We introduce a geometrically frustrated classical Ising model, dubbed the “spin-vorticity model,” whose ground-state manifold is a classical spin liquid: a 2-form Coulomb phase. We study the thermodynamics of this model both analytically and numerically, exposing the presence of algebraically decaying correlations and an extensive ground-state entropy, and give a comprehensive account of its ground-state properties and excitations. Each classical ground state may be decomposed into collections of closed two-dimensional membranes, supporting fractionalized string excitations attached to the edges of open membranes. At finite temperature, the model can then be described as a gas of closed strings in a background of fluctuating membranes. The emergent gauge structure of this spin liquid is naturally placed in the language of 2-form electrodynamics, which describes one-dimensional charged strings coupled to a rank-2 antisymmetric gauge field. After establishing the classical spin-vorticity model, we consider perturbing it with quantum exchange interactions, deriving an effective “membrane exchange” model of the quantum dynamics, analogous to ring exchange in quantum spin ice. We demonstrate the existence of a Rokhsar-Kivelson point where the quantum ground state is an equal-weight superposition of all classical ground-state configurations, i.e., a quantum spin liquid. The quantum aspects of this spin liquid are exposed by mapping the membrane exchange model to a strongly coupled frustrated 2-form U(1) lattice gauge theory. We further demonstrate how to quantize the string excitations by coupling a 1-form string field to the 2-form U(1) gauge field, thus mapping a quantum spin model to a 2-form U(1) gauge-Higgs model. We discuss the stability of the gapless deconfined phase of this gauge theory and the possibility of realizing a class of quantum phases of matter: 2-form U(1) quantum spin liquids.
We introduce a geometrically frustrated classical Ising model, dubbed the “spin-vorticity model,” whose ground-state manifold is a classical spin liquid: a 2-form Coulomb phase. We study the thermodynamics of this model both analytically and numerically, exposing the presence of algebraically decaying correlations and an extensive ground-state entropy, and give a comprehensive account of its ground-state properties and excitations. Each classical ground state may be decomposed into collections of closed two-dimensional membranes, supporting fractionalized string excitations attached to the edges of open membranes. At finite temperature, the model can then be described as a gas of closed strings in a background of fluctuating membranes. The emergent gauge structure of this spin liquid is naturally placed in the language of 2-form electrodynamics, which describes one-dimensional charged strings coupled to a rank-2 antisymmetric gauge field. After establishing the classical spin-vorticity model, we consider perturbing it with quantum exchange interactions, deriving an effective “membrane exchange” model of the quantum dynamics, analogous to ring exchange in quantum spin ice. We demonstrate the existence of a Rokhsar-Kivelson point where the quantum ground state is an equal-weight superposition of all classical ground-state configurations, i.e., a quantum spin liquid. The quantum aspects of this spin liquid are exposed by mapping the membrane exchange model to a strongly coupled frustrated 2-form U(1) lattice gauge theory. We further demonstrate how to quantize the string excitations by coupling a 1-form string field to the 2-form U(1) gauge field, thus mapping a quantum spin model to a 2-form U(1) gauge-Higgs model. We discuss the stability of the gapless deconfined phase of this gauge theory and the possibility of realizing a class of quantum phases of matter: 2-form U(1) quantum spin liquids.
The U(1) Dirac spin liquid might realize an exotic phase of matter whose low-energy properties are described by quantum electrodynamics in 2+1 dimensions, where gapless modes exist but spinons and gauge fields are strongly coupled. Its existence was proposed in frustrated Heisenberg models in the presence of frustrating superexchange interactions by the (Abrikosov) fermionic representation of the spin operators [Wen, Phys. Rev. B 65, 165113 (2002)], supplemented by the Gutzwiller projection. Here, we construct charge-Q monopole excitations in the Heisenberg model on the triangular lattice with nearest-neighbor (J1) and next-nearest-neighbor (J2) couplings. In the highly frustrated regime, singlet and triplet monopoles with Q=1 become gapless in the thermodynamic limit; in addition, the energies for generic Q agree with field-theoretical predictions, obtained for a large number of gapless fermion modes. Finally, we consider localized gauge excitations, in which magnetic $\ensuremath{\pi}$ fluxes are concentrated in the triangular plaquettes (in analogy with Z2 visons), showing that these kinds of states do not play a relevant role at low energies. All our findings lend support to a stable U(1) Dirac spin liquid in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice.
We explore the ground-state physics of two-dimensional spin-1/2 U(1) quantum link models, one of the simplest nontrivial lattice gauge theories with fermionic matter within experimental reach for quantum simulations. Whereas in the large mass limit we observe Neél-like vortex-antivortex and striped crystalline phases, for small masses there is a transition from the striped phases into a disordered phase whose properties resemble those at the Rokhsar-Kivelson point of the quantum dimer model. This phase is characterized on ladders by boundary Haldane-like properties, such as vanishing parity and finite string ordering. Moreover, from studies of the string tension between gauge charges, we find that, whereas the striped phases are confined, the novel disordered phase present clear indications of being deconfined. Our results open exciting perspectives of studying highly nontrivial physics in quantum simulators, such as spin-liquid behavior and confinement-deconfinement transitions, without the need of explicitly engineering plaquette terms.
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.
Algebraic spin liquids are quantum disordered phases of insulating magnets that exhibit fractionalized gapless excitations and power-law correlations. Quantum spin liquids in this category include the experimentally established one-dimensional (1D) Luttinger liquid, as well as the U(1) Dirac spin liquid (DSL) that has been a focus of recent candidate materials searches. Most notably, several exchange-frustrated Heisenberg materials on the triangular lattice have shown evidence of the U(1) DSL. In this work, we measure the algebraic correlations of spin-singlet excitations in the J 1 − J 2 antiferromagnetic Heisenberg model on the triangular lattice, prompting a detailed investigation of this model’s stability under spin-lattice coupling with dynamical phonons using variational Monte Carlo. As seen before in 1D spin chains, we observe a low-temperature transition from a U(1) DSL to valence bond order and predict the parameter regime where the model realizes a stable DSL ground state. To achieve this, we employ a series of finite-size scaling Ansätze inspired by the low-energy DSL’s conformal description in terms of quantum electrodynamics and show that emergent monopole operators drive the instability. We compare the physics of this transition to the 1D Luttinger liquid throughout our analysis. We derive the regime of stability against spin-Peierls ordering and argue that the DSL ground state might still be achievable in candidate materials, despite its tendency to valence bond solid ordering.
We study Rokhsar-Kivelson (RK) dimer and spin ice models realizing U(1)-lattice gauge theories in a wide class of quasi-one-dimensional settings, which define a setup for the study of few quantum strings (closed electric field lines) interacting with themselves and each other. We discover a large collection of mappings of these models onto three quantum chains: the spin-1/2 XXZ chain, a spin-1 chain, and a kinetically constrained fermion chain whose configurations are best described in terms of tilings of a rectangular strip. We show that the twist of boundary conditions in the chains maps onto the transverse momentum of the electric field string, and their Drude weight to the inverse of the string mass per unit length. We numerically determine the phase diagrams for these spin chains, employing DMRG simulations and find global similarities but also many interesting new features in comparison to the full 2D problems. For example, the spin-1 chain we obtain features a continuous family of degenerate ground states at its RK point analogous to a Bloch sphere, but without an underlying microscopic global SU(2) symmetry. We also argue for the existence of a (stable) Landau-forbidden gapless critical point away from the RK point in one of the models we study using bosonization and numerics. This is surprising given that the full 2D problem is generically gapped away from the RK point. The same model also displays extensively many local conserved quantities which fragment the Hilbert space, arising as a consequence of destructive resonances between the electric field lines. Our findings highlight spin-chain mappings as a potent technique for the exploration of unusual dynamics, exotic criticality, and low-energy physics in lattice gauge theories.
Antiferromagnetic Ising models on frustrated lattices can realize classical spin liquids, with highly degenerate ground states and, possibly, fractionalized excitations and emergent gauge fields. Motivated by the recent interest in many-body systems in negatively curved space, we study hyperbolic frustrated Ising models. Specifically, we consider nearest-neighbor Ising models on tesselations with odd-length loops in two-dimensional hyperbolic space. For finite systems with open boundaries we determine the ground-state degeneracy exactly, and we perform extensive finite-temperature Monte Carlo simulations to obtain thermodynamic data as well as correlation functions. We show that the shape of the boundary, constituting an extensive part of the system, can be used to control low-energy states: Depending on the boundary, we find ordered or disordered ground states. Our results demonstrate how geometric frustration acts in curved space to produce classical spin liquids.