⇦Chinese Physics Letters, 2021, Vol. 38, No. 9, Article code 097502Express Letter Learning the Effective Spin Hamiltonian of a Quantum Magnet Sizhuo Yu (于思拙)1†, Yuan Gao (高源)1†, Bin-Bin Chen (陈斌斌)1, and Wei Li (李伟)2,1* Affiliations 1School of Physics, Beihang University, Beijing 100191, China 2CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China Received 10 August 2021; accepted 21 August 2021; published online 27 August 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 11974036 and 11834014).
†These authors contributed equally to this work.
*Corresponding author. Email: w.li@itp.ac.cn Citation Text: Yu S Z, Gao Y, Chen B B, and Li W 2021 Chin. Phys. Lett. 38 097502    Abstract To understand the intriguing many-body states and effects in the correlated quantum materials, inference of the microscopic effective Hamiltonian from experiments constitutes an important yet very challenging inverse problem. Here we propose an unbiased and efficient approach learning the effective Hamiltonian through the many-body analysis of the measured thermal data. Our approach combines the strategies including the automatic gradient and Bayesian optimization with the thermodynamics many-body solvers including the exact diagonalization and the tensor renormalization group methods. We showcase the accuracy and powerfulness of the Hamiltonian learning by applying it firstly to the thermal data generated from a given spin model, and then to realistic experimental data measured in the spin-chain compound copper nitrate and triangular-lattice magnet TmMgGaO$_4$. The present automatic approach constitutes a unified framework of many-body thermal data analysis in the studies of quantum magnets and strongly correlated materials in general. DOI:10.1088/0256-307X/38/9/097502 © 2021 Chinese Physics Society Article Text The correlated quantum matter and materials harboring novel quantum phases constitute an active research field of condensed matter physics. Among others, an intriguing topic is the materialization of quantum spin liquids (QSLs) with topologically ordered ground states and long-pursued anyonic excitations.[1–4] Some prominent QSL candidates include the kagome,[5–7] triangular,[8–12] and Kitaev materials.[13–18] However, the lack of precise knowledge on the effective spin Hamiltonians of these frustrated magnets hinders the search for possible QSLs therein. For example, to understand the quantum states in the renowned Kitaev material $\alpha$-RuCl$_3$,[16–25] various spin models have been proposed, yet none of them could satisfactorily explain the major experimental observations.[26] The difficulty is two-fold. Firstly, to solve the spin Hamiltonian and compute experiment-relevant properties like thermodynamics is by no means an easy task, as there is an “exponential barrier” in the many-body Hilbert space to overcome. Secondly, even worse, inferring the effective spin Hamiltonian from experimental data constitutes an inverse many-body problem.[27–29] Recently, progresses in thermal tensor networks have been swift. Various tensor renormalization group algorithms have been proposed, which enable us to perform efficient and accurate calculations of the thermodynamic properties of large-scale 1D and 2D systems.[30–40] Nevertheless, these thermal tensor network calculations generically demand considerable computational resources for low-temperature simulations. Therefore, to determine the spin Hamiltonian of a realistic magnetic material, grid search in the parameter space is a very laborious and even unfeasible in practice for Hamiltonians with more than, say, 5 or 6 parameters. On the other hand, machine learning has brought very helpful methodology and new perspectives into the community of quantum many-body computations recently. For example, it has been proposed that the artificial neural network can serve as a powerful variational many-body wavefunction ansatz,[41] and the differentiable programming of tensor networks helps to design novel tensor renormalization group algorithms.[42,43] Reversely, the tensor networks are also found very useful in machine learning applications in, e.g., the matrix product state and tree tensor-network based supervised learning,[44,45] the Bayesian tensor-network probabilistic learning,[46] and many others.[47–49] In this work, we propose an automatic parameter searching approach for determining the effective spin Hamiltonian, i.e., the quantum magnetism genome, from analyzing thermal data of magnetic quantum materials with many-body solvers. Our method exploits techniques from machine learning to explore the parameter space efficiently, including the gradient optimization by automatic differentiation (called auto-gradient) and the global Bayesian optimization schemes. The automatic Hamiltonian searching, either auto-gradient or Bayesian, is very flexible approach and can be combined with various many-body methods ranging from small-size exact diagonalization (ED) as a high-$T$ solver to large-scale thermal tensor networks as low-$T$ solvers.[36,37,39,40] Our automatic approach helps reduce the human bias and can locate the optimal parameter point accurately. Moreover, the learnt loss landscape in the parameter space presents valuable information on the robustness and uniqueness in the model parameter fittings, which are also very precious in the studies of quantum materials. Thermodynamics Many-Body Solvers. When only high-$T$ thermal data are involved, the ED calculations can be employed to compute the spin lattice model with limited system sizes. As the effective thermal correlation length is usually short at high temperature, ED serves as a natural high-$T$ solver. Together with properly chosen optimizers, it can help extract the valuable correlations and thus interaction information “hidden” in the quantitative details of the thermodynamic curves that are though featureless to human eyes. Moreover, to unambiguously determine the spin Hamiltonian, we employ large-scale tensor network methods as the low-$T$ thermodynamics solver. Linearized tensor renormalization group (LTRG)[36,37] can efficiently contract the thermal tensor networks in Fig. 1(c) and compute the thermodynamics accurately. Beyond the 1D system, thermal tensor network methods[36,39,40,50] can be used to compute large-scale 2D systems. In the present work, we exploit the exponential tensor renormalization group (XTRG),[39,40] to solve frustrated 2D lattice model, which can also be conveniently combined with the auto-gradient and Bayesian optimization schemes (see the Supplementary Materials). Auto-Gradient, Bayesian Optimization, Random Grid Search, and Others. The objective loss function of the thermal data fitting reads $$ \mathcal{L}({\boldsymbol x}) = \sum_{\alpha}\frac{1}{N_\alpha} \lambda_{\alpha} \left(\frac{O^\mathrm{{\exp}}_\alpha-O^{{\rm sim},{\boldsymbol x}}_\alpha}{O^{\exp}_\alpha}\right)^2,~~ \tag {1} $$ where $O^{\exp}_\alpha$ and $O^{{\rm sim},{\boldsymbol x}}_\alpha$ are, respectively, the experimental and simulated quantities with given parameters ${\boldsymbol x}$ (with $\alpha$ labeling different physical quantities); $\lambda_{\alpha}$ is an empirical weight coefficient (set to unity by default), which can be used to adjust different observables' contributions to the loss in an evenly fashion. The parameter vector ${\boldsymbol x}$ contains various model parameters, including $J$, $\varDelta$ and $g$, which span a parameter space $\mathcal{X}$. $N_\alpha$ is the data point number of quantity $O_\alpha$, and thus $1/N_\alpha$ normalizes the fitting loss per data point.[51] An efficient optimizer that minimizes the loss function $\mathcal{L}$ in the parameter space $\mathcal{X}$ plays an indispensable role in the automatic Hamiltonian searching. In this work, we have employed two machine-learning inspired methods including the auto-gradient and Bayesian optimizers, which are compared to a plain random grid search as well as the Nelder–Mead simplex and simulated annealing methods.
cpl-38-9-097502-fig1.png
Fig. 1. The workflow of automatic Hamiltonian searching. (a) The HAFC model with coupling $J$, ratio $\alpha$ and magnetic anisotropy $\varDelta$, and (b) a triangular-lattice model with nearest-neighboring coupling $J_1$ and next-nearest-neighboring $J_2$. With the thermal tensor network solvers in (c), we can compute the (d) loss function $\mathcal{L}$, and feed it to the (e) auto-gradient, (f) Bayesian, or (g) the random grid optimizers. The optimizer proposes a new trial parameter set for the next iteration until convergence is reached. In panel (c), $\tau$ is a small inverse-temperature, $A_{i(j)}$ the local tensor, and $h_{i,j}$ local spin interaction term [cf. Eq. (2)]. In panel (f) the objective (obj.) loss as well as the predicted mean $\mu$ and uncertainty $\sigma$ are plotted, with the observed (obs.) points and acquisition $\alpha_{_{\scriptstyle \rm EI}}$ also indicated.
Inspired by the backpropagation in deep learning,[52] differentiable programming has been introduced into the tensor network methods.[42,43] In this work, to obtain the gradient information that greatly facilitates the parameter searching, we implement the automatic differentiation in the thermodynamics solver, i.e., making the latter fully differentiable. The basic idea is as follows: we store the derivatives between intermediate variables of adjacent calculation steps in the forward process (of many-body solver) all the way to the final loss function $\mathcal{L}$, and then the derivatives of the loss function respective to the model parameters, $\bar{\boldsymbol x} = \partial \mathcal{L}/\partial{{\boldsymbol x}}$, can be computed automatically following the derivative chain rule in the backward propagations. The derivative $\bar{\boldsymbol x}$ can be further employed to find the optimal model parameters ${\boldsymbol x}$ via gradient-based optimizer. Within the framework of differentiable programming, no additional measurements are needed and the gradients are computed efficiently and without any numeric differentiation errors. Since the loss $\mathcal{L}$ is generically non-convex (see Fig. 2), we need to restart and perform the auto-gradient search for multiple times, in order to guarantee a convergence to the global minimum. The Bayesian optimization constitutes a powerful and highly efficient technique in hyper-parameter tuning of deep neural networks, and have been widely used and incorporated in the active and reinforce learnings, etc.[53–55] As many-body calculations are usually expensive, it is then essential to make fully exploitation of the information on measured parameter points and determine where to perform the calculations next. In practice, the Bayesian optimization minimizes our loss function $\mathcal{L}$ by iteratively updating a statistical model, namely Gaussian process $\mathcal{GP} : \mathcal{X} \rightarrow \mu, \sigma$ over the entire parameter space $\mathcal{X}$, where $\mu, \sigma$ represent the predicted value and uncertainty as shown in Fig. 1(f). The parameter candidate in the next ($n+1$) iteration is determined by maximizing the expected-improvement acquisition function, i.e., ${\boldsymbol x}_{n+1} = \arg \max \alpha_{_{\scriptstyle \rm EI}}({\boldsymbol x}) = \arg \max \mathbb{E}[\mathcal{L}_{n,{\mathrm{{\min}}}} - \mu_n({\boldsymbol x})]$, where $\mu_n$ and $\mathcal{L}_{n,{\mathrm{{\min}}}}$ denote respectively the predicted mean and minimal loss function found in the $n$-th iteration. In practice, such acquisition criteria can elegantly balance the optimization efficiency and the exploration of parameter space $\mathcal{X}$.
cpl-38-9-097502-fig2.png
Fig. 2. (a) The scatters indicate the evaluated queries of 150 iterations of multi-restart gradient optimization processes, and the background, $\mathcal{L}$ landscape, is obtained via a grid search. (b) Landscape of $\mathcal{L}$ predicted by the Gaussian process of the Bayesian optimizer with an evaluated query of 80 parameter points, and the loss along the vertical and horizontal dash lines can be found in Supplementary Figs. S4(b) and S4(c). [(c), (d)] Solid lines indicate the lowest $\mathcal{L}$ found in the multi-restart auto-gradient and Bayesian optimizations, and the scatters represent the evaluated function value $\mathcal{L}({\boldsymbol x}_i)$ at iteration $i$.
Benchmarks on XXZ Spin Chain. We start with “experimental” thermal data generated from the XXZ Heisenberg antiferromagnetic chain (HAFC) with a given parameter, and feed the data to the automatic Hamiltonian searching approach to see if we can retrieve the correct model parameters. Below, we stick to the ED thermodynamics solver, and make a comparison between various optimizers. The testing thermodynamic data are computed from the spin-1/2 HAFC Hamiltonian $$ H = \sum_{\langle i,j \rangle} h_{ij} = \sum_{\langle i,j \rangle} J_{xy} (S_i^{x} S_j^{x} + S_i^y S_j^y) + J_z S_i^z S_j^z,~~ \tag {2} $$ where $\langle i, j \rangle$ represents a nearest-neighboring pair of sites. We employ LTRG to generate the thermal data of an infinite HAFC system with $J_{xy}=1$ and $J_z =1.5$ (for cases with different spin couplings, see Supplementary Fig. S4). Gaussian noises $\mathcal{N} (0, 0.01 \times E_i)$ are introduced to each data point with mean value $E_i$ (see Fig. 3), to mimic the random measurement errors in real experiments. We show below that the smart optimizers and the high-$T$ ED solver can cooperate and do a surprisingly good job to “learn” the accurate model parameters.
cpl-38-9-097502-fig3.png
Fig. 3. (a) The in-plane $\chi_{xy}^{\,}$ and out-of-plane $\chi_{z}^{\,}$ of the testing data generated by an infinitely-long XXZ chain with $J_z = 1.5, J_{xy} = 1$ (hollow symbols) and a best fitting based on 10-site ED calculations with the determined model parameters $J_z = 1.49(1), J_{xy}= 1.02(1)$ (solid line). Only “experimental” data with temperatures higher than $T_{\rm cut}$ are used in the fittings. Below the temperature scale $T_{\rm high}$, the susceptibility $\chi$ deviates from the Curie–Weiss behaviors marked with the green dashed line. (b) Magnetic specific heat $C_m$ and their optimal ED fitting. (c) The box plot indicates the maximum, 1/4 and 3/4 percentile, median, minimum and extreme values of best $\mathcal{L}$ found at $n$-th iteration of 100 independent experiments with five different optimization schemes.
As shown in Fig. 2(a), the loss function landscape scanned throughout the whole parameter space $\mathcal{X}$ is found to have a global minima at around $J_{xy}=1$ and $J_z=1.5$, exactly at the input model parameter point, which delivers a key information that one can, in principle, locate the correct model parameters even from the high-$T$ thermodynamics. Indeed, both the auto-gradient and Bayesian optimizers can retrieve the original parameters efficiently and accurately. The latter can also reproduce the correct loss landscape, see Figs. 2(a) and 2(b). In the automatic parameter searching, as the ED solver can only simulate relatively high-$T$ properties, we introduce a cut-off temperature $T_{\rm cut}$ in the fitting. As shown in Figs. 3(a) and 3(b), we only fit thermal data with $T \gtrsim T_{\rm cut} \simeq O(1)$, which are chosen as the peak positions of magnetic susceptibility $\chi_z$ and specific heat $C_m$ curves, respectively. Systematic studies on the dependence of inferred model parameters on the various choices of $T_{\rm cut}$ and the number of data points used in the fittings, as well as the robustness of fitting against data noises, can be found in the Supplementary Materials. In the box plot of Fig. 3(c), we note that, through 100 independent experiments, both the Bayesian and auto-gradient approaches clearly outperforms the other three methods including the Nelder–Mead simplex, simulated annealing, and the random grid search in both efficiency and accuracy. Although the auto-gradient method can lead to very accurate estimate in the “lucky” case (see Fig. 2), it also has a good chance to be trapped in the local minima, especially when the optimization iteration number is relatively small. On the other hand, the Bayesian optimization is mostly stable amongst all schemes, and it finds the optimal parameters very efficiently. For this reason, and also that the Bayesian optimization can provide comprehensive landscape information and is more flexible to be combined with various many-body solvers, in the following we mainly adopt the Bayesian approach and apply it to study realistic magnetic quantum materials.
cpl-38-9-097502-fig4.png
Fig. 4. (a) Magnetic specific heat $C_m /R$ of copper nitrate at various fields of 0, 0.87, 2.82, and 3.57 T. The dashed lines represent the ED fittings and the solid lines are LTRG calculations. (b) The $\mathcal{L}$ landscape within the $J$–$\alpha$ plane obtained after 300 Bayesian iterations with 10-site ED solver. The $J$–$\varDelta$ (c) and $J$–$\alpha$ (d) landscapes are obtained after 400 iterations of LTRG calculations. The estimated loss landscapes in (b)–(d) are fitted with the Gaussian process based on calculated samples. The estimated optimal parameter points, i.e., the hollow (ED) and solid asterisks (LTRG), are compared to previous studies.[56–58]
Quantum Spin-Chain Material Copper Nitrate. Given the successful benchmark calculations on the testing data set, we now move on to a realistic spin-chain material copper nitrate, Cu(NO$_3$)$_2\cdot$2.5H$_2$O, whose magnetic interactions are described by the alternating Heisenberg $XXZ$ chain [see Fig. 1(a)],[57–60] i.e., $$\begin{alignat}{1} H ={}&J \sum_{n=1}^{L/2}[(S_{2n-1}^x S_{2n}^x +S_{2n-1}^y S_{2n}^y + \varDelta S_{2n-1}^z S_{2n}^z)\\ + & \alpha (S_{2n}^x S_{2n+1}^x +S_{2n}^y S_{2n+1}^y + \varDelta S_{2n}^zS_{2n+1}^z)].~~ \tag {3} \end{alignat} $$ Therefore, the problem is to search for the minimal loss $\mathcal{L}$ within a four-dimensional parameter space spanned by the coupling $J$, ratio $\alpha$, magnetic anisotropy $\varDelta$, and the Landé factor $g$ (when coupled to magnetic fields). In Fig. 4, we employ the ED and LTRG solvers as our high- and low-$T$ thermodynamics solvers, and find the model parameters automatically by fitting the specific heat and magnetic susceptibility data. With the ED solver, we find the estimated $J$–$\alpha$ landscape [see Fig. 4(c)] has a relatively narrow distribution in $J$ while a large uncertainty in the alternating ratio $\alpha$. With the large-scale LTRG solver, we find greatly improved resolution and an optimal parameter set as $J = 5.16(2)$ K, $\alpha = 0.227(3)$, $\varDelta = 1.01(1)$, $g=2.237(8)$, which are very close to the hand-tuned optimal parameters in Ref. [58], and have a slightly smaller loss $\mathcal{L} = 7.4 \times 10^{-4}$. In particular, in the learnt ED landscape in Fig. 4(b), we fix $\varDelta=1$ as it has been widely believed (though not carefully examined before) that the magnetic interactions in copper nitrate are of isotropic Heisenberg type.[57,60] With the present automatic parameter searching approach, we resolve this problem and show the results in Fig. 4(c), where $\varDelta$ lies within a very narrow regime around 1 with no essential XXZ anisotropy. Triangular-Lattice Quantum Ising Magnet TmMgGaO$_4$. Our approach can also be applied to investigate 2D quantum magnets. In Fig. 5, we take the triangular-lattice rare-earth magnet TmMgGaO$_4$ as an example.[61–66] The precise determination of the spin Hamiltonian has played an indispensable role in the studies of topological Berezinskii–Kosterlitz–Thouless transition in this quantum magnet.[62,65,66] The effective low-energy spin model of TmMgGaO$_4$ has been found to be described by a triangular-lattice Ising Hamiltonian: $$\begin{alignat}{1} H ={}& J_1 \sum_{\langle i,j \rangle}S_i^z S_j^z + J_2 \sum_{\langle\langle i, j'\rangle\rangle} S_i^z S_{j'}^z \\ &- \varDelta \sum_i S_i^x - g \mu_B B \sum_i S_i^z.~~ \tag {4} \end{alignat} $$ Here, $J_1$ and $J_2$ are, respectively, the nearest-neighboring and next-nearest-neighboring Ising couplings [see Fig. 1(b)], $\varDelta$ is the intrinsic transverse field in the material (due to fine crystal-field splitting), and $g$ is the effective Landé factor. The $\mathcal{L}$-landscape of TmMgGaO$_4$ fittings are shown in Fig. 5, employing the high-$T$ ED and low-$T$ XTRG solvers. From Fig. 5, we see the optimal parameter points, hollow and solid asterisks from the ED (left figure in each panels) and XTRG solvers (right figure), respectively, are in very good agreements with two sets of model parameters from the previous works,[61,62] but different from that obtained from the linear spin-wave fittings.[63] Moreover, the XTRG solver can provide higher resolution in determining the optimal parameters than the ED solver, as the former can compute thermodynamics accurately till lower temperatures.
cpl-38-9-097502-fig5.png
Fig. 5. Loss landscape in (a) $J_1$–$\varDelta$, (b) $J_1$–$J_2$, and (c) $J_1$–$g$ planes, fitted by the Gaussian process with 300 (with ED) and 150 (XTRG) iterations of Bayesian optimization. The estimated model parameters found with 9-site ED (on a $3\times3$ cluster) and 54-site XTRG solvers (on a $6\times9$ cylinder, see the Supplementary Materials) are $J_1=11.5(1)$ K, $J_2=0.89(7)$ K, $\varDelta=5.32(6)$ K, $g=13.00(3)$ and $J_1=10.4(2)$ K, $J_2=0.58(8)$ K, $\varDelta=6.0(1)$ K, and $g=12.95(7)$, labeled by the solid and hollow asterisk, respectively. The estimated parameters from the previous works are labeled as the blue up-triangles,[61] cyan solid circles,[62] and green down-triangles.[63]
Discussion and Outlook. The accurate determination of effective spin Hamiltonian paves the way towards understanding the exotic quantum states in magnetic materials. As widely recognized, solving the many-body problem, i.e., computing the ground-state, thermodynamic, and dynamical properties from a spin lattice model constitutes a challenging problem. Therefore, at a first glance, the inverse problem, i.e., learning the microscopic model from macroscopic measurements, seems a problem intractable. Here we show, through solving the benchmark and realistic problems, that the inverse many-body problem can be elegantly resolved by combining the thermodynamics many-body solvers and machine learning inspired optimizers. The mystery lies in the fact that one actually does not need to solve a full many-body problem, but only a finite-temperature one (at relatively high or intermediate temperature) that is numerically much easier to simulate. We find the high-$T$ ED solver that can already help to find the valuable interaction information with guidance from smart optimizers. When using the large-scale thermal tensor network as the low-$T$ solver, we gain significantly improved resolution in the automatic parameter searching. Our approach, in particular the efficient combination of thermal tensor networks and the Bayesian optimization, can provide a very promising tool in studying quantum magnets and uncovering novel quantum states and effects therein. For example, it can be exploited to explore the family of rare-earth Chalcogenides AReCh$_2$ (A stands for alkali or monovalent ions, Re for rare earth, and Ch is O, S, or Se) which share a similar form of Hamiltonian while with different coupling parameters.[12] As there are abundant thermal data available from experiments,[10–12] the approach established here allows us to search for the most promising quantum spin liquid candidates. Moreover, with necessary prior knowledge from symmetry and crystal field analysis as well as first-principle calculations,[67] our approach enables the possibility to build up a quantum magnetism genome library. This is of great importance for the applications of magnetic quantum materials, e.g., as quantum critical refrigerants[68–72] and spin-chain quantum information data bus.[73,74] With the automatic Hamiltonian learning framework established here, and from proof-of-principle to realistic material examples tested, the exciting exploration of correlated quantum materials with precise many-body solvers and intelligent optimizers can be started from here. In Supplementary Materials, we describe five optimizers employed in Section A. In particular, the automatic differentiation and Bayesian optimization are discussed with more details in Sections B and C, respectively. Quantum many-body solvers in this work are described in more details in Section D. Sections E and F are devoted to loss function adaptation and robustness of the automatic parameter searching with fewer data points and/or measurement noises. More fitting results on the $XXZ$ Heisenberg spin chain and TmMgGaO$_4$ systems are presented in Sections G and H. Acknowledgments. W.L. thanks Shi-Ju Ran for the introduction to active learning and Bayesian optimization, and Lei Wang for stimulating discussions on the differentiable programming. The authors are also indebted to Han Li for the helpful discussion on TmMgGaO$_4$ and $\alpha$-RuCl$_3$. Source code relevant to this work is available at the https URL. References Resonating valence bonds: A new kind of insulator?Anyons in an exactly solved model and beyondQuantum spin liquid statesSpin liquids in frustrated magnetsFractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnetEvidence for a gapped spin-liquid ground state in a kagome Heisenberg antiferromagnetFrom Claringbullite to a New Spin Liquid Candidate Cu 3 Zn(OH) 6 FClSpin Liquid State in an Organic Mott Insulator with a Triangular LatticeHighly Mobile Gapless Excitations in a Two-Dimensional Candidate Quantum Spin LiquidRare-Earth Chalcogenides: A Large Family of Triangular Lattice Spin Liquid CandidatesMott Transition and Superconductivity in Quantum Spin Liquid Candidate NaYbSe 2Effective magnetic Hamiltonian at finite temperatures for rare-earth chalcogenidesMott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev ModelsKitaev-Heisenberg Model on a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides A 2 IrO 3 Direct evidence of a zigzag spin-chain structure in the honeycomb lattice: A neutron and x-ray diffraction investigation of single-crystal Na 2 IrO 3 Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnetNeutron scattering in the proximate quantum spin liquid α-RuCl 3Ferromagnetic Kitaev interaction and the origin of large magnetic anisotropy in α-RuCl3Excitations in the field-induced quantum spin liquid state of α-RuCl3Successive magnetic phase transitions in α − RuCl 3 : XY-like frustrated magnet on the honeycomb latticeMajorana fermions in the Kitaev quantum spin system α-RuCl3Thermodynamic evidence of fractionalized excitations in α − RuC l 3 Anisotropic susceptibilities in the honeycomb Kitaev system α − RuCl 3 Magnetic Properties of Restacked 2D Spin 1/2 honeycomb RuCl 3 NanosheetsMonoclinic crystal structure of α − RuCl 3 and the zigzag antiferromagnetic ground stateDynamical and thermal magnetic properties of the Kitaev spin liquid candidate α-RuCl3Hamiltonian Learning and Certification Using Quantum ResourcesQuantum approximate Bayesian computation for NMR model inferenceAutomatic design of HamiltoniansThe density matrix renormalization group for a quantum spin chain at non-zero temperatureTransfer-matrix density-matrix renormalization-group theory for thermodynamics of one-dimensional quantum systemsThermodynamics of quantum Heisenberg spin chainsFinite-temperature density matrix renormalization using an enlarged Hilbert spaceMinimally Entangled Typical Quantum States at Finite TemperatureMinimally entangled typical thermal state algorithmsLinearized Tensor Renormalization Group Algorithm for the Calculation of Thermodynamic Properties of Quantum Lattice ModelsBilayer linearized tensor renormalization group approach for thermal tensor networksSeries-expansion thermal tensor network approach for quantum lattice modelsExponential Thermal Tensor Network Approach for Quantum Lattice ModelsThermal tensor renormalization group simulations of square-lattice quantum spin modelsSolving the quantum many-body problem with artificial neural networksDifferentiable Programming Tensor NetworksAutomatic differentiation for second renormalization of tensor networksMachine learning by unitary tensor network of hierarchical tree structureBayesian Tensor Network with Polynomial Complexity for Probabilistic Machine LearningTensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future PerspectivesUnsupervised Generative Modeling Using Matrix Product StatesFermionic projected entangled pair states at finite temperatureDeep learningTaking the Human Out of the Loop: A Review of Bayesian OptimizationActive learning machine learns to create new quantum experimentsAlternating linear-chain antiferromagnetism in copper nitrate Cu ( N O 3 ) 2 .2.5 H 2 OTriplet Waves in a Quantum Spin LiquidCriticality-Enhanced Magnetocaloric Effect in Quantum Spin Chain Material Copper NitrateMagnetic Susceptibility of Cu ( N O 3 ) 2 ·2.5 H 2 O at Low TemperatureHigh-Field Magnetic Phase Transition in Cu ( N O 3 ) 2 · 2 1 2 H 2 O Kosterlitz-Thouless melting of magnetic order in the triangular quantum Ising material TmMgGaO4Partial Up-Up-Down Order with the Continuously Distributed Order Parameter in the Triangular Antiferromagnet TmMgGaO 4 Intertwined dipolar and multipolar order in the triangular-lattice magnet TmMgGaO4Anisotropic magnetic properties of the triangular plane lattice material TmMgGaO4Evidence of the Berezinskii-Kosterlitz-Thouless phase in a frustrated magnetNeutron scattering investigation of proposed Kosterlitz-Thouless transitions in the triangular-lattice Ising antiferromagnet TmMgGaO 4 PASP: Property analysis and simulation package for materialsEnhanced magnetocaloric effect in frustrated magnetsMagnetocaloric effect in one-dimensional antiferromagnetsSign change of the Grüneisen parameter and magnetocaloric effect near quantum critical pointsMagnetocaloric effect and magnetic cooling near a field-induced quantum-critical pointGrüneisen parameter studies on heavy fermion quantum criticalitySpin chains as perfect quantum state mirrorsSimulations of Information Transport in Spin Chains
[1] Anderson P W 1973 Mater. Res. Bull. 8 153
[2] Kitaev A 2006 Ann. Phys. 321 2
[3] Zhou Y, Kanoda K, and Ng T K 2017 Rev. Mod. Phys. 89 025003
[4] Balents L 2010 Nature 464 199
[5] Han T H, Helton J S, Chu S, Nocera D G, Rodriguez-Rivera J A, Broholm C, and Lee Y S 2012 Nature 492 406
[6] Fu M, Imai T, Han T H, and Lee Y S 2015 Science 350 655
[7] Feng Z, Yi W, Zhu K, Wei Y, Miao S, Ma J, Luo J, Li S, Meng Z Y, and Shi Y 2019 Chin. Phys. Lett. 36 017502
[8] Shimizu Y, Miyagawa K, Kanoda K, Maesato M, and Saito G 2003 Phys. Rev. Lett. 91 107001
[9] Yamashita M, Nakata N, Senshu Y, Nagata M, Yamamoto H M, Kato R, Shibauchi T, and Matsuda Y 2010 Science 328 1246
[10] Liu W, Zhang Z, Ji J, Liu Y, Li J, Wang X, Lei H, Chen G, and Zhang Q 2018 Chin. Phys. Lett. 35 117501
[11] Jia Y T, Gong C S, Liu Y X, Zhao J F, Dong C, Dai G Y, Li X D, Lei H C, Yu R Z, Zhang G M, and Jin C Q 2020 Chin. Phys. Lett. 37 097404
[12] Zhang Z, Li J, Liu W, Zhang Z, Ji J, Jin F, Chen R, Wang J, Wang X, Ma J, and Zhang Q 2021 Phys. Rev. B 103 184419
[13] Jackeli G and Khaliullin G 2009 Phys. Rev. Lett. 102 017205
[14] Chaloupka J, Jackeli G, and Khaliullin G 2010 Phys. Rev. Lett. 105 027204
[15] Ye F, Chi S, Cao H, Chakoumakos B C, Fernandez-Baca J A, Custelcean R, Qi T F, Korneta O B, and Cao G 2012 Phys. Rev. B 85 180403(R)
[16] Banerjee A, Bridges C A, Yan J Q, Aczel A A, Li L, Stone M B, Granroth G E, Lumsden M D, Yiu Y, Knolle J, Bhattacharjee S, Kovrizhin D L, Moessner R, Tennant D A, Mandrus D G, and Nagler S E 2016 Nat. Mater. 15 733
[17] Banerjee A, Yan J, Knolle J, Bridges C A, Stone M B, Lumsden M D, Mandrus D G, Tennant D A, Moessner R, and Nagler S E 2017 Science 356 1055
[18] Sears J A, Chern L E, Kim S, Bereciartua P J, Francoual S, Kim Y B, and Kim Y J 2020 Nat. Phys. 16 837
[19] Banerjee A, Lampen-Kelley P, Knolle J, Balz C, Aczel A, Winn B, Liu Y, Pajerowski D, Yan J, Bridges C A, Savici A T, Chakoumakos B C, Lumsden M D, Tennant D A, Moessner R, Mandrus D G, and Nagler S E 2018 npj Quantum Mater. 3 8
[20] Kubota Y, Tanaka H, Ono T, Narumi Y, and Kindo K 2015 Phys. Rev. B 91 094422
[21] Do S H, Park S Y, Yoshitake J, Nasu J, Motome Y, Kwon Y S, Adroja D T, Voneshen D J, Kim K, Jang T H, Park J H, Choi K Y, and Ji S 2017 Nat. Phys. 13 1079
[22] Widmann S, Tsurkan V, Prishchenko D A, Mazurenko V G, Tsirlin A A, and Loidl A 2019 Phys. Rev. B 99 094415
[23] Lampen-Kelley P, Rachel S, Reuther J, Yan J Q, Banerjee A, Bridges C A, Cao H B, Nagler S E, and Mandrus D 2018 Phys. Rev. B 98 100403
[24] Weber D, Schoop L M, Duppel V, Lippmann J M, Nuss J, and Lotsch B V 2016 Nano Lett. 16 3578
[25] Johnson R D, Williams S C, Haghighirad A A, Singleton J, Zapf V, Manuel P, Mazin I I, Li Y, Jeschke H O, Valentí R, and Coldea R 2015 Phys. Rev. B 92 235119
[26] Laurell P and Okamoto S 2020 npj Quantum Mater. 5 2
[27] Wiebe N, Granade C, Ferrie C, and Cory D G 2014 Phys. Rev. Lett. 112 190501
[28] Sels D, Dashti H, Mora S, Demler O, and Demler E 2020 Nat. Mach. Intell. 2 396
[29] Pakrouski K 2020 Quantum 4 315
[30] Bursill R J, Xiang T, and Gehring G A 1996 J. Phys.: Condens. Matter 8 L583
[31] Wang X and Xiang T 1997 Phys. Rev. B 56 5061
[32] Xiang T 1998 Phys. Rev. B 58 9142
[33] Feiguin A E and White S R 2005 Phys. Rev. B 72 220401(R)
[34] White S R 2009 Phys. Rev. Lett. 102 190601
[35] Stoudenmire E M and White S R 2010 New J. Phys. 12 055026
[36] Li W, Ran S J, Gong S S, Zhao Y, Xi B, Ye F, and Su G 2011 Phys. Rev. Lett. 106 127202
[37] Dong Y L, Chen L, Liu Y J, and Li W 2017 Phys. Rev. B 95 144428
[38] Chen B B, Liu Y J, Chen Z, and Li W 2017 Phys. Rev. B 95 161104(R)
[39] Chen B B, Chen L, Chen Z, Li W, and Weichselbaum A 2018 Phys. Rev. X 8 031082
[40] Li H, Chen B B, Chen Z, von Delft J, Weichselbaum A, and Li W 2019 Phys. Rev. B 100 045110
[41] Carleo G and Troyer M 2017 Science 355 602
[42] Liao H J, Liu J G, Wang L, and Xiang T 2019 Phys. Rev. X 9 031041
[43] Chen B B, Gao Y, Guo Y B, Liu Y, Zhao H H, Liao H J, Wang L, Xiang T, Li W, and Xie Z Y 2020 Phys. Rev. B 101 220409(R)
[44]Stoudenmire E and Schwab D J 2016 Advances in Neural Information Processing Systems 29 (Curran Associates, Inc.) p 4799
[45] Liu D, Ran S J, Wittek P, Peng C, García R B, Su G, and Lewenstein M 2019 New J. Phys. 21 073059
[46] Ran S J 2019 arXiv:1912.12923 [stat.ML]
[47] Cichocki A, Phan A H, Zhao Q, Lee N, Oseledets I, Sugiyama M, and Mandic D P 2017 Found. Trends$^{\rm\circledR}$ Mach. Learn. 9 431
[48] Han Z Y, Wang J, Fan H, Wang L, and Zhang P 2018 Phys. Rev. X 8 031012
[49]Glasser I, Sweke R, Pancotti N, Eisert J, and Cirac J I 2019 33rd Conference on Neural Information Processing Systems, in Advances in Neural Information Processing Systems ed Wallach H, Larochelle H, Beygelzimer A et al. (Vancouver, Canada: Curran Associates, Inc.) vol 32
[50] Czarnik P and Dziarmaga J 2014 Phys. Rev. B 90 035144
[51]In Figs. 2 and 3, we first exploit a loss function without the denominator $1/ O^{\exp}_\alpha$, i.e., $\mathcal{L}({\boldsymbol x}) = \sum_{\alpha} \sum_{T > T_{\rm cut}} \lambda_{\alpha} [O^{\exp}_\alpha(T)-O^{{\rm sim},{\boldsymbol x}}_\alpha(T)]^2$, where $\lambda_{\alpha}^{-1/2} = \max_{_{\scriptstyle T>T_{\rm cut}}} [O^{\exp}_\alpha(T), O^{{\rm sim},{\boldsymbol x}}_\alpha(T)]$. Then in Figs. 4 and 5 we follow exactly the loss definition in Eq. (1), and observe that both schemes work well.
[52] LeCun Y, Bengio Y, and Hinton G 2015 Nature 521 436
[53] Shahriari B, Swersky K, Wang Z, Adams R P, and de Freitas N 2016 Proc. IEEE 104 148
[54] Melnikov A A, Poulsen N H, Krenn M, Dunjko V, Tiersch M, Zeilinger A, and Briegel H J 2018 Proc. Natl. Acad. Sci. USA 115 1221
[55]Lizotte D J 2008 PhD Dissertation (Edmonton: University of Alberta)
[56] Bonner J C, Friedberg S A, Kobayashi H, Meier D L, and Blöte H W J 1983 Phys. Rev. B 27 248
[57] Xu G, Broholm C, Reich D H, and Adams M A 2000 Phys. Rev. Lett. 84 4465
[58] Xiang J S, Chen C, Li W, Sheng X L, Su N, Cheng Z H, Chen Q, and Chen Z Y 2017 Sci. Rep. 7 44643
[59] Berger L, Friedberg S A, and Schriempf J T 1963 Phys. Rev. 132 1057
[60] van Tol M W, Henkens L S J M, and Poulis N J 1971 Phys. Rev. Lett. 27 739
[61] Li H, Liao Y D, Chen B B, Zeng X T, Sheng X L, Qi Y, Meng Z Y, and Li W 2020 Nat. Commun. 11 1111
[62] Li Y, Bachus S, Deng H, Schmidt W, Thoma H, Hutanu V, Tokiwa Y, Tsirlin A A, and Gegenwart P 2020 Phys. Rev. X 10 011007
[63] Shen Y, Liu C, Qin Y, Shen S, Li Y D, Bewley R, Schneidewind A, Chen G, and Zhao J 2019 Nat. Commun. 10 4530
[64] Cevallos F A, Stolze K, Kong T, and Cava R J 2018 Mater. Res. Bull. 105 154
[65] Hu Z, Ma Z, Liao Y D, Li H, Ma C, Cui Y, Shangguan Y, Huang Z, Qi Y, Li W, Meng Z Y, Wen J, and Yu W 2020 Nat. Commun. 11 5631
[66] Dun Z, Daum M, Baral R, Fischer H E, Cao H, Liu Y, Stone M B, Rodriguez-Rivera J A, Choi E S, Huang Q, Zhou H, Mourigal M, and Frandsen B A 2021 Phys. Rev. B 103 064424
[67] Lou F, Li X Y, Ji J Y, Yu H Y, Feng J S, Gong X G, and Xiang H J 2021 J. Chem. Phys. 154 114103
[68] Zhitomirsky M E 2003 Phys. Rev. B 67 104421
[69] Zhitomirsky M E and Honecker A 2004 J. Stat. Mech.: Theory Exp. 2004 P07012
[70] Garst M and Rosch A 2005 Phys. Rev. B 72 205129
[71] Wolf B, Tsui Y, Jaiswal-Nagar D, Tutsch U, Honecker A, Remović-Langer K, Hofmann G, Prokofiev A, Assmus W, Donath G, and Lang M 2011 Proc. Natl. Acad. Sci. USA 108 6862
[72] Gegenwart P 2016 Rep. Prog. Phys. 79 114502
[73] Karbach P and Stolze J 2005 Phys. Rev. A 72 030301(R)
[74] Cappellaro P, Ramanathan C, and Cory D G 2007 Phys. Rev. Lett. 99 250506