Reconstructing spectral densities from Euclidean lattice correlators requires an inverse Laplace transform, which is inherently ill-conditioned when applied to numerical data with statistical uncertainties. The maximum amount of information that can be extracted from the imaginary-time dependence of correlators can be characterized by the singular value decomposition (SVD) of the kernel function $\exp(-\omega t)$ defined on discrete sets of imaginary times $t$ and energies $\omega$. The SVD provides orthogonal basis functions in both the $t$- and $\omega$-spaces, while the singular values determine the magnitude of their contributions to the correlators. By retaining only the components associated with the largest singular values, for which the correlator data remain statistically significant, one can reconstruct smeared spectral functions with controlled uncertainties. The systematic error arising from the truncation can also be bounded under reasonable assumptions. In the limit where the ranges of $t$ and $\omega$ become infinitely large and continuous, the SVD basis approaches the Mellin transform, allowing a representation of the smeared spectrum that is independent of the details of the lattice parameters.

Wegner's $\mathbb{Z}_2$ gauge model is the earliest formulation of pure lattice gauge theory and predicts the topological nature of the confinement-deconfinement transition. In three dimensions ($D=3$), the equilibrium critical behavior of the model is understood in terms of geometrically defined objects, namely loop excitations and Fortuin-Kasteleyn (FK) clusters. This work investigates the critical quench dynamics of this model from a geometric perspective, following quenches from both a high-temperature percolation phase and the zero-temperature ground state. Using time-dependent finite-size scaling analysis, we find that the critical non-equilibrium relaxation of the percolation order parameter is governed by a dynamical exponent $z_{\rm p} \simeq 2.6$, consistent with that associated with the energy density, $z_{\rm c}$. Importantly, the value of $z_{\rm p}$ is robust with respect to the initial quench condition and the choice of geometrical objects. Furthermore, we provide a detailed characterization of the kinetics of different geometrical objects during the evolution from the percolation phase. Notably, we observe that the quench dynamics obeys dynamic scaling in terms of a growing lengthscale, $\xi_{\rm p}(t) \sim t^{1/z_{\rm p}}$, despite the absence of a local order parameter.

Monte Carlo methods are essential across diverse scientific fields, yet their efficiency is frequently hampered by critical slowing down-a sharp increase in autocorrelation times near phase transitions. Although deep learning approaches, such as neural-network-based samplers, have been proposed to alleviate this issue, they face another serious problem: the difficulty of training the models. This difficulty partially stems from the overly general nature of original machine-learning architectures, which often ignore underlying physical symmetries and force networks to relearn them from scratch. In this paper, we demonstrate that incorporating physical priors into the model significantly enhances performance. Building upon existing strategies that integrate spin-spin interactions, we propose a framework that utilizes a prior probability distribution as a starting point for training. Our results for the Ising model, as well as for the Edwards-Anderson spin glass model, suggest that moving away from `blank slate' models in favor of physics-informed priors reduces the training burden and facilitates the simulation of larger system sizes in discrete spin models.

We make rigorous the physics prediction that lattice Yang-Mills theories with gauge groups which have trivial centers do not satisfy Wilson's criterion for quark confinement. Specifically we prove that $\mathrm{SO}(3)$ lattice Yang-Mills theory does not satisfy Wilson's criterion in a strong coupling regime.

Ground-state energy and matrix element are reconstructed from correlators in lattice QCD by diagonalizing transfer matrix $\hat{T}$ within the Krylov subspace spanned by $\hat{T}^n|\chi\rangle$, where $|\chi\rangle$ is a state generated by an interpolating field on the lattice. In numerical applications, this strategy is spoiled by statistical noise. To circumvent the problem, we introduce a low-rank approximation based on a singular-value decomposition of a matrix made of the correlators. The associated bias is eliminated by an extrapolation to the limit of vanishing variance of energy eigenvalue. The strategy is tested using a set of mock data as well as real data of $K$ and $D_s$ meson correlators.

We present the first systematic study of the radiative decays of charmed mesons using $2+1$-flavor clover fermion gauge ensembles generated by the CLQCD collaboration. One of the ensembles is at the physical pion mass, and one has a fine lattice spacing $a\sim 0.05 ~\text{fm}$. We determine the coupling constants to be $g_{D^{\ast+} D^+ \gamma} = -0.204(22)$ GeV$^{-1}$, $g_{D^{\ast0} D^0 \gamma} = 1.73(37)$ GeV$^{-1}$, and $g_{D_s^{\ast+} D_s^+ \gamma} =-0.120(14)$ GeV$^{-1}$, respectively. Compared with previous studies, our results demonstrate significant improvements in precision. In particular, we carefully estimate the systematic uncertainty arising from matrix element fits, momentum transfer extrapolations, and chiral and continuum limit extrapolations, which are included in the reported total uncertainties. These couplings yield the following predictions of decay widths: $\Gamma_{D^{\ast+} \rightarrow D^+ \gamma} = 0.253(55)$ keV, $\Gamma_{D^{\ast0} \rightarrow D^0 \gamma} = 18.2(7.8)$ keV, and $\Gamma_{D_s^{\ast+}\rightarrow D_s^+ \gamma} = 0.094(22)$ keV. This work establishes first-principles results of the charmed meson radiative transitions and provides inputs for understanding the structure and properties of heavy-light mesons.