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arXiv:1301.3470 · Physical Review B

Mott insulators of ultracold fermionic alkaline earth atoms in three dimensions

Hao Song, Michael Hermele

Published Apr 29, 2013• 12 citations
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Abstract

We study a class of $\text{SU}(N)$ Heisenberg models, describing Mott insulators of fermionic ultracold alkaline earth atoms on the three-dimensional simple cubic lattice. Based on an earlier semiclassical analysis, magnetic order is unlikely, and we focus instead on a solvable large-$N$ limit designed to address the competition among nonmagnetic ground states. We find a rich phase diagram as a function of the filling parameter $k$, composed of a variety of ground states spontaneously breaking lattice symmetries, and in some cases also time-reversal symmetry. One particularly striking example is a state spontaneously breaking lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state.

DOI
10.1103/physrevb.87.144423

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We study a class of SU(N)\text{SU}(N)SU(N) Heisenberg models, describing Mott insulators of fermionic ultracold alkaline earth atoms on the three-dimensional simple cubic lattice. Based on an earlier semiclassical analysis, magnetic order is unlikely, and we focus instead on a solvable large-NNN limit designed to address the competition among nonmagnetic ground states. We find a rich phase diagram as a function of the filling parameter kkk, composed of a variety of ground states spontaneously breaking lattice symmetries, and in some cases also time-reversal symmetry. One particularly striking example is a state spontaneously breaking lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state.
We study Mott insulators of fermionic alkaline earth atoms, described by Heisenberg spin models with enhanced SU(N) symmetry. In dramatic contrast to SU(2) magnetism, more than two spins are required to form a singlet. On the square lattice, the classical ground state is highly degenerate and magnetic order is thus unlikely. In a large-N limit, we find a chiral spin liquid ground state with topological order and Abelian fractional statistics. We discuss its experimental detection. Chiral spin liquids with non-Abelian anyons may also be realizable with alkaline earth atoms.
We numerically study the quantum magnetism of ultracold alkali and alkaline-earth fermion systems with large hyperfine spin F=32F=\frac{3}{2}F=23​, which are characterized by a generic Sp(N)Sp(N)Sp(N) symmetry with N=4N=4N=4. The methods of exact diagonalization (ED) and density matrix renormalization group are employed for the large size one-dimensional (1D) systems, and the ED method is applied to a two-dimensional (2D) square lattice on small sizes. We focus on the magnetic exchange models in the Mott-insulating state at quarter-filling. Both 1D and 2D systems exhibit rich phase diagrams depending on the ratio between the spin exchanges J0{J}_{0}J0​ and J2{J}_{2}J2​ in the bond spin singlet and quintet channels, respectively. In one dimension, the ground states exhibit a long-range-ordered dimerization with a finite spin gap at J0/J2>1{J}_{0}/{J}_{2}>1J0​/J2​>1 and a gapless spin-liquid state at ${J}_{0}/{J}_{2}\ensuremath{\leqslant}1$, respectively. In the former and latter cases, the correlation functions exhibit the two-site and four-site periodicities, respectively. In two-dimensions, various spin-correlation functions are calculated up to the size of $4\ifmmode\times\else\texttimes\fi{}4$. The N\'eel-type spin correlation dominates at large values of J0/J2{J}_{0}/{J}_{2}J0​/J2​, while a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette correlation is prominent at small values of this ratio. Between them, a columnar spin-Peierls dimerization correlation peaks. We infer the competition among the plaquette ordering, the dimer ordering, and the N\'eel ordering in the 2D system.
We study the zero-temperature phase diagram of a class of two-dimensional SU(N)(N)(N) antiferromagnets. These models are characterized by having the same type of SU(N)(N)(N) spin placed at each site of the lattice, and share the property that, in general, more than two spins must be combined to form a singlet. An important motivation to study these systems is that they may be realized naturally in Mott insulators of alkaline-earth atoms placed on optical lattices; indeed, such Mott insulators have already been obtained experimentally, although the temperatures are still high compared to the magnetic exchange energy. We study these antiferromagnets in a large-NNN limit, finding a variety of ground states. Some of the models studied here have a valence-bond solid ground state, as was found in prior studies, yet we find that many others have a richer variety of ground states. Focusing on the two-dimensional square lattice, in addition to valence cluster states (which are analogous to valence-bond states), we find both Abelian and non-Abelian chiral spin liquid ground states, which are magnetic counterparts of the fractional quantum Hall effect. We also find a ``doubled'' chiral spin liquid ground state that preserves time-reversal symmetry. These results are based on a combination of rigorous lower bounds on the large-NNN ground-state energy and a systematic numerical ground-state search. We conclude by discussing whether experimentally relevant SU(N)(N)(N) antiferromagnets, away from the large-NNN limit, may be chiral spin liquids.
The interplay between charge and spin degrees of freedom in strongly correlated fermionic systems, in particular of Dirac fermions, is a long-standing problem in condensed matter physics. We investigate the competing orders in the half-filled SU(2N)\mathrm{SU}(2N)SU(2N) Hubbard model on a honeycomb lattice, which can be accurately realized in optical lattices with ultracold large-spin alkaline-earth fermions. Employing large-scale projector determinant quantum Monte Carlo simulations, we have explored quantum phase transitions from the gapless Dirac semimetals to the gapped Mott insulating phases in the SU(4) and SU(6) cases. Both of these Mott insulating states are found to be columnar valence bond solid (cVBS) and to be absent of the antiferromagnetic N\'eel ordering and the loop current ordering. Inside the cVBS phases, the dimer ordering is enhanced by increasing fermion components and behaves nonmonotonically as the interaction strength increases. Although the transitions generally should be of first order due to a cubic invariance possessed by the cVBS order, the coupling to gapless Dirac fermions can soften the transitions to second order through a nonanalytic term in the free energy. Our simulations provide important guidance for the experimental explorations of novel states of matter with ultracold alkaline-earth fermions.
We propose to realize one-dimensional topological phases protected by SU(N)\mathrm{SU}(N)SU(N) symmetry using alkali or alkaline-earth atoms loaded into a bichromatic optical lattice. We derive a realistic model for this system and investigate it theoretically. Depending on the parity of NNN, two different classes of symmetry-protected topological (SPT) phases are stabilized at half-filling for physical parameters of the model. For even NNN, the celebrated spin-1 Haldane phase and its generalization to SU(N)\mathrm{SU}(N)SU(N) are obtained with no local symmetry breaking. In stark contrast, at least for N=3N=3N=3, a new class of SPT phases, dubbed chiral Haldane phases, that spontaneously break inversion symmetry, emerges with a twofold ground-state degeneracy. The latter ground states with open-boundary conditions are characterized by different left and right boundary spins, which are related by conjugation. Our results show that topological phases are within close reach of the latest experiments on cold fermions in optical lattices.
We develop a self-consistent variant of the constrained path quantum Monte Carlo approach which ensures its independence of the trial wave function, and apply the method to compute ground-state correlations in the two-dimensional SU(3) Fermi-Hubbard Hamiltonian at 13\frac{1}{3}31​ filling, modeling fermions with three possible spin flavors moving on a square lattice with an average of one particle per site. We provide clear evidence of a quantum critical point separating a nonmagnetic uniform metallic phase from a regime where long-range ``spin'' order is present. This discovery of multiple successive transitions to magnetic states with regular, long-range alternation of the different flavors, whose symmetry changes as the interaction strength increases, significantly extends previous work in the Heisenberg limit to itinerant fermions. In addition to the rich quantum magnetism, this important physical system allows one to study integer filling and the associated Mott transition disentangled from nesting, in contrast to the usual SU(2) model, while preserving the square-lattice geometry. Our results also provide a significant step towards the interpretation of present and future experiments on fermionic alkaline-earth atoms, and other realizations of SU(N)\mathrm{SU}(N)SU(N) physics.