## Abstract

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.

Frustrated spin systems belong to the most demanding problems of magnetism and condensed matter physics (*1*, *2*). The simplest realization of geometrical spin frustration is the triangular lattice (Fig. 1) with antiferromagnetic interactions: The spins cannot order in the favored antiparallel fashion and instead must compromise. The rich variety of possible spin configurations in the quantum spin case arising from the competition between interactions and the geometry of the lattice has been studied in many different contexts (*3*, *4*). Classical frustrated spin systems also show intriguing properties (*5*–*7*), such as highly degenerate ground states, and emergent phenomena, such as artificial magnetic fields and monopoles observed in spin ice.

Despite the interest in magnetically frustrated systems, their experimental realization and characterization in “natural” solid-state devices still poses a major challenge. Recently, there have been considerable advances in the direction of simulating quantum magnetism (*8*–*15*). We report on a versatile simulator for large-scale classical magnetism on a two-dimensional (2D) triangular optical lattice (*16*) by exploiting the motional degrees of freedom of ultracold bosons (*17*). The cornerstone of our simulation is the independent tuning of the nearest-neighbor coupling elements *J* and *J*′ (Fig. 1) by introducing a fast oscillation of the lattice (*18*). In particular, we can even control the sign of these elements (*19*, *20*), thus allowing for ferromagnetic or antiferromagnetic coupling schemes. Hence, we gain access to the whole diversity of expected complex magnetic phases in our 2D triangular system and can study large-system phase transitions as well as spontaneous symmetry-breaking caused by frustration. With our approach, the easily achievable Bose-Einstein condensate (BEC) temperatures are sufficient to observe Néel-ordered and spin-frustrated states. This is an advantage when compared with systems based on superexchange interaction (*10*), which demand much lower temperatures.

For weak interactions, ultracold bosonic atoms in an optical lattice form a superfluid state [in our 2D array of tubes: lattice depth is 5.6*E*_{r} (where *E*_{r} is the recoil energy of the lattice), on-site interaction *U* = 0.004*E*_{r}, single-particle tunneling , and a maximum of 250 particles per tube]. In this case, the atoms at each site *i* of the lattice have a well-defined local phase θ* _{i}* that can, as a central concept here, be identified with a classical vector spin (see also Fig. 1). Long-range order of these local phases (spins) is imprinted by the minimization of the energy(1)where the sum extends over all pairs of neighboring lattice sites. Note that we study large systems of ~1000 populated lattice sites. As a second central concept, the tunneling matrix elements

*J*assume the role of the “spin-spin” coupling parameters between neighboring lattice sites: Positive

_{ij}*J*correspond to ferromagnetic interaction, and negative

_{ij}*J*are consistent with antiferromagnetic interaction. The most important feature of our approach is the independent tuning of the tunneling parameters

_{ij}*J*and

*J*′ along two directions (Fig. 1) via an elliptical shaking of the lattice (

*17*). This leads to various ferromagnetic, antiferromagnetic, and mixed-spin configurations (Fig. 2). In the situation where all tunneling parameters are positive (

*J*,

*J*′ > 0), the spins align parallel, and we associate this with a fully ferromagnetically ordered phase. This is identical to the ordering observed without shaking. When, for example, the signs of the

*J*′ couplings are inverted (

*J*> 0,

*J*′ < 0), the new ground state of the system is of rhombic order: Along the direction of negative coupling, the spins arrange in antiferromagnetic order, whereas the coupling in

*J*direction remains ferromagnetic. The other configurations shown in Fig. 2 (spiral and chain order) can be explained in a similar fashion. Each of these spin configurations has its own, unique quasi-momentum distribution, which serves as a clear signature for identification via standard time-of-flight imaging techniques (

*18*). The experimental data obtained for the different cases are presented in Fig. 2.

The rich variety of spin orders as a function of the control parameters *J* and *J*′ can be mapped into the phase diagram (Fig. 3A). The background colors are meant to guide the eye and indicate the different spin configurations as expected from the minimization of the energy function (Eq. 1). We assign a symbol, representing the respective phase, to each data point by comparing the measured momentum distribution with the one obtained from theoretical calculations (*17*). The measured data matches very well with theory (*18*). The phase diagram has several interesting features that can be understood from the energy function (Eq. 1): First, the ferromagnetic phase (F) on the right-hand side (*J*′ > 0) extends into regions where the *J* coupling already favors antiferromagnetic order. The same behavior is observed for the rhombic phase (R) (*J*′ < 0), which extends to regions (*J*, *J*′ < 0) with purely antiferromagnetic couplings. For *J* < –|*J*′|/2, frustration finally breaks the (anti)parallel spin order and leads to phases characterized by spiral spin configurations (Sp1, Sp2). In this region, the system possesses two energetically degenerate ground states, which we will discuss in more detail below. Second, the transitions between the phases are even of different nature. The transition from F to R is of first order. The experimental signature is a sudden change of the momentum distribution when crossing the phase boundary. In consequence, with limited experimental resolution, we see a collection of interference peaks belonging to both neighboring phases directly at the phase boundary in the averaged data of consecutive experimental runs (Fig. 2). The phase transitions into the spiral region (R to Sp1 and F to Sp2) are of second order. Finally, within the spiral region of the phase diagram, the spin configurations smoothly evolve. As a consequence, the staggered chain order found around *J*′ = 0 is stable and experimentally well observable (Fig. 2). The gray shaded region in the center indicates that for small values of |*J*| and |*J*′| the long range spin order is lost, and the momentum distribution has no clear interference-peak structure.

We experimentally characterize the phase transition from R to Sp1 order as continuous (that is, second order) by following the evolution of the state when crossing the phase boundary (Fig. 3B). We observe a single momentum peak in the rhombic region of the phase diagram that smoothly splits into two once the trajectory enters the spiral region. This is in full agreement with the calculated dispersion relation also shown in the figure. The development of these two peaks marks the existence of two degenerate ground states. In this situation, the system is expected to randomly choose one of the possible states to thus exhibit spontaneous symmetry-breaking. We consider this important feature in more detail for the particular case of isotropic antiferromagnetic couplings (*J* = *J*′ < 0). These two possible ground-state spin configurations (Fig. 4A) are mirror images of each other and can thus be distinguished by the chiral-order parameter (*18*) of upward- and downward-pointing plaquettes. Their different momentum distributions (Fig. 4B) allow for a direct distinction in the experiment. For consecutive experimental runs, each of the two spin configurations appears entirely random (Fig. 4C), which is clear evidence for the spontaneous nature of the symmetry-breaking between the two ground states. In most cases (Fig. 4D), one of the modes clearly dominates. However, in some cases (below 10%), both configurations are almost equally populated at the same time. The simultaneous observation of two modes might be due to excitations like the formation of spin domains. Hence, for the first time, our system allows detailed studies on the mechanism of symmetry-breaking in large-scale “magnetic” systems.

A different aspect of our system is revealed when one recalls that the spin configurations actually correspond to local phases of a BEC at different sites of the triangular lattice. This provides new insight into unconventional superfluidity (*21*, *22*): For all phases except the ferromagnetic one, the state corresponds to a superfluid at nonzero quasi-momentum and, thus, nontrivial long-range phase order. Moreover, the observed spiral configurations spontaneously break time-reversal symmetry by showing circular bosonic currents around the triangular plaquettes of the lattice. Clockwise and counterclockwise currents are found in a staggered fashion from plaquette to plaquette. These resemble the currents of the staggered flux state thought to play a role in explaining the pseudogap phase of high-temperature cuprate superconductors (*23*).

Our results demonstrate the realization of a simulator for classical magnetism in a triangular lattice. Let us point out here that these results are obtained simply by using spinless bosons. Due to the high degree of controllability, we succeeded in observing all of the various magnetic phases and phase transitions of first and second order, as well as frustration-induced spontaneous symmetry-breaking. It now becomes possible to quench systems on variable time scales from ferromagnetic to antiferromagnetic couplings and to study the complex relaxation dynamics. Furthermore, extending the studies to the strongly correlated regime promises to give a deeper insight into the understanding of quantum spin models and spin-liquid–like phases (*24*).

## Supporting Online Material

## References and Notes

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**Acknowledgments:**We thank A. Rosch, P. Hauke, and D.-S. Lühmann for stimulating discussions and the Deutsche Forschungsgemeinschaft (FOR 801, GRK 1355) and the Landesexzellenzinitiative Hamburg, which is supported by the Joachim Herz Stiftung, for funding. A.E. and M.L. are grateful for support through the Spanish Ministerio de Ciencia y Innovación grant TOQATA, European Research Council grant QUAGATUA, and European Union grants AQUTE and NAMEQUAM.