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 β pl 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. Their numerical implementation requires truncations but there is no sign problem for arbitrary values of µ. We show that the time continuum limit of the blocked transfer matrix can be obtained numerically and, in the limit of infinite β pl and with 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 onsite repulsion. We extend this procedure for finite β pl 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.

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 either introducing 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 multi-leg 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 show that the Polyakov loop of the two-dimensional lattice Abelian Higgs model can be calculated using the tensor renormalization group approach. We check the accuracy of the results using standard Monte Carlo simulations. We show that the energy gap produced by the insertion of the Polyakov loop obeys universal finite-size scaling which persists in the time continuum limit. We briefly discuss the relevance of these results for quantum simulations.

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 two percent 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 where we expect a CFT with c = 1. This can be accomplished with a very small hopping parameter J, in contrast to existing studies with density one where a much larger J is needed. 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.PACS numbers: 05.10. Cc, 11.15.Ha, 11.25.Hf, 37.10.Jk, 67.85.Hj, 75.10.Hk The concept of universality provides a unified approach to the critical behavior of lattice models studied in condensed matter, lattice gauge theory (LGT) and experimentally accessible systems of cold atoms trapped in optical lattices. Conformal field theory (CFT) [1, 2] offers many interesting examples of universal behavior that can be observed for lattice models in two [3][4][5], three [6], and four [7,8] dimensions. Practical simulations for these models unavoidably involve a finite volume that breaks explicitly the conformal invariance. However, this symmetry breaking follows definite patterns dictated by the restoration of the symmetry at infinite volume and allows us to identify the universality class. In view of the rich collection of interesting CFTs, it would be highly desirable to study their universality classes using quantum simulations. In order to start this ambitious program, one needs a simple concrete example to demonstrate the feasibility of the idea.In this Letter, we propose to use the setup of ongoing cold atom experiments to quantum simulate the O(2) model with a chemical potential and check the predictions of CFT for the growth of the entanglement entropy with the size of the system corresponding to the universality class of the superfluid (SF) phase. The O(2) model is an extension of the Ising model where the spin is allowed to move on a circle, making an angle θ with respect to a direction of reference. This model can be used to describe easy plane ferromagnetism and the compactness of θ leads to topological configurations called vortices. Their unbinding provides a prime example of a Berezinski-Kosterlitz-Thouless transition [9,10] in a way that has also been advocated to apply for gauge theories near the boundary of the conformal window [11]. When space and Euclidean time are treated isotropically, this model has important common ...