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In the 1980s, Comtet et al. found that a modified version of the WKB quantization condition yields exact eigenvalues for all exactly solvable potentials that were known at the time. This intriguing property prompted investigations into the underlying reasons for such exact solvability. In this paper, we trace the journey that reveals shape invariance as the fundamental cause of this exactness and identifies the set of potentials for which it holds. We demonstrate that while shape invariance ensures this exactness in conjunction with an additional condition, it alone is not sufficient.

Comparative public health research makes wide use of self-report instruments. For example, research identifying and explaining health disparities across demographic strata may seek to understand the health effects of patient attitudes or private behaviors. Such personal attributes are difficult or impossible to observe directly and are often best measured by self-reports. Defensible use of self-reports in quantitative comparative research requires not only that the measured constructs have the same meaning across groups, but also that group comparisons of sample estimates (eg, means and variances) reflect true group differences and are not contaminated by group-specific attributes that are unrelated to the construct of interest. Evidence for these desirable properties of measurement instruments can be established within the confirmatory factor analysis (CFA) framework; a nested hierarchy of hypotheses is tested that addresses the cross-group invariance of the instrument's psychometric properties. By name, these hypotheses include configurai, metric (or pattern), strong (or scalar), and strict factorial invariance. The CFA model and each of these hypotheses are described in nontechnical language. A worked example and technical appendices are included.

Recent years have seen remarkable development in open quantum systems effectively described by non-Hermitian Hamiltonians. A unique feature of non-Hermitian topological systems is the skin effect, anomalous localization of an extensive number of eigenstates driven by nonreciprocal dissipation. Despite its significance for non-Hermitian topological phases, the relevance of the skin effect to quantum entanglement and critical phenomena has remained unclear. Here, we find that the skin effect induces a nonequilibrium quantum phase transition in the entanglement dynamics. We show that the skin effect gives rise to a macroscopic flow of particles and suppresses the entanglement propagation and thermalization, leading to the area law of the entanglement entropy in the nonequilibrium steady state. Moreover, we reveal an entanglement phase transition induced by the competition between the unitary dynamics and the skin effect even without disorder or interactions. This entanglement phase transition accompanies nonequilibrium quantum criticality characterized by a nonunitary conformal field theory whose effective central charge is extremely sensitive to the boundary conditions. We also demonstrate that it originates from an exceptional point of the non-Hermitian Hamiltonian and the concomitant scale invariance of the skin modes localized according to the power law. Furthermore, we show that the skin effect leads to the purification and the reduction of von Neumann entropy even in Markovian open quantum systems described by the Lindblad master equation. Our work opens a way to control the entanglement growth and establishes a fundamental understanding of phase transitions and critical phenomena in open quantum systems far from thermal equilibrium.

The modern description of elementary particles, as formulated in the standard model of particle physics, is built on gauge theories 1 . Gauge theories implement fundamental laws of physics by local symmetry constraints. For example, in quantum electrodynamics Gauss’s law introduces an intrinsic local relation between charged matter and electromagnetic fields, which protects many salient physical properties, including massless photons and a long-ranged Coulomb law. Solving gauge theories using classical computers is an extremely arduous task 2 , which has stimulated an effort to simulate gauge-theory dynamics in microscopically engineered quantum devices 3 – 6 . Previous achievements implemented density-dependent Peierls phases without defining a local symmetry 7 , 8 , realized mappings onto effective models to integrate out either matter or electric fields 9 – 12 , or were limited to very small systems 13 – 16 . However, the essential gauge symmetry has not been observed experimentally. Here we report the quantum simulation of an extended U(1) lattice gauge theory, and experimentally quantify the gauge invariance in a many-body system comprising matter and gauge fields. These fields are realized in defect-free arrays of bosonic atoms in an optical superlattice of 71 sites. We demonstrate full tunability of the model parameters and benchmark the matter–gauge interactions by sweeping across a quantum phase transition. Using high-fidelity manipulation techniques, we measure the degree to which Gauss’s law is violated by extracting probabilities of locally gauge-invariant states from correlated atom occupations. Our work provides a way to explore gauge symmetry in the interplay of fundamental particles using controllable large-scale quantum simulators. Quantum simulation in a 71-site optical lattice certifies gauge invariance, showing how this essential property of lattice gauge theories can be maintained across a quantum phase transition.

In quantum electrodynamics, the choice of gauge influences the form of light–matter interactions. However, gauge invariance implies that all physical results should be independent of this formal choice. The Rabi model, a widespread description for the dipolar coupling between a two-level atom and a quantized electromagnetic field, seemingly violates this principle in the presence of ultrastrong light–matter coupling, a regime that is now experimentally accessible in many physical systems. This failure is attributed to the finite-level truncation of the matter system, an approximation that enters the derivation of the Rabi model. Here, we identify the source of gauge violation and provide a general method for the derivation of light–matter Hamiltonians in truncated Hilbert spaces that produces gauge-invariant physical results, even for extreme light–matter interaction regimes. This is achieved by compensating the non-localities introduced in the construction of the effective Hamiltonians. The resulting quantum Rabi Hamiltonian in the Coulomb gauge differs significantly in form from the standard one, but provides the same physical results obtained by using the dipole gauge. These results shed light on gauge invariance in the non-perturbative and extreme-interaction regimes, and solve long-lasting controversies arising from gauge ambiguities in the quantum Rabi and Dicke models.The principle of gauge invariance in quantum electrodynamics may be violated by approximate models in the presence of strong light–matter interactions. A general approach solves gauge ambiguities and offers a way to construct gauge-invariant Hamiltonians.

Gauge theories constitute the basis of the Standard Model and provide useful descriptions of various phenomena in condensed matter. Realizing gauge theories on tunable tabletop quantum devices such as cold-atom quantum simulators offers the possibility to study their dynamics from first principles and to probe effects that are out of reach of dedicated particle colliders, such as deviations from gauge invariance. These quantum simulators can potentially provide insights into high-energy and nuclear physics questions, while also serving as a versatile tool for the exploration of topological phases and ergodicity-breaking mechanisms relevant to low-energy many-body physics. Recent years have seen substantial progress in the implementation of (1 + 1)D Abelian gauge theories using ultracold atoms. In this Review, we chronicle these advances, highlighting key developments in stabilizing gauge invariance and scaling up from basic building blocks to large-scale realizations where gauge-theory phenomena can be probed. We offer an outlook on future directions and the requirements for advancing this technology to the next level. Large-scale quantum simulations of gauge theories are relevant to high-energy and condensed matter physics. This Review covers recent developments in simulating lattice gauge theories using cold atoms.

In this talk, we discuss how gauge symmetries broken explicitly by a Poincare-breaking UV cutoff can be restored. We show that gauge symmetries can be restored by the introduction of affine curvature in reminiscence to the Higgs field. In fact, gauge symmetries get restored and general relativity emerges at the extremum of the metric-affine action. As per this point, we show emergence of the general relativity, reveal how its parameters relate to the flat spacetime loops, elucidate the new particle spectrum it brings along, and discuss its salient signatures. We show that the resulting field-theoretic plus gravitational setup can be probed via various phenomena ranging from collider experiments to black holes.

The study of topological materials possessing nontrivial band structures enables exploitation of relativistic physics and development of a spectrum of intriguing physical phenomena. However, previous studies of Weyl physics have been limited exclusively to semimetals. Here, via systematic magnetotransport measurements, two representative topological transport signatures of Weyl physics, the negative longitudinal magnetoresistance and the planar Hall effect, are observed in the elemental semiconductor tellurium. More strikingly, logarithmically periodic oscillations in both the magnetoresistance and Hall data are revealed beyond the quantum limit and found to share similar characteristics with those observed in ZrTe₅ and HfTe₅. The log-periodic oscillations originate from the formation of two-body quasi-bound states formed between Weyl fermions and opposite charge centers, the energies of which constitute a geometric series that matches the general feature of discrete scale invariance (DSI). Our discovery reveals the topological nature of tellurium and further confirms the universality of DSI in topological materials. Moreover, introduction of Weyl physics into semiconductors to develop “Weyl semiconductors” provides an ideal platform for manipulating fundamental Weyl fermionic behaviors and for designing future topological devices.

A bstract Both celestial and momentum space amplitudes in four dimensions are beset by divergences resulting from spacetime translation and sometimes scale invariance. In this paper we consider a (linearized) marginal deformation of the celestial CFT for Yang-Mills theory which preserves 2D conformal invariance but breaks both spacetime translation and scale invariance and involves a chirally coupled massive scalar. The resulting MHV celestial amplitudes are completely finite (apart from the usual soft and collinear divergences and isolated poles in the sum of the weights) and take the canonical CFT form. Moreover, we show they can be simply rewritten in terms of AdS 3 -Witten contact diagrams which evaluate to the well-known D -functions, thereby forging a direct connection between flat and AdS holography.

This study addresses the limitations of Transformer models in image feature extraction, particularly their lack of inductive bias for visual structures. Compared to Convolutional Neural Networks (CNNs), the Transformers are more sensitive to different hyperparameters of optimizers, which leads to a lack of stability and slow convergence. To tackle these challenges, we propose the Convolution-based Efficient Transformer Image Feature Extraction Network (CEFormer) as an enhancement of the Transformer architecture. Our model incorporates E-Attention, depthwise separable convolution, and dilated convolution to introduce crucial inductive biases, such as translation invariance, locality, and scale invariance, into the Transformer framework. Additionally, we implement a lightweight convolution module to process the input images, resulting in faster convergence and improved stability. This results in an efficient convolution combined Transformer image feature extraction network. Experimental results on the ImageNet1k Top-1 dataset demonstrate that the proposed network achieves better accuracy while maintaining high computational speed. It achieves up to 85.0% accuracy across various model sizes on image classification, outperforming various baseline models. When integrated into the Mask Region-Convolutional Neural Network (R-CNN) framework as a backbone network, CEFormer outperforms other models and achieves the highest mean Average Precision (mAP) scores. This research presents a significant advancement in Transformer-based image feature extraction, balancing performance and computational efficiency.