The renormalization group (RG) method developed by Ken Wilson more than four decades ago has revolutionized the way we think about problems involving a broad range of energy scales such as phase transitions, turbulence, continuum limits and bifurcations in dynamical systems. The Theme Issue provides articles reviewing recent progress made using the RG method in atomic, condensed matter, nuclear and particle physics. In the following, we introduce these articles in a way that emphasizes common themes and the universal aspects of the method.

Lattice gauge theories are fundamental to our understanding of high-energy physics. Nevertheless, the search for suitable platforms for their quantum simulation has proven difficult. We show that the Abelian Higgs model in 1+1 dimensions is a prime candidate for an experimental quantum simulation of a lattice gauge theory. To this end, we use a discrete tensor reformulation to smoothly connect the space-time isotropic version used in most numerical lattice simulations to the continuous-time limit corresponding to the Hamiltonian formulation. The eigenstates of the Hamiltonian are neutral for periodic boundary conditions, but we probe the nonzero charge sectors by introducing either a Polyakov loop or an external electric field. In both cases we obtain universal functions relating the mass gap, the gauge coupling, and the spatial size, which are invariant under the deformation of the temporal lattice spacing. We propose to use a physical multileg ladder of atoms trapped in optical lattices and interacting with Rydberg-dressed interactions to quantum simulate the model and check the universal features. Our results provide a path to the analog quantum simulation of lattice gauge theories with atoms in optical lattices.

We calculate the von Neumann and Rényi bipartite entanglement entropy of the O(2) model with a chemical potential on a 1+1-dimensional Euclidean lattice with open and periodic boundary conditions. We show that the Calabrese-Cardy conformal field theory predictions for the leading logarithmic scaling of these entropies are consistent with a central charge c=1. This scaling survives the time-continuum limit and truncations of the microscopic degrees of freedom, modifications which allow us to connect the Lagrangian formulation to quantum Hamiltonians. At half-filling, the forms of the subleading corrections imposed by conformal field theory allow the determination of the central charge with an accuracy better than 2% for moderately sized lattices. We briefly discuss the possibility of estimating the central charge using quantum simulators.

We demonstrate that current experiments using cold bosonic atoms trapped in one-dimensional optical lattices and designed to measure the second-order Rényi entanglement entropy S2 can be used to verify detailed predictions of conformal field theory (CFT) and estimate the central charge c. We discuss the adiabatic preparation of the ground state at half filling and small hopping parameter J/U, where we expect a CFT with c=1. We provide two complementary methods to estimate and subtract the classical entropy generated by the experimental preparation and imaging processes. We compare numerical calculations for the classical O(2) model with a chemical potential on a (1+1)-dimensional lattice, and the quantum Bose-Hubbard Hamiltonian implemented in the experiments. S2 is very similar for the two models and follows closely the Calabrese-Cardy scaling, (c/8)ln(Ns), for Ns sites with open boundary conditions, provided that the large subleading corrections are taken into account.

We present a gauge-invariant effective action for the Abelian-Higgs model (scalar electrodynamics) with a chemical potential μ on a (1+1)-dimensional lattice. This formulation provides an expansion in the hopping parameter κ which we test with Monte Carlo simulations for a broad range of the inverse gauge coupling bpl=1/g2 and small values of the scalar self-coupling λ. In the opposite limit of infinitely large λ, the partition function can be written as a traced product of local tensors which allows us to write exact blocking formulas. Gauss's law is automatically satisfied and the introduction of μ has consequences only if we have an external electric field, g2=0 or an explicit gauge symmetry breaking. The time-continuum limit of the blocked transfer matrix can be obtained numerically and, for g2=0 and a spin-1 truncation, the small volume energy spectrum is identical to the low energy spectrum of a two-species Bose-Hubbard model in the limit of large on-site repulsion. We extend this procedure for finite bpl and derive a spin-1 approximation of the Hamiltonian. It involves new terms corresponding to transitions among the two species in the Bose-Hubbard model. We propose an optical lattice implementation involving a ladder structure.

We connect explicitly the classical O(2) model in 1+1 dimensions, a model sharing important features with U(1) lattice gauge theory, to physical models potentially implementable on optical lattices and evolving at physical time. Using the tensor renormalization-group formulation, we take the time continuum limit and check that finite-dimensional projections used in recent proposals for quantum simulators provide controllable approximations of the original model. We propose two-species Bose-Hubbard models corresponding to these finite-dimensional projections at strong coupling and discuss their possible implementations on optical lattices using a Rb87 and K41 Bose-Bose mixture.