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The detection of gravitational waves has ushered in a new era of observing the universe. Quantum resource advantages offer significant enhancements to the sensitivity of gravitational wave observatories. While squeezed states for ground-based gravitational wave detection have received marked attention, the generation of squeezed states suitable for mid-to-low-frequency detection has remained unexplored. To address the gap in squeezed state optical fields at ultra-low frequencies, we report on the first direct observation of a squeezed vacuum field until Fourier frequency of 4 millihertz with the quantum noise reduction of up to 8 dB, by the employment of a multiple noise suppression scheme. Our work provides quantum resources for future gravitational wave observatories, facilitating the development of quantum precision measurement., Comment: 17 pages,4 figures,1 table and 50 Refs. To be published in Light: Science & Applications

Topological Dirac nodal-line semimetals host topologically nontrivial electronic structure with nodal-line crossings around the Fermi level, which could affect the photocarrier dynamics and lead to novel relaxation mechanisms. Herein, by using time- and angle-resolved photoemission spectroscopy, we reveal the previously-inaccessible linear dispersions of the bulk conduction bands above the Fermi level in a Dirac nodal-line semimetal PtSn$_4$, as well as the momentum and temporal evolution of the gapless nodal lines. A surprisingly ultrafast relaxation dynamics within a few hundred femtoseconds is revealed for photoexcited carriers in the nodal line. Theoretical calculations suggest that such ultrafast carrier relaxation is attributed to the multichannel scatterings among the complex metallic bands of PtSn$_4$ via electron-phonon coupling. In addition, a unique dynamic relaxation mechanism contributed by the highly anisotropic Dirac nodal-line electronic structure is also identified. Our work provides a comprehensive understanding of the ultrafast carrier dynamics in a Dirac nodal-line semimetal.

Despite the success in specific scenarios, existing foundation agents still struggle to generalize across various virtual scenarios, mainly due to the dramatically different encapsulations of environments with manually designed observation and action spaces. To handle this issue, we propose the General Computer Control (GCC) setting to restrict foundation agents to interact with software through the most unified and standardized interface, i.e., using screenshots as input and keyboard and mouse actions as output. We introduce Cradle, a modular and flexible LMM-powered framework, as a preliminary attempt towards GCC. Enhanced by six key modules, Cradle can understand input screenshots and output executable code for low-level keyboard and mouse control after high-level planning, so that Cradle can interact with any software and complete long-horizon complex tasks without relying on any built-in APIs. Experimental results show that Cradle exhibits remarkable generalizability and impressive performance across four previously unexplored commercial video games, five software applications, and a comprehensive benchmark, OSWorld. Cradle is the first to enable foundation agents to follow the main storyline and complete 40-minute-long real missions in the complex AAA game Red Dead Redemption 2 (RDR2). Cradle can also create a city of a thousand people in Cities: Skylines, farm and harvest parsnips in Stardew Valley, and trade and bargain with a maximal weekly total profit of 87% in Dealer's Life 2. Cradle can not only operate daily software, like Chrome, Outlook, and Feishu, but also edit images and videos using Meitu and CapCut. Cradle greatly extends the reach of foundation agents by enabling the easy conversion of any software, especially complex games, into benchmarks to evaluate agents' various abilities and facilitate further data collection, thus paving the way for generalist agents.

We have performed electrochemical treatment of the van der Waals antiferromagnetic materials FePS$_3$ and NiPS$_3$ with the ionic liquid EMIM-BF$_4$, achieving significant molecular intercalation. Mass analysis of the intercalated compounds, EMIM$_x$-FePS$_3$ and EMIM$_x$-NiPS$_3$, indicated respective intercalation levels, $x$, of approximately 27\% and 37\%, and X-ray diffraction measurements demonstrated a massive (over 50\%) enhancement of the $c$-axis lattice parameters. To investigate the consequences of these changes for the magnetic properties, we performed magnetic susceptibility and $^{31}$P nuclear magnetic resonance (NMR) studies of both systems. For EMIM$_x$-FePS$_3$, intercalation reduces the magnetic ordering temperature from $T_N = 120$~K to 78~K, and we find a spin gap in the antiferromagnetic phase that drops from 45~K to 30~K. For EMIM$_x$-NiPS$_3$, the ordering temperature is almost unaffected (changing from 148~K to 145~K), but a change towards nearly isotropic spin fluctuations suggests an alteration of the magnetic Hamiltonian. Such relatively modest changes, given that the huge extension of the $c$ axes is expected to cause a very strong suppression any interlayer interactions, point unequivocally to the conclusion that the magnetic properties of both parent compounds are determined solely by two-dimensional (2D), intralayer physics. The changes in transition temperatures and low-temperature spin dynamics in both compounds therefore indicate that intercalation also results in a significant modulation of the intralayer magnetic interactions, which we propose is due to charge doping and localization on the P sites. Our study offers chemical intercalation with ionic liquids as an effective method to control not only the interlayer but also the intralayer interactions in quasi-2D magnetic materials.

In this paper, we focus on estimating measure upper bounds of nodal sets of solutions to the following boundary value problem \begin{equation*} \left\{ \begin{array}{lll} \Delta u+Vu=0\quad \mbox{in}\ \Omega,\\[2mm] u=0\quad \mbox{on}\ \partial\Omega, \end{array}\right. \end{equation*} where $V\in W^{1,\infty}(\Omega)$ is a potential and $\Omega\subset \mathbb{R}^n (n\geq2)$ is a bounded domain. We show that upper bounds on the $(n-1)$-dimensional Hausdorff measure of the nodal sets of $u$ in $\Omega$ is less than or equal to $$C\Big(1+\log\left(\|\nabla V\|_{L^{\infty}(\Omega)}+1\right)\Big)\cdot\left(\|V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+|\nabla V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+1\right),$$ provided $\partial\Omega$ is $C^{2}$-smooth and $V$ is analytic. Here $C$ is a positive constant depending only on $n$ and $\Omega$. In particular, if $\|\nabla V\|_{L^{\infty}(\Omega)}$ is small, the measure upper bound of the of nodal set of $u$ is less than or equal to $C\left(\|V\|^{\frac{1}{2}}_{L^{\infty}(\Omega)}+1\right)$., Comment: 35pages

Unsupervised image Anomaly Detection (UAD) aims to learn robust and discriminative representations of normal samples. While separate solutions per class endow expensive computation and limited generalizability, this paper focuses on building a unified framework for multiple classes. Under such a challenging setting, popular reconstruction-based networks with continuous latent representation assumption always suffer from the "identical shortcut" issue, where both normal and abnormal samples can be well recovered and difficult to distinguish. To address this pivotal issue, we propose a hierarchical vector quantized prototype-oriented Transformer under a probabilistic framework. First, instead of learning the continuous representations, we preserve the typical normal patterns as discrete iconic prototypes, and confirm the importance of Vector Quantization in preventing the model from falling into the shortcut. The vector quantized iconic prototype is integrated into the Transformer for reconstruction, such that the abnormal data point is flipped to a normal data point.Second, we investigate an exquisite hierarchical framework to relieve the codebook collapse issue and replenish frail normal patterns. Third, a prototype-oriented optimal transport method is proposed to better regulate the prototypes and hierarchically evaluate the abnormal score. By evaluating on MVTec-AD and VisA datasets, our model surpasses the state-of-the-art alternatives and possesses good interpretability. The code is available at https://github.com/RuiyingLu/HVQ-Trans.

Structural investigations conducted using transmission electron microscopy (TEM) on LuH2 synthesized under atmospheric pressure (AP-LuH2) and nitrogen-doped LuH2 synthesized under high pressure (HP-LuH2) have revealed numerous microstructural phenomena. Both materials show a clear superstructure modulation with wave vector, q^* = 1/4 (2-20), and this modulation can be well interpreted by the displacements of Lu atoms. Further investigations on the nitrogen-doped HP-LuH2 materials reveal the appearance of high-density antiphase boundaries, in particular, domain walls of a few atomic layer thickness without structural modulation can be observed, suggesting possible interface properties could be detected in this system. In-situ TEM observations of AP-LuH2 suggest that no evident structural phase transition occurs between 94 K and 673 K., Comment: 8 pages, 7 figures

In this paper, we focus on the quantitative unique continuation property of solutions to \begin{equation*} \Delta^2u=Vu, \end{equation*} where $V\in W^{1,\infty}$. We show that the maximal vanishing order of the solutions is not large than \begin{equation} C\left(\|V\|^{\frac{1}{4}}_{L^{\infty}}+\|\nabla V\|_{L^{\infty}}+1\right). \end{equation} Our key argument is to lift the original equation to that with a positive potential, then decompose the resulted fourth-order equation into a special system of two second-order equations. Based on the special system, we define a variant frequency function with weights and derive its almost monotonicity to establishing some doubling inequalities with explicit dependence on the Sobolev norm of the potential function., Comment: 21pages

Transductive Few-Shot Learning (TFSL) has recently attracted increasing attention since it typically outperforms its inductive peer by leveraging statistics of query samples. However, previous TFSL methods usually encode uniform prior that all the classes within query samples are equally likely, which is biased in imbalanced TFSL and causes severe performance degradation. Given this pivotal issue, in this work, we propose a novel Conditional Transport (CT) based imbalanced TFSL model called {\textbf P}rototypes-oriented {\textbf U}nbiased {\textbf T}ransfer {\textbf M}odel (PUTM) to fully exploit unbiased statistics of imbalanced query samples, which employs forward and backward navigators as transport matrices to balance the prior of query samples per class between uniform and adaptive data-driven distributions. For efficiently transferring statistics learned by CT, we further derive a closed form solution to refine prototypes based on MAP given the learned navigators. The above two steps of discovering and transferring unbiased statistics follow an iterative manner, formulating our EM-based solver. Experimental results on four standard benchmarks including miniImageNet, tieredImageNet, CUB, and CIFAR-FS demonstrate superiority of our model in class-imbalanced generalization., Comment: Accepted by ICCV2023

In this paper, we investigate the quantitative unique continuation, propagation of smallness and measure bounds of nodal sets of solutions to the Buckling type equation $\triangle^2u+\lambda\triangle u-k^2u=0$ in a bounded analytic domain $\Omega\subseteq\mathbb{R}^n$ with the homogeneous boundary conditions $u=0$ and $\frac{\partial u}{\partial\nu}=0$ on $\partial\Omega$, where $\lambda,\ k$ are nonnegative real constants, and $\nu$ is the outer unit normal vector on $\partial\Omega$. We obtain that, the upper bounds for the maximal vanishing order of $u$ and the $n-1$ dimensional Hausdorff measure of the nodal set of $u$ are both $C(\sqrt{\lambda}+\sqrt{k}+1)$, where $C$ is a positive constant only depending on $n$ and $\Omega$. Moreover, we also give a quantitative result of the propagation of smallness of $u$.

Optical microscopy image plays an important role in scientific research through the direct visualization of the nanoworld, where the imaging mechanism is described as the convolution of the point spread function (PSF) and emitters. Based on a priori knowledge of the PSF or equivalent PSF, it is possible to achieve more precise exploration of the nanoworld. However, it is an outstanding challenge to directly extract the PSF from microscopy images. Here, with the help of self-supervised learning, we propose a physics-informed masked autoencoder (PiMAE) that enables a learnable estimation of the PSF and emitters directly from the raw microscopy images. We demonstrate our method in synthetic data and real-world experiments with significant accuracy and noise robustness. PiMAE outperforms DeepSTORM and the Richardson-Lucy algorithm in synthetic data tasks with an average improvement of 19.6\% and 50.7\% (35 tasks), respectively, as measured by the normalized root mean square error (NRMSE) metric. This is achieved without prior knowledge of the PSF, in contrast to the supervised approach used by DeepSTORM and the known PSF assumption in the Richardson-Lucy algorithm. Our method, PiMAE, provides a feasible scheme for achieving the hidden imaging mechanism in optical microscopy and has the potential to learn hidden mechanisms in many more systems.

Few-shot classification aims to learn a classifier to recognize unseen classes during training, where the learned model can easily become over-fitted based on the biased distribution formed by only a few training examples. A recent solution to this problem is calibrating the distribution of these few sample classes by transferring statistics from the base classes with sufficient examples, where how to decide the transfer weights from base classes to novel classes is the key. However, principled approaches for learning the transfer weights have not been carefully studied. To this end, we propose a novel distribution calibration method by learning the adaptive weight matrix between novel samples and base classes, which is built upon a hierarchical Optimal Transport (H-OT) framework. By minimizing the high-level OT distance between novel samples and base classes, we can view the learned transport plan as the adaptive weight information for transferring the statistics of base classes. The learning of the cost function between a base class and novel class in the high-level OT leads to the introduction of the low-level OT, which considers the weights of all the data samples in the base class. Experimental results on standard benchmarks demonstrate that our proposed plug-and-play model outperforms competing approaches and owns desired cross-domain generalization ability, indicating the effectiveness of the learned adaptive weights.

Superconductivity becomes more interesting when it encounters dimensional constraint or topology, because it is of importance for exploring exotic quantum phenomena or developing superconducting electronics. Here we report the coexistence of naturally formed surface superconducting state and three-dimensional topological Dirac state in single crystals of BaMg$_2$Bi$_2$. The electronic structure obtained from the first-principles calculations demonstrates that BaMg$_2$Bi$_2$ is an ideal Dirac material, in which the Dirac point is very close to the Fermi level and no other energy band crosses the Fermi level. Superconductivity up to 4.77 K can be observed under ambient pressure in the measurements of resistivity. The angle dependent magnetoresistance reveals the two-dimensional characteristic of superconductivity, indicating that superconductivity occurs on the surface of the sample and is absent in the bulk state. Our study not only provides BaMg$_2$Bi$_2$ as a suitable platform to study the interplay between superconductivity and topological Dirac state, but also indicates that MgBi-based materials may be a promising system for exploring new superconductors.

High-Curie-temperature ferromagnets are promising candidates for designing new spintronic devices. Here we have successfully synthesized a single-crystal sample of the itinerant ferromagnet Mn$ _{5}$Ge$_{3}$ used flux method and its critical properties were investigated by means of bulk dc-magnetization at the boundary between the ferromagnetic (FM) and paramagnetic (PM) phase. Critical exponents $ \beta=0.336 \pm 0.001 $ with a critical temperature $ T_{c}=300.29 \pm 0.01 $ K and $ \gamma=1.193 \pm 0.003 $ with $ T_{c} = 300.15 \pm 0.05 $ K are obtained by the modified Arrott plot, whereas $ \delta = 4.61 \pm 0.03 $ is deduced by a critical isotherm analysis at $ T_{c} = 300 $ K. The self-consistency and reliability of these critical exponents are verified by the Widom scaling law and the scaling equations. Further analysis reveals that the spin coupling in Mn$ _{5}$Ge$_{3}$ exhibits three-dimensional Ising-like behavior. The magnetic exchange is found to decay as $ J(r)\approx r^{-4.855} $ and the spin interactions are extended beyond the nearest neighbors, which may be related to different set of Mn--Mn interactions with unequal magnitude of exchange strengths. Additionally, the existence of noncollinear spin configurations in Mn$ _{5} $Ge$ _{3} $ results in a small deviation of obtained critical exponents from those for standard 3D-Ising model., Comment: 8 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1803.05905 by other authors

We perform a detailed analysis of the magnetotransport and de Haas-van Alphen (dHvA) oscillations in crystal PdGa which is predicted to be a typical chiral Fermion semimetal from CoSi family holding a large Chern number. The unsaturated quadratic magnetoresistance (MR) and nonlinear Hall resistivity indicate that PdGa is a multi-band system without electron-hole compensation. Angle-dependent resistivity in PdGa shows weak anisotropy with twofold or threefold symmetry when the magnetic field rotates within the (1$\bar{1}$0) or (111) plane perpendicular to the current. Nine or three frequencies are extracted after the fast Fourier-transform analysis (FFT) of the dHvA oscillations with B//[001] or B//[011], respectively, which is confirmed to be consistent with the Fermi surfaces (FSs) obtained from first-principles calculations with spin-orbit coupling (SOC) considered., Comment: 8 pages, 7 figures

We present a study on the magnetization, anomalous Hall effect (AHE) and novel longitudinal resistivity in layered antiferromagnet Co$_{0.29}$TaS$_{2}$. Of particular interests in Co$_{0.29}$TaS$_{2}$ are abundant magnetic transitions, which show that the magnetic structures are tuned by temperature or magnetic field. With decreasing temperature, Co$_{0.29}$TaS$_{2}$ undergoes two transitions at T$_{t1}\sim$ 38.3 K and T$_{t2}\sim$ 24.3 K. Once the magnetic field is applied, another transition T$_{t3}\sim$ 34.3 K appears between 0.3 T and 5 T. At 2 K, an obvious ferromagnetic hysteresis loop within H$_{t1}\sim\pm$ 6.9 T is observed, which decreases with increasing temperature and eventually disappears at T$_{t2}$. Besides, Co$_{0.29}$TaS$_{2}$ displays step-like behavior as another magnetic transition around H$_{t2}\sim\pm$ 4 T, which exists until $\sim$ T$_{t1}$. These characteristic temperatures and magnetic fields mark complex magnetic phase transitions in Co$_{0.29}$TaS$_{2}$, which are also evidenced in transport results. Large AHE dominates in the Hall resistivity with the conspicuous value of R$_{s}$/R$_{0}\sim 10^{5}$, considering that the tiny net magnetization (0.0094$\mu_{B}$/Co) alone would not lead to this value, thus the contribution of Berry curvature is necessary. The longitudinal resistivity illustrates a prominent irreversible behavior within H$_{t1}$. The abrupt change at H$_{t2}$ below T$_{t1}$, corresponding to the step-like magnetic transitions, is also observed. Synergy between the magnetism and topological properties, both playing a crucial role, may be the key factor of large AHE in antiferromagnet, which also offers a new perspective in magnetic topological materials with the platform of Co$_{0.29}$TaS$_{2}$., Comment: 5 pages,4 figures

Squeezing transformation as an essential component, gives rise to the possibility to perform various tasks of quantum information processing. However, the reported squeezing gate with best performance so far is a conditional realization at the expense of low success probability, while the performance of the deterministic ones is currently circumscribed by the limited squeezing degree of the non-classical ancilla. To address this issue, we develop and demonstrate a new scheme of deterministic and universal quantum squeezing gate with the property of non-local operation attributed to the teleportation-like protocol. We demonstrate high fidelity squeezing operation, even when the level of target squeezing being up to 10 dB, where a squeezed state with non-classical noise reduction of 6.5 dB is directly observed. Moreover, we perform a high-fidelity complex operation including a Fourier transformation and a phase squeezing gate, exploring the potential of implementing the complex task of quantum information processing with the presented functional unit. Our method can be applied to distributed quantum processor, the creation of exotic non-classical states, and quantum error correction., Comment: 10 pages, 4 figures, artical

Revealing the ultrafast dynamics of three-dimensional (3D) Dirac fermions upon photoexcitation is critical for both fundamental science and device applications. So far, how the cooling of 3D Dirac fermions differs from that of two-dimensional (2D) Dirac fermions and whether there is population inversion are fundamental questions that remain to be answered. Here we reveal the ultrafast dynamics of Dirac fermions in a model 3D Dirac semimetal Cd$_3$As$_2$ by ultrafast time- and angle-resolved photoemission spectroscopy (TrARPES) with a tunable probe photon energy from 5.3 - 6.9 eV. The energy- and momentum-resolved relaxation rate shows a linear dependence on the energy, suggesting Dirac fermion cooling through intraband relaxation. Moreover, a population inversion is reported based on the observation of accumulated photoexcited carriers in the conduction band with a lifetime of $\tau_n$ = 3.0 ps. Our work provides direct experimental evidence for a long-lived population inversion in a 3D Dirac semimetal, which is in contrast to 2D graphene where the interband relaxation occurs on a much faster timescale.

Learning from set-structured data is a fundamental problem that has recently attracted increasing attention, where a series of summary networks are introduced to deal with the set input. In fact, many meta-learning problems can be treated as set-input tasks. Most existing summary networks aim to design different architectures for the input set in order to enforce permutation invariance. However, scant attention has been paid to the common cases where different sets in a meta-distribution are closely related and share certain statistical properties. Viewing each set as a distribution over a set of global prototypes, this paper provides a novel prototype-oriented optimal transport (POT) framework to improve existing summary networks. To learn the distribution over the global prototypes, we minimize its regularized optimal transport distance to the set empirical distribution over data points, providing a natural unsupervised way to improve the summary network. Since our plug-and-play framework can be applied to many meta-learning problems, we further instantiate it to the cases of few-shot classification and implicit meta generative modeling. Extensive experiments demonstrate that our framework significantly improves the existing summary networks on learning more powerful summary statistics from sets and can be successfully integrated into metric-based few-shot classification and generative modeling applications, providing a promising tool for addressing set-input and meta-learning problems.

MnBi$_8$Te$_{13}$ is an intrinsic ferromagnetic (FM) topological insulator with different complex surface terminations. Resolving the electronic structures of different termination surfaces and manipulation of the electronic state are important. Here, by using micrometer spot time- and angle-resolved photoemission spectroscopy ($\mu$-TrARPES), we resolve the electronic structures and reveal the ultrafast dynamics upon photoexcitation. Photoinduced filling of the surface state hybridization gap is observed for the Bi$_2$Te$_3$ quintuple layer directly above MnBi$_2$Te$_4$ accompanied by a nontrivial shift of the surface state, suggesting light-tunable interlayer interaction. Relaxation of photoexcited electrons and holes is observed within 1-2 ps. Our work reveals photoexcitation as a potential control knob for tailoring the interlayer interaction and surface state of MnBi$_8$Te$_{13}$.

The interplay between time-reversal symmetry (TRS) and band topology plays a crucial role in topological states of quantum matter. In time-reversal-invariant (TRI) systems, the inversion of spin-degenerate bands with opposite parity leads to nontrivial topological states, such as topological insulators and Dirac semimetals. When the TRS is broken, the exchange field induces spin splitting of the bands. The inversion of a pair of spin-splitting subbands can generate more exotic topological states, such as quantum anomalous Hall insulators and magnetic Weyl semimetals. So far, such topological phase transitions driven by the TRS breaking have not been visualized. In this work, using angle-resolved photoemission spectroscopy, we have demonstrated that the TRS breaking induces a band inversion of a pair of spin-splitting subbands at the TRI points of Brillouin zone in EuB$_6$, when a long-range ferromagnetic order is developed. The dramatic changes in the electronic structure result in a topological phase transition from a TRI ordinary insulator state to a TRS-broken topological semimetal (TSM) state. Remarkably, the magnetic TSM state has an ideal electronic structure, in which the band crossings are located at the Fermi level without any interference from other bands. Our findings not only reveal the topological phase transition driven by the TRS breaking, but also provide an excellent platform to explore novel physical behavior in the magnetic topological states of quantum matter., Comment: 22 pages, 7 figures, accepted by Phys. Rev. X

We demonstrate experimentally the simultaneous generation and detection of two types of continuous variable nonclassical states from one type-0 phase-matching optical parametric amplification (OPA) and subsequent two ring filter cavities (RFCs). The output field of the OPA includes the baseband {\omega}0 and sideband modes {\omega}0+/-n{\omega}f subjects to the cavity resonance condition, which are separated by two cascaded RFCs. The first RFC resonates with half the pump wavelength {\omega}0 and the transmitted baseband component is a squeezed state. The reflected fields of the first RFC, including the sideband modes {\omega}0+/-{\omega}f, are separated by the second RFC, construct Einstein-Podolsky-Rosen entangled state. All freedoms, including the filter cavities for sideband separation and relative phases for the measurements of these sidebands, are actively stabilized. The noise variance of squeezed states is 10.2 dB below the shot noise limit (SNL), the correlation variances of both quadrature amplitude-sum and quadrature phase-difference for the entanglement state are 10.0 dB below the corresponding SNL., Comment: 7 pages, 3 figures

A small in-plane external uniaxial pressure has been widely used as an effective method to acquire single domain iron pnictide BaFe$_2$As$_2$, which exhibits twin-domains without uniaxial strain below the tetragonal-to-orthorhombic structural (nematic) transition temperature $T_s$. Although it is generally assumed that such a pressure will not affect the intrinsic electronic/magnetic properties of the system, it is known to enhance the antiferromagnetic (AF) ordering temperature $T_N$ ($

Plasmonic structures are renowned for their capability to efficiently convert light into heat at the nanoscale. However, despite the possibility to generate deep sub-wavelength electromagnetic hot spots, the formation of extremely localized thermal hot spots is an open challenge of research, simply because of the diffusive spread of heat along the whole metallic nanostructure. Here we tackle this challenge by exploiting single gold nanocones. We theoretically show how these structures can indeed realize extremely high temperature gradients within the metal, leading to deep sub-wavelength thermal hot spots, owing to their capability of concentrating light at the apex under resonant conditions even under continuous wave illumination. A three-dimensional Finite Element Method model is employed to study the electromagnetic field in the structure and subsequent thermoplasmonic behaviour, in terms of the three-dimensional temperature distribution. We show how the latter is affected by nanocone size, shape, and composition of the surrounding environment. Finally, we anticipate the use of photoinduced temperature gradients in nanocones for applications in optofluidics and thermoelectrics or for thermally induced nanofabrication.

The conductive domain wall (CDW) is extensively investigated in ferroelectrics, which can be considered as a quasi-two-dimensional reconfigurable conducting channel embedded into an insulating material. Therefore, it is highly important for the application of ferroelectric nanoelectronics. Hitherto, most CDW investigations are restricted in oxides, and limited work has been reported in non-oxides to the contrary. Here, by successfully synthesizing the non-oxide ferroelectric Sn2P2S6 single crystal, we observed and confirmed the domain wall conductivity by using different scanning probe techniques which origins from the nature of inclined domain walls. Moreover, the domains separated by CDW also exhibit distinguishable electrical conductivity due to the interfacial polarization charge with opposite signs. The result provides a novel platform for understanding electrical conductivity behavior of the domains and domain walls in non-oxide ferroelectrics.

We report the magnetoresistance(MR), de Haas-van Alphen (dHvA) oscillations and the electronic structures of single crystal PtGa. The large unsaturated MR is observed with the magnetic field B//[111]. Evident dHvA oscillations with B//[001] configuration have been observed, from which twelve fundamental frequencies are extracted and the spin-orbit coupling effect (SOC) induced band splitting is revealed. The light cyclotron effective masses are extracted from the fitting by the thermal damping term of the Lifshitz-Kosevich (LK) formula. Combining with the calculated frequencies from the first-principles calculations, the dHvA frequencies F$_1$/F$_3$ and F$_{11}$/F$_{12}$ are confirmed to originate from the electron pockets at $\Gamma$ and R, respectively. The first-principles calculations also reveal the existence of spin-3/2 RSW fermion and time-reversal (TR) doubling of the spin-1 excitation at $\Gamma$ and R with large Chern number $\pm4$ when SOC is included., Comment: 5 pages, 4 figures

We report the doping effects of cobalt on van der Waals (vdW) magnet Fe$_5$GeTe$_2$. A series of (Fe$_{1-x}$Co$_x$)$_5$GeTe$_2$ (0$\leq$x$\leq$0.44) single crystals have been successfully grown, their structural, magnetic and transport properties are investigated. For x=0.20, The Curie temperature $T_C$ increases from 276~K to 337~K. Moreover, the magnetic easy-axis is reoriented to the $ab$-plane from the $c$-axis in undoped Fe$_5$GeTe$_2$ with largely enhanced magnetic anisotropy. These magnetic properties would make (Fe$_{0.8}$Co$_{0.2}$)$_5$GeTe$_2$ more effective in stabilizing magnetic order in the two-dimensional limit. A complex magnetic phase diagram is identified on the higher doping side. The x=0.44 crystal first orders ferromagnetically at $T_C$=363~K then undergoes an antiferromagnetic transition at $T_N$=335~K. Furthermore magnetic-field-induced spin-flop transitions are observed for the AFM ground state. Our work reveals (Fe$_{1-x}$Co$_x$)$_5$GeTe$_2$ as promising candidates for developing new spin-related applications and proposes a method to engineer the magnetic properties of vdW magnet., Comment: 5 pages, 5 figures

We use neutron scattering to study vanadium (hole)-doped LiFe$_{1-x}$V$_x$As. In the undoped state, LiFeAs exhibits superconductivity at $T_c=18$ K and transverse incommensurate spin excitations similar to electron overdoped iron pnictides. Upon vanadium-doping to form LiFe$_{0.955}$V$_{0.045}$, the transverse incommensurate spin excitations in LiFeAs transform into longitudinally elongated in a similar fashion as that of potassium (hole) doped Ba$_{0.7}$K$_{0.3}$Fe$_2$As$_2$, but with dramatically enhanced magnetic scattering and elimination of superconductivity. This is different from the suppression of the overall magnetic excitations in hole doped BaFe$_2$As$_2$ and the enhancement of superconductivity near optimal hole doping. These results are consistent with density function theory plus dynamic mean field theory calculations, suggesting that vanadium-doping in LiFeAs may induce an enlarged effective magnetic moment $S_{eff}$ with a spin crossover ground state arising from the inter-orbital scattering of itinerant electrons., Comment: Accepted by npj Quantum Materials

Transport-of-intensity equation (TIE) is one of the most well-known approaches for phase retrieval and quantitative phase imaging. It directly recovers the quantitative phase distribution of an optical field by through-focus intensity measurements in a noninterferometic, deterministic manner. Nevertheless, the accuracy and validity of state-of-the-art TIE solvers depend on restrictive preknowledge or assumptions, including appropriate boundary conditions, a well-defined closed region, and quasi-uniform in-focus intensity distribution, which, however, cannot be strictly satisfied simultaneously under practical experimental conditions. In this Letter, we propose a universal solution to TIE with the advantages of high accuracy, convergence guarantee, applicability to arbitrarily-shaped regions, and simplified implementation and computation. With the "maximum intensity assumption", we firstly simplified TIE as a standard Possion equation to get an initial guess of the solution. Then the initial solution is further refined iteratively by solving the same Possion equation, and thus, the instability associated with the division by zero/small intensity values and large intensity variations can be effectively bypassed. Simulations and experiments with arbitrary phase, arbitrary aperture shapes, and nonuniform intensity distributions verify the effectiveness and universality of the proposed method.

We report the magnetoresistance, Hall effect, de Haas-van Alphen (dHvA) oscillations and the electronic structures of single crystal RhSn, which is a typical material of CoSi family holding a large Chern number. The large unsaturated magnetoresistance is observed with B//[001]. The Hall resistivity curve indicates that RhSn is a multi-band system with high mobility. Evident quantum oscillations have been observed, from which the light effective masses are extracted. Ten fundamental frequencies are extracted after the fast Fourier transform analysis of the dHvA oscillations with B//[001] configuration. The two low frequencies F$_1$ and F$_2$ do not change obviously and the two high frequencies F$_9$ and F$_{10}$ evolve into four when B rotates from B//[001] to B//[110], which is consistent with the band structure in the first-principles calculations with spin-orbit coupling (SOC). The extracted Berry phases of the relative pockets show a good agreement with the Chern number $\pm4$ (with SOC) in the first-principles calculations. Above all, our studies indicate that RhSn is an ideal platform to study the unconventional chiral fermions and the surface states., Comment: 6 pages, 7 figures, accepted by Physical Review B

In materials chiral fermions such as Weyl fermions are characterized by nonzero chiral charges, which are singular points of Berry curvature in momentum space. Recently, new types of chiral fermions beyond Weyl fermions have been discovered in structurally chiral crystals CoSi, RhSi and PtAl. Here, we have synthesized RhSn single crystals, which have opposite structural chirality to the CoSi crystals we previously studied. Using angle-resolved photoemission spectroscopy, we show that the bulk electronic structures of RhSn are consistent with the band calculations and observe evident surface Fermi arcs and helical surface bands, confirming the existence of chiral fermions in RhSn. It is noteworthy that the helical surface bands of the RhSn and CoSi crystals have opposite handedness, meaning that the chiral fermions are reversed in the crystals of opposite structural chirality. Our discovery establishes a direct connection between chiral fermions in momentum space and chiral lattices in real space., Comment: 18 pages, 4 figures

We present experimental evidence of an intriguing phase transition between distinct topological states in the type-II Weyl semimetal MoTe2. We observe anomalies in the Raman phonon frequencies and linewidths as well as electronic quasielastic peaks around 70 K, which, together with structural, thermodynamic measurements, and electron-phonon coupling calculations, demonstrate a temperature-induced transition between two topological phases previously identified by contrasting spectroscopic measurements. An analysis of experimental data suggests electron-phonon coupling as the main driving mechanism for the change of key topological characters in the electronic structure of MoTe2.We also find the phase transition to be sensitive to sample conditions distinguished by synthesis methods. These discoveries of temperature and material condition-dependent topological phase evolutions and transitions in MoTe2 advance the fundamental understanding of the underlying physics and enable an effective approach to tuning Weyl semimetal states for technological applications., Comment: 6 pages, 4 figures

We use inelastic neutron scattering to study spin excitation anisotropy in mechanically detwinned Ba(Fe1-xCox)2As2 with x = 0.048 and 0.054. Both samples exhibit a tetragonal-to-orthorhombic structural transition at Ts, a collinear static antiferromagnetic (AF) order at wave vector Q1 = QAF = (1, 0) below the Neel temperature TN, and superconductivity below Tc (Ts > TN > Tc). In the high temperature paramagnetic tetragonal phase (T > Ts), spin excitations centered at Q1 and Q2 = (0, 1) are gapless and have four-fold (C4) rotational symmetry. On cooling to below TN but above Tc, spin excitations become highly anisotropic, developing a gap at Q2 but still are gapless at Q1. Upon entering into the superconducting state, a neutron spin resonance appears at Q1 with no magnetic scattering at Q2. By comparing these results with those from angle resolved photoemission spectroscopy experiments, we conclude that the anisotropic shift of the dyz and dxz bands in detwinned Ba(Fe1-xCox)2As2 below Ts is associated with the spin excitation anisotropy, and the superconductivity-induced resonance arises from the electron-hole Fermi surface nesting of quasiparticles with the dyz orbital characters.

Quantum limit is quite easy to achieve once the band crossing exists exactly at the Fermi level ($E_F$) in topological semimetals. In multilayered Dirac fermion system, the density of Dirac fermions on the zeroth Landau levels (LLs) increases in proportion to the magnetic field, resulting in intriguing angle- and field-dependent interlayer tunneling conductivity near the quantum limit. BaGa$_2$ is an example of multilayered Dirac semimetal with anisotropic Dirac cone close to $E_F$, providing a good platform to study its interlayer transport properties. In this paper, we report the negative interlayer magnetoresistance (NIMR, I//c and B//c) induced by the tunneling of Dirac fermions on the zeroth LLs of neighbouring Ga layers in BaGa$_2$. When the field deviates from the c-axis, the interlayer resistivity $\rho_{zz}(\theta)$ increases and finally results in a peak with the field perpendicular to the c-axis. These unusual interlayer transport properties (NIMR and resistivity peak with B$\perp$c) are observed together for the first time in Dirac semimetal under ambient pressure and are well explained by the model of tunneling between Dirac fermions in the quantum limit., Comment: 12 pages, 8 figures, with supplementary materials

For bidirectional joint image-text modeling, we develop variational hetero-encoder (VHE) randomized generative adversarial network (GAN), a versatile deep generative model that integrates a probabilistic text decoder, probabilistic image encoder, and GAN into a coherent end-to-end multi-modality learning framework. VHE randomized GAN (VHE-GAN) encodes an image to decode its associated text, and feeds the variational posterior as the source of randomness into the GAN image generator. We plug three off-the-shelf modules, including a deep topic model, a ladder-structured image encoder, and StackGAN++, into VHE-GAN, which already achieves competitive performance. This further motivates the development of VHE-raster-scan-GAN that generates photo-realistic images in not only a multi-scale low-to-high-resolution manner, but also a hierarchical-semantic coarse-to-fine fashion. By capturing and relating hierarchical semantic and visual concepts with end-to-end training, VHE-raster-scan-GAN achieves state-of-the-art performance in a wide variety of image-text multi-modality learning and generation tasks., Comment: ICLR 2020

We report the effects of cobalt doping on the magnetic properties of two-dimensional van der Waals ferromagnet Fe3GeTe2. Single crystals of (Fe{1-x}Cox)3GeTe2 with x=0-0.78 were successfully synthesized and characterized with x-ray diffraction, energy dispersive x-ray spectroscopy and magnetization measurements. Both the Curie-Weiss temperature and ferromagnetic (FM) ordered moment of Fe3GeTe2 are gradually suppressed upon Co doping. A kink in zero-field-cooling low field M(T) curve which is previously explained as an antiferromagnetic transition is observed for samples with x=0-0.58. Our detailed magnetization measurements and theoretical calculations strongly suggest that this kink is originated from the pinning of magnetic domain walls. The domain pinning effects are suddenly enhanced when the doping concentration of cobalt is around 50%, both the coercive field Hc and the magnetic remanence to saturated magnetization ratio MR/MS are largely improved and a hard magnetic phase emerges in bulk single crystal samples. The strong doping dependent magnetic properties suggest more spintronic applications of Fe3GeTe2., Comment: 6 pages, 4 figures, accepted by PRB

We report an investigation of the structural, magnetic and electronic properties of Ba(Fe(1-x)V(x))2As2 using x-ray, transport, magnetic susceptibility and neutron scattering measurements. The vanadium substitutions in Fe sites are possible up to 40\%. Hall effect measurements indicate strong hole-doping effect through V doping, while no superconductivity is observed in all samples down to 2K. The antiferromagnetic and structural transition temperature of BaFe2As2 is gradually suppressed to finite temperature then vanishes at x=0.245 with the emergence of spin glass behavior, suggesting an avoided quantum critical point (QCP). Our results demonstrate that the avoided QCP and spin glass state which were previously reported in the superconducting phase of Co/Ni-doped BaFe2As2 can also be realized in non-superconducting Ba(Fe(1-x)V(x))2As2., Comment: 5 pages, 6 figures

We use inelastic neutron scattering to study energy and wave vector dependence of spin fluctuations in SrCo$_2$As$_2$, derived from SrFe$_{2-x}$Co$_x$As$_2$ iron pnictide superconductors. Our data reveals the coexistence of antiferromagnetic (AF) and ferromagnetic (FM) spin fluctuations at wave vectors $\textbf{Q}_{\rm AF}$=(1,0) and $\textbf{Q}_{\rm FM}$=(0,0)/(2,0), respectively. By comparing neutron scattering results with those of dynamic mean field theory calculation and angle-resolved photoemission spectroscopy experiments, we conclude that both AF and FM spin fluctuations in SrCo$_2$As$_2$ are closely associated with a flat band of the $e_g$ orbitals near the Fermi level, different from the $t_{2g}$ orbitals in superconducting SrFe$_{2-x}$Co$_x$As$_2$. Therefore, Co-substitution in SrFe$_{2-x}$Co$_x$As$_2$ induces a $t_{2g}$ to $e_g$ orbital switching, and is responsible for FM spin fluctuations detrimental to the singlet pairing superconductivity., Comment: 6 pages, 4 figures, supplemental material available upon request, to be published in PRL

We report the synthesis and magneto-transport measurements on the single crystal of Dirac semimetal PdTe$_2$. The de Haas-van Alphen oscillations with multiple frequencies have been clearly observed, from which the small effective masses and nontrivial Berry phase are extracted, implying the possible existence of the Dirac fermions in PdTe$_2$. The planar Hall effect and anisotropic longitudinal resistivity originating from the chiral anomaly and nontrivial Berry phase are observed, providing strong evidence for the nontrivial properties in PdTe$_2$. With the increase of temperature up to 150 K, planar Hall effect still remains. The possible origin of mismatch between experimental results and theoretical predictions is also discussed., Comment: 4 pages, two column

We report the magneto-transport properties of CaAl$_4$ single crystals with $C2/m$ structure at low temperature. CaAl$_4$ exhibits large unsaturated magnetoresistance $\sim$3000$\%$ at 2.5 K and 14 T. The nonlinear Hall resistivity is observed, which indicates the multi-band feature. The first-principles calculations show the electron-hole compensation and the complex Fermi surface in CaAl$_4$, to which the two-band model with over-simplified carrier mobility can't completely apply. Evident quantum oscillations have been observed with B//c and B//ab configurations, from which the nontrivial Berry phase is extracted by the multi-band Lifshitz-Kosevich formula fitting. An electron-type quasi-2D Fermi surface is found by the angle-dependent Shubnikov-de Haas oscillations, de Haas-van Alphen oscillations and the first-principles calculations. The calculations also elucidate that CaAl$_4$ owns a Dirac nodal line type band structure around the $\Gamma$ point in the $Z$-$\Gamma$-$L$ plane, which is protected by the mirror symmetry as well as the space inversion and time reversal symmetries. Once the spin-orbit coupling is included, the crossed nodal line opens a negligible gap (less than 3 meV). The open-orbit topology is also found in the electron-type Fermi surfaces, which is believed to help enhance the magnetoresistance observed., Comment: 6 pages, two column

The sources, which generate atom-photon quantum correlations or entanglement based on quantum memory, are basic blocks for building quantum repeaters (QRs). For achieving highly entanglement-generation rates in ensemble-based QRs, spatial-, temporal- and spectral-multimode memories are needed. The previous temporal-multimode memories are based on rephrasing mechanisms in inhomogeneous-broadened media. Here, by applying a train of multi-direction write pulses into a homogeneous-broadened atomic ensemble to induce Duan-Lukin-Cirac-Zoller-like Raman processes, we prepare up to 19 pairs of modes, namely, a spin-wave mode and a photonic time bin. Spin-wave-photon (i.e., atom-photon) entanglement is probabilistically produced in the mode pairs. As a proof-in-principle demonstration, we show that the multiplexed source using all the spin-wave modes and feed-forward-control readout increases spin-wave-photon entanglement generation rate by a factor of $\sim$18.3, compared to non-multiplexed source using the individual modes. The measured Bell parameter for the multiplexed source is $2.30\pm 0.02$ combined with a memory lifetime of $30\mu s$.

We report the magnetotransport properties of HoSb, a semimetal with antiferromagnetic ground state. HoSb shows extremely large magnetoresistance (XMR) and Shubnikov-de Haas (SdH) oscillation at low temperature and high magnetic field. Different from previous reports in other rare earth monopnictides, kinks in $\rho(B)$ and $\rho_{xy}(B)$ curves and the field dependent resistivity plateau are observed in HoSb, which result from the magnetic phase transitions. The fast Fourier transform analysis of the SdH oscillation reveals the split of Fermi surfaces induced by the nonsymmetric spin-orbit interaction. The Berry phase extracted from SdH oscillation indicates the possible nontrivial electronic structure of HoSb in the presence of magnetic field. The Hall measurements suggest that the XMR originates from the electron-hole compensation and high mobility., Comment: 7 pages, 5 figures

In this paper, we consider the nodal set of a bi-harmonic function $u$ on an $n$ dimensional $C^{\infty}$ Riemannian manifold $M$, that is, $u$ satisfies the equation $\triangle_M^2u=0$ on $M$, where $\triangle_M$ is the Laplacian operator on $M$. We first define the frequency function and the doubling index for the bi-harmonic function $u$, and then establish their monotonicity formulae and doubling conditions. With the help of the smallness propagation and partitions, we show that, for some ball $B_r(x_0)\subseteq M$ with $r$ small enough, an upper bound for the measure of nodal set of the bi-harmonic function $u$ can be controlled by $N^\alpha$, that is, \mathcal{H}^{n-1}\left(\left\{x\in B_{r/2}(x_0)|u(x)=0\right\}\right)\leq CN^{\alpha}r^{n-1}, where $N=\max\left\{C_0,N(x_0,r)\right\}$, $\alpha$, $C$ and $C_0$ both are positive constants depending only on $n$ and $M$. Here $N(x_0,r)$ is the frequency function of $u$ centered at $x_0$ with radius $r$. Furthermore, we derive that an upper measure for nodal sets of eigenfunctions of the bi-harmonic operator on a $C^{\infty}$ compact Riemannian manifold without boundary can be controlled by $\lambda^\beta$ for the corresponding eigenvalue $\lambda^2$ and some positive constant $\beta$., Comment: There is a mistake in the proof

In this paper, we obtain the upper bounds for the Hausdorff measures of nodal sets of eigenfunctions with the Robin boundary conditions, i.e., \begin{equation*} {\left\{\begin{array}{l} \triangle u+\lambda u=0,\quad in\quad \Omega,\\ u_{\nu}+\mu u=0,\quad on\quad\partial\Omega, \end{array} \right.} \end{equation*} where the domain $\Omega\subseteq\mathbb{R}^n$, $u_{\nu}$ means the derivative of $u$ along the outer normal direction of $\partial\Omega$. We show that, if $\Omega$ is bounded and analytic, and the corresponding eigenvalue $\lambda$ is large enough,then the measure upper bounds for the nodal sets of eigenfunctions are $C\sqrt{\lambda}$, where $C$ is a positive constant depending only on $n$ and $\Omega$ but not on $\mu$ We also show that, if $\partial\Omega$ is $C^{\infty}$ smooth and $\partial\Omega\setminus\Gamma$ is piecewise analytic, where $\Gamma\subseteq\partial\Omega$ is a union of some $n-2$ dimensional submanifolds of $\partial\Omega$, $\mu>0$, and $\lambda$ is large enough, then the corresponding measure upper bounds for the nodal sets of $u$ are $C(\sqrt{\lambda}+\mu^{\alpha}+\mu^{-c\alpha})$ for some positive number $\alpha$, where $c$ is a positive constant depending only on $n$, and $C$ is a positive constant depending on $n$, $\Omega$, $\Gamma$ and $\alpha$., Comment: 23pages

In this paper, we mainly investigate the set of critical points associated to solutions of mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce that mean curvature equation has exactly one nondegenerate critical point in a smooth, bounded and strictly convex domain of $\mathbb{R}^{n}(n\geq2)$. Secondly, we study the geometric structure about the critical set $K$ of solutions $u$ for the constant mean curvature equation in a concentric (respectively an eccentric) spherical annulus domain of $\mathbb{R}^{n}(n\geq3)$, and deduce that $K$ exists (respectively does not exist) a rotationally symmetric critical closed surface $S$. In fact, in an eccentric spherical annulus domain, $K$ is made up of finitely many isolated critical points ($p_1,p_2,\cdots,p_l$) on an axis and finitely many rotationally symmetric critical Jordan curves ($C_1,C_2,\cdots,C_k$) with respect to an axis., Comment: 13 pages, 5 figures

In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain $\Omega$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local Chen & Huang's comparison technique and the geometric properties of approximate surfaces at the non-degenerate critical points. Secondly, we deduce the uniqueness and non-degeneracy of the critical points of solutions in a rotationally symmetric domain of $\mathbb{R}^{n}(n\geq3)$ by the projection of higher dimensional space onto two dimensional plane., Comment: 15pages, 4figures

In this paper, we mainly investigate the critical points associated to solutions $u$ of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain $\Omega$ in $\mathbb{R}^2$. Based on the fine analysis about the distribution of connected components of a super-level set $\{x\in \Omega: u(x)>t\}$ for any $\mathop {\min}_{\partial\Omega}u(x)

We report the magneto-transport properties and the electronic structure of TmSb. TmSb exhibits extremely large transverse magnetoresistance and Shubnikov-de Haas (SdH) oscillation at low temperature and high magnetic field. Interestingly, the split of Fermi surfaces induced by the nonsymmetric spin-orbit interaction has been observed from SdH oscillation. The analysis of the angle-dependent SdH oscillation illustrates the contribution of each Fermi surface to the conductivity. The electronic structure revealed by angle-resolved photoemission spectroscopy (ARPES) and first-principles calculations demonstrates a gap at $X$ point and the absence of band inversion. Combined with the trivial Berry phase extracted from SdH oscillation and the nearly equal concentrations of electron and hole from Hall measurements, it is suggested that TmSb is a topologically trivial semimetal and the observed XMR originates from the electron-hole compensation and high mobility., Comment: 6 pages, 5 figures

In this article, we consider eigenfunctions $u$ of the bi-harmonic operator, i.e., $\triangle^2u=\lambda^2u$ on $\Omega$ with some homogeneous linear boundary conditions. We assume that $\Omega\subseteq\mathbb{R}^n$ ($n\geq2$) is a $C^{\infty}$ bounded domain, $\partial\Omega$ is piecewise analytic and $\partial\Omega$ is analytic except a set $\Gamma\subseteq\partial\Omega$ which is a finite union of some compact $(n-2)$ dimensional submanifolds of $\partial\Omega$. The main result of this paper is that the measure upper bounds of the nodal sets of the eigenfunctions is controlled by $\sqrt{\lambda}$. We first define a frequency function and a doubling index related to these eigenfunctions. With the help of establishing the monotonicity formula, doubling conditions and various a priori estimates, we obtain that the $(n-1)$ dimensional Hausdorff measures of nodal sets of these eigenfunctions in a ball are controlled by the frequency function and $\sqrt{\lambda}$. In order to further control the frequency function with $\sqrt{\lambda}$, we first establish the relationship between the frequency function and the doubling index, and then separate the domain $\Omega$ into two parts: a domain away from $\Gamma$ and a domain near $\Gamma$, and develop iteration arguments to deal with the two cases respectively.

We report the magneto-transport properties and electronic structures of BaZnBi$_2$. BaZnBi$_2$ is a quasi-two-dimensional (2D) material with metallic behavior. The transverse magnetoresistance (MR) depends on magnetic field linearly and exhibits Shubnikov-de Haas (SdH) oscillation at low temperature and high field. The observed linear MR may originate from the disorder in samples or the edge conductivity in compensated two-component systems. The first-principles calculations reveal the absence of stable gapless Dirac fermion. Combining with the trivial Berry phase extracted from SdH oscillation, BaZnBi$_2$ is suggested to be a topologically trivial semimetal. Nearly compensated electron-like Fermi surfaces (FSs) and hole-like FSs coexist in BaZnBi$_2$., Comment: 6 pages, 6 figures