Inspired by recent observations of the $1/3$ magnetization plateau in kagome-based magnets, we investigate the $J_1-J_2$ Heisenberg model on the kagome lattice under the influence of an external magnetic field. Although the classical ground state at zero field depends on the sign of $J_2$, we find a robust $1/3$ semiclassical magnetization plateau in both cases. The mechanism that stabilizes this plateau is analogous to that observed in the triangular lattice, where quantum fluctuations select a collinear state from the degenerate classical manifold. We calculate the plateau width, which shows a weak dependence on $J_2$, using nonlinear spin-wave theory. Additionally, we find that a straightforward approach based on linear spin-wave yields quantitatively accurate results even for $S=1/2$. Furthermore, we identify a magnetization jump at the saturation field when $J_2=0$; this jump is related to the presence of a flat band and disappears for $J_2 eq 0$. Our study demonstrates that a semiclassical approach effectively captures the $1/3$ plateau in the kagome lattice and serves as a valuable tool for interpreting experimental and numerical results alike.
The barren plateau phenomenon; where cost function gradients vanish exponentially with system size; remains a fundamental obstacle to training variational quantum circuits (VQCs) at scale. We demonstrate, both theoretically and numerically, that embedding partial differential equation (PDE) constraints into the VQC loss function provides a natural and effective mitigation mechanism against barren plateaus. We derive analytical gradient variance lower bounds showing that physics-constrained loss functions composed of local PDE residuals evaluated at spatial collocation points inherit the favorable polynomial scaling of local cost functions, while additionally benefiting from constraint-induced landscape narrowing that concentrates gradient information. Systematic numerical experiments on the one-dimensional heat equation, Burgers' equation, and the Saint-Venant shallow water equations quantify the gradient variance across 4-8 qubits and 1-5 layer depths, comparing global cost, local cost, PDE-constrained, and PDE-constrained with structured ansatz configurations. We find that PDE-constrained circuits exhibit favorable gradient variance scaling with system size, with the physics cons
The plateau phase in the radio afterglows has been observed in very few gamma-ray bursts (GRBs), and 27 radio light curves with plateau phase were acquired from the published literature in this article. We obtain the related parameters of the radio plateau, such as temporal indexes during the plateau phase ($α_1$ and $α_2$), break time ($\Tbz$) and the corresponding radio flux ($F_{\rm b}$). The two parameter Dainotti relation between the break time of the plateau and the corresponding break luminosity ($\Lbz$) in radio band is $\Lbz \propto \Tbz^{-1.20\pm0.24}$. Including the isotropic energy $\Eiso$ and the peak energy $\Epi$, the three parameter correlations for the radio plateaus are written as $\Lbz \propto \Tbz^{-1.01 \pm 0.24} \Eiso^{0.18 \pm 0.09}$ and $\Lbz \propto \Tbz^{-1.18 \pm 0.27} \Epi^{0.05 \pm 0.28}$, respectively. The correlations are less consistent with that of X-ray and optical plateaus, implying that radio plateaus may have a different physical mechanism. The typical frequencies crossing the observational band may be a reasonable hypothesis that causes the breaks of the radio afterglows. We calibrate GRBs empirical luminosity correlations as standard candle fo
Antiferromagnets exhibiting the anomalous Hall effect represent a fascinating convergence of magnetism, topology, and electronic structure. Identifying antiferromagnets with large and tunable anomalous Hall effects is crucial for the development of spintronic applications. Here, we report a strain-tunable anomalous Hall plateau in CoNb$_3$S$_6$, which is a prime candidate for altermagnetism. The plateau emerges as a flat extended intermediate step of the anomalous Hall hysteresis loop with a controllable step height with temperature and strain. The remarkable tunability of the plateau position is in contrast with typical magnetic plateau associated with a field-induced metastable magnetic structure, but indicates the existence of a hidden phase transition that significantly alters the magnetic anisotropy energy without changing the magnetic order. The symmetry analysis of the strain tuning suggests that the hidden phase preserves the rotational symmetry of the ab-plane. Our results show the plateau reflects the phase coexistence during the hidden transition, and anomalous Hall resistivity of the plateau is thus non-volatile, enabling a novel four-state switching of the anomalous Ha
A stellar-mass black hole (BH) or a millisecond magnetar is believed to be born as the central engine of Gamma-ray bursts (GRBs). The presence of plateaus in the X-ray extended emission or afterglow of GRBs is widely accepted as an indicator of magnetar central engine, particularly those with a sharp decay (faster than $t^{-3}$), so-called internal plateau. However, an alternative model, by taking the evolution of the magnetic flux at the BH horizon into account, suggests that an internal plateau can also arise from a Blandford-Znajek (BZ) mechanism powered jet (hereafter referred to as the BZ jet). In this study, we propose that a precessional BZ jet would manifest a Quasi-Periodic Oscillation (QPO) signature on the internal plateau and the subsequent sharp decay. Such lightcurves cannot be readily explained by the activity of a short-lived, supermassive magnetar, thus favoring a Kerr BH as the central engine. The X-ray afterglow of GRB 050904, comprising nine flares, is characterized by a QPO-modulated plateau and sharp decay, which can be well reproduced by a precessional BZ jet model. Therefore, one potential clue for distinguishing between these two engines lies in whether QPO
This manuscript presents a multilevel analysis of gamma-ray bursts (GRBs). We focus on the plateau phase, which is often observed in the light curves (LCs) of GRBs. We discuss its observational properties and then thoroughly examine possible theoretical models to explain them. Inspired by the limitations of many currently known models, we introduce a novel scenario of an LC powered by the kinetic energy of a rotating black hole (BH). We investigate observational correlations between the properties of GRBs across the gamma, X-ray, and optical bands during the prompt and plateau phases of their LCs. Our analysis includes all GRBs with known redshifts detected by the Neil Gehrels Swift Observatory (Swift) and the Fermi Gamma-ray Space Telescope (Fermi), as well as ground-based optical telescopes. We identify a tight correlation with the R^2 coefficient of ~0.89 for the three-dimensional Dainotti relation between the luminosity at the end of the plateau, its duration measured by Swift, and the peak luminosity measured by Fermi in the 10-1000 keV band. When accounting for redshift evolution, we achieve very small intrinsic scatter $σ_{int}=0.25\pm0.04$ (~43% reduction compared to the pr
Two main challenges preventing efficient training of variational quantum algorithms and quantum machine learning models are local minima and barren plateaus. Typically, barren plateaus are associated with deep circuits, while shallow circuits have been shown to suffer from suboptimal local minima. We point out a simple mechanism that creates exponentially many poor local minima specifically in the barren plateau regime. These local minima are trivial solutions, optimizing only a few terms in the loss function, leaving the rest on their barren plateaus. More precisely, we show the existence of approximate local minima, optimizing a single loss term, and conjecture the existence of exact local minima, optimizing only a logarithmic fraction of all loss function terms. One implication of our findings is that simply yielding large gradients is not sufficient to render an initialization strategy a meaningful solution to the barren plateau problem.
The multi-stage phenomenon in the training loss curves of neural networks has been widely observed, reflecting the non-linearity and complexity inherent in the training process. In this work, we investigate the training dynamics of neural networks (NNs), with particular emphasis on the small initialization regime, identifying three distinct stages observed in the loss curve during training: the initial plateau stage, the initial descent stage, and the secondary plateau stage. Through rigorous analysis, we reveal the underlying challenges contributing to slow training during the plateau stages. While the proof and estimate for the emergence of the initial plateau were established in our previous work, the behaviors of the initial descent and secondary plateau stages had not been explored before. Here, we provide a more detailed proof for the initial plateau, followed by a comprehensive analysis of the initial descent stage dynamics. Furthermore, we examine the factors facilitating the network's ability to overcome the prolonged secondary plateau, supported by both experimental evidence and heuristic reasoning. Finally, to clarify the link between global training trends and local par
We introduce a diffused interface formulation of the Plateau problem, where the Allen--Cahn energy $\mathcal{AC}_\varepsilon$ is minimized under a volume constraint $v$ and a spanning condition on the level sets of the densities. We discuss two singular limits of these Allen--Cahn Plateau problems: when $\varepsilon\to 0^+$, we prove convergence to the Gauss' capillarity formulation of the Plateau problem with positive volume $v$; and when $\varepsilon\to 0^+$, $v\to 0^+$ and $\varepsilon/v\to 0^+$, we prove convergence to the classical Plateau problem (in the homotopic spanning formulation of Harrison and Pugh). As a corollary of our analysis we resolve the incompatibility between Plateau's laws and the Allen--Cahn equation implied by a regularity theorem of Tonegawa and Wickramasekera. In particular, we show that Plateau-type singularities can be approximated by energy minimizing solutions of the Allen--Cahn equation with a volume Lagrange multiplier and a transmission condition on a spanning free boundary.
To search for a conceptual picture describing the magnetization plateau phenomenon, we surveyed the crystal structures and the spin lattices of those magnets exhibiting plateaus in their magnetization vs. magnetic field curves by probing the three questions: (a) why only certain magnets exhibit magnetization plateaus, (b) why there occur several different types of magnetization plateaus, and (c) what controls the widths of magnetization plateaus. We show that the answers to these questions lie in how the magnets under field absorb Zeeman energy hence changing their magnetic structures. The magnetic structure of a magnet insulator is commonly described in terms of its spin lattice, which requires the determination of the spin exchanges nonnegligible strengths between the magnetic ions. Our work strongly suggests that a magnet under magnetic field partitions its spin lattice into antiferromagnetic (AFM) or ferrimagnetic fragments by breaking its weak magnetic bonds. Our supposition of the field-induced partitioning of spin lattice into magnetic fragments is supported by the anisotropic magnetization plateaus of Ising magnets and by the highly anisotropic width of the 1/3-magnetizatio
The frustrated quantum magnet SrCu_2(BO_3)_2 shows a remarkably rich phase diagram in an external magnetic field including a sequence of magnetization plateaux. The by far experimentally most studied and most prominent magnetization plateau is the 1/8 plateau. Theoretically, one expects that this material is well described by the Shastry-Sutherland model. But recent microscopic calculations indicate that the 1/8 plateau is energetically not favored. Here we report on a very simple microscopic mechanism which naturally leads to a 1/8 plateau for realistic values of the magnetic exchange constants. We show that the 1/8 plateau with a diamond unit cell benefits most compared to other plateau structures from quantum fluctuations which to a large part are induced by Dzyaloshinskii-Moriya interactions. Physically, such couplings result in kinetic terms in an effective hardcore boson description leading to a renormalization of the energy of the different plateaux structures which we treat in this work on the mean-field level. The stability of the resulting plateaux are discussed. Furthermore, our results indicate a series of stripe structures above 1/8 and a stable magnetization plateau a
The existence of a plateau in the short-duration tail of the observed distribution of cosmological Long-soft Gamma Ray Bursts (LGRBs) has been argued as the first direct evidence of Collapsars. A similar plateau in the short-duration tail of the observed duration distribution of Short-hard Gamma Ray Bursts (SGRBs) has been suggested as evidence of compact binary mergers. We present an equally plausible alternative interpretation for this evidence, which is purely statistical. Specifically, we show that the observed plateau in the short-duration tail of the duration distribution of LGRBs can naturally occur in the statistical distributions of strictly positive physical quantities, exacerbated by the effects of mixing with the duration distribution of SGRBs, observational selection effects, and data aggregation (e.g., binning) methodologies. The observed plateau in the short-duration tail of the observed distributions of SGRBs can similarly result from a combination of sample incompleteness and inhomogeneous binning of data. The observed plateau in the short-duration tail of the observed distributions of SGRBs can similarly result from a combination of sample incompleteness and inhom
Two legged skewed spin-$\frac{1}{2}$ ladders are frustrated and exhibit exotic quantum phases in ground state due to strong quantum fluctuations and competing spin exchanges. Here, we study ground state properties of a spin-$\frac{1}{2}$ Heisenberg model on 3/4, 3/5 and 5/5 skewed ladders in the presence of a Zeeman magnetic field, $B$, using exact diagonalization and the density matrix renormalization group method. We note the existence of plateaus at $m =$ 1/3 and 2/3 for 3/4 skewed ladder, at $m =$ 1/4, 1/2, and 3/4 for 3/5 skewed ladder, and at $m =$ 0, 1/3, and 2/3 for 5/5 skewed ladder, where $m$ is the ratio of the observed magnetization ($M$) to the saturated magnetization (${M_\mathrm{max}}$). The plateau state is always a gapped state and the plateau width depends on the gap in the system. Surprisingly, the 3/4 and 5/5 skewed ladders show interesting quadrupolar or n-type spin nematic phases below the 1/3$^{rd}$ plateau, i.e, at very low magnetic fields. These two systems are unique as they host both a plateau and a quadrupolar phase at low magnetic fields. The linear variation of pitch angle of the spin with magnetization and behavior of binding energy of magnon pairs as
The phenomenon of magnetization plateaus in antiferromagnets under magnetic field has always been an important topic in magnetism. We propose to probe the elusive physics of plateaus in 2-d by considering hole-doped antiferromagnet and studying the signatures of magnetization plateaus in terms of the properties of holes, coupled to an effective gauge field generated by the spin sector. The latter mediates interaction between the holes, found to be algebraically decaying long-ranged with both Coulombic and dipolar forms outside plateau and short-ranged (local) inside plateau. The resulting hole spectral weight is significantly broadened off-plateau, while it remains sharply-peaked on-plateau. We also extend the result obtained for 1-d system where finite hole doping gives rise to a shift in the magnetization value of the plateaus.
We show that every compactly supported smoothly calibrated integral current with connected $C^{3,α}$ boundary is the unique solution to the oriented Plateau problem for its boundary data. The same holds true for compactly supported $``$continuously calibrated$"$ integral flat chains. This is proved as a consequence of the boundary regularity theory for area-minimizing currents and a unique continuation argument in the spirit of Frank Morgan. In codimension one, the argument yields a sufficient condition for uniqueness in the oriented Plateau problem expressed in terms of the regularity of the calibrating form.
A review on the classical Plateau problem is presented. Then, the state of the art about the Kirchhoff-Plateau problem is illustrated as well as some possible future directions of research.
The quantum Hall effect (QHE) is a topologically protected phenomenon which has been observed in various systems. In experiments, the size of Hall bar device to realize the QHE is generally much larger than the phase coherence length, in which the quantum coherence of electrons is destroyed. Here, we theoretically study the influence of dephasing effect on the quantized Hall (QH) plateaus. We find that the QH plateau disappears in perfectly quantum coherent systems if the coupling between leads and central region is imperfect. The Hall resistance is very large and strongly oscillates instead of presenting the QH plateau in this case. However, by introducing the dephasing, the Hall resistance decreases and the QH plateau appears gradually. Similar results can also be observed for the quantum anomalous Hall effect. Our results propose that dephasing effect promotes the appearance of QH plateaus, which opens a new topic of the dephasing effect on topological systems.
We present an interesting dynamical temporal localization of a strongly driven two-level system (TLS), a plateau with quantized oscillation, by an analytical and transparent method, the counter-rotating-hybridized rotating-wave (CHRW) method. This approach, which is based on unitary transformations with a single parameter, treats the rotating and counter-rotating terms on equal footing. In the unitarily transformed representation, we find that it is the multiple-harmonic terms shown in the transformed Hamiltonian that make a crucial contribution to the generation of the exotic plateau phenomenon. By comparing the results of the numerically exact calculation and several other methods, we show that the CHRW results obtained by analytical formalism involving the collective effects of multiple harmonics are in good agreement with the numerical results, which illustrates not only the general tendency of the dynamical evolution of strongly driven TLS, but also the interesting phenomena of plateaus. The developed CHRW method reveals two kinds of plateau patterns: zigzag plateau and armchair plateau. The plateau phenomenon has a periodical pattern whose frequency is double the driving freq
Plateau's problem is to show the existence of an area minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau's problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator to solve Plateau's problem for orientable, embedded surfaces. But integral currents cannot represent surfaces such as the Moebius strip or surfaces with triple junctions. In the class of varifolds, there are no existence theorems for a general Plateau problem because of a lack of a boundary operator. We use the chain complex of differential chains with its continuous boundary operator to solve a general version of Plateau's problem. We find the first solution which minimizes area taken from a collection of surfaces that includes all previous special cases, as well as all smoothly immersed surfaces of any genus type, both orientable and nonorientable, and surfaces w
We discuss the ground-state degeneracy of spin-$1/2$ kagome-lattice quantum antiferromagnets on magnetization plateaus by employing two complementary methods: the adiabatic flux insertion in closed boundary conditions and a 't Hooft anomaly argument on inherent symmetries in a quasi-one-dimensional limit. The flux insertion with a tilted boundary condition restricts the lower bound of the ground-state degeneracy on $1/9$, $1/3$, $5/9$, and $7/9$ magnetization plateaus under the $\mathrm{U(1)}$ spin-rotation and the translation symmetries: $3$, $1$, $3$, and $3$, respectively. This result motivates us further to develop an anomaly interpretation of the $1/3$ plateau. Taking advantage of the insensitivity of anomalies to spatial anisotropies, we examine the existence of the unique gapped ground state on the $1/3$ plateau from a quasi-one-dimensional viewpoint. In the quasi-one-dimensional limit, kagome antiferromagnets are reduced to weakly coupled three-leg spin tubes. Here, we point out the following anomaly description of the $1/3$ plateau. While a simple $S=1/2$ three-leg spin tube cannot have the unique gapped ground state on the $1/3$ plateau because of an anomaly between a $\m