Condensed matter exhibits a wide variety of exotic emergent phenomena such as the fractional quantum Hall effect and the low temperature cooperative behavior of highly frustrated magnets. I consider the classical Hamiltonian dynamics of spins of the latter phenomena using a method introduced by Dirac in the 1950s by assuming they are constrained to their lowest energy configurations as a simplifying measure. Focusing on the kagome antiferromagnet as an example, I find it is a gauge system with topological dynamics and non-locally connected edge states for certain open boundary conditions similar to doubled Chern–Simons electrodynamics expected of a Z 2 spin liquid. These dynamics are also similar to electrons in the fractional quantum Hall effect. The classical theory presented here is a first step toward a controlled semi-classical description of the spin liquid phases of many pyrochlore and kagome antiferromagnets and toward a description of the low energy classical dynamics of the corresponding unconstrained Heisenberg models.
Condensed matter exhibits a wide variety of exotic emergent phenomena such as the fractional quantum Hall effect and the low temperature cooperative behavior of highly frustrated magnets. I consider the classical Hamiltonian dynamics of spins of the latter phenomena using a method introduced by Dirac in the 1950s by assuming they are constrained to their lowest energy configurations as a simplifying measure. Focusing on the kagome antiferromagnet as an example, I find it is a gauge system with topological dynamics and non-locally connected edge states for certain open boundary conditions similar to doubled Chern–Simons electrodynamics expected of a Z 2 spin liquid. These dynamics are also similar to electrons in the fractional quantum Hall effect. The classical theory presented here is a first step toward a controlled semi-classical description of the spin liquid phases of many pyrochlore and kagome antiferromagnets and toward a description of the low energy classical dynamics of the corresponding unconstrained Heisenberg models.
Abstract. Condensed matter exhibits a wide variety of exotic emergent phenomena such as the fractional quantum Hall effect and the low temperature cooperative behavior of highly frustrated magnets. I consider the classical Hamiltonian dynamics of spins of the latter phenomena using a method introduced by Dirac in the 1950s by assuming they are constrained to their lowest energy configurations as a simplifying measure. Focusing on the kagome antiferromagnet as an example, I find it is a gauge system with topological dynamics and non-locally connected edge states for certain open boundary conditions similar to doubled Chern-Simons electrodynamics expected of a Z2 spin liquid. These dynamics are also similar to electrons in the fractional quantum Hall effect. The classical theory presented here is a first step towards a controlled semi-classical description of the spin liquid phases of many pyrochlore and kagome antiferromagnets and towards a description of the low energy classical dynamics of the corresponding unconstrained Heisenberg models. ar X iv
Interactions between the microscopic constituents of a solid---a many-body system--- can lead to novel phases and exotic physical phenomena like fractionalization, topological order, quantum spin liquids, emergent gauge field, etc.. The concept of frustration provides a ground for such exotic phenomena. Frustration can prevent a many-body system from establishing long-range order down to the lowest temperatures due to competing interactions. Instead, competing interactions may result in disordered and liquid-like phases of matter that provide the vacuum for fractional excitations. The absence of any order parameter in strongly frustrated systems---due to not breaking any symmetry spontaneously--- immediately raises the question about possible experimental probes of spin-liquids and their fractional excitations. Dynamic probes, like inelastic neutron scattering or Raman scattering, provide an experimental method to detect signatures of fractionalised quasiparticles. The energy and momentum transferred in a scattering event is split between the fractional quasiparticles. On the theory side, computing such dynamical signatures beyond one spatial dimension is generally a difficult task. In this thesis, numerical methods like density matrix renormalisation group and matrix product states are used to study strongly frustrated magnets and their dynamics in a non-perturbative way. This thesis covers two physical models in the context of frustration and emergent gauge fields. Firstly, the Kitaev model of spin-1/2 degrees of freedom subject to strongly anisotropic spin exchange. The Kitaev model features quantum spin liquid ground states with fractionalization of spins into Majorana fermions and Z_2-fluxes---the visons of an emergent Z_2 gauge theory. The main questions addressed here concern the stability of the quantum spin liquid phase upon adding perturbations relevant in magnetic compounds such as Heisenberg or the symmetric-offdiagonal Gamma exchange. Applying a magnetic field drives the Kitaev model into a topologically ordered phase. The excitations and dynamical signatures within the spin liquid, the topologically ordered phase, and within ordered phases are studied. Secondly, a classical minimal model of the proton configuration in water ice is studied. The ice rules, a local constraint describing the low energy manifold, result in emergent Maxwell's equation. Upon applying an external electric field along certain axis, a polarization plateau occurs in which the remaining degrees of freedom can be described by dimers on two-dimensional lattices.
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.
Frustrated spin systems on Kagome lattices have long been considered to be a promising candidate for realizing exotic spin liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of experimental materials such as Volborthite and Herbertsmithite that have Kagome like structures. In this thesis I will focus on studying frustrated spin systems on the Kagome lattice using a spin-1/2 antiferromagnetic XXZ Heisenberg model in the presence of an external magnetic field as well as other perturbations. Such a system is expected to give rise to magnetization platueaus which can exhibit topological characteristics in certain regimes. We will first develop a flux-attachment transformation that maps the Heisenberg spins (hard-core bosons) onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Using this newly developed mapping we analyse the phases/magnetization plateaus that arise at the mean-field level and also consider the effects of adding fluctuations to various mean-fi eld states. Along the way, we show how to discretize an abelian Chern-Simons gauge theory on generic 2D planar lattices that satisfy certain conditions. We find that as long as there exists a one-to-one correspondence between the vertices and plaquettes defined on the graph, one can write down a discretized lattice version of the abelian Chern-Simons gauge theory. Using the newly developed flux attachment transformation, we show the existence of chiral spin liquid states for various magnetization plateaus for certain range of parameters in the XXZ Heisenberg model in the presence of an external magnetic field. Speci cally, in the regime of XY anisotropy the ground states at the 1/3 and 2/3 plateau are equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction 1/2 and that the 5/9 plateau is equivalent to the first bosonic Jain daughter state at filling fraction 2/3. Next, we also consider the effects of several perturbations: a) a chirality term, b) a Dzyaloshinskii-Moriya term, and c) a ring-exchange type term on the bowties of the kagome lattice, and inquire if they can also support chiral spin liquids as ground states. We find that the chirality term leads to a chiral spin liquid even in the absence of an uniform magnetic field, with an effective spin Hall conductance of 1/2 in the regime of XY anisotropy. The Dzyaloshinkii-Moriya term also leads a similar chiral spin liquid but only when this term is not too strong. An external magnetic field when combined with some of the above perturbations also has the possibility of giving rise to additional plateaus which also behave like chiral spin liquids in the XY regime. Under the in influence of a ring-exchange term we find that provided its coupling constant is large enough, it may trigger a phase transition into a chiral spin liquid by the spontaneous breaking of time-reversal invariance. Finally, we also present some numerical results based on some exact diagonalization studies. Here, we specifically focus on the 2/3-magnetization plateau which we previously argued should be a chiral spin liquid with a spin hall conductance of 1/2 . Such a topological state has a non-trivial ground state degeneracy and it excitations are described by semionic quasiparticles. In the numerical analysis, we analyse the ground state degeneracy structure on various Kagome clusters of different sizes. We compute modular matrices from the resultant minimally entangled states as well as the Chern numbers of various eigenstates all of which provide strong evidence that the 2/3-magnetization plateau very closely resembles a chiral spin liquid state with the expected characteristics.
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.
The history of modern condensed matter physics may be regarded as the competition and reconciliation between Stoner’s and Anderson’s physical pictures, where the former is based on momentum–space descriptions focusing on long wave-length fluctuations while the latter is based on real-space physics emphasizing emergent localized excitations. In particular, these two view points compete with each other in various nonperturbative phenomena, which range from the problem of high [Formula: see text] superconductivity, quantum spin liquids in organic materials and frustrated spin systems, heavy-fermion quantum criticality, metal-insulator transitions in correlated electron systems such as doped silicons and two-dimensional electron systems, the fractional quantum Hall effect, to the recently discussed Fe-based superconductors. An approach to reconcile these competing frameworks is to introduce topologically nontrivial excitations into the Stoner’s description, which appear to be localized in either space or time and sometimes both, where scattering between itinerant electrons and topological excitations such as skyrmions, vortices, various forms of instantons, emergent magnetic monopoles, and etc. may catch nonperturbative local physics beyond the Stoner’s paradigm. In this review paper, we discuss nonperturbative effects of topological excitations on dynamics of correlated electrons. First, we focus on the problem of scattering between itinerant fermions and topological excitations in antiferromagnetic doped Mott insulators, expected to be relevant for the pseudogap phase of high [Formula: see text] cuprates. We propose that nonperturbative effects of topological excitations can be incorporated within the perturbative framework, where an enhanced global symmetry with a topological term plays an essential role. In the second part, we go on to discuss the subject of symmetry protected topological states in a largely similar light. While we do not introduce itinerant fermions here, the nonperturbative dynamics of topological excitations is again seen to be crucial in classifying topologically nontrivial gapped systems. We point to some hidden links between several effective field theories with topological terms, starting with one-dimensional physics, and subsequently finding natural generalizations to higher dimensions.