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arXiv:1301.5636 · Molecular Physics

Kitaev honeycomb and other exotic spin models with polar molecules

Alexey Vyacheslavovich Gorshkov, Kaden R. A. Hazzard, Ana María Rey

Published Jun 13, 2013• 65 citations
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Abstract

We show that ultracold polar molecules pinned in an optical lattice can be used to access a variety of exotic spin models, including the Kitaev honeycomb model. Treating each molecule as a rigid rotor, we use DC electric and microwave fields to define superpositions of rotational levels as effective spin degrees of freedom, while dipole–dipole interactions give rise to interactions between the spins. In particular, we show that, with sufficient microwave control, the interaction between two spins can be written as a sum of five independently controllable Hamiltonian terms proportional to the five rank-2 spherical harmonics Y 2, q (θ, φ), where (θ, φ) are the spherical coordinates of the vector connecting the two molecules. To demonstrate the potential of this approach beyond the simplest examples studied in [S.R. Manmana et al., Phys. Rev. B. 87, 081106 (2013)], we focus on the realisation of the Kitaev honeycomb model, which can support exotic non-Abelian anyonic excitations. We also discuss the possibility of generating spin Hamiltonians with arbitrary spin S, including those exhibiting SU(N=2S+1) symmetry.

DOI
10.1080/00268976.2013.800604

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We show that ultracold polar molecules pinned in an optical lattice can be used to access a variety of exotic spin models, including the Kitaev honeycomb model. Treating each molecule as a rigid rotor, we use DC electric and microwave fields to define superpositions of rotational levels as effective spin degrees of freedom, while dipole–dipole interactions give rise to interactions between the spins. In particular, we show that, with sufficient microwave control, the interaction between two spins can be written as a sum of five independently controllable Hamiltonian terms proportional to the five rank-2 spherical harmonics Y 2, q (θ, φ), where (θ, φ) are the spherical coordinates of the vector connecting the two molecules. To demonstrate the potential of this approach beyond the simplest examples studied in [S.R. Manmana et al., Phys. Rev. B. 87, 081106 (2013)], we focus on the realisation of the Kitaev honeycomb model, which can support exotic non-Abelian anyonic excitations. We also discuss the possibility of generating spin Hamiltonians with arbitrary spin S, including those exhibiting SU(N=2S+1) symmetry.
We show how to use polar molecules in an optical lattice to engineer quantum spin models with arbitrary spin $S\ensuremath{\ge}1/2$ and with interactions featuring a direction-dependent spin anisotropy. This is achieved by encoding the effective spin degrees of freedom in microwave-dressed rotational states of the molecules and by coupling the spins through dipolar interactions. We demonstrate how one of the experimentally most accessible anisotropies stabilizes symmetry protected topological phases in spin ladders. Using the numerically exact density matrix renormalization group method, we find that these interacting phases---previously studied only in the nearest-neighbor case---survive in the presence of long-range dipolar interactions. We also show how to use our approach to realize the bilinear-biquadratic spin-1 and the Kitaev honeycomb models. Experimental detection schemes and imperfections are discussed.
In this chapter, we consider Kitaev's honeycomb lattice model (Kitaev, 2006). This is an analytically tractable spin model that gives rise to quasiparticles with Abelian as well as non-Abelian statistics. Some of its properties are similar to the fractional quantum Hall effect, which has been studied experimentally in great detail even though it evades exact analytical treatment (Moore and Read, 1991). Due to its simplicity, the honeycomb lattice model is likely to be the first topological spin model to be realised in the laboratory, e.g., with optical lattice technology (Micheli et al. , 2006). Understanding its properties can facilitate its physical realisation and can provide a useful insight into the mechanisms underlining topological insulators and the fractional quantum Hall effect. The honeycomb lattice model comprises interacting spin-½ particles arranged on the sites of a honeycomb lattice. It is remarkable that such a simple model can support a rich variety of topological behaviours. For certain values of its couplings, Abelian anyons emerge that behave like the toric code anyons. For another coupling regime, non-Abelian anyons emerge that correspond to the Ising anyons. The latter are manifested as vortex-like configurations of the original spin model that can effectively be described by Majorana fermions. These are fermionic fields that are antiparticles of themselves. They were first introduced in the context of high-energy physics (Majorana, 1937) and become increasingly important in the analysis of solid state phenomena (Wilczek, 2009).
We present a technique for engineering quantum magnets via ultracold polar molecules in optical lattices and explore exotic interplay between its spin superfluidity and solidity. The molecular ground and first excited rotational states are resonantly coupled by a linearly polarized microwave field. The spin-up (spin-down) states are presented by molecular rotational states of electric dipole moment along (against) the coupling field. By controlling the angle between the lattice direction and the coupling field, the inter-site interaction can be tuned from antiferromagnetic to ferromagnetic. Furthermore, the dipole-dipole interaction induces an exotic interplay between spin superfluidity and solidity, and spin supersolid phases may appear in mediate regions.
There is growing interest to investigate states of matter with topological order, which support excitations in the form of anyons, and which underly topological quantum computing. Examples of such systems include lattice spin models in two dimensions. Here we show that relevant Hamiltonians can be systematically engineered with polar molecules stored in optical lattices, where the spin is represented by a single electron outside a closed shell of a heteronuclear molecule in its rotational ground state. Combining microwave excitation with the dipole-dipole interactions and spin-rotation couplings allows us to build a complete toolbox for effective two-spin interactions with designable range and spatial anisotropy, and with coupling strengths significantly larger than relevant decoherence rates. As an illustration we discuss two models: a 2D square lattice with an energy gap providing for protected quantum memory, and another on stacked triangular lattices leading to topological quantum computing.
The exactly solvable Kitaev honeycomb lattice model is realized as the low-energy effect Hamiltonian of a spin-1/2 model with spin rotation and time-reversal symmetry. The mapping to low-energy effective Hamiltonian is exact without truncation errors in traditional perturbation series expansions. This model consists of a honeycomb lattice of clusters of four spin-1/2 moments and contains short-range interactions up to six-spin (or eight-spin) terms. The spin in the Kitaev model is represented not as these spin-1/2 moments but as pseudospin of the two-dimensional spin-singlet sector of the four antiferromagnetically coupled spin-1/2 moments within each cluster. Spin correlations in the Kitaev model are mapped to dimer correlations or spin-chirality correlations in this model. This exact construction is quite general and can be used to make other interesting spin-1/2 models from spin-rotation invariant Hamiltonians. We discuss two possible routes to generate the high-order spin interactions from more natural couplings, which involves perturbative expansions thus breaks the exact mapping, although in a controlled manner.
We show that dipolar interactions between ultracold polar alkali-metal dimers in optical lattices can be used to realize a highly tunable generalization of the ttt-JJJ model, which we refer to as the ttt-JJJ-VVV-WWW model. The model features long-range spin-spin interactions Jz{J}_{z}Jz​ and ${J}_{\ensuremath{\perp}}$ of XXZXXZXXZ type, long-range density-density interaction VVV, and long-range density-spin interaction WWW, all of which can be controlled in both magnitude and sign independently of each other and of the tunneling ttt. The ``spin'' is encoded in the rotational degree of freedom of the molecules, while the interactions are controlled by applied static electric and continuous-wave microwave fields. Furthermore, we show that nuclear spins of the molecules can be used to implement an additional (orbital) degree of freedom that is coupled to the original rotational degree of freedom in a tunable way. The presented system is expected to exhibit exotic physics and to provide insights into strongly correlated phenomena in condensed-matter systems. Realistic experimental imperfections are discussed.