We study antiferromagnetic systems with enhanced “nearly-SU(N)” symmetry, which can be realized in systems of ultracold spinor atoms in optical lattices. Examples of N = 3 (for S = 1 bosons) and N = 4 (for S=32 fermions) are considered. Near the SU(N) point, the low-energy physics can be described by the CPN−1 model with an additional symmetry-breaking term lowering the symmetry down to SU(2) and favoring the Néel ordering. We show that the effective theory of such systems can be cast in the form of a nonlinear sigma model with the SO(3) matrix-valued field, which is typically obtained for frustrated magnets with non-collinear order. Further, we show that those systems possess a peculiar effect of topological binding: for a system with the underlying spin S, lowering of the symmetry from SU(2S + 1) to SU(2) leads to binding of topological unit-charge excitations of the CP2S model (skyrmions for space dimension d = 2, instantons for d = 1, and hedgehogs for d = 3) into 2S-multiplets.
We study antiferromagnetic systems with enhanced “nearly-SU(N)” symmetry, which can be realized in systems of ultracold spinor atoms in optical lattices. Examples of N = 3 (for S = 1 bosons) and N = 4 (for S=32 fermions) are considered. Near the SU(N) point, the low-energy physics can be described by the CPN−1 model with an additional symmetry-breaking term lowering the symmetry down to SU(2) and favoring the Néel ordering. We show that the effective theory of such systems can be cast in the form of a nonlinear sigma model with the SO(3) matrix-valued field, which is typically obtained for frustrated magnets with non-collinear order. Further, we show that those systems possess a peculiar effect of topological binding: for a system with the underlying spin S, lowering of the symmetry from SU(2S + 1) to SU(2) leads to binding of topological unit-charge excitations of the CP2S model (skyrmions for space dimension d = 2, instantons for d = 1, and hedgehogs for d = 3) into 2S-multiplets.
We examine large-N limits of the nearest-neighbor SU(N) quantum antiferromagnets on bipartite lattices in d=1,2. In d=2 the model displays a transition from a N\'eel to a disordered phase. The properties of the disordered phase close to the phase boundary are crucially dependent upon the nature of ``hedgehog''-like instanton tunneling events. We calculate the Berry phases of the instantons and show that, at scales larger than the spin-correlation length, the system can be described by a Coulomb plasma of instantons with complex fugacities. The properties of the Coulomb plasma vary periodically with the ``spin'' nc of the states at each site, with periodicity given by the coordination number Z of the lattice [nc=2S for SU(2)]. For nc\ensuremath{\ne}0 (mod Z) the disordered phase has a broken lattice symmetry with spin-Peierls order, while for nc=0 (mod Z), the ground state is a valence-bond solid state with no broken symmetry. Related topological effects for the d=1 chain lead to spin-Peierls order for odd nc. These results are for a class of models which have, at sites of the A sublattice, representations of SU(N) described by a Young tableau with a single row, and the conjugate on the B sublattice. Similar results are also obtained for representations with m rows, using U(m) gauge theory.
In this thesis, we investigate strongly correlated phenomena in quantum spin systems. In the first part of this work, we study geometrically frustrated antiferromagnets (AFMs). Generalizing the SU(2) Heisenberg Hamiltonian to Sp(N) symmetry, we obtain, in the large-N limit, the mean-field phase diagrams for the planar pyrochlore and cubic AFMs. We then use gauge theories to consider fluctuation effects about their respective mean-field configurations. We find, in addition to conventional Neel states, a plethora of novel magnetically disordered phases: two kinds of spin liquids, Z2 in 2+1D and U(1)in 3+1D, and several valence bond solids such as two and three-dimensional plaquette and columnar singlet states. We use the same approach to study the diamond lattice AFM which possesses extended classical ground state degeneracy. We demonstrate that quantum and entropic fluctuations lift this degeneracy in different ways. In the second part of the thesis, we study ultracold spinor atoms confined in optical lattices. We first demonstrate the feasibility of experimental realization of rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice. We show that the ground state of such disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense, and suggest an imaging experiment to detect the biaxial nematicity in such systems. Finally, using variational wavefunction methods, we study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with n = 1 and n = 2 atoms per lattice site. We show that the long-range dipolar interaction stabilizes a state characterized by antiferromagnetic chains made of ferromagnetically aligned spins. We also obtain the mean-field phase boundary for the superfluid-insulator transition, and show that inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition.
The progress of ultracold atoms renders numerous possibilities to investigate exotic magnetism and superfluidity, which are rarely observed in solid state systems. In this thesis, we will introduce two novel physical descriptions: "frustrated Cooper pairing" and "large-hyperfine spin physics". Geometric frustration in quantum magnetism refers to which magnetic interactions on different bonds cannot be simultaneously minimized, and usually Cooper pairing favors uniform phases among different lattice sites. Here, we introduce "frustration" in Cooper pairing in a fermionic p-orbital model. By mean- field calculations, we show that the system exhibits behavior analogous to frustrated magnetism, and an unconventional supersolid state with the f-wave symmetry. Next, we introduce large-spin physics. In usual condensed matter systems, large spin is not intriguing because large values of spin suppress quantum fluctuations. In contrast, in ultracold fermion systems, large-hyperfine spin enhances quantum fluctuations and brings exotic quantum magnetism. Here the simplest large-spin fermionic system, a spin-3/2 exchange system is proposed, which can be characterized by an Sp(4)/SO(5) symmetry. In one dimension, the ground states exhibit either a dimerized state with a finite spin gap or a gapless spin liquid state by means of the density matrix renormalization group method. In the latter case, the spin-spin correlation functions are identified to have 4-site periodicities, which behaves similarly to the SU(4) chain. In two dimension, we infer that there exist three competing phases: Neel ordering, columnar dimerization and 2 x 2 plaquette formation, in the thermodynamic limit by exact diagonalization calculation on small sizes. Finally we perform the projector Quantum Monte Carlo method to study another large-spin system: the half-filled SU(N) Hubbard model. We show that at half-filling there is no sign problem such that our simulations are accurate. By finite size scaling, it is clearly found that the magnetic Neel ordering can exist not only for N = 2 but also in the N = 4 case at strong interactions. For N ̲> 6 or N = 4 at small U, the numerical results do not have any prominent signal that the long-range ordering exists in the thermodynamic limit. Due to strong finite size effects and finite numerical accuracy, however, we are unable to make any conclusion to identify the physics in the regimes
It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a $\ensuremath{\Theta}$ term at $\ensuremath{\Theta}=2\ensuremath{\pi}$. In this work we study a three-dimensional (3d) SPT phase of a SU(2N) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold M. Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for N>1, this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2N) spin symmetry, or its subgroup SU(N)$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2N) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.
Rapid advancements in the experimental capabilities with ultracold alkaline-earth-like atoms (AEAs) bring to a surprisingly near term the prospect of performing quantum simulations of spin models and lattice field theories exhibiting SU( N ) symmetry. Motivated in particular by recent experiments preparing high density samples of strongly interacting 87 Sr atoms in a three-dimensional optical lattice, we develop a low-energy effective theory of fermionic AEAs which exhibits emergent multi-body SU( N )-symmetric interactions, where N is the number of atomic nuclear spin levels. Our theory is limited to the experimental regime of (i) a deep lattice, with (ii) at most one atom occupying each nuclear spin state on any lattice site. The latter restriction is a consequence of initial ground-state preparation. We fully characterize the low-lying excitations in our effective theory, and compare predictions of many-body interaction energies with direct measurements of many-body excitation spectra in an optical lattice clock. Our work makes the first step in enabling a controlled, bottom-up experimental investigation of multi-body SU( N ) physics.
In this thesis, we study the topological phases of quantum spin systems. One project is to investigate a class of anti-ferromagnetic SU(N) Heisenberg models, describing Mott insulators of fermionic ultra-cold alkaline earth atoms on the three-dimensional simple cubic lattice. Our large-N analysis maps a rich phase diagram. One particularly striking state we found spontaneously breaks lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state. In the other projects, we study the phenomenon of symmetry fractionalization on anyons as a tool to characterize two-dimensional symmetry enriched topological phases. In particular, we focus on how crystalline symmetries may fractionalize in gapped Z2 spin liquids. If the system has the symmetry of the square lattice, then there are 2080 symmetry fractionalization patterns possible. With exactly solvable models, we realize 487 of these in strictly two-dimensional systems. In addition, we succeed to understand why the remaining patterns cannot be found in the family of models we construct. Some can only appear on the surface of three-dimensional systems with non-trivial point group symmetry protected topological (pgSPT) order, whose boundary degrees of freedom transform non-locally under the symmetries. We construct a simple toy model to show this anomalous crystalline symmetry fractionalization phenomenon associated with a reflection. Moreover, our approach establishes the connection between the pgSPT phases and the topological phases with on-site symmetries in lower dimensions. This insight is very useful for classification of pgSPT orders in general.