Motivated by experiments on ultracold atoms which have realized the Haldane model for a Chern insulator, we consider its strongly correlated Mott limit with spin-1/2 fermions. A slave rotor mean field theory suggests the appearance of gapped or gapless chiral spin liquid Mott insulators. To study competing magnetic orders, we consider the strong coupling effective spin Hamiltonian which includes chiral three-spin exchange. We obtain its classical phase diagram, uncovering various chiral magnetic orders including tetrahedral, cone, and noncoplanar spiral states which can compete with putative chiral quantum spin liquids. We study the effect of thermal fluctuations on these states, identifying crossovers in the spin chirality, and phase transitions associated with lattice symmetry breaking. We also discuss analogous effective spin Hamiltonians for correlated spin-1/2 bosons.
Motivated by experiments on ultracold atoms which have realized the Haldane model for a Chern insulator, we consider its strongly correlated Mott limit with spin-1/2 fermions. A slave rotor mean field theory suggests the appearance of gapped or gapless chiral spin liquid Mott insulators. To study competing magnetic orders, we consider the strong coupling effective spin Hamiltonian which includes chiral three-spin exchange. We obtain its classical phase diagram, uncovering various chiral magnetic orders including tetrahedral, cone, and noncoplanar spiral states which can compete with putative chiral quantum spin liquids. We study the effect of thermal fluctuations on these states, identifying crossovers in the spin chirality, and phase transitions associated with lattice symmetry breaking. We also discuss analogous effective spin Hamiltonians for correlated spin-1/2 bosons.
Motivated by experiments on ultracold atoms which have realized the Haldane model for a Chern insulator, we consider its strongly correlated Mott limit with spin-21 fermions. We find that slave rotor mean-field theory yields gapped or gapless chiral spin liquid Mott insulators. To study competing magnetic orders, we consider the strong coupling effective spin Hamiltonian which includes chiral three-spin exchange. We obtain its classical phase diagram, uncovering various chiral magnetic orders including tetrahedral, cone, and noncoplanar spiral states which can compete with putative chiral quantum spin liquids. We study the effect of thermal fluctuations on these states, identifying crossovers in the spin chirality, and phase transitions associated with lattice symmetry breaking. We also discuss analogous effective spin Hamiltonians for correlated spin-21 bosons. Finally, we point out possible experimental implications of our results for cold atom experiments.
Motivated by cold atom experiments on Chern insulators, we study the honeycomb lattice Haldane-Hubbard Mott insulator of spin-1/2 fermions using exact diagonalization and density matrix renormalization group methods. We show that this model exhibits various chiral magnetic orders including a wide regime of triple-Q tetrahedral order. Incorporating third-neighbor hopping frustrates and ultimately melts this tetrahedral spin crystal. From analyzing the low energy spectrum, many-body Chern numbers, entanglement spectra, and modular matrices, we identify the molten state as a chiral spin liquid (CSL) with gapped semion excitations. We formulate and study the Chern-Simons-Higgs field theory of the exotic CSL-to-tetrahedral spin crystallization transition.
The phase diagram of the strongly correlated Hubbard model with intrinsic spin-orbit coupling on the honeycomb lattice is explored here. We obtain the low-energy effective model describing the spin degree of freedom. The resulting model is then studied by the Schwinger boson and Schwinger fermion approaches. The Schwinger boson method elucidates the boundary between the spin liquid phase and the magnetically ordered phases, Neel order, and incommensurate Neel order. Increasing the strength of the spin-orbit coupling is shown to narrow the width of the spin liquid region. The Schwinger fermion approach sheds further light on the nature of the spin liquid phase. We obtained three different candidates for the spin liquid phase within the mean-field approximation, namely, the gapless spin liquid, topological Mott insulator (fractionalized topological insulator), and chiral spin liquid phases. However, we argue that the gauge fluctuations and the instanton effect may suppress the first two spin liquids, while the chiral spin liquid is stable against gauge fluctuations due to its nontrivial topology.
Recently, the frustrated XY model for spins 1/2 on the honeycomb lattice has attracted a lot of attention in relation with the possibility to realize a chiral spin liquid state. This model is relevant to the physics of some quantum magnets. Using the flexibility of ultracold atom setups, we propose an alternative way to realize this model through the Mott regime of the bosonic Kane-Mele-Hubbard model. The phase diagram of this model is derived using bosonic dynamical mean-field theory. Focusing on the Mott phase, we investigate its magnetic properties as a function of frustration. We do find an emergent chiral spin state in the intermediate frustration regime. Using exact diagonalization we study more closely the physics of the effective frustrated XY model and the properties of the chiral spin state. This gapped phase displays a chiral order, breaking time-reversal and parity symmetry, but is not topologically ordered (the Chern number is zero).
In recent years, topology has become increasingly prevalent in condensed matter physics. It has allowed us to understand, and even predict, a variety of striking and remarkable physical phenomena. The study of strongly interacting systems has similarly lavished us with a diverse range of exotic phases and unconventional transitions, many of which are still poorly understood. In this thesis we will explore the interplay between topology and interactions in an effort to uncover new and novel phases. First we study how interactions impact the quantum phase transition between a topologically non-trivial phase and a trivial phase. The combination of interactions and the low-energy degrees of freedom associated with the transition leads to the emergence of a dome of lattice-symmetry breaking nematic order. Such behaviour is reminiscent of a number of strongly correlated electronic systems. We move on to study the strongly interacting limit of one of the earliest and best-known non-interacting topological phases, Haldane's model of a Chern insulator. Recently realized with ultracold atoms in a shaken optical lattice, the model has a non-trivial topological invariant associated with its band structure. In the strongly interacting limit the spin degrees of freedom are all that survive and we find a rich phase diagram of magnetically ordered phases, using a combination of both classical and quantum techniques. Supplementing the model with an additional term we can `quantum-melt' one of these ordered states to produce a disordered, liquid state that we positively identify as a chiral spin liquid, a highly entangled state of matter with fractionalised excitations. We generalise this mechanism to other two dimensional lattices, uncovering a possible unifying framework with which to understand the emergence of chiral spin liquids in lattice spin models. Finally, motivated by groundbreaking experiments in the ultracold atoms community, we investigate a model of two-component bosons with an artificial spin-orbit coupling. The interplay between the lattice, interactions and spin-orbit coupling produces a variety of unusual superfluid phases. Using a novel Monte Carlo technique we reveal the finite temperature phase diagram that appears close to the Mott transition.
In this thesis, the Hubbard model with hopping amplitude, t, and on-site interaction, U, in the Mott-insulating regime is explored for unconventional magnetic phases at T = 0 K. These phases are captured within effective low-energy descriptions, and are triggered by quantum fluctuations due to finite hopping. We aim for unconventional phases from two mechanisms, an increased symmetry SU(N) with N > 2 compared to the most common SU(2), and highly frustrated interactions. For SU(N)-symmetric fermions in artificial gauge fields, chiral spin liquid phases (CSLs) are investigated. They are determined within effective spin models, which we derive in order six in t/U for the honeycomb lattice for general values of N, and study in detail for the triangular lattice up to order five in t/U for SU(3) and SU(7) by exact diagonalisations and variational Monte Carlo simulations. On the triangular lattice, the third-order effective model is the J -K model with nearest-neighbour coupling, J ∈ R, and ring exchange, K ∈ C. For SU(3) and Im K = 0, the three-sublattice long-range ordered phase is ruled out by a spontaneously time-reversal symmetry breaking π/3-flux CSL through � nite hopping. Further, for Im K = 0, a lattice nematic, a 120◦ long-ranged ordered, and the ferromagnetically ordered phase occur. The CSL is found to be extended to Re K = 0. It is also present in the fourth- and fifth-order model in the most relevant π-flux case, where time-reversal symmetry is spontaneously broken. Together with the estimated metal-insulator transition point at larger hoppings, this CSL is therefore expected to be realised in the SU(3) Hubbard model. The analogous CSL is also predicted for SU(7)-symmetric fermions. Experimental realisations with ultra-cold atoms in optical lattices are described. SrCu2(BO3)2 is a highly frustrated quantum magnet. At ambient pressure it is captured by the spin-1/2 Shastry-Sutherland model, which yields the effective strong-coupling description of the associated SU(2) Hubbard model. To describe SrCu2(BO3)2 under higher pressure, we extend the Shastry-Sutherland model by asymmetric coupling strengths, and find that asymmetries of a few percent are sufficient to replace the empty plaquette singlet phase (EPP) of the Shastry-Sutherland model with the filled plaquette singlet phase (FPP) that is found to be identical to a quasi one-dimensional Haldane phase. These results are achieved by series expansions and infinite projected entangled-pair states. The dispersions and dynamic structure factors of both plaquette phases are derived. The EPP naturally hosts one low-lying triplon, whereas the FPP has two. Therefore, the FPP appears to be the likely candidate for SrCu2(BO3)2 under pressure. The excitations of both phases show some agreement with experimental data if the exchange amplitudes change more than expected. Further measurements are suggested. At last, an orthogonal-plaquette model of spins-1/2 is introduced and proven to host two exact valence bond crystals based on plaquette singlets with an extensive degeneracy at the phase transition. This model exhibits an extensive number of conserved quantities, which is used for further investigations of the phase diagram.