obtained using previous methods.

Magnetic moments arranged at the corners of a honeycomb lattice are predicted to form a novel state of matter, Kitaev quantum spin liquid, under the influence of frustration effects between bond-dependent Ising interactions. Some layered honeycomb iridates and related materials, such as Na$_{2}$IrO$_{3}$ and $\alpha$-RuCl$_{3}$, are proximate to Kitaev quantum spin liquid, but bosonic spin-wave excitations associated with undesirable antiferromagnetic long-range order mask the inherent properties of Kitaev Hamiltonian. Here, we use $^{63}$Cu nuclear quadrupole resonance to uncover the low energy spin excitations in the nearly ideal honeycomb lattice of effective spin $S = 1/2$ at the Ir$^{4+}$ sites in Cu$_{2}$IrO$_{3}$. We demonstrate that, unlike Na$_{2}$IrO$_{3}$, Ir spin fluctuations exhibit no evidence for critical slowing down toward magnetic long range order in zero external magnetic field. Moreover, the low energy spin excitation spectrum is dominated by a mode that has a large excitation gap comparable to the Ising interactions, a signature expected for Majorana fermions of Kitaev quantum spin liquid., Comment: 18 figs

The kagome Heisenberg antiferromagnet formed by frustrated spins arranged in a lattice of corner-sharing triangles is a prime candidate for hosting a quantum spin liquid (QSL) ground state consisting of entangled spin singlets. But the existence of various competing states makes a convincing theoretical prediction of the QSL ground state difficult, calling for experimental clues from model materials. The kagome lattice materials Zn-barlowite ZnCu$_{3}$(OD)$_{6}$FBr and herbertsmithite ZnCu$_{3}$(OD)$_{6}$Cl$_2$ do not exhibit long range order, and they are considered the best realizations of the kagome Heisenberg antiferromagnet known to date. Here we use $^{63}$Cu nuclear quadrupole resonance combined with the inverse Laplace transform (ILT) to probe locally the inhomogeneity of delicate quantum ground states affected by disorder. We present direct evidence for the gradual emergence of spin singlets with spatially varying excitation gaps, but even at temperatures far below the super-exchange energy scale their fraction is limited to approximately 60\% of the total spins. Theoretical models need to incorporate the role of disorder to account for the observed inhomogeneously gapped behaviour., Comment: 4 Figs

We use $^{79}$Br nuclear quadrupole resonance (NQR) to demonstrate that ultra slow lattice dynamics set in below the temperature scale set by the Cu-Cu super-exchange interaction $J$~($\simeq160$~K) in the kagome lattice Heisenberg antiferromagnet Zn-barlowite. The lattice completely freezes below 50~K, and $^{79}$Br NQR lineshapes become twice broader due to increased lattice distortions. Moreover, the frozen lattice exhibits an oscillatory component in the transverse spin echo decay, a typical signature of pairing of nuclear spins by indirect nuclear spin-spin interaction. This indicates that some Br sites form structural dimers via a pair of kagome Cu sites prior to the gradual emergence of spin singlets below $\sim30$~K. Our findings underscore the significant roles played by subtle structural distortions in determining the nature of the disordered magnetic ground state of the kagome lattice., Comment: Phys. Rev. Lett. (in press)

There has been considerable recent progress in identifying candidate materials for the transverse-field Ising chain (TFIC), a paradigmatic model for quantum criticality. Here, we study the local spin dynamical structure factor of different spin components in the quantum disordered region of the TFIC. We show that the low-frequency local dynamics of the spins in the Ising- and transverse-field directions have strikingly distinctive temperature dependencies. This leads to the thermal-activation gap for the secular term of the NMR $1/T_2^{\prime}$ relaxation rate to be half of that for the $1/T_1$ relaxation rate. Our findings reveal a new surprise in the nonzero-temperature dynamics of the venerable TFIC model and uncover a means to evince the material realization of the TFIC universality., Comment: 17 pages, 5 figures

We investigate the structural and magnetic properties of a Kitaev spin liquid candidate material Ag$_3$LiIr$_2$O$_6$ based on $^7$Li nuclear magnetic resonance line shape, Knight shift and spin-lattice relaxation rate $1/T_1$. The first sample A shows signatures of magnetically ordered spins, and exhibits one sharp $^7$Li peak with FWHM increasing significantly below 14~K. $1/T_1^{stretch}$ of this sample displays a broad local maximum at 40~K, followed by a very sharp peak at $T_N = 9\pm1$~K due to critical slowing down of Ir spin fluctuations, a typical signature of magnetic long range order. In order to shed light on the position-by-position variation of $1/T_1$ throughout the sample, we use Inverse Laplace Transform $T_1$ analysis based on Tikhonov regularization to deduce the density distribution function $P(1/T_1)$. We demonstrate that $\sim 60\%$ of Ir spins are statically ordered at the NMR measurement timescale but the rest of the sample volume remains paramagnetic even at 4.2~K, presumably because of structural disorder induced primarily by stacking faults. In order to further investigate the influence of structural disorder, we compare these NMR results with those of a second sample B, which has been shown by transmission electron microscope to have domains with unwanted Ag inclusion at Li and Ir sites within the Ir honeycomb planes. The sample B displays an additional NMR peak with relative intensity of $\sim 17\%$. The small Knight shift and $1/T_1$ of these defect-induced $^7$Li sites and the enhancement of bulk susceptibility at low temperatures suggest that these defects generate domains of only weakly magnetic Ir spins accompanied by free spins, leading to a lack of clear signatures of long-range order. The apparent lack of long-range order could be easily misinterpreted as evidence for the realization of a spin liquid ground state in highly disordered Kitaev lattice., Comment: 10 figures

This paper presents a phase description of chaotic dynamics for the study of chaotic phase synchronization. A prominent feature of the proposed description is that it systematically incorporates the dynamics of the non-phase variables inherent in the system. Taking these non-phase dynamics into account is essential for capturing the complicated nature of chaotic phase synchronization, even in a qualitative manner. We numerically verified the validity of the proposed description in application to the R\"{o}ssler and Lorenz oscillators, and we found that our method provides an accurate description of the characteristic distorted shapes of the synchronization regions for these chaotic oscillators. Furthermore, the proposed description allows us to systematically identify and describe the origin of this distortion.

Local spectral functions and Nuclear Magnetic Relaxation (NMR) rates, 1/T1, for the spin-half Heisenberg antiferromagnet on the Kagome Lattice are calculated using Moriyas Gaussian approximation, as well as through an extrapolation of multiple frequency moments. The temperature dependence of the calculated rates is compared with the oxygen 1/T1 NMR data in Herbertsmithite. We find that the Gaussian approximation for 1/T1 shows behavior qualitatively similar to experiments with a sharp drop in rates at low temperatures, consistent with a spin-gapped behavior. However, this approximation significantly underestimates the magnitude of 1/T1 even at room temperature. Rates obtained from extrapolation of multiple frequency moments give very good agreement with the room temperature NMR data with J = 195 +/- 20 K and hyperfine couplings determined independently from other measurements. The use of multiple frequency moments also leads to additional low frequency weight in the local structure factors. The convergence of our calculations with higher frequency moments breaks down at low temperatures suggesting the existence of longer range dynamic correlations in the system despite the very short-range static correlations., Comment: 5 pages 4 figures

We report $^{133}$Cs NMR and $^{75}$As Nuclear Quadrupole Resonance (NQR) measurements on the normal metallic state above $T_c$ of a quasi-one-dimensional superconductor Cs$_2$Cr$_3$As$_3$ ($T_c < 1.6$~K). From the $^{133}$Cs NMR Knight shift $^{133}K$ measured at the Cs1 site, we show that the uniform spin susceptibility $\chi_{spin}$ increases from 295~K to $\sim$ 60~K, followed by a mild suppression; $\chi_{spin}$ then levels off below $\sim$10~K. In contrast, a vanishingly small magnitude of $^{133}K$ indicates that Cs2 sites contribute very little to electrical conduction and the exchange interactions between 3d electrons at Cr sites. Low frequency Cr spin dynamics, reflected on $^{75}$As $1/T_1T$ (the nuclear spin-lattice relaxation rate $1/T_1$ divided by temperature $T$), shows an analogous trend as $\chi_{spin}$. Comparison with the results of $1/T_1T$ near $T_c$ with Rb$_2$Cr$_3$As$_3$ ($T_c=6.1$~K) and Rb$_2$Cr$_3$As$_3$ ($T_c=4.8$~K) establishes a systematic trend that substitution of K$^{+}$ ions with larger alkali ions progressively suppresses Cr spin fluctuations together with $T_c$., Comment: Final accepted version with 11 Figures

We demonstrate that the anisotropy R of the paramagnetic spin fluctuations grows toward Tc at 75As sites in the optimally electron-doped superconductor Ba[(Fe0.92Co0.08)2]2As2, with stronger spin fluctuations along the c-axis. Our finding is in remarkable contrast with the case of high T$_c$ cuprates, where R is independent of temperature above Tc., Comment: 2 Figures