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This article gives an introduction to the realization of effective quantum magnetism with ultracold molecules in an optical lattice, reviews experimental and theoretical progress, and highlights future opportunities opened up by ongoing experiments. Ultracold molecules offer capabilities that are otherwise difficult or impossible to achieve in other effective spin systems, such as long-ranged spin-spin interactions with controllable degrees of spatial and spin anisotropy and favorable energy scales. Realizing quantum magnetism with ultracold molecules provides access to rich many-body behaviors, including many exotic phases of matter and interesting excitations and dynamics. Far-from-equilibrium dynamics plays a key role in our exposition, just as it did in recent ultracold molecule experiments realizing effective quantum magnetism. In particular, we show that dynamical probes allow the observation of correlated many-body spin physics, even in polar molecule gases that are not quantum degenerate. After describing how quantum magnetism arises in ultracold molecules and discussing recent observations of quantum magnetism with polar molecules, we survey prospects for the future, ranging from immediate goals to long-term visions.
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In the last decade, quantum simulators, and in particular cold atoms in optical lattices, have emerged as a valuable tool to study strongly correlated quantum matter. These experiments are now reaching regimes that are numerically difficult or impossible to access. In particular they have started to fulfill a promise which has contributed significantly to defining and shaping the field of cold atom quantum simulations, namely the exploration of doped and frustrated quantum magnets and the search for the origins of high-temperature superconductivity in the fermionic Hubbard model. Despite many future challenges lying ahead, such as the need to further lower the experimentally accessible temperatures, remarkable studies have already emerged. Among them, spin-charge separation in one-dimensional systems has been demonstrated, extended-range antiferromagnetism in two-dimensional systems has been observed, connections to modern day large-scale numerical simulations were made, and unprecedented comparisons with microscopic trial wavefunctions have been carried out at finite doping. In many regards, the field has acquired new realms, putting old ideas to a new test and producing new insights and inspiration for the next generation of physicists. In the first part of this paper, we review the results achieved in cold atom realizations of the Fermi-Hubbard model in recent years. In the second part of this paper, with the stage set and the current state of the field in mind, we propose a new direction for cold atoms to explore: namely mixed-dimensional bilayer systems, where the charge motion is restricted to individual layers which remain coupled through spin-exchange. We propose a novel, strong pairing mechanism in these systems, which puts the formation of hole pairs at experimentally accessible, elevated temperatures within reach.
The exchange coupling between quantum mechanical spins lies at the origin of quantum magnetism. We report on the observation of nearest-neighbor magnetic spin correlations emerging in the many-body state of a thermalized Fermi gas in an optical lattice. The key to obtaining short-range magnetic order is a local redistribution of entropy within the lattice structure. This is achieved in a tunable-geometry optical lattice, which also enables the detection of the magnetic correlations. We load a low-temperature two-component Fermi gas with repulsive interactions into either a dimerized or an anisotropic simple cubic lattice. For both systems the correlations manifest as an excess number of singlets as compared to triplets consisting of two atoms with opposite spins. For the anisotropic lattice, we determine the transverse spin correlator from the singlet-triplet imbalance and observe antiferromagnetic correlations along one spatial axis. Our work paves the way for addressing open problems in quantum magnetism using ultracold fermions in optical lattices as quantum simulators.
We review the concepts and the present state of theoretical studies of spin-imbalanced superfluidity, in particular the elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in the context of ultracold quantum gases. The comprehensive presentation of the theoretical basis for the FFLO state that we provide is useful also for research on the interplay between magnetism and superconductivity in other physical systems. We focus on settings that have been predicted to be favourable for the FFLO state, such as optical lattices in various dimensions and spin-orbit coupled systems. These are also the most likely systems for near-future experimental observation of the FFLO state. Theoretical bounds, such as Blochs and Luttingers theorems, and experimentally important limitations, such as finite-size effects and trapping potentials, are considered. In addition, we provide a comprehensive review of the various ideas presented for the observation of the FFLO state. We conclude our review with an analysis of the open questions related to the FFLO state, such as its stability, superfluid density, collective modes and extending the FFLO superfluid concept to new types of lattice systems.
We show that the interplay between antiferromagnetic interaction and hole motion gives rise to a topological superconducting phase. This is captured by the one dimensional anisotropic $t-J$ model which can be experimentally achieved with ultracold polar molecules trapped onto an optical lattice. As a function of the anisotropy strength we find that different quantum phases appear, ranging from a gapless Luttinger liquid to spin gapped conducting and superconducting regimes. In presence of appropriate $z$-anisotropy, we also prove that a phase characterized by non-trivial topological order takes place. The latter is described uniquely by a finite non local string parameter and presents robust edge spin fractionalization. These results allow to explore quantum phases of matter where topological superconductivity is induced by the interaction.
The superfluidity and pairing phenomena in ultracold atomic Fermi gases have been of great interest in recent years, with multiple tunable parameters. Here we study the BCS-BEC crossover behavior of balanced two-component Fermi gases in a one-dimensional optical lattice, which is distinct from the simple three-dimensional (3D) continuum and a fully 3D lattice often found in a condensed matter system. We use a pairing fluctuation theory which includes self-consistent feedback effects at finite temperatures, and find widespread pseudogap phenomena beyond the BCS regime. As a consequence of the lattice periodicity, the superfluid transition temperature $T_c$ decreases with pairing strength in the BEC regime, where it approaches asymptotically $T_c = pi an/2m$, with $a$ being the $s$-wave scattering length, and $n$ ($m$) the fermion density (mass). In addition, the quasi-two dimensionality leads to fast growing (absolute value of the) fermionic chemical potential $mu$ and pairing gap $Delta$, which depends exponentially on the ratio $d/a$. Importantly, $T_c$ at unitarity increases with the lattice constant $d$ and hopping integral $t$. The effect of the van Hove singularity on $T_c$ is identified. The superfluid density exhibits $T^{3/2}$ power laws at low $T$, away from the extreme BCS limit. These predictions can be tested in future experiments.
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