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The recurrent neural network has been greatly developed for effectively solving time-varying problems corresponding to complex environments. However, limited by the way of centralized processing, the model performance is greatly affected by factors like the silos problems of the models and data in reality. Therefore, the emergence of distributed artificial intelligence such as federated learning (FL) makes it possible for the dynamic aggregation among models. However, the integration process of FL is still server-dependent, which may cause a great risk to the overall model. Also, it only allows collaboration between homogeneous models, and does not have a good solution for the interaction between heterogeneous models. Therefore, we propose a Distributed Computation Model (DCM) based on the consortium blockchain network to improve the credibility of the overall model and effective coordination among heterogeneous models. In addition, a Distributed Hierarchical Integration (DHI) algorithm is also designed for the global solution process. Within a group, permissioned nodes collect the local models' results from different permissionless nodes and then sends the aggregated results back to all the permissionless nodes to regularize the processing of the local models. After the iteration is completed, the secondary integration of the local results will be performed between permission nodes to obtain the global results. In the experiments, we verify the efficiency of DCM, where the results show that the proposed model outperforms many state-of-the-art models based on a federated learning framework.

In underdoped cuprate superconductors, a rich competition occurs between superconductivity and charge density wave (CDW) order. Whether rotational symmetry breaking (nematicity) occurs intrinsically and generically or as a consequence of other orders is under debate. Here we employ resonant x-ray scattering in stripe-ordered (La,M)_2CuO_4 to probe the relationship between electronic nematicity of the Cu 3d orbitals, structure of the (La,M)_2O_2 layers, and CDW order. We find distinct temperature dependences for the structure of the (La,M)_2O_2 layers and the electronic nematicity of the CuO_2 planes, with only the latter being enhanced by the onset of CDW order. These results identify electronic nematicity as an order parameter that is distinct from a purely structural order parameter in underdoped striped cuprates., Comment: 23 pages including supplementary information, 3 main figures, 4 supplementary figures

We consider a system of spins on the sites of a three-dimensional pyrochlore lattice of corner-sharing tetrahedra interacting with a predominant effective $xy$ exchange. In particular, we investigate the selection of a long-range ordered state with broken discrete symmetry induced by thermal fluctuations near the critical region. At the standard mean-field theory (s-MFT) level, in a region of the parameter space of this Hamiltonian that we refer to as $\Gamma_5$ region, the ordered state possesses an accidental $U(1)$ degeneracy. In this paper, we show that fluctuations beyond s-MFT lift this degeneracy by selecting one of two states (so-called $\psi_2$ and $\psi_3$) from the degenerate manifold, thus exposing a certain form of order-by-disorder (ObD). We analytically explore this selection at the microscopic level and close to criticality by elaborating upon and using an extension of the so-called TAP method, originally developed by Thouless, Anderson and Palmer to study the effect of fluctuations in spin glasses. We also use a single-tetrahedron cluster-mean-field theory (c-MFT) to explore over what minimal length scale fluctuations can lift the degeneracy. We find the phase diagrams obtained by these two methods to be somewhat different since c-MFT only includes the shortest-range fluctuations. General symmetry arguments used to construct a Ginzburg-Landau theory to lowest order in the order parameters predict that a weak magnetic moment, $m_z$, along the local $\langle 111 \rangle$ (${\hat z}$) direction is generically induced for a system ordering into a $\psi_2$ state, but not so for $\psi_3$ ordering. Both E-TAP and c-MFT calculations confirm this weak fluctuation-induced $m_z$ moment. Using a Ginzburg-Landau theory, we discuss the phenomenology of multiple phase transitions below the paramagnetic phase transition and within the $\Gamma_5$ long-range ordered phase., Comment: 17 pages, 8 figures. ArXiv version essentially identical to the version published in Physical Review B

We carry out an analytical study of quantum spin ice, a U$(1)$ quantum spin liquid close to the classical spin ice solution for an effective spin $1/2$ model with anisotropic exchange couplings $J_{zz}$, $J_{\pm}$ and $J_{z\pm}$ on the pyrochlore lattice. Starting from the quantum rotor model introduced by Savary and Balents in Phys. Rev. Lett. 108, 037202 (2012), we retain the dynamics of both the spinons and gauge field sectors in our treatment. The spinons are described by a bosonic representation of quantum XY rotors while the dynamics of the gauge field is captured by a phenomenological Hamiltonian. By calculating the one-loop spinon self-energy, which is proportional to $J_{z\pm}^2$, we determine the stability region of the U$(1)$ quantum spin liquid phase in the $J_{\pm}/J_{zz}$ vs $J_{z\pm}/J_{zz}$ zero temperature phase diagram. From these results, we estimate the location of the boundaries between the spin liquid phase and classical long-range ordered phases., Comment: 15 pages, 6 figures. The published version

Actively shought since the turn of the century, two-dimensional quantum spin liquids (QSLs) are exotic phases of matter where magnetic moments remain disordered even at extremely low temperatures. Despite ongoing searches, QSLs remain elusive, due to a lack of concrete knowledge of the microscopic mechanisms that inhibit magnetic order in real materials. Here, we study a theoretical model for a broad class of frustrated magnetic rare-earth pyrochlore materials called "quantum spin ices". When subject to an external magnetic field along the [111] crystallographic direction, the resulting spin interactions contain a mix of geometric frustration and quantum fluctuations in decoupled two-dimensional kagome planes. Using large-scale quantum Monte Carlo simulations, we identify a simple set of interactions sufficient to promote a groundstate with no magnetic long-range order, and a gap to excitations, consistent with a $Z_2$ spin liquid phase. This suggests a systematic experimental procedure to search for two-dimensional QSLs within the broader class of three-dimensional pyrochlore quantum spin ice materials., Comment: 11 pages, 13 figures. Includes supplemental information

The imminent realization of topologically-protected qubits in fabricated systems will provide not only an elementary implementation of fault-tolerant quantum computing architecture, but also an experimental vehicle for the general study of topological order. The simplest topological qubit harbors what is known as a Z$_2$ liquid phase, which encodes information via a degeneracy depending on the system's topology. Elementary excitations of the phase are fractionally charged objects called {\it spinons}, or Ising flux vortices called {\it visons}. At zero temperature a Z$_2$ liquid is stable under deformations of the Hamiltonian until spinon or vison condensation induces a quantum phase transition destroying the topological order. In this paper, we use quantum Monte Carlo to study a vison-induced transition from a Z$_2$ liquid to a valence-bond solid in a quantum dimer model on the kagome lattice. Our results indicate that this critical point is controlled by a new universality class beyond the standard Landau paradigm., Comment: 5 pages, 4 figures. Published version

Motivated by recent experimental and theoretical progress on the Er2Ti2O7 pyrochlore XY antiferromagnet, we study the problem of quantum order-by-disorder in pyrochlore XY systems. We consider the most general nearest-neighbor pseudo spin-1/2 Hamiltonian for such a system characterized by anisotropic spin-spin couplings J_e = [J_\pm, J_{\pm\pm}, J_{z\pm}, J_{zz}] and construct zero-temperature phase diagrams. Combining symmetry arguments and spin-wave calculations, we show that the ground state phase boundaries between the two candidate ground states of the \Gamma_5 irreducible representation, the \psi_2 and \psi_3 (basis) states, are rather accurately determined by a cubic equation in J_{\pm}J_{\pm\pm})/J_{z\pm}^2. Depending on the value of J_{zz}, there can be one or three phase boundaries that separate alternating regions of \psi_2 and \psi_3 states. In particular, we find for sufficiently small J_{zz}/J_{\pm} a narrow \psi_2 sliver sandwiched between two \psi_3 regions in the J_{\pm\pm}/J_\pm vs J_{z\pm}/J_\pm phase diagram. Our results further illustrate the very large potential sensitivity of the ground state of XY pyrochlore systems to minute changes in their spin Hamiltonian. Using the experimentally determined J_3 and g-tensor values for Er2Ti2O7, we show that the heretofore neglected long-range 1/r^3 magnetostatic dipole-dipole interactions do not change the conclusion that Er2Ti2O7 has a \psi_2 ground state induced via a quantum order-by-disorder mechanism. We propose that the CdDy2Se4 chalcogenide spinel, in which the Dy^{3+} ions form a pyrochlore lattice and may be XY-like, could prove interesting to investigate., Comment: 8 pages, 3 figures. Version is the same as the published one, within epsilon

We study the problem of partially ordered phases with periodically arranged disordered (paramagnetic) sites on the pyrochlore lattice, a network of corner-sharing tetrahedra. The periodicity of these phases is characterized by one or more wave vectors k=(1/2 1/2 1/2). Starting from a general microscopic Hamiltonian including anisotropic nearest-neighbor exchange, long-range dipolar interactions and second- and third-nearest neighbor exchange, we identify using standard mean-field theory (s-MFT) an extended range of interaction parameters that support partially ordered phases. We demonstrate that thermal fluctuations ignored in s-MFT are responsible for the selection of one particular partially ordered phase, e.g. the "4-k" phase over the "1-k" phase. We suggest that the transition into the 4-k phase is continuous with its critical properties controlled by the cubic fixed point of a Ginzburg-Landau theory with a 4-component vector order-parameter. By combining an extension of the Thouless-Anderson-Palmer method originally used to study fluctuations in spin glasses with parallel-tempering Monte-Carlo simulations, we establish the phase diagram for different types of partially ordered phases. Our results elucidate the long-standing puzzle concerning the origin of the 4-k partially ordered phase observed in the Gd2Ti2O7 dipolar pyrochlore antiferromagnet below its paramagnetic phase transition temperature., Comment: 13 pages, 4 figures

Motivated by recent numerical and experimental studies of the spin-1/2 Heisenberg antiferromagnet on kagome, we formulate a many-body model for fermionic spinons introduced by us earlier [Phys. Rev. Lett. 103, 187203 (2009)]. The spinons interact with an emergent U(1) gauge field and experience strong short-range attraction in the S=0 channel. The ground state of the model is generically a $Z_2$ liquid. We calculate the edge of the two-spinon continuum and compare the theory to the slave-fermion approach to the Heisenberg model., Comment: 7 figures

We propose a novel interaction-based route to half-metal state for interacting electrons on two-dimensional lattices. Magnetic field applied parallel to the lattice is used to tune one of the spin densities to a particular commensurate with the lattice value in which the system spontaneously `locks in' via van Hove enhanced density wave state. Electrons of opposite spin polarization retain their metallic character and provide for the half-metal state which, in addition, supports magnetization plateau in a finite interval of external magnetic field. Similar half-metal state is realized in the finite-U Hubbard model on a triangular lattice at 1/3 of the maximum magnetization., Comment: 7 pages, 8 figures

Many unconventional quantum phases host special non-magnetic excitations such as photons and visons. We discuss two possible ways to detect these excitations experimentally. Firstly, spin-lattice coupling mixes the excitations with phonons. The phonon spectral function acquires new features that can be detected by neutron or X-ray scattering. Secondly, valence-bond fluctuations translate into charge density fluctuations on non-bipartite lattices. Such charge fluctuations can be characterized by conventional spectroscopies such as Terahertz spectroscopy. Observation of exotic singlet excitations would provide positive identification of unconventional quantum phases in frustrated antiferromagnets., Comment: 3 figures

A small value of the spin gap in quantum antiferromagnets with strong frustration makes them susceptible to nominally small deviations from the ideal Heisenberg model. One of such perturbations, the anisotropic Dzyaloshinskii-Moriya interaction, is an important perturbation for the $S=1/2$ kagome antiferromagnet, one of the current candidates for a quantum-disordered ground state. We study the influence of the DM term in a related one-dimensional system, the sawtooth chain that has valence-bond order in its ground state. Through a combination of analytical and numerical methods, we show that a relatively weak DM coupling, $0.115J$, is sufficient to destroy the valence-bond order, close the spin gap, and turn the system into a Luttinger liquid with algebraic spin correlations. A similar mechanism may be at work in the kagome antiferromagnet., Comment: 11 pages. References added. Revisions made as requested by referees

The ground state of the S=1/2 Heisenberg antiferomagnet on kagome can be viewed as a collection of fermionic spinons bound into small, heavy singlet pairs. Low-energy magnetic excitations in this system correspond to breaking the pairs into individual spinons. We calculate the structure factor for inelastic neutron scattering from independent spinon pairs., Comment: 9 pages. Revised in response to referee reports. References added

We present an analytical model of orbital ordering in vanadium spinel MnV2O4. The model is based on recent first-principles calculation indicating a strong trigonal distortion at the vanadium sites of this compound [Phys. Rev. Lett. 102, 216405 (2009)]. At the single-ion level, the trigonal crystal field leaves a doubly degenerate atomic ground state and breaks the approximate rotational symmetry of t2g orbitals. We find that the effective interaction between the low-energy doublets is described by a quantum antiferromagnetic 120-degree model on the pyrochlore lattice. We obtain the classical ground state and show its stability against quantum fluctuations. The corresponding orbital order consisting of two inequivalent orbital chains is consistent with the experimentally observed tetragonal symmetry. A periodic modulation of electron density function along orbital chains is shown to arise from the staggering of local trigonal axes. In the presence of orbital order, single-ion spin anisotropy arising from relativistic spin-orbit interaction stabilizes the experimentally observed orthogonal magnetic structure., Comment: 10 pages, 9 figures, references added

Spin excitations in an ordered Heisenberg magnet are magnons--bosons with spin 1. That may change when frustration and quantum fluctuations suppress the order and restore the spin-rotation symmetry. We show that spin excitations in the $S=1/2$ Heisenberg antiferromagnet on kagome are spinons--fermions with spin 1/2. In the ground state the system can be described as a collection of small, heavy pairs of spinons with spin 0. A magnetic excitation of lowest energy amounts to breaking up a pair into two spinons at a cost of $0.06 J$., Comment: 4 pages, 4 figures, accepted version of the manuscript

We re-analyzed the issue whether the resonance peak observed in neutron scattering experiments on the cuprates is an exciton, a \pi-resonance, or a magnetic plasmon. We considered a toy model with on-cite Hubbard U and nearest-neighbor interactions in both charge and spin channels. We found that the resonance is predominantly an exciton, even if magnetic interaction is absent and d-wave pairing originates from attractive density-density interaction. Our results indicate that one cannot distinguish between spin and charge-mediated pairing interactions by just looking at the resonance peak in the dynamic spin susceptibility., Comment: 5 pp, 5 figures, submitted to PRB