Quantum Link Hamiltonians with exact gauge invariance on the lattice

In this talk, we introduce a lattice Hamiltonian which realizes a microscopic U(1) gauge invariance exactly with a discrete Hilbert space via quantum links as gauge degrees of freedom residing on the links joining lattice sites. These so-called "Quantum Link models" are an extension of Wilson construction of lattice gauge theories, and show many novel properties. Using a combination of analytic and newly developed numerical methods, we study the phase transition of the model on a square lattice, and establish crystalline confined phases, where confinement is realized through fractional fluxes joining static charges. A connection with the extensively studied quantum dimer model (on a square lattice) is also established, and results on the phase diagram using a novel Monte Carlo method is also described. Non-Abelian extensions of quantum link models are already available, and are very crucial in realizing quantum simulators for gauge theories on cold atoms in optical lattices.