We perform a projective symmetry group (PSG) classification of symmetric quantum spin liquids with different gauge groups on the square-octagon lattice. Employing the Abrikosov fermion representation for spin $\frac{1}{2}$, we obtain $32\phantom{\rule{4pt}{0ex}}\text{SU}(2), 1808\phantom{\rule{4pt}{0ex}}\text{U}(1),$ and $384\phantom{\rule{4pt}{0ex}}{\mathbb{Z}}_{2}$ algebraic PSGs. Constraining ourselves to mean-field parton ans\"atze with short-range amplitudes, the classification reduces to a limited number, with 4 $\text{SU}(2)$, 24 $\text{U}(1),$ and 36 ${\mathbb{Z}}_{2}$, distinct phases. We discuss their ground-state properties and spinon dispersions within a self-consistent treatment of the Heisenberg Hamiltonian with frustrating couplings.
We perform a projective symmetry group (PSG) classification of symmetric quantum spin liquids with different gauge groups on the square-octagon lattice. Employing the Abrikosov fermion representation for spin 21, we obtain 32SU(2),1808U(1), and 384Z2 algebraic PSGs. Constraining ourselves to mean-field parton ans"atze with short-range amplitudes, the classification reduces to a limited number, with 4 SU(2), 24 U(1), and 36 Z2, distinct phases. We discuss their ground-state properties and spinon dispersions within a self-consistent treatment of the Heisenberg Hamiltonian with frustrating couplings.
Projective symmetry groups beyond spin-1/2: Classifying integer-spin quantum spin liquids
Wen's projective symmetry group (PSG), introduced a quarter of a century ago, reduced the classification of symmetric quantum spin liquids to an algebraic problem and has guided the study of spin-1/2 models and materials ever since. Many current candidate materials and quantum simulators, however, have spin-1 or larger moments, and no comparable classification exists for them. We formulate the PSG for fermionic partons of arbitrary spin S. Integer and half-odd-integer spins already differ in their gauge structure. Half-odd-integer spins give the familiar SU(2), U(1), and Z2 invariant gauge groups, while integer spins give a different hierarchy that includes U(1)⋊Z2 and Z2×Z2; the latter has no spin-1/2 counterpart. Three lattice-independent results follow for integer spin. Singlet pairing of the partons is odd under site exchange. A lattice symmetry can exchange the particle and hole representations of the local constraint, which forces symmetry-related sites into opposite constraint sectors. And a time-reversal-symmetric U(1) state with the same constraint on every site can be gapped only if the number of sites per unit cell is a multiple of 2S+1. We apply the formalism to the spin-1 diamond lattice, realized in frustrated A-site spinels, and find ten fully symmetric Ansätze up to second neighbors. In the J1-J2 Heisenberg model at the mean-field level, the ground state depends on how the constraint is imposed. With the particle and hole representations, the lowest states are fully gapped U(1) states whose two sublattices sit in opposite constraint sectors, and one of them is gauge equivalent to a paired Ansatz. The classification provides a starting point for integer-spin quantum spin liquids on other lattices and in other materials.
Motivated by recent numerical studies reporting putative quantum paramagnetic behavior in spin-1/2 Heisenberg models on the maple-leaf lattice, we classify Abrikosov fermion mean-field Ans\"atze of fully symmetric U(1) and Z2 quantum spin liquids within the framework of projective symmetry groups. We obtain a total of 17U(1) and 12Z2 algebraic PSGs, and, upon restricting their realization via mean-field Ans\"atze with nearest-neighbor amplitudes (relevant to the studied models), only 12 U(1) and 8 Z2 distinct phases are obtained. We present both singlet and triplet fields for all Ans\"atze up to third nearest-neighbor bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. We further assess the effects of Gutzwiller projection on the equal-time spin structure factors.
The ruby lattice has been the subject of much interest recently due its realization in Rydberg atom arrays, where its rich variety of frustrated interactions gives rise to topologically ordered quantum spin liquids. Similarly, numerical studies of ruby-lattice spin models, with both isotropic and anisotropic interactions, have provided evidence of gapped and gapless spin-liquid ground states with different low-energy gauge structures. Motivated by these findings, we perform a projective symmetry group (PSG) classification of U(1) and Z2 fermionic spinon mean-field theories$\unicode{x2014}$respecting space-group and time-reversal symmetries$\unicode{x2014}$for S=1/2 spins. We obtain a total of 50 U(1) and 64 Z2 PSGs, and upon restricting their realization via mean-field Ansa¨tze with up to second-nearest-neighbor singlet amplitudes (relevant to the models studied here), only 8 U(1) and 18 Z2 distinct states are obtained. We present the singlet fields for all Ansa¨tze up to third-nearest-neighboring bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. Building on this information, we also obtain the phase diagram of the Heisenberg model in the presence of first (J1), second (J1′), and third (J2) neighbor antiferromagnetic couplings within a self-consistent mean-field approximation.
We perform extensive classifications of Z2 quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic (fcc) lattices using a spin-rotation-invariant fermionic projective symmetry group approach. Taking into account that all three lattices share the same point group Oh, we apply an efficient gauge where the classification for the simple cubic lattice can be partially carried over to the other two lattices. We identify hundreds of projective representations for each of the three lattices, however, when constructing short-range mean-field models for the fermionic partons (spinons) these phases collapse to only very few relevant cases. We self-consistently calculate the corresponding mean-field parameters for frustrated Heisenberg models on all three lattices with up to third-neighbor spin interactions and discuss the spinon dispersions, ground-state energies, and dynamical spin structure factors. Our results indicate that phases with nonuniform spinon hopping or pairing amplitudes are energetically favored. An unusual situation is identified for the fcc lattice where the spinon dispersion minimizing the mean-field energy features a network of symmetry-protected linelike zero modes in reciprocal space. We further discuss characteristic fingerprints of these phases in the dynamical spin structure factor which may help to identify and distinguish them in future numerical or experimental studies.
Quantum spin liquids are novel phases of matter whose physical properties cannot be captured within the conventional framework of phase transitions. These quantum states are characterized by excitations, called spinons, carrying a fractional value of the integer spin. A description based on a mean-field approximation leads to effective gauge theory. Due to this gauge freedom symmetry group operations act via projective symmetry group (PSG) representations, which can be used to classify quantum states beyond the conventional method. This introducing discussion is found in Chapters 2-6. A preceding PSG classification on the square lattice, which took spin rotational symmetry breaking terms into account, showed that the resulting mean-field states could exhibit topologically non-trivial spinon bands. In Chapter 7, only those el- ements of this Classification are studied, which lead to short-ranged mean-field models. The mutual effect of spinons and visons is scrutinized by adding static configurations of a background gauge field. Spinons and visons form composite ob- jects, which in the case of topological spinon bands, are described by Majorana zero modes. An exact mapping corroborates the relation to topological supercon- ductors and vortices. The effect of multiple visons is investigated, and a possible experimental signature is exposed. Chapter 8 explains how spinon mean-field theory can be used to derive a phe- nomenological effective model for the compound Ca 10 Cr 7 O 28 . A material that was recently proposed as a quantum spin liquid. This low-energy theory consists of spinon hopping processes. The underlying bilayer kagome structure can be simpli- fied to an effective honeycomb lattice. The dynamical spin structure factor is calcu- lated and compared to neutron scattering data. It is discussed how f -wave pairing of spinons can explain small deviations from a linear temperature dependency observed in heat capacity measurements at the low temperatures. This pairing mechanism breaks the U (1) invariance group of the pure hopping model and renders a coherent picture of Ca 10 Cr 7 O 28 being a Z 2 quantum spin liquid. Chapter 9 carries out a PSG classification for the simple-, body centered-, and face centered cubic lattices. Despite a large number of possible PSG representations, only very few describe short-ranged mean-field states. These mean-field models are constructed up to third neighbor couplings and calculated in a self-consistent manner. It is found that the energy is lower for non-trivial sign patterns induced by non-trivial representations of the symmetry generators. Higher neighbor extensions, including pairing terms, yield lower mean-field ground state energies compared to extensions with only hopping terms. For the fcc lattice, a first neighbor state could be identified, which exhibits symmetry protected zero-energy modes.
Projective symmetry group classification of fermionic Z2 spin liquids on the dipolar-octupolar pyrochlore magnets
We construct and classify Z2 quantum spin liquids (QSLs) of fermionic partons on the dipolar-octupolar (DO) pyrochlore lattice. The DO pseudospin shares the space-group and time-reversal algebra of the effective spin-1/2 Kramers doublet, its projective symmetry group (PSG) yields the same 48 symmetric Z2 classes as the generic fermionic classification. We show, the distinct DO symmetry action, encoded in a set of sublattice-independent operators, selects a physically different set of symmetry-allowed mean-field channels, so that the resulting ansatze, spinon spectra, and observables differ from both the effective spin-1/2 case and the bosonic DO classification. Decomposing the anisotropic XYZ exchange model into all symmetric singlet and triplet hopping and pairing channels and minimizing the mean-field energy self-consistently, we obtain phase diagrams in the (Jx~,Jz~) plane in which only a few Z2 states, several supporting gapless or flat-band spinon spectra, are stabilized. For these states we compute the spinon band structures, the low-temperature specific heat, and the dynamical spin structure factor, and we argue that the power-law specific-heat exponent is a diagnostic of the spinon nodal structure. A near-linear-in-temperature specific heat together with broad continua in the structure factor gives a fermionic-parton account of the gapless, fractionalized thermodynamics reported in DO materials such as Nd2ScNbO7, complementing the bosonic quantum-spin-ice picture of the Ce-based compounds. The work extends the fermionic-parton classification of DO pyrochlore QSLs beyond the U(1) sector, where Z2 topological order is realized equally naturally by fermionic and bosonic spinons.