The presence of long-range quantum spin correlations underlies a variety of physical phenomena in condensed matter systems, potentially including high-temperature superconductivity [1,2]. However, many properties of exotic strongly correlated spin systems (e.g., spin liquids) have proved difficult to study, in part because calculations involving N-body entanglement become intractable for as few as N ∼ 30 particles [3]. Feynman divined that a quantum simulator -a special-purpose "analog" processor built using quantum particles (qubits) -would be inherently adept at such problems [4,5]. In the context of quantum magnetism, a number of experiments have demonstrated the feasibility of this approach [6][7][8][9][10][11][12][13][14]. However, simulations of quantum magnetism allowing controlled, tunable interactions between spins localized on 2D and 3D lattices of more than a few 10's of qubits have yet to be demonstrated, owing in part to the technical challenge of realizing large-scale qubit arrays. Here we demonstrate a variable-range Ising-type spin-spin interaction J i, j on a naturally occurring 2D triangular crystal lattice of hundreds of spin-1/2 particles ( 9 Be + ions stored in a Penning trap), a computationally relevant scale more than an order of magnitude larger than existing experiments. We show that a spindependent optical dipole force can produce an antiferromagnetic interaction J i, j ∝ d −a i, j , where a is tunable over 0 < a < 3; d i, j is the distance between spin pairs. These power-laws correspond physically to infinite-range (a = 0), Coulomb-like (a = 1), monopole-dipole (a = 2) and dipole-dipole (a = 3) couplings. Experimentally, we demonstrate excellent agreement with theory for 0.05 a 1.4. This demonstration coupled with the high spin-count, excellent quantum control and low technical complexity of the Penning trap brings within reach simulation of interesting and otherwise computationally intractable problems in quantum magnetism.A challenge in condensed matter physics is the fact that many quantum-magnetic interactions cannot currently be modeled in a meaningful way. A canonical example is the spin-liquid postulated by Anderson [1]. He suggested this exotic state would arise in a collection of spin-1/2 particles residing on a triangular lattice and coupled by a nearest-neighbor antiferromagnetic Heisenberg interaction. The spin-liquid's Figure 1. The Penning trap confines hundreds of spin-1/2 particles (qubits) on a two-dimensional (2D) triangular lattice. Each qubit is the valence electron spin of a 9 Be + ion. (lower) A Penning trap confines ions by use of a combination of static electric and magnetic fields. The trap parameters are configured so that laser-cooled ions form a triangular 2D crystal. A general spin-spin interactionĤ I is generated by a spin-dependent excitation of the transverse (alongẑ) motional modes of the ion crystal. This coupling is implemented with an optical dipole force (ODF) due to a pair of off-resonance laser beams (left side) with angular separation θ R and dif...

A quantum simulator is a well-controlled quantum system that can follow the evolution of a prescribed model whose behaviour may be difficult to determine. A good example is the simulation of a set of interacting spins, where phase transitions between various spin orders can underlie poorly understood concepts such as spin liquids. Here we simulate the emergence of magnetism by implementing a fully connected non-uniform ferromagnetic quantum Ising model using up to 9 trapped 171 Yb + ions. By increasing the Ising coupling strengths compared with the transverse field, the crossover from paramagnetism to ferromagnetic order sharpens as the system is scaled up, prefacing the expected quantum phase transition in the thermodynamic limit. We measure scalable order parameters appropriate for large systems, such as various moments of the magnetization. As the results are theoretically tractable, this work provides a critical benchmark for the simulation of intractable arbitrary fully connected Ising models in larger systems.

We study the spinless and spinful extended Hubbard models with repulsive interactions on the kagome and the decorated honeycomb ("star") lattice. Using Hartree-Fock mean-field theory, we show that interaction-driven insulating phases with non-trivial topological invariants (Chern number or Z2 invariant) exist for an experimentally reasonable range of parameters. These phases occur at filling fractions which involve either Dirac points or quadratic band crossing points in the non-interacting limit. We present comprehensive mean-field phase diagrams for these lattices and discuss the competition between topologically non-trivial phases and numerous other ordered states, including various charge, spin, and bond orderings. Our results suggest that Z2 topological insulators should be found in a number of systems with either little or no intrinsic spin-orbit coupling.

We demonstrate spectroscopy and thermometry of individual motional modes in a mesoscopic 2D ion array using entanglement-induced decoherence as a method of transduction. Our system is a ∼400 µm-diameter planar crystal of several hundred 9 Be + ions exhibiting complex drumhead modes in the confining potential of a Penning trap. Exploiting precise control over the 9 Be + valence electron spins, we apply a homogeneous spin-dependent optical dipole force to excite arbitrary transverse modes with an effective wavelength approaching the interparticle spacing (∼20 µm). Center-of-mass displacements below 1 nm are detected via entanglement of spin and motional degrees of freedom. Studies of quantum physics at the interface of microscopic and mesoscopic regimes have recently focused on the observation of quantum coherent phenomena in optomechanical systems [1][2][3]. The realization of quantum coherence in mechanical oscillations involving many particles behaving approximately as a continuum provides exciting insights into the quantum-classical transition. Previous work has shown that crystals of cold, trapped ions behave as atomic-scale nanomechanical oscillators [4][5][6], with the benefits of in-situ tunable motional modes and exploitable single-particle quantum degrees of freedom (e.g. valence electron spin). Our system of hundreds of crystallized ions in a Penning trap provides a bottom-up approach to studying mesoscopic quantum coherence. In this context, the relevant particle numbers are sufficiently small to permit excellent quantum control without sacrificing continuum mechanical features. Beyond these capabilites, trapped ions have long provided a laboratory platform for studying diverse physical phenomena including: strongly-coupled one-component plasmas (OCPs) [7,8]; quantum computation [9,10] and simulation [11][12][13][14][15]; dynamical decoupling [16]; and atomic clocks and precision measurement [17].In this Letter, we present an experimental and theoretical study of motional drumhead modes in a 2D crystal of 9 Be + ions confined within a Penning trap. We excite inhomogeneous modes of arbitrary wavelength (see Fig. 1(a)) through application of a homogeneous, spinstate-dependent optical dipole force (ODF) to a largescale spin superposition. Distinct drumhead modes are entangled with the 9 Be + valence electron spins by tuning a beat frequency (µ R ) between two ODF lasers near a mode resonance. This spin-motion entanglement is de-

We derive the normal modes for a rotating Coulomb ion crystal in a Penning trap, quantize the motional degrees of freedom, and illustrate how they can by driven by a spin-dependent optical dipole force to create a quantum spin simulator on a triangular lattice with hundreds of spins. The analysis for the axial modes (oscillations perpendicular to the two-dimensional crystal plane) follow a standard normal-mode analysis, while the remaining planar modes are more complicated to analyze because they have velocity-dependent forces in the rotating frame. After quantizing the normal modes into phonons, we illustrate some of the different spin-spin interactions that can be generated by entangling the motional degrees of freedom with the spin degrees of freedom via a spindependent optical dipole force. In addition to the well-known power-law dependence of the spin-spin interactions when driving the axial modes blue of phonon band, we notice certain parameter regimes in which the level of frustration between the spins can be engineered by driving the axial or planar phonon modes at different energies. These systems may allow for the analog simulation of quantum spin glasses with large numbers of spins.