A quantum simulator of U(1) lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

The (2 + 1)-d U (1) quantum link model is a gauge theory, amenable to quantum simulation, with a spontaneously broken SO(2) symmetry emerging at a quantum phase transition. Its low-energy physics is described by a (2 + 1)-d RP (1) effective field theory, perturbed by a dangerously irrelevant SO(2) breaking operator, which prevents the interpretation of the emergent pseudo-Goldstone boson as a dual photon. At the quantum phase transition, the model mimics some features of deconfined quantum criticality, but remains linearly confining. Deconfinement only sets in at high temperature.Quantum link models (QLMs) are lattice gauge theories formulated in terms of discrete quantum degrees of freedom. U (1) and SU (2) QLMs were first constructed by Horn in 1981 [1], and further investigated in [2]. In [3] QLMs were introduced as an alternative nonperturbative regularization of Abelian and non-Abelian gauge theories, in which ordinary gauge fields emerge dynamically from the dimensional reduction of discrete quantum link variables. Dimensional reduction of discrete variables is a generic phenomenon in asymptotically free theories, which gives rise to the D-theory formulation of quantum field theory [4]. In the D-theory formulation of 4-d Quantum Chromodynamics (QCD), the confining gluon field emerges by dimensional reduction from a deconfined Coulomb phase of a (4 + 1)-d SU (3) QLM [5]. Chiral quarks arise naturally as domain wall fermions located at the two 4-d sides of a (4 + 1)-d slab. The (2 + 1)-d U (1) QLM has also been investigated in the context of quantum spin liquids [6]. With staggered background charges ±1, it is equivalent to a quantum dimer model [7][8][9]. Furthermore, Kitaev's toric code [10] is a Z(2) QLM. In contrast to Wilson's lattice gauge theory [11], QLMs have a finite-dimensional Hilbert space per link, which makes them ideally suited for the construction of atomic quantum simulators for dynamical Abelian [12][13][14][15][16] and non-Abelian gauge theories [17][18][19][20]. A long-term goal of this research is to quantum simulate QCD in the D-theory formulation with ultracold matter, in order to address the real-time evolution of strongly interacting systems in nuclear and particle physics, as well as their dynamics at non-zero baryon density.In this paper, we investigate the (2+1)-d U (1) QLM, in order to demonstrate that, despite its structural simplicity, it displays highly non-trivial dynamics, and thus is ideally suited to demonstrate the power of gauge theory quantum simulators. We consider the model with a plaquette coupling J and a Rokhsar-Kivelson (RK) coupling λ. The phase diagram is sketched in Fig.1. At zero temperature, the model is confining for λ < 1. At finite temperature T , it has a deconfinement phase transition above which there is a massless mode transforming non-trivially under the U (1) center symmetry. Due to the Mermin- FIG. 1. [Color online]Schematic sketch of the λ-T phase diagram. The insets indicate the location of the peaks in the probability distribution of the order p...

The quantum dimer model on the square lattice is equivalent to a U (1) gauge theory. Quantum Monte Carlo calculations reveal that, for values of the Rokhsar-Kivelson (RK) coupling λ < 1, the theory exists in a confining columnar phase. The interfaces separating distinct columnar phases display plaquette order, which, however, is not realized as a bulk phase. Static "electric" charges are confined by flux tubes that consist of multiple strands, each carrying a fractionalized flux 1 4 . A soft pseudo-Goldstone mode emerges around λ ≈ 0, long before one reaches the RK point at λ = 1.

We present detailed analytic calculations of finite-volume energy spectra, mean-field theory, as well as a systematic low-energy effective field theory for the square lattice quantum dimer model. An emergent approximate spontaneously broken SO(2) symmetry gives rise to a pseudo-Goldstone boson. Remarkably, this soft phononlike excitation, which is massless at the Rokhsar-Kivelson (RK) point, exists far beyond this point. The Goldstone physics is captured by a systematic low-energy effective field theory. We determine its low-energy parameters by matching the analytic effective field theory with exact diagonalization results. This confirms that the model exists in the columnar (and not in a plaquette or mixed) phase all the way to the RK point.

We show that exotic phases arise in generalized lattice gauge theories known as quantum link models in which classical gauge fields are replaced by quantum operators. While these quantum models with discrete variables have a finite-dimensional Hilbert space per link, the continuous gauge symmetry is still exact. An efficient cluster algorithm is used to study these exotic phases. The (2 + 1)-d system is confining at zero temperature with a spontaneously broken translation symmetry. A crystalline phase exhibits confinement via multi-stranded strings between chargeanti-charge pairs. A phase transition between two distinct confined phases is weakly first order and has an emergent spontaneously broken approximate SO(2) global symmetry. The low-energy physics is described by a (2 + 1)-d RP(1) effective field theory, perturbed by a dangerously irrelevant SO(2) breaking operator, which prevents the interpretation of the emergent pseudoGoldstone boson as a dual photon. This model is an ideal candidate to be implemented in quantum simulators to study phenomena that are not accessible using Monte Carlo simulations such as the real-time evolution of the confining string and the real-time dynamics of the pseudo-Goldstone boson.