Adrien Florio

We present a discussion on lattice techniques for the simulation of non-canonical field theory circumstances, complementing our previous monograph [1] on canonical cases. We begin by reviewing basic aspects of lattice field theory, including symplectic and non-symplectic evolution algorithms. We then introduce lattice implementations of non-canonical interactions, considering scalars with a non-minimal coupling to gravity, ϕ ² R , non-minimal scalar kinetic theories, 𝒢 _ab ({ ϕ _c }) ∂ _μ ϕ ^a ∂ ^μ ϕ ^b , and axion-like particle (ALP) interactions with Abelian gauge fields, ϕF _μν F̃ ^μν . Next, we discuss methods to set up special field configurations, including the creation of cosmic defect networks towards scaling (e.g. cosmic strings and domain walls), field configurations based on arbitrary power spectra or spatial profiles, and probabilistic methods as required e.g. for thermal configurations. We further extend the notion of non-canonical theories, discussing the discretization of scalar field dynamics in d + 1 dimensions, with d ≠ 3. Unrelated to non-canonical aspects, we also discuss implementation(s) of gravitational wave (GW) dynamics on the lattice. This document represents the theoretical basis for the non-canonical field theory aspects (interactions, initial conditions, dimensionality) and GW dynamics implemented in 𝒞 osmo ℒ attice v2.0 to be released in 2026.

Motivated by axion-like inflation and its warm embedding within the Standard Model, we study the early stages of the energy transfer between an axion condensate and an SU(2) gauge ensemble, by employing non-linear classical real-time lattice simulations. The discretized equations of motion are worked out, elaborating on Gauss constraints. A numerical solution is implemented on the CosmoLattice platform. Adopting a quadratic potential, and omitting universe expansion, we establish initial exponential growth of the low-momentum gauge modes; damping of axion oscillations after some delay; and subsequent energy equipartition between axion and gauge ensembles. A clear difference between the SU(2) and U(1) dynamics is observed, likely associated with non-Abelian self-interactions. We elaborate on what this implies for the possible thermalization of the SU(2) ensemble.

A bstract: Understanding real-time dynamics of interacting quantum fields in curved space-time remains a major theoretical challenge. We employ tensor network methods to study such dynamics using interacting scalar and gauge theories in 1+1 spacetime dimensions, subject to a quench modeling a homogeneously expanding gravitational background. The models considered are the scalar λϕ ⁴ theory and the Schwinger model, i.e. a Dirac fermion coupled to a U(1) gauge field which is equivalent via bosonization to a scalar field with a cosine self-interaction. In the free scalar limit, both theories reproduce known analytical results, providing a nontrivial numerical validation of bosonization in curved spacetime for the Schwinger model. Our central finding is that self-interactions lead to a suppression of gravitational particle production compared to the free-field case, as evidenced by two-point functions and the spectra of produced particles. We further examine the behavior of entanglement generation and find that interactions suppress entanglement growth in the λϕ ⁴ theory, while in the Schwinger model, the interplay between suppressed particle production and enhanced inter-particle correlations leads to more complex entanglement behavior. Our results pave the way for further explorations of nonperturbative quantum real-time dynamics of interacting scalar and gauge theories in arbitrary gravitational backgrounds.

We investigate the emergence of thermalization in a quantum-field-theoretic model mimicking the production of jets in QCD: the massive lattice Schwinger model coupled to external sources. Specifically, we compute the expectation values of local operators as functions of time and compare them to their thermal counterparts, quantify the overlap between the evolving density matrix and the thermal one, and compare the dynamics of the energy-momentum tensor to predictions from relativistic hydrodynamics. Through these studies, we find that the system approaches thermalization at late times and elucidate the mechanisms by which quantum entanglement drives thermalization in closed field-theoretic systems. Our results show how thermodynamic behavior emerges in real time from unitary quantum dynamics.

The entanglement asymmetry has emerged in recent years as a practical quantity to study phases of matter. We present the first study of entanglement asymmetry in gauge theories by considering the chiral anomaly of the analytically solvable massless Schwinger model at both zero and finite temperatures. At zero temperature, we find the asymmetry exhibits logarithmic growth with system size. At finite temperature, we show that it is parametrically more sensitive to chiral symmetry-breaking than the corresponding local order parameter: while the chiral condensate decays exponentially, the asymmetry decreases only logarithmically. This establishes the entanglement asymmetry as a promising tool to probe (finite-temperature) phase transitions in gauge theories.

In this work, we investigate the potential connection between entanglement and thermalization, as well as the dynamics of hadronization in QCD physics. Through simulations of quantum dynamics on classical hardware, we analyze the real-time response of the massive Schwinger model coupled to external sources, which serves as an analog for the production and fragmentation of quark jets. This analogy is justified by the shared properties of confinement and chiral symmetry breaking between the Schwinger model and QCD. We observe the growth of time-dependent entanglement between the produced jets, which is driven by the increasing number of contributing eigenstates of the reduced density matrix with sufficiently large and closely spaced eigenvalues. This growth is intrinsically linked to the process of thermalization. The emergent eigenstates are identified as meson-like bound states, and by tracking their temporal evolution, we are able to observe real-time hadronization. At mid-rapidity, the long-time values of local observables approach approximately constant values, indicating the onset of equilibrium and the approach to thermalization.

We investigate the emergence of thermalization in a quantum field-theoretic model mimicking the production of jets in QCD -- the massive Schwinger model coupled to external sources. Specifically, we compute the expectation values of local operators as functions of time and compare them to their thermal counterparts, quantify the overlap between the evolving density matrix and the thermal one, and compare the dynamics of the energy-momentum tensor to predictions from relativistic hydrodynamics. Through these studies, we find that the system approaches thermalization at late times and elucidate the mechanisms by which quantum entanglement drives thermalization in closed field-theoretic systems. Our results show how thermodynamic behavior emerges in real time from unitary quantum dynamics.