Projective Symmetry Group Research Articles

Characterization of quantum spin liquids and their spinon band structures via functional renormalization

We combine the pseudofermion functional renormalization group (PFFRG) method with a self-consistent Fock-like mean-field scheme to calculate low-energy effective theories for emergent spinon excitations in spin-1/2 quantum spin liquids. Using effective spin interactions from PFFRG as an input for the Fock equation and allowing for the most general types of free spinon ans\"atze as classified by the projective symmetry group (PSG) method, we are able to systematically determine spinon band structures for spin-liquid candidate systems beyond mean-field theory. We apply this approach to the antiferromagnetic $J_1$-$J_2$ Heisenberg model on the square lattice and to the antiferromagnetic nearest-neighbor Heisenberg model on the kagome lattice. For the $J_1$-$J_2$ model, we find that in the regime of maximal frustration a SU(2) $\pi$-flux state with Dirac spinons yields the largest mean-field amplitudes. For the kagome model, we identify a gapless $\mathbb{Z}_2$ spin liquid with a small circular spinon Fermi surface and approximate Dirac-cones at low but finite energies.

Projective Symmetry Group of a Finite Frame

We define the projective symmetry group of a finite sequence of vectors (a frame) in a natural way as a group of permutations on the vectors (or their indices). This definition ensures that the projective symmetry group is the same for a frame and its complement. We give an algorithm for computing the projective symmetry group from a small set of projective invariants when the underlying field is a subfield of which is closed under conjugation. This algorithm is applied in a number of examples including equiangular lines (in particular SICs), MUBs, and harmonic frames.

Spinon Magnetic Resonance of Quantum Spin Liquids.

We describe electron spin resonance in a quantum spin liquid with significant spin-orbit coupling. We find that the resonance directly probes spinon continuum, which makes it an efficient and informative probe of exotic excitations of the spin liquid. Specifically, we consider spinon resonance of three different spinon mean-field Hamiltonians, obtained with the help of projective symmetry group analysis, which model a putative quantum spin liquid state of the triangular rare-earth antiferromagnet YbMgGaO_{4}. The band of absorption is found to be very broad and exhibit strong van Hove singularities of single spinon spectrum as well as pronounced polarization dependence.

Projective symmetry group classification of Z3 parafermion spin liquids on a honeycomb lattice

To study exotic excitations described by parafermions in the possible spin liquid states of SU($n$) spin systems, we introduce a parafermion parton approach. The SU($n$) spin operators can be represented by clock and shift matrices, which are shown to be the polynomials of parafermion operators in the parafermion representation. We find that SU($n$) spins can be decomposed into $n$ parafermion matrices of degree one. In this decomposition, the spin has a $\{\bigotimes{\rm SU}(n)\}^{n-1}$ gauge symmetry. As an application, we study the one-dimensional three-state clock model and generalized Kitaev model by a mean-field theory, both of them have been proved to be related to parafermion excitations. We find that with the symmetries of translations, $6$-fold rotation and combination of parity and time reversal, there are $9$ types and $102$ solutions for two-dimensional $Z_3$ parafermion spin liquids on the honeycomb lattice. On the contrast, there are $9$ types and $36$ solutions if both parity and time-reversal symmetries are present. Our results provide a novel route for the systematic search of new types of spin liquids with exotic anyon excitations.

Fermionic spin liquid analysis of the paramagnetic state in volborthite

Recently, thermal Hall effect has been observed in the paramagnetic state of Volborthite, which consists of distorted Kagome layers with $S=1/2$ local moments. Despite the appearance of a magnetic order below $1 \, \mathrm{K}$, the response to external magnetic field and unusual properties of the paramagnetic state above $1 \, \mathrm{K}$ suggest possible realization of exotic quantum phases. Motivated by these discoveries, we investigate possible spin liquid phases with fermionic spinon excitations in a non-symmorphic version of the Kagome lattice, which belongs to the two-dimensional crystallographic group $p2gg$. This non-symmorphic structure is consistent with the spin model obtained in the density functional theory (DFT) calculation. Using projective symmetry group (PSG) analysis and fermionic parton mean field theory, we identify twelve distinct $\mathbb{Z}_2$ spin liquid states, four of which are found to have correspondence in the eight Schwinger boson spin liquid states we classified earlier. We focus on the four fermionic states with bosonic counterpart and find that the spectrum of their corresponding root $U(1)$ states feature spinon Fermi surface. The existence of spinon Fermi surface in candidate spin liquid states may offer a possible explanation of the finite thermal Hall conductivity observed in Volborthite.

Spinon Fermi surface U(1) spin liquid in the spin-orbit-coupled triangular-lattice Mott insulator YbMgGaO4

Motivated by the recent progress on the spin-orbit-coupled triangular lattice spin liquid candidate YbMgGaO4, we carry out a systematic projective symmetry group analysis and mean-field study of candidate U(1) spin liquid ground states. Due to the spin-orbital entanglement of the Yb moments, the space group symmetry operation transforms both the position and the orientation of the local moments, and hence brings different features for the projective realization of the lattice symmetries from the cases with spin-only moments. Among the eight U(1) spin liquids that we find with the fermionic parton construction, only one spin liquid state, that was proposed and analyzed in Yao Shen, et al, Nature 540, 559-562 (2016) and labeled as U1A00 in the present work, stands out and gives a large spinon Fermi surface and provides a consistent explanation for the spectroscopic results in YbMgGaO4. Further connection of this spinon Fermi surface U(1) spin liquid with YbMgGaO4 and the future directions are discussed. Finally, our results may apply to other spin-orbit-coupled triangular lattice spin liquid candidates, and more broadly, our general approach can be well extended to spin-orbit-coupled spin liquid candidate materials.

Quantum spin liquid and magnetic order in a two-dimensional nonsymmorphic lattice: Considering the distorted kagome lattice of volborthite

The Kagome-lattice-based material, Volborthite, $\mathrm{Cu_3 V_2 O_7 (OH)_2 \cdot 2 H_2 O}$, has been considered as a promising platform for discovery of unusual quantum ground states due to the frustrated nature of spin interactions. Here we explore possible quantum spin liquid and magnetically ordered phases in a two-dimensional non-symmorphic lattice described by $p2gg$ layer space group, which is consistent with the spatial anisotropy of the spin model derived from density functional theory (DFT) for Volborthite. Using the projective symmetry group (PSG) analysis and Schwinger boson mean field theory, we classify possible spin liquid phases with bosonic spinons and investigate magnetically ordered phases connected to such states. It is shown, in general, that only translationally invariant mean field states are allowed in two-dimensional non-symmorphic lattices, which simplifies the classification considerably. The mean field phase diagram of the DFT-derived spin model is studied and it is found that possible quantum spin liquid phases are connected to two types of magnetically ordered phases, a coplanar incommensurate $(q,0)$ spiral order as the ground state and a closely competing coplanar commensurate $(\pi,\pi)$ spin density wave order. In addition, periodicity enhancement of the two-spinon continuum, a signature of symmetry fractionalization, is found in the spin liquid phases connected to the $(\pi,\pi)$ spin density wave order. We discuss relevance of these results to recent and future experiments on Volborthite.

Quantum spin liquid in a breathing kagome lattice

Motivated by recent experiments on the vanadium oxyfluoride material DQVOF, we examine possible spin liquid phases on a breathing kagome lattice of S=1/2 spins. By performing a projective symmetry group analysis, we determine the possible phases for both fermionic and bosonic $\mathbb{Z}_2$ spin liquids on this lattice, and establish the correspondence between the two. The nature of the ground state of the Heisenberg model on the isotropic kagome lattice is a hotly debated topic, with both $\mathbb{Z}_2$ and U(1) spin liquids argued to be plausible ground states. Using variational Monte Carlo techniques, we show that a gapped $\mathbb{Z}_2$ spin liquid emerges as the clear ground state in the presence of this breathing anisotropy. Our results suggest that the breathing anisotropy helps to stabilize this spin liquid ground state, which may aid us in understanding the results of experiments and help to direct future numerical studies on these systems.

Interplay of nonsymmorphic symmetry and spin-orbit coupling in hyperkagome spin liquids: Applications to Na4Ir3O8

Na$_4$Ir$_3$O$_8$ provides a material platform to study three-dimensional quantum spin liquids in the geometrically frustrated hyperkagome lattice of Ir$^{4+}$ ions. In this work, we consider quantum spin liquids on hyperkagome lattice for generic spin models, focusing on the effects of anisotropic spin interactions. In particular, we classify possible $\mathbb{Z}_2$ and $U(1)$ spin liquid states, following the projective symmetry group analysis in the slave-fermion representation. There are only three distinct $\mathbb{Z}_2$ spin liquids, together with 2 different $U(1)$ spin liquids. The non-symmorphic space group symmetry of hyperkagome lattice plays a vital role in simplifying the classification, forbidding "$\pi$-flux" or "staggered-flux" phases in contrast to symmorphic space groups. We further prove that both $U(1)$ states and one $Z_2$ state among all 3 are symmetry-protected gapless spin liquids, robust against any symmetry-preserving perturbations. Motivated by the "spin-freezing" behavior recently observed in Na$_4$Ir$_3$O$_8$ at low temperatures, we further investigate the nearest-neighbor spin model with dominant Heisenberg interaction subject to all possible anisotropic perturbations from spin-orbit couplings. We found a $U(1)$ spin liquid ground state with spinon fermi surfaces is energetically favored over $Z_2$ states. Among all spin-orbit coupling terms, we show that only Dzyaloshinskii-Moriya (DM) interaction can induce spin anisotropy in the ground state when perturbing from the isotropic Heisenberg limit. Our work paves the way for a systematic study of quantum spin liquids in various materials with a hyperkagome crystal structure.

Spin structure factors of chiral quantum spin liquids on the kagome lattice

We calculate dynamical spin structure factors for gapped chiral spin liquid states in the spin-1/2 Heisenberg antiferromagnet on the kagome lattice using Schwinger-boson mean-field theory. In contrast to static (equal-time) structure factors, the dynamical structure factor shows clear signatures of time-reversal symmetry breaking for chiral spin liquid states. In particular, momentum inversion $\bm{k} \to -\bm{k}$ symmetry as well as the six-fold rotation symmetry around the $\Gamma$-point are lost. We highlight other interesting features, such as a relatively flat onset of the two-spinon continuum for the \emph{cuboc1} state. Our work is based on the projective symmetry group classification of time-reversal symmetry breaking Schwinger-boson mean-field states by Messio, Lhuillier, and Misguich.

Schwinger boson spin-liquid states on square lattice

We study possible spin liquids on square lattice that respect all lattice symmetries and time-reversal symmetry within the framework of Schwinger boson (mean-field) theory. Such spin liquids have spin gap and emergent Z_2 gauge field excitations. We classify them by the projective symmetry group method, and find six spin liquid states that are potentially relevant to the J_1-J_2 Heisenberg model. The properties of these states are studied under mean-field approximation. Interestingly we find a spin liquid state that can go through continuous phase transitions to either the N\'eel magnetic order or magnetic orders of the wavevector at Brillouin zone edge center. We also discuss the connection between our results and the Abrikosov fermion spin liquids.

Low-Energy Spin Dynamics of the Honeycomb Spin Liquid Beyond the Kitaev Limit

We investigate the generic features of the low energy dynamical spin structure factor of the Kitaev honeycomb quantum spin liquid perturbed away from its exact soluble limit by generic symmetry-allowed exchange couplings. We find that the spin gap persists in the Kitaev-Heisenberg model, but generally vanishes provided more generic symmetry-allowed interactions exist. We formulate the generic expansion of the spin operator in terms of fractionalized Majorana fermion operators according to the symmetry enriched topological order of the Kitaev spin liquid, described by its projective symmetry group. The dynamical spin structure factor displays power-law scaling bounded by Dirac cones in the vicinity of the Γ, K, and K^{'} points of the Brillouin zone, rather than the spin gap found for the exactly soluble point.

Confinement transition to density wave order in metallic doped spin liquids

Insulating quantum spin liquids can undergo a confinement transition to a valence bond solid via the condensation of topological excitations of the associated gauge theory. We extend the theory of such transitions to fractionalized Fermi liquids (FL*): These are metallic doped spin liquids in which the Fermi surfaces only have gauge neutral quasiparticles. Using insights from a duality transform on a doped quantum dimer model for the U(1)-FL* state, we show that projective symmetry group of the theory of the topological excitations remains unmodified, but the Fermi surfaces can lead to additional frustrating interactions. We propose a theory for the confinement transition of ${\mathbb{Z}}_{2}$-FL* states via the condensation of visons. A variety of confining, incommensurate density wave states are possible, including some that are similar to the incommensurate $d$-form factor density wave order observed in several recent experiments on the cuprate superconductors.

Projective symmetry group classification of chiral spin liquids

We present a general review of the projective symmetry group classification of fermionic quantum spin liquids for lattice models of spin $S=1/2$. We then introduce a systematic generalization of the approach for symmetric $\mathbb{Z}_2$ quantum spin liquids to the one of chiral phases (i.e., singlet states that break time reversal and lattice reflection, but conserve their product). We apply this framework to classify and discuss possible chiral spin liquids on triangular and kagome lattices. We give a detailed prescription on how to construct quadratic spinon Hamiltonians and microscopic wave functions for each representation class on these lattices. Among the chiral $\mathbb{Z}_2$ states, we study the subset of U(1) phases variationally in the antiferromagnetic $J_1$-$J_2$-$J_d$ Heisenberg model on the kagome lattice. We discuss static spin structure factors and symmetry constraints on the bulk spectra of these phases.

Projective symmetry of partons in the Kitaev honeycomb model

Low-energy states of quantum spin liquids are thought to involve partons living in a gauge-field background. We study the spectrum of Majorana fermions of Kitaev's honeycomb model on spherical clusters. The gauge field endows the partons with half-integer orbital angular momenta. As a consequence, the multiplicities reflect not the point-group symmetries of the cluster, but rather its projective symmetries, operations combining physical and gauge transformations. The projective symmetry group of the ground state is the double cover of the point group.

Classification of spin liquids on the square lattice with strong spin-orbit coupling

Spin liquids represent exotic types of quantum matter that evade conventional symmetry-breaking order even at zero temperature. Exhaustive classifications of spin liquids have been carried out in several systems, particularly in the presence of full SU(2) spin-rotation symmetry. Real magnetic compounds, however, generically break SU(2) spin symmetry as a result of spin-orbit coupling - which in many materials provides an `order one' effect. We generalize previous works by using the projective symmetry group method to classify $Z_2$ spin liquids on the square lattice when SU(2) spin symmetry is maximally lifted. We find that, counterintuitively, the lifting of spin symmetry actually results in vastly more spin liquid phases compared to SU(2)-invariant systems. A generic feature of the SU(2)-broken case is that the spinons naturally undergo $p+ip$ pairing; consequently, many of these $Z_2$ spin liquids feature a topologically nontrivial spinon band structure supporting gapless Majorana edge states. We study in detail several spin-liquid phases with varying numbers of gapless edge states and discuss their topological protection. The edge states are often protected by a combination of time reversal and lattice symmetries and hence resemble recently proposed topological crystalline superconductors.

Quantum spin liquids in the absence of spin-rotation symmetry: Application to herbertsmithite

It has been suggested that the nearest-neighbour (NN) antiferromagnetic Heisenberg (HAF) model on the Kagome lattice may be a good starting point to understand the quantum spin-liquid (QSL) behaviour discovered in Herbertsmithite. We investigate possible QSL phases in the presence of experimentally relevant spin-rotation symmetry-breaking perturbations such as Dzyaloshinskii-Moriya and Ising interactions, as well as second-neighbour antiferromagnetic Heisenberg interactions. We use the projective symmetry group analysis within the slave-fermion framework of QSL phases and systematically classify possible QSLs in the presence of these perturbations. The dynamical spin-structure factor for relevant QSLs is computed and their effects are studied. Our calculations reveal dispersive features in the spin structure factor embedded in a generally diffuse background due to the existence of fractionalized S=1/2 excitations called spinons. For two of the previously proposed $Z_2$ states, the dispersive features are almost absent, and diffuse scattering dominates over a large energy window throughout the Brillouin zone. This resembles the structure factor observed in recent inelastic neutron scattering experiments on singlet crystals of Herbertsmithite. Furthermore, one of the $Z_2$ states with the spin structure factor with mostly diffuse scattering is gapped, and it may be adiabatically connected to the gapped QSL state observed in recent DMRG calculations for the NN HAF. The above perturbations are found to enhance the diffuse nature of the spin structure factor and reduce the momentum dependencies of the spin gap. We also calculate the electron spin resonance absorption spectra that further characterize the role of spin-rotation symmetry breaking perturbations, and can shed more light into the nature of the ground state in Herbertsmithite.

Spin-orbital liquids in non-Kramers magnets on the kagome lattice

Localized magnetic moments with crystal-field doublet or pseudospin-$\frac{1}{2}$ may arise in correlated insulators with even number of electrons and strong spin-orbit coupling. Such a non-Kramers pseudospin-$\frac{1}{2}$ is the consequence of crystalline symmetries as opposed to the Kramers doublet arising from time-reversal invariance, and is necessarily a composite of spin and orbital degrees of freedom. We investigate possible spin-orbital liquids with fermionic spinons for such non-Kramers pseudospin-$\frac{1}{2}$ systems on the kagome lattice. Using the projective symmetry group analysis, we find 10 phases that are not allowed in the corresponding Kramers systems. These phases are allowed due to unusual action of the time-reversal operation on non-Kramers pseudospins. We compute the spin-spin dynamic structure factor that shows characteristic features of these non-Kramers spin-orbital liquids arising from their unusual coupling to neutrons, which is therefore relevant for neutron scattering experiments. We also point out possible anomalous broadening of Raman scattering intensity that may serve as a signature experimental feature for gapless non-Kramers spin-orbital liquids.

Three-dimensional nematic spin liquid in a stacked triangular lattice 6H-B structure

Recently, a number of experiments indicate the possible presence of spin-liquid phases in quantum magnets with spin-$\frac{1}{2}$ and spin-1 moments sitting on triangular-lattice-based structures in Ba${}_{3}$CuSb${}_{2}$O${}_{9}$ and Ba${}_{3}$NiSb${}_{2}$O${}_{9}$, respectively. In relation to these experiments, several theoretical proposals have been made for spin-liquid phases and spin-liquid-like behaviors on the stacked triangular lattice. While the crystal structures of these materials are currently under debate, it is nonetheless interesting to understand possible spin-liquid phases on such frustrated lattices. In this work, we apply Schwinger boson mean-field theory and projective symmetry group analysis to investigate spin-liquid phases on the fully three-dimensional 6$H$-B structure, in contrast to previous works that considered two-dimensional systems. We find that a nematic ${Z}_{2}$ spin-liquid phase, where the lattice-rotational symmetry is spontaneously broken, is the most promising spin-liquid phase that is consistent with spiral magnetic ordering in the classical limit. We discuss the implications of our results to future theoretical and experimental works.

Finite-size effects in theZ2spin liquid on the kagome lattice

Motivated by the recent discovery of the Z_2 quantum spin liquid state in the nearest neighbor Heisenberg model on the kagome lattice, we investigate the "even-odd" effect occuring when this state is confined to infinitely long cylinders of finite circumference. We pursue a dual analysis, where we map the effective Z_2 gauge theory from the kagome lattice to a frustrated Ising model on the dice lattice. Unexpectedly, we find that the latter theory, if restricted to nearest neighbor interactions, is insufficient to capture this effect. We provide an explanation of why further neighbor interactions are needed via a high-temperature expansion of the effective Hamiltonian. We then carry out projective symmetry group analysis to understand which second neighbor interactions can be introduced while respecting the lattice symmetries. Finally, we qualitatively compare our results to numerics by computing the dimerization operator within our theory. Systems with odd circumferences exhibit a non-vanishing dimerization that decays exponentially with circumference.