While Hamiltonian Lattice Gauge Theory is gaining traction, today's limited numerical capacity leaves simulations affected by discretization errors. This motivates the implementation of renormalization group (RG) techniques to find discretization-error-free operators. To this end, we introduce VAPOR, a variational quantum algorithm that decomposes operators into Pauli strings, identifies RG flow orbits, and determines fixed points of a naively discretized operator. We illustrate this using a toy model of a kinematic operator in a symmetry-restricted SU(2) Yang-Mills theory.

Euclidean lattice correlators determine spectral functions only through a smoothing integral transform, so a nominal uncertainty band on a reconstructed spectrum need not have a coverage interpretation for a physical summary. We formulate this as a target-wise adequacy test for reported spectral uncertainties. For a chosen summary \(T[\rho]\), the reported interval is tested on Euclidean-admissible mock correlators with known truth using empirical coverage, simulation-based calibration ranks, physical diagnostics, and stress tests. The test is conditional, but it is a useful falsification tool: passing it does not prove that a reconstruction is the QCD truth, while failing it shows that the reported uncertainty law is not adequate for the chosen functional under the stated mock extension. In a generic benchmark, peak locations are substantially better calibrated than peak heights or low-frequency weights, reflecting different degrees of functional identifiability under the Euclidean kernel. We then apply the same logic to a finite-temperature shear correlator. A family of BG-style reconstructions is compatible with the Euclidean data at \(\chi^2/N_\tau\simeq 1.3\). Within the scanned grid and stated observable-matched mock extension, a \(W_{\rm low}\)-calibrated representative can be identified, whereas pointwise peak-height intervals are not certified for the tested BG-style uncertainty law. Thus Euclidean compatibility is a necessary consistency check, but not a sufficient adequacy criterion for spectral uncertainty claims.

A central issue in lattice formulations of chiral gauge theories is how the anomaly cancellation mechanism of the continuum theory can be realized at finite lattice spacing. In the present paper, based on non-Abelian bosonization, we propose a lattice formulation of the bosonic theory corresponding to a two-dimensional non-Abelian chiral gauge theory. In the continuum theory, the gauge anomaly of chiral fermions is represented, in the bosonized description, as anomaly inflow from a three-dimensional Chern--Simons-type bulk contribution contained in a gauged Wess--Zumino--Witten model. Motivated by this structure, we introduce gauge-neutral spectator fermions and use the resulting bosonized description. We then construct a lattice counterpart of the gauged Wess--Zumino--Witten model with a three-dimensional bulk extension under appropriate smoothness conditions. A salient feature of this lattice formulation is the cancellation of the left and right bulk contributions in the exponentiated action. This cancellation occurs even before taking the continuum limit when the anomaly-free condition is satisfied, namely when the left and right representations have identical quadratic indices. Thus, the present construction realizes the anomaly-cancellation mechanism at finite lattice spacing via the bosonized description of two-dimensional anomaly-free chiral gauge theories. Establishing the desired continuum limit remains an important open problem.

These lecture notes are intended as a coherent introduction to conformal field theory in general, and composite operators in particular, through a semiclassical framework for computing scaling dimensions, with emphasis on operators of the form $\phi^n$. In doing so, they aim to fill a gap in the literature and to help decode some of the relevant concepts. The physical idea is that at large $n$ an (heavy) operator creates a highly occupied state. Through the state-operator correspondence, this state lives on the cylinder $\mathbb{R}\times S^{d-1}$, and its scaling dimension is the corresponding energy of the theory on the cylinder. The notes are organized as a self-contained route from conformal symmetry to semiclassical dynamics. Part I reviews the conformal group, primary operators, radial quantization, the state-operator correspondence, and operator mixing. Part II builds the semiclassical framework, first in the free scalar theory, where the dimension of $\phi^n$ is recovered in three independent ways, and then through the double-scaling limit, the action variable, and Bohr-Sommerfeld quantization. Part III develops the general machinery of periodic saddles, Floquet theory, fluctuation determinants, the Gel'fand-Yaglom method, and the Gutzwiller trace formula. Part IV applies the framework to the $O(N)$ $\phi^4$ theory in $d=4-\epsilon$ at the Wilson-Fisher fixed point, deriving the classical elliptic solution, the Lamé fluctuation spectrum, the zero modes, and the one-loop contribution to the large-$n$ scaling dimensions. Beyond the explicit computation, the notes emphasize the role of composite operators as probes of collective sectors of quantum field theory, with extensions to gauge theories, conformal windows, and asymptotically safe field theories.

We investigate the phase structure of the single-flavor Gross--Neveu model with Wilson fermions using the Grassmann corner transfer matrix renormalization group (CTMRG). The path integral is formulated as a two-dimensional Grassmann tensor network and approximately contracted by the Grassmann CTMRG algorithm. We investigate the phase diagram by varying the fermion mass and the four-fermion coupling, using the pseudoscalar condensate as an order parameter for the $\mathbb{Z}_{2}$ parity symmetry breaking phase. The universality classes of the phase boundaries are identified through the central charge $c$ obtained via scaling analysis of the entanglement entropy. Furthermore, we extract the quantity related to the entanglement spectrum from the converged CTMRG environments, allowing us to distinguish the topological insulator phase and the trivial phase. The resulting phase structure suggests that the Aoki phase is separated from the other phases by critical lines characterized by $c=1/2$, while the critical lines with $c=1$ separate the topological insulating and trivial phases. Our numerical results also indicate that the Aoki phase does not persist in the strong-coupling regime for the single-flavor theory.

A key challenge in normalizing flows is finding expressive invertible scalar bijections. Existing approaches face trade-offs: affine transformations are smooth and analytically invertible but lack expressivity; monotonic splines offer local control but are only piecewise smooth and act on bounded domains; residual flows achieve smoothness but need numerical inversion. We introduce three families of analytic bijections that are globally smooth ($C^\infty$), defined on all of $\mathbb{R}$, and analytically invertible in closed form, combining the favorable properties of prior approaches. Beyond serving as drop-in replacements in coupling flows, where they match or exceed spline performance, we develop radial flows: a novel architecture using direct parametrization that transforms the radial coordinate while preserving angular direction. Radial flows exhibit exceptional training stability, produce geometrically interpretable transformations, and on targets with radial structure can achieve comparable quality to coupling flows with $1000\times$ fewer parameters. We provide comprehensive evaluation on 1D and 2D benchmarks, and demonstrate applicability to higher-dimensional physics problems through experiments on $\phi^4$ lattice field theory, where our bijections outperform affine baselines and enable problem-specific designs that address mode collapse.