The hallmark of highly frustrated systems is the presence of many states close in energy to the ground state. Fluctuations between these states can preclude the emergence of any form of order and lead to the appearance of spin liquids. Even on the classical level, spin liquids are not all alike: they may have algebraic or exponential correlation decay, and various forms of long wavelength description, including vector or tensor gauge theories. Here, we introduce a classification scheme, allowing us to fit the diversity of classical spin liquids (CSLs) into a general framework as well as predict and construct new kinds. CSLs with either algebraic or exponential correlation-decay can be classified via the properties of the bottom flat band(s) in their soft-spin Hamiltonians. The classification of the former is based on the algebraic structures of gapless points in the spectra, which relate directly to the emergent generalized Gauss's laws that control the low-temperature physics. The second category of CSLs, meanwhile, are classified by the fragile topology of the gapped bottom band(s). Utilizing the classification scheme we construct new models realizing exotic CSLs, including one with anisotropic generalized Gauss's laws and charges with subdimensional mobility, one with a network of pinch-line singularities in its correlation functions and a series of fragile topological CSLs connected by zero-temperature transitions. Published by the American Physical Society 2024
The hallmark of highly frustrated systems is the presence of many states close in energy to the ground state. Fluctuations between these states can preclude the emergence of any form of order and lead to the appearance of spin liquids. Even on the classical level, spin liquids are not all alike: they may have algebraic or exponential correlation decay, and various forms of long wavelength description, including vector or tensor gauge theories. Here, we introduce a classification scheme, allowing us to fit the diversity of classical spin liquids (CSLs) into a general framework as well as predict and construct new kinds. CSLs with either algebraic or exponential correlation-decay can be classified via the properties of the bottom flat band(s) in their soft-spin Hamiltonians. The classification of the former is based on the algebraic structures of gapless points in the spectra, which relate directly to the emergent generalized Gauss's laws that control the low-temperature physics. The second category of CSLs, meanwhile, are classified by the fragile topology of the gapped bottom band(s). Utilizing the classification scheme we construct new models realizing exotic CSLs, including one with anisotropic generalized Gauss's laws and charges with subdimensional mobility, one with a network of pinch-line singularities in its correlation functions and a series of fragile topological CSLs connected by zero-temperature transitions. Published by the American Physical Society 2024
Classical spin liquids (CSL) lack long-range magnetic order and are characterized by an extensive ground-state degeneracy. We propose a classification scheme of CSLs based on the structure of the flat bands of their Hamiltonians. Depending on absence or presence of the gap from the flat band, the CSL are classified as algebraic or fragile topological, respectively. Each category is further classified: the algebraic case by the nature of the emergent Gauss's law at the gap-closing point(s), and the fragile topological case by the homotopy of the eigenvector winding around the Brillouin zone. Previously identified models of CSLs fit snugly into our scheme, on a landscape where algebraic CSLs are located at transitions between fragile topological ones. It also allows us to present new families of models illustrating this landscape, which hosts both fragile topological and algebraic CSLs, as well as transitions between them. Published by the American Physical Society 2024
A concept---quantum order---is introduced to describe a new kind of orders that generally appear in quantum states at zero temperature. Quantum orders that characterize the universality classes of quantum states (described by complex ground-state wave functions) are much richer than classical orders that characterize the universality classes of finite-temperature classical states (described by positive probability distribution functions). Landau's theory for orders and phase transitions does not apply to quantum orders since they cannot be described by broken symmetries and the associated order parameters. We introduced a mathematical object---projective symmetry group---to characterize quantum orders. With the help of quantum orders and projective symmetry groups, we construct hundreds of symmetric spin liquids, which have SU(2), U(1), or Z2 gauge structures at low energies. We found that various spin liquids can be divided into four classes: (a) Rigid spin liquid---spinons (and all other excitations) are fully gapped and may have bosonic, fermionic, or fractional statistics. (b) Fermi spin liquid---spinons are gapless and are described by a Fermi liquid theory. (c) Algebraic spin liquid---spinons are gapless, but they are not described by free fermionic-bosonic quasiparticles. (d) Bose spin liquid---low-lying gapless excitations are described by a free-boson theory. The stability of those spin liquids is discussed in detail. We find that stable two-dimensional spin liquids exist in the first three classes (a)--(c). Those stable spin liquids occupy a finite region in phase space and represent quantum phases. Remarkably, some of the stable quantum phases support gapless excitations even without any spontaneous symmetry breaking. In particular, the gapless excitations in algebraic spin liquids interact down to zero energy and the interaction does not open any energy gap. We propose that it is the quantum orders (instead of symmetries) that protect the gapless excitations and make algebraic spin liquids and Fermi spin liquids stable. Since high-Tc superconductors are likely to be described by a gapless spin liquid, the quantum orders and their projective symmetry group descriptions lay the foundation for a spin liquid approach to high-Tc superconductors.
Spin liquids form fluctuating magnetic textures which have to obey certain rules imposed by frustration. These rules can often be written in the form of a Gauss law, indicating the local conservation of an emergent electric field. In reciprocal space, these emergent Gauss laws appear as singularities known as pinch points, that are accessible to neutron-scattering measurements. But more exotic forms of electromagnetism have been stabilized in spin liquids, and in a few rare instances, these zero-dimensional singularities have been extended into one-dimensional pinch lines. Here we propose a simple framework for the design of pinch-line spin liquids in a layered structure of two-dimensional algebraic spin liquids. A plethora of models can be build within this framework, as exemplified by several concrete examples where our theory is confirmed by simulations, and where the rank of the tensorial gauge field is continuously varied along the pinch line, opening new avenues in fractonic matter. Then we use our framework to understand how the evolution of the singularity pinch point along the pinch line can be understood as the interference pattern of two emergent electric fields. Finally, we apply our intuition on these emergent electric fields in real space to generic pinch line models beyond our layered framework, and revisit the recently proposed pinch line model on the octochlore lattice.
Coulomb spin liquids are topological magnetic states obeying an emergent Gauss law. Little distinction has been made between different kinds of Coulomb liquids. Here we show how a series of distinct Coulomb liquids can be generated straightforwardly by varying the constraints on a classical spin system. This leads to pair creation, and coalescence, of topological defects of an underlying vector field. The latter makes higher-rank spin liquids, of recent interest in the context of fracton theories, with attendant multifold pinch points in the structure factor, appear naturally. New Coulomb liquids with an abundance of pinch points also arise. We thus establish a new and general route to uncovering exotic Coulomb liquids, via the manipulation of topological defects in momentum space.
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 tetratrillium lattice has recently been proposed as being responsible for the dynamical properties observed in the S = 1 langbeinite compound K 2 Ni 2 ( SO 4 ) 3 . Here, we study in detail the classical spin liquid properties of this lattice of tricoordinated tetrahedra using classical Monte Carlo and large- N theory calculations. In the large- N limit, we find that the system presents a gapped spectrum with flat bottom bands, giving rise to a fragile spin liquid with exponentially decaying correlations according to the classification of classical spin liquids. We confirm that this scenario also holds in the more realistic Ising and Heisenberg cases, for which the system does not exhibit any finite-temperature phase transition, and the low-temperature spin structure factors exhibit excellent quantitative agreement with the large- N theory. We also provide insight into the quantum S = 1 / 2 limit by performing pseudo-Majorana functional renormalization-group calculations at finite temperatures, and we discuss the possible phases that can arise in the ground state due to quantum fluctuations.