Three dimensional frustrated magnets realized in the rare-earth pyrochlores R_2B_2O_7, R=rare-earth ion, B transition metal ion) have attracted much interest since the discovery of spin ice in Ho_2Ti_2O_7 by Harris et al. Since then, a number of interesting phenomena in pyrochlore magnets have been observed and the interest in these systems, from an experimental and theoretical point of view, has nothing but increased. Perhaps the most appealing and unique possibility is that these systems may realize a quantum disordered phase; a so-called quantum spin ice. Theoretically, it has been predicted by Hermele et al. that a U(1) quantum spin liquid is realized in the XXZ quantum spin ice (with global U(1)xZ_2 symmetry) in the perturbative regime and more recently Savary and Balents have introduced a quantum rotor model with bosonic spinons coupled to $U(1)$ gauge fields to describe non-peturbatively the deconfined phase predicted. They studied an anisotropic quantum spin ice (QSI) model which corresponds to the XXZ QSI with an additional anisotropic coupling. They solved the quantum rotor model using a mean-field approximation combined with a large-N approximation and predicted the possibility of a U(1) liquid in addition to an exotic ferromagnetic phase. Further work by Lee et al. has addressed the more complicated fully frustrated XXZ quantum spin ice regime, where all components of the model are frustrated, by using some insights provided by the previous perturbative results. \n \nIn this thesis, we will address the non-perturbative regime of the fully frustrated XXZ QSI beyond the simplest perturbative treatment. We will begin by mapping the quantum spin ice model to a compact U(1) gauge theory with deconfined electric and magnetic monopoles. Neglecting the interaction between the gapless photon excitation and the electric charges, we will study the matter sector with only electric charges, the so-called spinons. Building up on insights provided by high order degenerate perturbation theory calculations, we will construct some symmetric spin liquid ansatz for the static part of the gauge fields. The two novel spin liquids that we construct break time reversal symmetry and thus have a non-trivial chirality. In order to solve the matter sector of the theory and determine the stability of the deconfined phases, we will introduce a novel representation of a XY quantum rotor in terms of ``exclusive" bosons, a representation which is dynamically implemented in the dilute limit. We will benchmark this mapping in different ways by comparing our results with Quantum Monte Carlo simulations and exact diagonalization in some limiting cases. Using this mapping we will compute the region of stability of the U(1) liquid phase of XXZ QSI. Within this mapping we will reproduce qualitatively the results of previous work by finding only the so-called zero and \\pi-flux U(1) liquids in the vicinity of the classical spin ice point, while improving greatly on the estimation of the exchange coupling at which the bosonic spinons condense to form a superfluid phase. In the fully frustrated XXZ QSI, we will provide evidence, using exact diagonalization of the quantum rotor model, that one of the chiral spin liquid may be realized in the strongly correlated and frustrated QSI regime, where all previous approximations break down.
Three dimensional frustrated magnets realized in the rare-earth pyrochlores R_2B_2O_7, R=rare-earth ion, B transition metal ion) have attracted much interest since the discovery of spin ice in Ho_2Ti_2O_7 by Harris et al. Since then, a number of interesting phenomena in pyrochlore magnets have been observed and the interest in these systems, from an experimental and theoretical point of view, has nothing but increased. Perhaps the most appealing and unique possibility is that these systems may realize a quantum disordered phase; a so-called quantum spin ice. Theoretically, it has been predicted by Hermele et al. that a U(1) quantum spin liquid is realized in the XXZ quantum spin ice (with global U(1)xZ_2 symmetry) in the perturbative regime and more recently Savary and Balents have introduced a quantum rotor model with bosonic spinons coupled to U(1) gauge fields to describe non-peturbatively the deconfined phase predicted. They studied an anisotropic quantum spin ice (QSI) model which corresponds to the XXZ QSI with an additional anisotropic coupling. They solved the quantum rotor model using a mean-field approximation combined with a large-N approximation and predicted the possibility of a U(1) liquid in addition to an exotic ferromagnetic phase. Further work by Lee et al. has addressed the more complicated fully frustrated XXZ quantum spin ice regime, where all components of the model are frustrated, by using some insights provided by the previous perturbative results. \n \nIn this thesis, we will address the non-perturbative regime of the fully frustrated XXZ QSI beyond the simplest perturbative treatment. We will begin by mapping the quantum spin ice model to a compact U(1) gauge theory with deconfined electric and magnetic monopoles. Neglecting the interaction between the gapless photon excitation and the electric charges, we will study the matter sector with only electric charges, the so-called spinons. Building up on insights provided by high order degenerate perturbation theory calculations, we will construct some symmetric spin liquid ansatz for the static part of the gauge fields. The two novel spin liquids that we construct break time reversal symmetry and thus have a non-trivial chirality. In order to solve the matter sector of the theory and determine the stability of the deconfined phases, we will introduce a novel representation of a XY quantum rotor in terms of ``exclusive" bosons, a representation which is dynamically implemented in the dilute limit. We will benchmark this mapping in different ways by comparing our results with Quantum Monte Carlo simulations and exact diagonalization in some limiting cases. Using this mapping we will compute the region of stability of the U(1) liquid phase of XXZ QSI. Within this mapping we will reproduce qualitatively the results of previous work by finding only the so-called zero and \pi-flux U(1) liquids in the vicinity of the classical spin ice point, while improving greatly on the estimation of the exchange coupling at which the bosonic spinons condense to form a superfluid phase. In the fully frustrated XXZ QSI, we will provide evidence, using exact diagonalization of the quantum rotor model, that one of the chiral spin liquid may be realized in the strongly correlated and frustrated QSI regime, where all previous approximations break down.
We carry out an analytical study of quantum spin ice, a U(1) quantum spin liquid close to the classical spin ice solution for an effective spin 1/2 model with anisotropic exchange couplings Jzz, J± and Jz± on the pyrochlore lattice. Starting from the quantum rotor model introduced by Savary and Balents in Phys. Rev. Lett. 108, 037202 (2012), we retain the dynamics of both the spinons and gauge field sectors in our treatment. The spinons are described by a bosonic representation of quantum XY rotors while the dynamics of the gauge field is captured by a phenomenological Hamiltonian. By calculating the one-loop spinon self-energy, which is proportional to Jz±2, we determine the stability region of the U(1) quantum spin liquid phase in the J±/Jzz vs Jz±/Jzz zero temperature phase diagram. From these results, we estimate the location of the boundaries between the spin liquid phase and classical long-range ordered phases.
The presence of multiple competing interactions in spin systems can lead to novel phases known as quantum spin liquids (QSLs). QSLs are disordered spin systems with long-range entanglement that realize the deconfined phase of emergent lattice gauge theories and host fractional excitations. A notable example of great experimental interest is quantum spin ice (QSI), a three-dimensional QSL that realizes an analog of quantum electrodynamics with emergent photons and interacting electric and magnetic charges. The current leading candidate materials for realizing QSI are dipolar-octupolar (DO) cerium-based pyrochlore compounds. These are believed to stabilize a U(1) QSL known as π-flux QSI, where translation symmetry acting on spinon excitations fractionalizes. In this thesis, we explore the physics of QSI, relying heavily on parton constructions, to make experimentally relevant predictions for DO materials. First, we explore the physics of symmetry fractionalization in QSI, whereby lattice symmetries are realized projectively when acting on fractional quasiparticles. Symmetry fractionalization is examined within the framework of gauge mean-field theory (GMFT), a parton construction designed explicitly for application to QSI. This study shows that the pyrochlore lattice space group strongly constrains possible symmetry fractionalization classes. It systematically explains how all space group transformations, beyond translation, fractionalize in π-flux QSI. Next, we apply this framework to DO materials and show how the equal-time structure factor predicted for π-flux QSI is consistent with recent energy-integrated inelastic neutron scattering experiments on certain candidate materials. We also predict that the dynamical spin structure factor of π-flux QSI is characterized by a broad continuum with three distinctive peaks of decreasing intensity that are approximately equally spaced. We argue that these peaks should be accessible with currently available experimental apparatus and offer a smoking-gun signature of π-flux QSI. We confirm those predictions by extending the exclusive boson representation and speculate on the thermodynamic behavior of cerium candidates. We close by commenting on recent experimental breakthroughs and possible future directions.
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 low-energy physics of quantum spin ice is known to support an emergent form of quantum electrodynamics (QED), where magnetic monopoles exist and the fine-structure constant is material dependent. In this article, we show how this QED is modified via a coupling to dynamical matter on the centered pyrochlore lattice, a structure that has recently been synthesized using metal-organic frameworks. Specifically, we study the low-energy properties of the S = 1 / 2 quantum XXZ model on the centered pyrochlore lattice, with a focus on the sign-problem-free region. At fourth order in degenerate perturbation theory, this model hosts a quantum spin liquid distinct from the well-known U(1) quantum spin ice on the pyrochlore because of the presence of dynamical matter in the ground state. Exact diagonalization results are consistent with this quantum spin liquid over an extended region of the ground-state phase diagram, although potential quantum critical points within this region could indicate a richer phase structure. Our work thus expands the physics of quantum spin ice in an experimentally motivated geometry, showing how the emergent QED can be coupled to dynamical matter at zero temperature.
We study the S=1/2 Heisenberg antiferromagnet on the pyrochlore lattice in the limit of strong easy-axis exchange anisotropy. We find, using only standard techniques of degenerate perturbation theory, that the model has a U(1) gauge symmetry generated by certain local rotations about the z axis in spin space. Upon addition of an extra local interaction in this and a related model with spins on a three-dimensional network of corner-sharing octahedra, we can write down the exact ground-state wave function with no further approximations. Using the properties of the soluble point we show that these models enter the U(1) spin liquid phase, a fractionalized spin liquid with an emergent U(1) gauge structure. This phase supports gapped Sz=1/2 spinons carrying the U(1) ``electric'' gauge charge, a gapped topological point defect or ``magnetic'' monopole, and a gapless ``photon,'' which in spin language is a gapless, linearly dispersing Sz=0 collective mode. There are power-law spin correlations with a nontrivial angular dependence, as well as U(1) topological order. This state is stable to all zero-temperature perturbations and exists over a finite extent of the phase diagram. Using a convenient lattice version of electric-magnetic duality, we develop the effective description of the U(1) spin liquid and the adjacent soluble point in terms of Gaussian quantum electrodynamics and calculate a few of the universal properties. The resulting picture is confirmed by our numerical analysis of the soluble point wave function. Finally, we briefly discuss the prospects for understanding this physics in a wider range of models and for making contact with experiments.
Quantum spin liquids are exotic phases of matter whose low-energy physics is described as the deconfined phase of an emergent gauge theory. With recent theory proposals and an experiment showing preliminary signs of Z 2 topological order [G. Semeghini , ], Rydberg atom arrays have emerged as a promising platform to realize a quantum spin liquid. In this work, we propose a way to realize a U(1) quantum spin liquid in three spatial dimensions, described by the deconfined phase of U(1) gauge theory in a pyrochlore lattice Rydberg atom array. We study the ground state phase diagram of the proposed Rydberg system as a function of experimentally relevant parameters. Within our calculation, we find that by tuning the Rabi frequency, one can access both the confinement-deconfinement transition driven by a proliferation of “magnetic” monopoles and the Higgs transition driven by a proliferation of “electric” charges of the emergent gauge theory. We suggest experimental probes for distinguishing the deconfined phase from ordered phases. This work serves as a proposal to access a confinement-deconfinement transition in three spatial dimensions on a Rydberg-based quantum simulator.