We present a variational approach to quantum field theory based on wavefunctions parameterized by neural networks. While variational methods have a celebrated history across many fields, their application to quantum field theory has been limited by well-known challenges. We show that neural-network wavefunctions, combined with modern machine-learning techniques, enable competitive variational calculations in nontrivial field theories. As a demonstration, we reproduce the essential features of the two-dimensional nonlinear $\sigma$-model: asymptotic freedom, dynamical mass generation and the model's step-scaling function.

We develop two implementations of the Event-Chain Monte Carlo (ECMC) algorithm for Yang-Mills $\mathrm{SU}(N)$ lattice gauge theories with the Wilson action. These algorithms consist in a succession of local ballistic updates intersped with stochastic events, resulting in an irreversible and rejection-free Markov process. The resulting dynamics satisfy global balance, ensuring the correct equilibrium distribution. The algorithms are formulated for general $\mathrm{SU}(N)$ Yang-Mills theories with Wilson action and implemented for the case $N=3$. Numerical tests on four-dimensional lattices show that standard gauge observables, such as the mean plaquette, agree with results obtained using conventional Monte Carlo algorithms. These results provide a first validation of ECMC as a viable sampling scheme for Yang-Mills lattice gauge theories.

A practical Hamiltonian approach to lattice gauge theories would provide access to several important areas of phenomenology that have been beyond the reach of conventional lattice methods. Quantum computers seem to be a natural platform for this approach. With near-term quantum computers in mind, our work considers a three-dimensional spatial lattice that can host fermions and non-Abelian gauge fields while needing fewer qubits than a simple cubic lattice. Specifically, the semi-simple cubic (ssc) lattice is obtained by removing half of the gauge links from a standard cubic lattice in such a way that every vertex becomes trivalent, which streamlines the handling of Gauss's law. The ssc lattice is topologically equivalent to the triamond lattice but, because the gauge links at each vertex span all three directions, the ssc lattice can accommodate a local fermion derivative. The case of staggered fermions with SU(2) gauge fields is presented here.

The proton's unpolarized $u(x)-d(x)$ parton distribution function (PDF) has been studied by a number of lattice QCD groups through large momentum expansion. However, due to lattice artifacts (excited state contaminations, unphysical pion masses, and discretization effects) and less-advanced theoretical analysis (renormalizations, large-distance extrapolations, and large-log resummations), the resulting PDFs cannot be compared strictly with experimental data. By using the state-of-the-art theoretical tools and mitigating the lattice artifacts empirically, we reanalyze the available datasets in the literature and find that the new PDF in the physical limits is consistent with global fittings within $\sim1\sigma$. This provides compelling evidence that large momentum expansion is capable of accurately predicting the $x$-dependence of the PDFs when ideal lattice data become available.

We investigate multi-particle states of the (1+1)d Ising Model using a spectroscopy scheme based on transfer matrix and tensor renormalization group method. The scheme begins with computing the energy spectrum of the system from the transfer matrix estimated by the coarse-grained tensor network. The quantum number and momentum of these energy eigenstates are not a priori known, thus we identify them using matrix elements of an interpolating operator that is numerically computed with an impurity tensor network. Furthermore, by observing the dependence of the energy as a function of system size, we identify the number of particles of the eigenstates and obtain one-, two-, and three-particle states for a specific quantum number and momentum. From the two-particle state sector, we compute the scattering phase shift using Lüscher's formula and wave function approach, and observe their consistency with theoretical prediction. Using the information of the two-particle scattering phase shift, we investigate the degeneracy of the two-particle states, the theoretical prediction of the three-particle finite volume energy and also the degeneracy in the three-particle states.

We use numerical minimization of the lattice action of trace-deformed Yang-Mills theory on $T^4$ with twisted boundary conditions to find the classical minimum action configurations of fractional topological charge. We vary the twists and ratios of torus periods to interpolate between different $R^{4-k} \times T^k$ geometries. This allows us to see how the corresponding minimum action saddle point configurations -- monopole-instantons ($k=1$), center vortices ($k=2$), and fractional instantons ($k=3,4$) -- morph into each other. We also study how the transition between them depends on the presence of a deformation potential. In particular, we argue that the recent analytic picture of chains of monopole-instantons collimating their flux into center-vortex sheets, while technically relying on the deformation potential, also holds in pure Yang-Mills theory, for tori whose shape causes the abelianization due to the deformation to align with the one due to the twists. Our results also indicate that with nonzero deformation potential, some transitions between different minimal-action fractional charge configurations may be discontinuous and involve level crossing.

We present a quantum algorithm for simulating Generalized Parton Distributions (GPDs) in the Schwinger model. Unlike the staggered fermions widely utilized in current quantum simulations, we employ Wilson fermions for lattice discretization. This choice is critical for the quantum computation of GPDs due to their strict preservation of charge conjugation symmetry. We construct a comprehensive algorithmic framework that includes the preparation of hadronic states with non-zero momentum and the measurement of light-cone correlation functions incorporating Wilson lines. We provide a complexity analysis, demonstrating that the resources required for our algorithm scale polynomially with both the number of qubits and the desired precision $\varepsilon$. Finally, we benchmark our approach using exact diagonalization, extracting mass spectra and GPDs (also parton distribution functions) that are consistent with theoretical expectations and fundamental physical constraints.

We present a combined angular analysis within and beyond the Standard Model (SM) of experimental measurements for the $B \to D^{*}\ell \bar{\nu}_{\ell}$ angular coefficients provided by the Belle collaboration, together with lattice-calculated hadronic form-factor data from the HPQCD, JLQCD, and FNAL/MILC collaborations. We focus on determining the CKM matrix element $|V_{cb}|$ and constraining a set of Wilson coefficients associated with new physics (NP) mediated by scalar and tensor currents. SM predictions for the angular coefficients are obtained using form-factor parameterisations based on the Boyd-Grinstein-Lebed (BGL) ansatz, with unitarity constraints imposed as Bayesian priors. Experimental and theoretical data are analysed jointly by considering the cases $\ell = e, \mu$ separately and comparing with the massless approximation. For the latter, we determine $|V_{cb}| = 0.03997(71)$, with no resolution of the exclusive-inclusive puzzle. Using the full expressions for the angular coefficients in the presence of scalar, vector, and tensor currents, the corresponding Wilson coefficients are constrained through a joint Bayesian fit to lattice and experimental data. By including the renormalisation group evolution of the Wilson coefficients in the SM effective field theory (SMEFT), these constraints translate into bounds on the effective scale of potential heavy NP at the TeV scale. We find, at the $68\%$ confidence level, that NP mediated by a scalar leptoquark and a vector leptoquark/colourless scalar boson are excluded at the effective scales 1.0 and 2.5 TeV, respectively.

Andreev reflection converts an incoming fermion into an outgoing hole and is usually tied to a superconducting interface. We show that an analogous charge-conjugating boundary condition emerges from a purely lattice duality defect. Starting from a Majorana representation of the transverse-field Ising chain, we construct a folded lattice model in which a boundary Majorana impurity implements a one-site translation of a staggered Majorana chain. In the continuum, this translation becomes a chiral fermion-parity defect: it flips the sign of the only left-moving Majorana mode while leaving the right-moving mode unchanged. When the two Majorana modes are recombined into a complex fermion in the folded geometry, this sign flip becomes the Andreev-like boundary condition. Our lattice formulation gives a microscopic interpretation of the Emery--Kivelson boundary of the two-channel Kondo problem and of Maldacena--Ludwig monopole scattering, while identifying the boundary as the interface between a Kitaev-chain SPT phase and a gapless chain. The same Majorana translation defect also provides a lattice realization of an axial $U(1)_A$-symmetric charge-flip boundary.

We study the color correlation between two static quarks in 3Q ($QQQ$) and 4Q ($QQ\bar Q\bar Q$) multiquark systems at $T=0$ based on the reduced two-body density matrices $\rho$ in color space. We perform quenched lattice QCD calculations with the Coulomb gauge adopting the standard Wilson gauge action, and the spatial volume is $L^3 = 32^3$ at $\beta = 5.8$, which corresponds to the lattice spacing $a=0.14$ fm and the system volume $L^3=4.5^3$ fm$^3$. We evaluate the two-body color density matrix $\rho$ of static quarks, and investigate the dependence of color correlations on the quarks' spatial configuration. As a result, we find that the color correlations depend on the minimal path length along a flux tube which connects two quarks under consideration. The color correlation between quarks quenches because of color leak into the gluon field (flux tube) and finally approaches the random color configuration in the large distance limit. We also find a ``universality'' in the flux-tube path length dependence of the color correlation for 2Q, 3Q, and 4Q ground-state systems. Our results show that the color correlations of end-point quarks can be a clue to clarify the internal structures of hadrons, including exotic (multiquark) hadrons.

The reliable treatment of highly boosted hadrons is crucial for many lattice QCD applications. For all these cases the kinematically enhanced interpolators promise very significant improvements and are, therefore, ever more often used in recent calculations, especially for highly boosted mesons like the pion. Motivated by, e.g., the physics program of the future Electron-Ion Collider (EIC) in the US and Electron-Ion collider in China (EIcC), we systematically benchmark our code for the unpolarized isovector nucleon quark matrix elements extracted at large source-sink separations, where excited state artifacts are significantly suppressed. We find that the precision of the renormalized nucleon matrix elements is typically improved by an order of magnitude at momentum $P_z\sim2.5$ GeV. By comparing the results from three CLS ensembles with different lattice spacings $a$ but the same pion mass, we observe no statistically significant dependence on $a$ in the renormalized matrix elements at nearly identical values of $P_z$. These encouraging results suggest that the use of kinematically improved operators is highly advantageous for parton physics calculations and can be extended to a broader class of baryon observables, making them a promising candidate for a standard component of modern lattice QCD.

A recent non-perturbative formulation of $2n$ dimensional chiral gauge theories relies on realizing chiral fermions on the $2n$ dimensional boundary of a $2n+1$ dimensional disk manifold. It also requires extending boundary gauge configurations into the interior of the disk using some flow prescription that preserves 2n dimensional gauge invariance. In this paper we propose a concrete realization of the equation of motion flow with the disk embedded on a square lattice. In addition, we couple the flow gauge field to fermions and demonstrate the mechanism of anomaly inflow and anomaly cancellation at work on the lattice.

We consider a four dimensional $SU(3)$ lattice gauge theory with 12 staggered fermions having identical masses and an unimproved action. Using sets of plaquette distributions for various inverse bare couplings $\beta$, we reconstruct the density of states with the Ferrenberg -Swendsen method and calculate the zeros of the partition in the complex $\beta$ plane with bare quark masses $m_q$ = 0.02, 0.06, 0.08 and 0.1 for hypercubes of linear size $L$= 4, 6, 8, 10, and 12. Our hypothesis is that there is a line of first order transitions in the $(m_q,\beta)$ plane ending at a second order phase transition. We expect this transition to be in the 4D Ising, mean field, universality class. We fit the $L$ dependence of the zeros with the lowest imaginary part using two ($y = bL^{-d}$) and three ($y = a + bL^{-d}$) parameter fits. For $m_q$ = 0.02 the results provide strong support for a first order phase transition ($d=3.98(6)$, and $a$ statistically compatible with 0). The results also indicate, with less statistical significance for $m_q=0.06$, that the three other masses are above the critical value $m_q^c$. In addition, we suggest that the infinite volume gap for the lowest zero $a$, can be represented as $a\simeq A(m_q-m_q^c)^{B}$ with $m_q^c\sim 0.05$ and $B\sim 1$. Given that there are only three data points with significant error bars, it is difficult to rule out the mean field value $B=3/2$. Combining this result with spectroscopic results by Jin and Mawhinney, indicates that the gap with real axis (Lee-Yang edge) scales roughly like $m_\sigma ^2$, where $m_\sigma $ is the mass of the $0^{++}$ scalar which is also the lowest excitation.

We perform the first simultaneous global QCD analysis of pion and kaon parton distribution functions (PDFs), constrained by pion- and kaon-induced Drell-Yan (DY) and leading neutron electroproduction data, together with lattice QCD data on pion and kaon PDF moments. The analysis indicates a softer valence $\bar u$ distribution in the $K^-$ than in the $\pi^-$, and a significantly more peaked valence $s$-quark density in $K^-$ compared with the $\bar u$. The effective exponent governing the high-$x$ behavior of the PDF is found to be larger for $\bar u$ in the kaon, $\beta_{\bar u}^{K^-}\!= 1.6(2)$, than in the pion, $\beta_{\bar u}^{\pi^-}\!= 1.16(4)$, in the range $0.7 \leq x \leq 0.95$. From the gluon momentum fractions we find the pion's gluon content accounts for $\approx 1/3$ of the mass budget of the pion at $\mu=2~{\rm GeV}$, but only $\approx 1/4$ for the kaon.

We investigate the gluon gravitational form factors of the $\pi$, $N$, $\rho$, and $\Delta$ using lattice QCD data at $m_\pi \approx 450 \text{MeV}$ and $m_\pi \approx 170 \text{MeV}$. We base the analysis on fits to a simple $\sigma/f_0(500)$-meson pole, supplemented by a polynomial background term. The fitted residues agree with predictions from dilaton effective theory, in which the $\sigma$-meson acts as the dilaton, the pseudo Goldstone boson of spontaneously broken scale symmetry. We derive new dilaton-based predictions for the $\rho$- and $\Delta$-gravitational form factors, and comment on the $\eta_{c}$- and $\eta_b$-form factors in the context of the dilaton interpretation. These results reinforce our earlier findings, based on lattice total (quark and gluon) gravitational form factors, and provide further evidence that QCD dynamics may be governed by an infrared fixed point.

We study QED$_2$ in de Sitter space as a minimal interacting gauge theory in which cosmological expansion directly competes with quantum dynamics. In cosmic time, the hopping redshifts as $1/a(t)$ while the electric term grows as $g^2 a(t)$, sweeping the spectrum through a moving narrow-gap region in the $(\tau,m)$ plane. Exact diagonalization shows that this defines a pseudo-critical line governing the loss of adiabaticity, excitation growth, and redshifted response. Using matrix-product states at a fixed mass, we separate the fixed-cutoff thermodynamic limit from the continuum extrapolation. The late-time dip survives in the infinite physical box size limit, and shifts to later $\tau$ as the lattice spacing goes to zero, with current data favoring $\tau_* \approx 3.1$, while the dip depth remains less controlled. For Gibbs initial states, the same mechanism produces an irreversibility front in the relative entropy that tracks the pseudo-critical line and is detectable via LOCC-accessible observables. These results identify de Sitter QED$_2$ as a controlled setting for linking curved-space gauge dynamics, near-critical spectral structure, and operational irreversibility.

We propose the gauge-invariant criteria of center-vortex condensation and monopole condensation using the $\mathbb{Z}_N^{[1]}$-symmetry twisted partition functions: The torus twisted partition function characterizes the center-vortex condensation, and the lens-space twisted partition function characterizes the monopole condensation. To justify our proposal, we study how these twisted partition functions behave in the adjoint Higgs phase and show that their leading nontrivial contributions come from the center vortex and monopole, respectively. Using the techniques of topological field theories, we uncover the relation between the center-vortex and monopole condensations, and in particular, we prove that the gapped phase with the center-vortex condensation necessarily shows the monopole condensation, too. We then study a center-vortex model with monopoles as an illustrative example, and the higher-charge monopole condensation gives an example of the symmetry fractionalization, which goes beyond the conventional Wilson-'t Hooft classification.