We evaluate the electrostatic self-energy of a uniformly charged regular polygon, treated as a planar charge distribution embedded in three-dimensional space and interacting through the Coulomb potential. By exploiting the piecewise-linear nature of the boundary, the four-dimensional Coulomb integral can be reduced exactly to a finite sum of elementary sector contributions that is valid for any number of sides. This construction leads to closed-form expressions for the uniformly charged triangle, square, pentagon, and hexagon. The pentagon result, although involving a quartic number field, is remarkably compact. As the number of sides increases, the energy decreases monotonically toward the corresponding value for a uniformly charged disk. A boundary-perturbation analysis gives the leading relative correction, which decreases as the inverse square of the number of sides. The second variation of the energy produces a nonanalytic correction proportional to the inverse fourth power of the number of sides multiplied by its logarithm. This term provides a direct signature of the polygonal corners. The arithmetic structure of the exact results also explains why compact closed forms cease to be available beyond the uniformly charged hexagon. All results are validated by high-precision numerical calculations and provide useful benchmarks for studies of confined two-dimensional Coulomb systems.

Finite-temperature topological diagnostics in quasiperiodic systems are subtle because quasiperiodicity removes the Bloch momentum torus on which mixed-state band indices are usually formulated. We study a quasiperiodic Qi-Wu-Zhang Chern insulator on finite Fibonacci approximants and use twisted boundary phases as an operational two-torus. A gap scan over $L=21,34,55$ shows that the fixed-boundary spectral gap can remain finite while extended low-gap regions appear elsewhere on the twist torus. Projected-Uhlmann diagnostics reveal a sharper finite-size result: The stability of the projected winding is not determined by the scalar minimum twist gap alone. A high-resolution $L=34$, $25\times 41$ audit gives operational stability scales $5\times {10}^{-4}$, ${10}^{-2}$, and $3\times {10}^{-2}$ for three cases whose minimum twist gaps are all of order ${10}^{-3}$. The same ranking is obtained with projection bases on the ${\theta }_x=0$ and exact ${\theta }_x=\pi$ slices. Direct phase-curve diagnostics show that collapse is associated with the loss or singularization of a $2\pi$ projected-Uhlmann phase winding, whereas stable cases preserve the winding despite small spectral gaps. Within the finite approximants studied here, these results identify phason-resolved boundary-phase geometry and projected-holonomy sensitivity, rather than the fixed-boundary gap or scalar minimum gap alone, as the mechanism controlling this finite-size projected diagnostic.

Spontaneous pattern formation at unstable fluid-fluid interfaces is a popular feature of nonequilibrium systems, often arising from the interplay between interfacial tension and hydrodynamic instabilities. One of the most prominent examples is viscous fingering in quasi-two-dimensional Hele-Shaw geometries, where the displacement of a more viscous fluid by a less viscous one produces highly branched, dendritic morphologies. In this work, we report the formation of nearly dendritic patterns in Langmuir Reverse Schaefer ($\mathcal{L}$-RSh) films of the nematic liquid crystal $N$-(4-methoxybenzylidene)-4-butylaniline (MBBA) upon incubation with silver nanocolloids (AgNcs) of average diameter $\sim 30$ nm, whereas linear stripelike structures are observed for larger nanoparticles ($\sim 64$ nm) on the same $\mathcal{L}$-RSh film of MBBA molecules. The origin of this size-dependent morphological transition is interpreted in terms of topological defects generated due to the incorporation of metal nanoparticles within the continuum $\mathcal{L}$-RSh liquid crystal matrix. A comprehensive statistical characterization of the patterned substrate has been carried out through parameters such as lateral correlation length (ξ), fractal dimension ($D$), Hurst exponent (β), and interface width (ω). Further insights into the complexity of the patterns reveal their multifractal nature, evaluated using two-dimensional multifractal detrended fluctuation analysis. To the best of our knowledge, this study appears to be the first report where thorough investigations on the surface properties of the as-prepared patterned substrate have been discussed in detail. Finally, the patterned substrate demonstrates the good performance as surface-enhanced Raman scattering platforms, enabling detection of rhodamine 6G and 1,10-phenanthroline molecules at extremely low concentrations.

For classical field theories with probabilistic initial conditions, the classical field observables are an idealization. Their arbitrarily precise values poorly reflect the characteristic uncertainty in the presence of substantial fluctuations. We propose to employ observables based on fluctuating fields. In terms of these “statistical observables,” the probabilistic classical field theory becomes a quantum field theory. Noncommuting operators are associated with observables. The quantum rules follow from the laws for classical probabilities. The “quantum” is part of the “classical.” A regularized functional integral guarantees the unitarity of the quantum field theory. We discuss in detail the classical relativistic Klein-Gordon equation with interactions.

Physics-informed neural networks and neural quantum states have consolidated a new paradigm to analyze and discover physical phenomena through constrained neural parametrizations. In this context, we investigate whether the semiclassical structure of the eigenfunctions of a quantum chaotic system can be unveiled through unsupervised learning. To this end, we train a “quantum dictionary,” formulated as an overcomplete autoencoder, that sparsely represents the eigenstates of the system, using as an illustration the quantum baker map. The only explicit physical information imposed on the dictionary atoms is their localization in phase space, without providing any kind of information about the periodic orbits of the corresponding classical system. The model achieves high fidelity in reconstructing held-out validation eigenstates not used in the gradient optimization of the dictionary. By comparing the learned atoms with an independently constructed “semiclassical dictionary,” we find that many of them develop scarlike Husimi-density concentrations around short periodic orbits. This result is interesting in two ways: Phase-space localization, combined with sparse reconstruction and dictionary diversity, can recover nontrivial semiclassical organization from spectral data and at the same time periodic orbits confirm their fundamental role in the structure of eigenfunctions of quantum systems with chaotic classical dynamics. More generally, our proposed architecture opens a route to learning representations whose atoms optimize other chosen physical properties.

IceCube observes a break near 30 TeV in the diffuse astrophysical neutrino spectrum. We interpret it as the photomeson ($p\gamma$) threshold. Accelerated protons interact with the soft x rays of accreting black holes through the $\mathrm{\Delta }\left(1232\right)$ resonance. One kinematic relation sets the break energy from the target field’s lowest photon energy, the seed scale ${\varepsilon }_0$. The measured break fixes ${\varepsilon }_0\simeq 0.13\mathrm{keV}$ and supplies the low-energy suppression no sum of falling power laws can produce. Building on the $\mathrm{\Delta }\left(1232\right)$ identification of Khatee Zathul, Fang, Halzen, and Hooper and the single-source-class photo-pion fit of Winter, Fiorillo, and Buson, we give closed-form relations that invert the four broken-power-law fit parameters to the source physics with no remaining freedom, namely, ${\gamma }_2=s$ and ${\gamma }_1=s-({\alpha }_x-1)$ ($s, {\alpha }_x$ the proton and target photon indices), and $E_{\mathrm{br}}\propto {\left[(1+\overline{z}){\varepsilon }_0\right]}^{-1}$. The break width then measures the population’s uniformity. The total smearing is bounded to ${\sigma }_s\le 0.235\mathrm{dex}$ (68% confidence level), no broader than redshift alone must make it, and the seed scale is the same across the sources to within a factor $\lesssim 1.5$, a constraint mean-spectrum fits do not provide. The model also predicts that below-break neutrinos come on average from higher redshifts than those above. The threshold also fixes the flavor and neutrino-antineutrino composition, including the energy-dependent ${\overline{\nu }}_e$ fraction and the Glashow-resonance discriminator computed in detail by Winter et al., matching current flavor data. Inverted source by source, the relation gives each resolved IceCube source its target photon scale, from the keV coronae of NGC 1068 and PKS $1424+240$ to the $\sim 20\mathrm{eV}$ jet of TXS $0506+056$. That two-decade spread means they cannot supply the homogeneous break. It requires a separate, fainter population. KM3-230213A points to a warm-dust target, ${\varepsilon }_0\simeq 0.02\mathrm{eV}$, unless cosmogenic.

We analyze a class of one-dimensional quantum systems characterized by a position-dependent kinetic term arising as the continuum limit of an inhomogeneous tight-binding model with spatially varying hopping amplitudes. In this limit, the Schrödinger equation takes the so-called BenDaniel-Duke form with an effective mass, scaling as $m_{\mathrm{eff}}\left(x\right)=m_{\mathrm{eff}}{\left|x\right|}^{\alpha }$, with $\alpha >0$, leading to a framework we term “sluggish quantum mechanics,” where particle motion is progressively suppressed at larger distances. For the case with no external confinement, we obtain the eigenfunctions and the quantum propagator exactly. In the presence of a confining potential $V_{\mathrm{ext}}\left(x\right)=\frac{1}{2}{m}_{\mathrm{eff}}\left(x\right){\omega }^2{x}^2=\frac{1}{2}{m}_{\mathrm{eff}}{\omega }^2{\left|x\right|}^{\alpha +2}$, we show that the model remains exactly solvable and obtain the spectral properties and the propagator analytically. We then investigate the problem of $N$ noninteracting spinless fermions in the trap, determining the many-body ground-state wavefunction and the joint probability density function of the positions of the $N$ fermions. We show that the many-body quantum probability density in the ground state forms a determinantal point process whose correlation kernel can be computed for any $N$, giving access to the average density as well as higher-order correlation functions for any finite $N$. Moreover, we analyze the scaling form of this kernel in the large-$N$ limit in the bulk, near the edge, and close to the origin. Our results show that the scaled average density profile for large $N$ has a finite support symmetric with respect to the origin, but has a nonmonotonic shape with a vanishing minimum at the origin for any $\alpha >0$. One of the key findings of our work is that the scaled kernel near the origin $x=0$ for $\alpha >0$ is neither the Bessel nor the Airy kernel (that are standard for trapped fermions), but is new, and is given by a sum of two Bessel kernels with different indices. Our results thus provide a framework relevant to engineered optical lattices with position-dependent tunneling.

We study a model system consisting of a tunneling barrier driven by an external harmonic field and coupled to two leads with finite bandwidth. Avoiding Floquet expansions, we derive simple expressions for the time-dependent tunneling current in the adiabatic regime. Our approach relates the barrier modulation to a measurable time delay in the steady-state periodic current. It provides a physically consistent definition of the tunneling time inside the barrier by subtracting the time delay associated with the leads from the total time delay. We find that the tunneling time always vanishes for wide/high barriers. Remarkably, the time delay persists even when the barrier becomes static, i.e., in the limit where the modulation frequency vanishes. This indicates that the time delay obtained through the introduction of an external periodic perturbation actually reflects an intrinsic property of the tunneling dynamics, rather than an effect of the external drive or of a particular system. We apply our results to the analysis of tunneling times in optical experiments and find good agreement with the experimental data.

Comparison of the measured spectral emission properties of single-crystal Cu(001) and W(111) photocathodes to established photoemission theories reveals evidence for an additional one-photon emission process predominantly affecting electron emission near and below the photoemission threshold. This additional photoemission process is postulated to be due to a momentum-resonant Franck-Condon mechanism mediated by inelastic Umklapp electron scattering. An initial first-principles simulation of this emission process (involving the electron thermal effective mass, the inelastic electron mean free path at the vacuum level, and the number of Fermi surfaces in the metal), when combined with a direct one-step band emission model, is consistent with the measured spectral dependencies of both the quantum efficiency and mean transverse energy of electron photoemission from the two single-crystal metal photocathodes.

Models of unification predict additional $Z^{\prime }$ boson, which contributes to parity nonconservation (PNC) in atoms. If $Z^{\prime }$ boson is light, ratio of $Z^{\prime }$-boson contribution to the standard model $Z$-boson contribution to atomic PNC increases with decreasing nuclear charge $Z$ faster than $1/Z^2$. This motivated our previous study of PNC in hydrogen and deuterium, where $s\text{−}p$ wave mixing induced by the weak interaction was considered. An even more pronounced relative enhancement occurs when higher-angular-momentum states are involved, such as in the $p\text{−}d$ weak matrix elements. A light boson mediates a long-range interaction, in contrast to the effectively contactlike interaction mediated by the standard model $Z$ boson. Consequently, higher-angular-momentum states provide a background-free system: Their suppressed overlap with the nucleus strongly reduces the standard model contribution while leaving the long-range $Z^{\prime }$ contribution unsuppressed.

Nonlinear oscillators are ubiquitous in physics and engineering, yet finding explicit analytical expressions for their amplitude-dependent frequency remains a fundamental challenge. The Duffing equation is the paradigmatic model for this class of problems, and despite decades of progress through perturbative, variational, and homotopic methods, general closed-form frequency relations are unavailable for most of its configurations. In this work, we investigate whether symbolic regression (SR)—a machine-learning technique that searches for interpretable mathematical expressions directly from data—can recover the closed-form amplitude-frequency relations produced by classical methods, and approximate them where no closed form is available. We generate reference datasets using four classical approaches: the Lindstedt-Poincaré method, first-harmonic balance, multiharmonic balance, and numerical integration via a Runge-Kutta 4–5 scheme. We stress at the outset that, for the analytical and semianalytical methods, the training labels are obtained by evaluating the corresponding closed-form expressions; recovering those expressions therefore constitutes symbolic rediscovery of a known formula from sampled data, rather than discovery of new physics. The only genuinely dynamics-based case is the quintic oscillator labeled with Runge-Kutta integration followed by spectral analysis. SR, implemented through PySR, is then trained on these datasets for six configurations spanning cubic, quartic, and quintic nonlinearities under both free and forced conditions. In all free-oscillator datasets the oscillation amplitude is fixed (normalized to $A=1$), so the learned relations are parameter-frequency relations at fixed amplitude. For free oscillators with known analytical solutions, SR reproduces the reference expressions to machine-level numerical precision ($R^2=1.0000$, mean squared error (MSE) $\sim {10}^{-17}$), returning the Lindstedt-Poincaré, harmonic-balance, and first-harmonic-balance expressions used to generate the labels. For forced oscillators, where the response amplitude is defined implicitly as the root of a cubic (or quintic) polynomial in $A^2$, SR captures the dominant resonance structure but cannot fully resolve the implicit-root behavior with the present algebraic operator set ($0.53\le R^2\le 0.83$). These results position SR as a practical complement to classical analytical approaches to nonlinear dynamics, with particular value in contexts where specialized perturbative methods are inaccessible or their validity is restricted to narrow parameter regimes. We are explicit about the limits of the present scope: amplitude dependence, robustness to noise, extrapolation beyond the training box, and run-to-run variability of the stochastic search are identified as the validation steps required before any claim of genuine discovery can be made.

In discrete models describing growing rough interfaces in the Kardar-Parisi-Zhang universality class, we examine height fluctuations at a fixed site as a function of time, measured in monolayer units. For small systems, we show that it is possible to reach the stationary state. We compute the two-time autocorrelation and power spectra independently. The correlation function remains nonexponential and vanishes after a correlation time that diverges with system size. As a result, the power spectra display a lower cutoff with constant power. In the nontrivial frequency regime, we observe $1/f^{\gamma }$-type scaling with the spectral exponent 5/3. Finite-size scaling reveals that the temporal correlation function follows dynamic scaling. Our findings, supported by scaling-theoretical arguments, establish that the fluctuations are wide-sense stationary, implying the applicability of the Wiener-Khinchin theorem. The scaling arguments also reveal a scaling relation $\gamma =1+2\alpha /z$, where $\alpha$ and $z$ denote the roughness and dynamical exponents.

Density cube theory extends the canonical quantum density matrix ${\rho }_{ij}$ with the addition of an extra index to ${\rho }_{ijk}$. The elements of the density cube with two different indices, ${\rho }_{iij}$ and ${\rho }_{ijj}$, correspond to the real and imaginary parts of the off-diagonal element ${\rho }_{ij}$ of the density matrix and describe double-path interference, while those with three different indices describe noncanonical triple-path interference. In this Letter, we propose an equation of motion for the density cube, obtained from the quantization of ternary Nambu dynamics, and find that pairs of triple-path interferences oscillate into each other.

The Casimir-Polder interaction energy between a unipolarizable point atom and a unipolarizable dielectric ring has been limited, until now, to the case when the atom is confined on the axis of symmetry of the ring. We find the generalized analytical expression for any position of the atom relative to the ring in terms of complete elliptic integrals. This is aided by the construction of a class of integrals of a Jacobian elliptic function as a linear combination of complete elliptic integrals. Our expression for the interaction energy allows us to investigate the instability of the atom even for the equilibrium points that exist off the axis of symmetry.

We find the ground-state energy of the Ising model using the cascaded variational quantum eigensolver algorithm with the guided-sampling ansatz using up to 63 qubits on a quantum computer. We study a heavy-hex lattice to match the qubit architecture, allowing us to perform calculations on a scale that is inaccessible to exact diagonalization. We study both a homogeneous and a random-coupling model. We locate the boundary of acceptable quantum errors as a function of the number of qubits and coupling strength. An entropic analysis is performed, giving insights into the quantum computing performance. A subspace analysis is performed, which suggests that the Ising model is especially suited for near-term quantum computing.

We investigate dissipative charging of a single particle in finite tight-binding lattices coupled to engineered nonequilibrium Markovian reservoirs. The engineered number-conserving bond jumps convert local antisymmetric bond modes into symmetric modes, thereby steering the system toward upper spectral-edge states with finite steady-state ergotropy. In an even periodic chain, this mechanism admits a simple bond-mode interpretation and gives ${\mathcal{E}}_{\mathrm{ss}}=4t$ in the clean limit, where $t$ denotes the uniform nearest-neighbor hopping amplitude. We further show that the same dissipative charging principle applies to a two-dimensional honeycomb tight-binding lattice. For on-site Anderson disorder, disorder-averaged calculations show that the mixed steady state becomes more spatially concentrated, while the final ergotropy decreases and the charging transient shortens over the examined parameter range. Additional local dephasing produces a similar finite-time trade-off in the displayed rate window. The finite-time metric $P_{0.99}$, defined by $P_{0.99}=0.99{\mathcal{E}}_{\mathrm{ss}}/{\tau }_{0.99}$, with ${\tau }_{0.99}$ the first time at which the ergotropy reaches $99\%$ of its steady-state value, is therefore interpreted as a battery-side charging measure rather than the net thermodynamic power or efficiency of the full reservoir-engineering apparatus.