We discuss a general framework for the realization of a family of Abelian lattice gauge theories, i.e., link models or gauge magnets, in optical lattices. We analyze the properties of these models that make them suitable for quantum simulations. Within this class, we study in detail the phases of a U(1)-invariant lattice gauge theory in 2+1 dimensions, originally proposed by P. Orland. By using exact diagonalization, we extract the low-energy states for small lattices, up to 4 × 4. We confirm that the model has two phases, with the confined entangled one characterized by strings wrapping around the whole lattice. We explain how to study larger lattices by using either tensor network techniques or digital quantum simulations with Rydberg atoms loaded in optical lattices, where we discuss in detail a protocol for the preparation of the ground-state. We propose two key experimental tests that can be used as smoking gun of the proper implementation of a gauge theory in optical lattices. These tests consist in verifying the absence of spontaneous (gauge) symmetry breaking of the ground-state and the presence of charge confinement. We also comment on the relation between standard compact U(1) lattice gauge theory and the model considered in this paper.

We propose an experimentally feasible setup with ultracold alkaline earth atoms to simulate the dynamics of U(1) lattice gauge theories in 2+1 dimensions with a Chern-Simons term. To this end we consider the ground state properties of spin-5/2 alkaline earth fermions in a honeycomb lattice. We use the Gutzwiller projected variational approach in the strongly repulsive regime in the case of filling 1/6. The ground state of the system is a chiral spin liquid state with 2π/3 flux per plaquette, which spontaneously violates time reversal invariance. We demonstrate that due to the breaking of time reversal symmetry the system exhibits quantum Hall effect and chiral edge states. We relate the experimentally accessible spin fluctuations to the emerging gauge field dynamics. We discuss also properties of the lowest energy competing orders.One of the main motivations of studying ultracold atoms in optical lattices is the high extent of experimental control. Such systems are very flexible and therefore are good candidates for simulating other quantum systems, where experimental control is more cumbersome. There is a vast number of proposals where ultracold quantum gases can serve as simulators of condensed matter, or even high energy physics systems (see for instance Ref. [1]). An important example of such proposals concerns the recent experimental realization of trapping and cooling of ultracold alkaline earth atoms [2][3][4], which could serve for quantum simulators of high symmetry magnetism [5]. Despite of these spectacular developments, one of the most important goals of quantum simulators remains still to be realized, namely the simulation of quantum gauge theories, which appear first of all in high energy physics, but arise naturally also in many areas of condensed matter physics, such as physics of frustrated systems, or of high temperature superconductors [6]. The main difficulty here is to map the many modes of the gauge field to those of an atomic ensemble. Very recent proposals use mixtures of fermionic and bosonic atoms, so that the bosons are the mediators of the gauge field [7,8].Here we propose another, somewhat simpler, scheme with only a single species of ultracold atoms to simulate a 2+1 dimensional U(1) lattice gauge theory with a Chern-Simons term. Our proposal is based on the observation that low energy excitations of certain Mott insulators can be described by lattice gauge theories [6,9]. The Mott insulator we consider here is formed by spin-5/2 alkaline earth atoms, such as 173 Yb, which, as was shown by Hermele et al.[10], can exhibit time reversal symmetry breaking, and have a so called chiral spin liquid (CSL) ground state in a square lattice. CSL states lack any kind of long range order, but due to the violation of time reversal invariance, they are stable also at low temperatures. The fluctuations above the CSL state are described by a U(1) gauge theory with a Chern-Simons term arising from the chiral (time reversal symmetry breaking) nature of the ground state [11]. Here we treat the case of ...

We study spin liquid phases of spin-5/2 alkaline earth atoms on a honeycomb lattice at finite temperatures. Our analysis is based on a Gutzwiller projection variational approach recast to a path-integral formalism. In the framework of a saddle-point approximation we determine spin liquid phases with lowest free energy and study their temperature dependence. We identify a critical temperature, where all the spin liquid phases melt and the system goes to the paramagnetic phase. We also study the stability of the saddle-point solutions and show that a time-reversal symmetry breaking state, a so called chiral spin liquid phase is realized even at finite temperatures. We also determine the spin structure factor, which, in principle, is an experimentally measurable quantity and is the basic tool to map the spectrum of elementary excitations of the system

We consider a generalization of the two-dimensional (2D) quantum Hall insulator to a noncompact, non-Abelian gauge group, the Heisenberg-Weyl group. We show that this kind of insulator is actually a layered three-dimensional (3D) insulator with nontrivial topology. We further show that nontrivial combinations of quantized transverse conductivities can be engineered with the help of a staggered potential. We investigate the robustness and topological nature of this conductivity and connect it to the surface modes of the system. We also propose a simple experimental realization with ultracold atoms in 3D confined to a 2D square lattice with the third dimension being mapped to a gauge coordinate.

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