We present a method to quantize free fermions which eliminates the doublers when implemented on the lattice in any number of dimensions and in the $m=0$ limit. The elimination of doublers is achieved by combining a second-order description of fermions, with the tools associated with non-Hermitian Hamiltonians. We identify a new Pseudo-Hermitian symmetry of the second-order fermion equation, and we identify the associated $U(1)$ symmetry which will become charge when shifted to a local gauge theory. We validated the methods numerically.

We provide the conditions for complete asymptotic freedom for chiral gauge theories including scalars, as motivated by grand unified models. These are generalised Georgi-Glashow and Bars-Yankielowicz theories that feature a scalar field transforming either in the fundamental or in the adjoint of the gauge group. In both scenarios, we consider the addition of multiple chiral fermion families. We systematically analyse the interplay between gauge, Yukawa, and quartic couplings required for all interactions to remain asymptotically free at short distances. We find that for both scalar representations, complete asymptotic free models can be obtained for a specific number of colours and multiplicity of vector-like and chiral families.

Loopy tensor networks exhibit internal correlations that often render their compression inefficient. We show that even local bond optimization can more effectively exploit locally available information about relevant loop correlations. By cutting a bond, we define a set of states whose linear dependence can be identified through a zero mode of the states' metric tensor and used to truncate the bond dimension. In the absence of an exact zero mode, a linear combination of a small number of the lowest modes can instead be optimized to provide the optimal approximation to a zero mode. The truncation does not require prior gauge fixing. The method is applied to the two-dimensional finite-temperature $Z_2$ lattice gauge theory, whose thermal-state purification is represented by an infinite projected entangled-pair state (iPEPS).

Several simple asymptotically-free chiral gauge theories are studied. The only ``free parameters'' of our models are the choice of the gauge group and the matter Weyl fermion representations, and the relative magnitudes of the renormalization-group-invariant scales $\Lambda_i$ associated with each gauge group. None of our models has nontrivial nonabelian global symmetries (``family''--like fermion representations). We rely on some recent theoretical developments on the dynamics of strongly--coupled chiral gauge theories, based on the generalized symmetries and associated new types of anomaly-matching consideration, but also on the solid knowledge on vectorlike gauge theories such as QCD and supersymmetric Yang-Mills theories. The structures of the infrared effective theories, the RG flows, and the light spectra found in these models are surprisingly rich and intriguing.

We propose HoloNet, a neural-network framework that unifies lattice QCD(LQCD) thermodynamics and holographic Einstein-Maxwell-Dilaton (EMD) theory within a data-to-holography pipeline. Instead of assuming specific functional forms, HoloNet learns the metric profile $A(z)$ and the gauge-dilaton coupling $f(z)$ directly from 2+1-flavor LQCD data at $\mu=0$. These learned functions are embedded into the EMD equations, enabling the model to reproduce the lattice equation of state and baryon number fluctuations with high fidelity. Once trained, HoloNet provides a fully data-driven holographic description of QCD that extends naturally to finite density, allowing us to map the phase diagram and estimate the location of the critical end point (CEP). The reconstructed potential $V(\phi)$ and coupling $f(\phi)$ agree quantitatively with those obtained from holographic renormalization, demonstrating that HoloNet can consistently bridge different holographic models.

I review recent lattice results on strongly interacting matter under extreme conditions, with emphasis on the finite-temperature QCD transition at $\mu_B=0$, its approach toward the chiral limit and the fate of the $U_A(1)$ anomaly, as well as recent constraints on the QCD phase boundary and the possible critical endpoint at $\mu_B>0$. I also discuss selected advances in lattice methods and in QCD thermodynamics under external conditions, in particular strong magnetic fields, isospin chemical potential, rotation, acceleration, and quark spin polarization.

We investigate the symmetry structure of the $3+1$ D staggered fermion Hamiltonian and its implications for anomalies. Since the spin and flavor degrees of freedom of Dirac fermions are distributed over the lattice, in addition to the standard on-site mass term, the staggered fermion system also admits one-, two-, and three-link bilinear terms within a unit cube as local, charge conserving mass terms with different spin and flavor dependence. We identify the spin flavor structures of all those bilinear mass terms and determine the symmetries preserved by each of them. Among them, one of the one-link mass terms preserves a larger residual symmetry associated with conserved charges that generate the Onsager algebra. Motivated by this structure, we consider a kink profile of the one-link mass and analyze the resulting domain-wall system. In the low-energy limit, the $3+1$ D bulk becomes gapped, while two-flavor massless Dirac fermions appear as localized modes on the $2+1$ D domain wall. We show that the bulk conserved charges act on the wall as generators of a flavor $\mathrm{SU}(2)$ symmetry, and that no symmetric mass gap is allowed for the boundary theory when this $\mathrm{SU}(2)$ symmetry and space reflection symmetry are both imposed. This realizes the parity anomaly of the boundary theory and shows that the boundary flavor symmetry and anomaly descend from the ultraviolet staggered-fermion Hamiltonian rather than emerging only in the infrared.

In the last decades, femtoscopic measurements from heavy-ion collisions have become a popular tool to investigate the strong interactions between hadrons. The key observables measured in such experiments are the two-hadron momentum correlations, which depend on the production mechanism of hadron pairs and the final-state interactions. Given the complexity of ultra-relativistic collision experiments, the source term describing the production mechanism can only be modeled phenomenologically based on numerous assumptions. The commonly employed approach for analyzing femtoscopic data relies on the Koonin-Pratt formula, which relates the measured correlation functions with the relative wave function of an outgoing hadron pair and a source term that is assumed to be universal. Here, we critically examine this universality assumption and show that for strongly interacting particles such as nucleons, the interpretation of femtoscopic measurements suffers from a potentially large intrinsic uncertainty. We also comment on the ongoing efforts to explore three-body interactions using this experimental technique.

The study of fermionic quantum field theories is an important problem for realizing the standard model of particle physics on a quantum computer. As a step towards this goal, we consider the massive Thirring and Gross--Neveu models with arbitrary number of fermion flavors, $N_f$, discretized on a spatial one-dimensional lattice of size $L$ in the Hamiltonian formulation. We compute the gate complexity using the higher-order product formula and using block-encoding/qubitization and quantum singular value transformations in the limit of large $N_f$ and $L$. We also prepare the ground states of both models with excellent fidelity for system sizes up to 20 qubits with $N_f = 1,2,3,4$ using the adaptive-variational quantum imaginary time algorithm. In addition, we also classify the dynamical Lie algebras of these relativistic fermionic models and show that they belong to the same isomorphism class. Our work is a concrete step towards the quantum simulation of real-time dynamics of large $N_f$ fermionic quantum field theories models relevant for chiral symmetry breaking, understanding dimensional transmutation, and exploring the conformal window of field theories on near-term and early fault-tolerant quantum computers.