Magnetism plays a key role in modern technology and stimulates research in several branches of condensed matter physics. Although the theory of classical magnetism is well developed, the demonstration of a widely tunable experimental system has remained an elusive goal. Here, we present the realization of a large-scale simulator for classical magnetism on a triangular lattice by exploiting the particular properties of a quantum system. We use the motional degrees of freedom of atoms trapped in an optical lattice to simulate a large variety of magnetic phases: ferromagnetic, antiferromagnetic, and even frustrated spin configurations. A rich phase diagram is revealed with different types of phase transitions. Our results provide a route to study highly debated phases like spin-liquids as well as the dynamics of quantum phase transitions.

We present a universal method to create a tunable, artificial vector gauge potential for neutral particles trapped in an optical lattice. The necessary Peierls phase of the hopping parameters between neighboring lattice sites is generated by applying a suitable periodic inertial force such that the method does not rely on any internal structure of the particles. We experimentally demonstrate the realization of such artificial potentials, which generate ground state superfluids at arbitrary non-zero quasi-momentum. We furthermore investigate possible implementations of this scheme to create tuneable magnetic fluxes, going towards model systems for strong-field physics.First introduced in electromagnetism, gauge fields play a central role in the description of interactions in physics, from particle physics to condensed matter. Currently there is a large interest to introduce gauge fields into model systems in order to study fundamental aspects of physics [1]. Especially the emulation of synthetic electric and magnetic fields for of ultracold atomic systems is crucial in order to extend their proven quantum simulation abilities further, e.g. to quantum Hall physics or topological insulators. In this context, the analogy between inertial and Lorentz forces triggered the simulation of homogeneous artificial magnetic fields using rapidly rotating trapped ultracold gases [2]. Recently, several proposals ([3, 4] and references therein) and experimental realizations focused on the simulation of a gauge vector potential A either in a bulk system [5,6] or in optical lattices [7,8]. The realized schemes exploit the Berry phase which arises when the atomic ground state is split in several space-dependent sublevels, as in the presence of an electromagnetic field. Hence they rely on the coupling between internal and external degrees of freedom induced by laser fields. Here we demonstrate the generation of artificial gauge potentials for neutral atoms in an optical lattice without any requirements on the specific internal structure. As the realized scheme only relies on the trapability of the particle, it can be very widely applied to many atomic systems, in principle also to molecules and other complex particles. It is particularly interesting for fermionic systems, where in a many-body state governed by Pauli principle, the use of internal degrees of freedom often lead to conflicts with the creation of gauge potentials. As an important additional benefit, the internal degrees of freedom of the particles can be addressed independently, e.g. by real magnetic fields or microwave excitations. In general, the presence of a gauge vector potential modifies the kinetic part of the Hamiltonian describing the system. In a lattice, an artificial field can then be simulated by engineering a complex tunneling parameter J = |J| · e iθ , where θ is the Peierls phase. The central approach here is to control this phase via a suitable forcing of the lattice potential, acting at the single-particle level. We describe the general scheme for the...

Periodicity is one of the most fundamental structural characteristics of systems occurring in nature. The properties of these systems depend strongly on the symmetry of the underlying periodic structure. In solid state materials -for example -the static and transport properties as well as the magnetic and electronic characteristics are crucially influenced by the crystal symmetry. In this context, hexagonal structures play an extremely important role and lead to novel physics like that of carbon nanotubes or graphene. Here we report on the first realization of ultracold atoms in a spin-dependent optical lattice with hexagonal symmetry. We show how combined effects of the lattice and interactions between atoms lead to a forced antiferromagnetic Néel order when two spin-components localize at different lattice sites. We also demonstrate that the coexistence of two components -one Mott-insulating and the other one superfluid -leads to the formation of a forced supersolid. Our observations are consistent with theoretical predictions using Gutzwiller mean-field theory.

Emulation of gauge fields for ultracold atoms provides access to a class of exotic states arising in strong magnetic fields. Here we report on the experimental realisation of tunable staggered gauge fields in a periodically driven triangular lattice. For maximal staggered magnetic fluxes, the doubly degenerate superfluid ground state breaks both a discrete Z 2 (Ising) symmetry and a continuous U (1) symmetry. By measuring an Ising order parameter, we observe a thermally driven phase transition from an ordered antiferromagnetic to an unordered paramagnetic state and textbook-like magnetisation curves. Both the experimental and theoretical analysis of the coherence properties of the ultracold gas demonstrate the strong influence of the Z 2 symmetry onto the condensed phase.Phase transitions in systems with combined continuous and discrete symmetries are fundamentally different from their purely continuous and discrete counterparts. The interplay between different types of excitations in the various degrees of freedom can lead to a complex behaviour and coupling of the associated order parameters [1][2][3][4][5]. A paradigm example is the fully frustrated XY model on a triangular lattice. It combines vector spin-type symmetries with discrete chiral degrees of freedom, which result in the famous spin-chirality coupling at low temperatures [6]. However, experimental studies in solid-state systems are challenging in view of implementing and isolating an XY model Hamiltonian [7][8][9].Ultracold bosonic quantum gases in optical lattices, on the other hand, constitute a highly versatile system with an extraordinary degree of control [10,11]. In particular, the recent experimental realisations of artificial gauge potentials for bulk [12][13][14][15] and optical lattice systems [16][17][18][19] allow for the investigation of new physical regimes, not realisable in condensed matter systems.Here, we demonstrate the realisation of a system with combined U (1) and Z 2 symmetries using ultracold atoms submitted to artificial gauge fields. Our experimental setup consists of an ultracold gas of 87 Rb atoms held in a two-dimensional triangular lattice [20] (see Fig. 1a). At each lattice site j with particle number N j , the weakly interacting superfluid gas can be described by the local order parameter a j = N j e iϕj . As a central aspect, the local phases ϕ j are mapped onto classical XY spins s j = (cos ϕ j , sin ϕ j ), where the tunneling matrix elements between neighbouring lattice sites correspond to the spinspin coupling parameters. Such classical spins possess a continuous degree of freedom. In presence of a long-range order, analogous to the onset of Bose-Einstein condensation (BEC), the order parameter assumes an arbitrary, but fixed phase, thus breaking the continuous U (1) symmetry [21].Beyond that, we experimentally engineer strong staggered gauge fields, which generate an additional discrete Z 2 symmetry in our system. The resulting magnetic flux induces cyclotron-like mass currents around each plaquette. The two poss...

Time-periodic driving like lattice shaking offers a low-demanding method to generate artificial gauge fields in optical lattices. We identify the relevant symmetries that have to be broken by the driving function for that purpose and demonstrate the power of this method by making concrete proposals for its application to two-dimensional lattice systems: We show how to tune frustration and how to create and control band touching points like Dirac cones in the shaken kagome lattice. We propose the realization of a topological and a quantum spin Hall insulator in a shaken spin-dependent hexagonal lattice. We describe how strong artificial magnetic fields can be achieved for example in a square lattice by employing superlattice modulation. Finally, exemplified on a shaken spin-dependent square lattice, we develop a method to create strong non-abelian gauge fields.