This thesis investigates the plausibility of producing a quantum spin liquid (QSL) with ultracold bosonic atoms optically confined to the Mott insulating state. QSLs have received a great deal of attention for being an antiferromagnetic groundstate with many exotic properties, including the absence of local order, long-range entanglement, and fractionalized excitations. However, the identification and characterization of these phases in solid state systems remains a great challenge. Here we outline an alternate route to uncovering the QSL phase, which from the nature of spin angular momentum for ultracold atoms encounters many properties unique to these systems along the way. This proposal is possible because the magnetic exchange interactions for Mott insulating ultracold atoms are mediated by the hopping of whole atoms. Whole-atom exchange—a unique property of cold atoms—allows large fluctuations between the quantized Zeeman sublevels of each atomic spin. As we demonstrate, these fluctuations increase when large-spin atoms are used, or when interactions are tuned via optical Feshbach resonance (OFR). These strong quantum spin fluctuations inhibit classical magnetic ordering, and lead to a QSL ground state. To illustrate the relationship between the spin magnitude, interaction strength, and QSL ground state, we present two distinct approaches to solving the relevant Hamiltonian. With mean field theory we find that for large spin (f > 2), and strong scattering through the spin-singlet channel, that magnetically-ordered Bose-Einstein condensates are unstable to the formation of a QSL. We then utilize Rayleigh-Schodinger perturbation theory to derive an effective Hamiltonian for our system in the Hilbert space of nearest-neighbor singlet coverings. At large spin this Hamiltonian produces a type of quantum dimer model (QDM), which are known to possess QSL phases. We derive the QDM parameters t, t′ and V as a function of spin on several lattices, finding they scale with the inverse number of Zeeman sublevels. We then determine the proximity of the physically accessible states to the QSL phase, and discuss how other regions of the phase diagram may be accessed. We then conclude by highlighting several advantages to studying QSLs and QDMs with ultracold bosons.
This thesis investigates the plausibility of producing a quantum spin liquid (QSL) with ultracold bosonic atoms optically confined to the Mott insulating state. QSLs have received a great deal of attention for being an antiferromagnetic groundstate with many exotic properties, including the absence of local order, long-range entanglement, and fractionalized excitations. However, the identification and characterization of these phases in solid state systems remains a great challenge. Here we outline an alternate route to uncovering the QSL phase, which from the nature of spin angular momentum for ultracold atoms encounters many properties unique to these systems along the way. This proposal is possible because the magnetic exchange interactions for Mott insulating ultracold atoms are mediated by the hopping of whole atoms. Whole-atom exchange—a unique property of cold atoms—allows large fluctuations between the quantized Zeeman sublevels of each atomic spin. As we demonstrate, these fluctuations increase when large-spin atoms are used, or when interactions are tuned via optical Feshbach resonance (OFR). These strong quantum spin fluctuations inhibit classical magnetic ordering, and lead to a QSL ground state. To illustrate the relationship between the spin magnitude, interaction strength, and QSL ground state, we present two distinct approaches to solving the relevant Hamiltonian. With mean field theory we find that for large spin (f > 2), and strong scattering through the spin-singlet channel, that magnetically-ordered Bose-Einstein condensates are unstable to the formation of a QSL. We then utilize Rayleigh-Schodinger perturbation theory to derive an effective Hamiltonian for our system in the Hilbert space of nearest-neighbor singlet coverings. At large spin this Hamiltonian produces a type of quantum dimer model (QDM), which are known to possess QSL phases. We derive the QDM parameters t, t′ and V as a function of spin on several lattices, finding they scale with the inverse number of Zeeman sublevels. We then determine the proximity of the physically accessible states to the QSL phase, and discuss how other regions of the phase diagram may be accessed. We then conclude by highlighting several advantages to studying QSLs and QDMs with ultracold bosons.
Mott insulating ultracold gases possess a unique whole-atom exchange interaction which enables large quantum fluctuations between the Zeeman sublevels of each atom. By strengthening this interaction---either through the use of large-spin atoms or by tuning the particle-particle interactions via optical Feshbach resonance---one may enhance fluctuations and facilitate the appearance of the long-sought-after quantum spin liquid phase---all in the highly tunable environment of cold atoms. To illustrate the relationship between the spin magnitude, interaction strength, and resulting magnetic phases, we present and solve a mean-field theory for bosons optically confined to the one-particle-per-site Mott state, using both analytic and numerical methods. We find on square and triangular lattices for bosons of hyperfine spin f>2 that making the repulsive s-wave scattering length through the singlet channel small---relative to the higher-order scattering channels---accesses a short-range resonating valence bond (s-RVB) spin liquid phase.
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.
Frustrated magnetism provides a rich platform for realizing exotic quantum phases in condensed matter systems. Among these, quantum spin liquids (QSLs) are a particularly intriguing class of states that exhibit no magnetic ordering and preserve all symmetries down to zero temperature. Just as conventional magnets can be thought of as spin crystals,QSLs represent their liquid analogs. This thesis explores two broad types of QSLs in two-dimensional magnets, with a focus on their dynamic properties and the nature of transitions from conventional magnetic states into QSL phases. QSLs are topological phases that host fractionalized excitations known as spinons. In these states, mobile spinons are coupled to an emergent gauge field, forming a highly entangled phase of matter. The first part of this thesis focuses on Dirac spin liquids (DSLs), a class of QSLs where spinons exhibit a gapless Dirac dispersion. To study the dynamicsof these spin liquids beyond leading-order mean-field theory, we develop a novel large-N fermionic limit of gapless spinons. In this limit, all gauge excitations are suppressed, and the leading-order corrections instead arise from order parameter fluctuations, which may condense and drive transitions into symmetry-broken phases from the parent DSL. Incorporating random phase approximation (RPA) corrections to the spin susceptibility, we find that sharp, spin-wave–like modes are recovered in the excitation spectrum. This methodology successfully captures characteristic features of the dynamics near phase transitions and in weakly symmetry-broken phases proximate to a DSL. We apply this approach to microscopic models, including the J1-J2 XXZ model on the triangular lattice and the J1-J3 XXZ model on the honeycomb lattice. These models closely represent a wide range of frustrated materials, such as triangular and honeycomb cobaltates and triangular ytterbium compounds. Our theoretical predictions show excellent agreement with experimentally measured inelastic neutron scattering spectra from these materials. Gapless spinons can also sometimes become gapped through chiral symmetry breaking, leading to the formation of a chiral spin liquid (CSL). The CSL is an archetypal bosonic realization of a fractional quantum Hall (FQH) state at half-filling, and has been proposed as a time-reversal–breaking ground state in a variety of frustrated spin models. The second part of this thesis addresses the intriguing possibility of realizing a CSL through the "melting" of a chiral spin crystal—a conventional magnet with non-coplanar ordering. Such a transition offers a physically appealing analogue to the melting of a Wigner crystal into a fermionic FQH liquid. Focusing on the kagome lattice, where chiral interactions are knownto stabilize a CSL ground state, we study the effects of additional long-range Heisenberg interactions and demonstrate a phase transition into a non-coplanar octahedral magnetic order. To describe the universal class of transition between a CSL and a non-coplanar anti-ferromagnetic order, we then develop a novel Chern-Simons–matter field theory framework, constrained by symmetry principles and symmetry fractionalization patterns. This thesis thus provides new insights into the dynamical signatures and phase transitions of quantum spin liquids, offering a bridge between theoretical models and experimental observations in frustrated magnetic systems.
The strong interest in strongly correlated systems in condensed matter physics has continued unabated for the past few decades. In recent years, the number of novel, exotic quantum phases found in theoretical studies has seen a phenomenal rise. Among those interesting quantum states are bose liquids and spin liquids, where strong quantum fluctuations have prevented the systems from developing a long range order. Our work in this thesis seeks to further the understanding of frustrated systems. In the study of a hard-core boson model with ring-only exchange interactions on a square lattice, we obtain concrete numerical realization of the unconventional Exciton Bose Liquid (EBL) phase, which possesses interesting properties such as a "Bose surface" which resembles the Fermi surface in a metal, as well as unusual thermodynamic properties such as a T Log T dependence for specific heat. An equally important result from this work is the demonstration that the widely used Gutzwiller projection on slave-particle wave functions may generally fail to capture the correct long wavelength physics in the respective systems. For the Heisenberg antiferromagnet on the kagome lattice, which is a promising candidate for realizing a spin-disordered ground state, our variational study shows that the projected Schwinger boson wave function is energetically better than the Dirac spin liquid wave function when a small antiferromagnetic second-neighbor spin coupling is added to the nearest-neighbor model. We also study the anisotropic triangular Heisenberg antiferromagnetic in magnetic field, and find simple, yet accurate wave functions for various regions of the surprisingly rich phase diagram, thus providing insights into the energetics of the competing phases in this interesting model. Finally, our work also highlights permanent-type wave functions as potentially useful constructions in variational studies of systems with short-ranged correlations, e.g., a Mott insulator and a gapped spin liquid.
We propose an experimental protocol for using cold atoms to create and probe quantum dimer models, thereby exploring the Pauling-Anderson vision of a macroscopic collection of resonating bonds. This process can allow the study of exotic crystalline phases, fractionalization, topological spin liquids, and the relationship between resonating dimers and superconductivity subjects which have been challenging to address in solid-state experiments. Our key technical development is considering the action of an off-resonant photoassociation laser on large spin atoms localized at the sites of a deep optical lattice. The resulting superexchange interaction favors nearest-neighbor singlets. We derive an effective Hamiltonian in terms of these dimer degrees of freedom, finding that it is similar to well-known quantum dimer models, which boast a rich variety of valence bond crystal and spin liquid phases. We numerically study the ground state, explain how to tune the parameters, and develop a protocol to directly measure the dimers and their resonating patterns.
The quantum nature of electrons emerges in various exotic ways when they are put together with quantum many-body interactions. In Mott insulating systems with strong electron correlation, frustration and strong quantum fluctuations promote localized electron spins to form nontrivial quantum paramagnetic phases such as quantum spin liquid and valence bond solid preventing the development of conventional magnetic orders. In this thesis, we study such exotic quantum phases in various different frustrated quantum magnets. First, encouraged by the spin liquid phenomenology observed in Ba3XSb2O9 (X=Cu, Ni), we investigate Z2 spin liquids on a three dimensional triangular lattice by employing the Schwinger boson mean-field theory with projective symmetry group analysis. It is found that a nematic spin liquid is energetically favorable on this lattice structure rather than symmetric spin liquids. Next, we study various valence bond solid phases that can appear in kagome lattice antiferromagnets by generalizing the recently proposed phenomenological Z2 gauge theory for the topological Z2 spin liquid state found in the nearest-neighbor kagome lattice Heisenberg model. This work provides a bridge between the topological spin liquid and valence bond solid phases previously proposed for kagome lattice antiferromagnets via vison condensation in the gauge theory. Then, we consider optical conductivity in Mott insulators as an experimental probe to detect quantum paramagnetic phases like valence bond solid. Applying this idea to deformed kagome lattice antiferromagnet Rb2Cu3SnF12 with a 12-site valence bond solid state, we estimate the size of optical conductivity in this quantum magnet. After this, another quantum paramagnet BiCu2PO6 is carefully investigated to clarify its puzzling ground state nature and quasiparticle excitations. Utilizing recent high resolution neutron scattering data, we construct a microscopic spin model and also an effective theory for the low energy spin excitations in the compound. We find significant influences of magnetic anisotropies on the ground state and quasiparticle dynamics. Lastly, we investigate emergent quantum phases in the thin film geometries of the pyrochlore iridates, where a number of exotic quantum ground states are proposed to occur in bulk materials as a result of the interplay between electron correlation and strong spin-orbit coupling. The fate of these bulk phases, as well as the novel quantum states that may arise only in a thin film setting, are studied via a theoretical model that allows layer-dependent magnetic structures.