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 ${\mathbb{Z}}_{2}$ 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 ${\mathbb{Z}}_{2}$ 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.
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 Z2 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 Z2 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.
We revisit the fermionic parton approach to S=1/2 quantum spin liquids with SU(2) spin-rotation symmetry, and the associated projective symmetry group (PSG) classification. We point out that the existing PSG classification is incomplete; upon completing it, we find spin-liquid states with S=1 and S=0 Majorana fermion excitations coupled to a deconfined Z2 gauge field. The crucial observation leading us to this result is that, like space group and time-reversal symmetries, spin rotations can act projectively on the fermionic partons; that is, a spin rotation may be realized by simultaneous SU(2) spin and gauge rotations. We show that there are only two realizations of spin rotations acting on fermionic partons: the familiar naive realization where spin rotation is not accompanied by any gauge transformation, and a single type of projective realization. We discuss the PSG classification for states with projective spin rotations. To illustrate these results, we show that there are four such PSGs on the two-dimensional square lattice. We study the properties of the corresponding states, finding that one---with gapless Fermi points---is a stable phase beyond mean-field theory. In this phase, depending on parameters, a small Zeeman magnetic field can open a partial gap for the Majorana fermion excitations. Moreover, there are nearby gapped phases supporting Z2 vortex excitations obeying non-Abelian statistics. We conclude with a discussion of various open issues, including the challenging question of where such S=1 Majorana spin liquids may occur in models and in real systems.
Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work, we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical Z2 gauge fields. Requiring U(1) spin conservation, time-reversal, and certain space-group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excitations are stable against any perturbations, as long as the required symmetries are preserved. Applying these gapless criteria to spin-21 systems on square, triangular, and kagome lattices, we show that all gapped symmetric Z2 spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped Z2 spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.
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.
Quantum spin liquids, systems in which quantum fluctuations prevent the magnetic ordering of the spin degrees of freedom down to zero temperature, have been the source of much recent theoretical and experimental interest. These systems are characterised by their long ranged entangled states, preserve symmetries down to zero Kelvin, and have been shown to exhibit fascinating properties such as a topological ground state degeneracy and fractionalised spin excitations. In this thesis, we study several of these phases. The first, Kitaev spin liquids, appear in strongly spin-orbit coupled systems, where the SU(2) spin symmetry is broken. Within our mean field approach, we study the quantum phase transition from the gapless Z 2 spin liquid to a magnetically ordered phase within the Heisenberg-Kitaev model. Beyond the mean field theory, we argue that the gauge structure of the spin liquid plays a crucial role in this transition, leading to a confinement of spinons and the generation of magnetic order. We also discuss the three-dimensional iridate compounds, where the Kitaev spin liquid phase has topologically protected bulk and surface excitations. We next discuss spin liquids appearing on the geometrically frustrated kagome lattice. First, we consider non-Kramers spin liquids, in which the spin structure of the theory is protected by lattice, rather than time-reversal, symmetry. As a result, more phases are allowed, which have different properties than those which transform as Kramers doublets. In addition, these may have positive experimental signatures in the Raman scattering intensity, offering a clear path to detect such a state. Finally, we examine possible spin liquid phases on a breathing kagome lattice, where we find a stable Z2 spin liquid to be the ground state. This may help to guide future numerical studies of the kagome lattice Heisenberg model, and may also be relevant to DQVOF, a recently discovered spin liquid candidate compound.
We elaborate a simple classification scheme of all rank-5 SU(2) spin rotational symmetric tensors according to (i) the onsite physical spin S, (ii) the local Hilbert space ${V}^{\ensuremath{\bigotimes}4}$ of the four virtual (composite) spins attached to each site, and (iii) the irreducible representations of the C4v point group of the square lattice. We apply our scheme to draw a complete list of all SU(2)-symmetric translationally and rotationally invariant projected entangled pair states (PEPS) with bond dimension $D\ensuremath{\le}6$. All known SU(2)-symmetric PEPS on the square lattice are recovered and simple generalizations are provided in some cases. More generally, to each of our symmetry class can be associated a $(\mathcal{D}\ensuremath{-}1)\text{-dimensional}$ manifold of spin liquids (potentially) preserving lattice symmetries and defined in terms of D-independent tensors of a given bond dimension D. In addition, generic (low-dimensional) families of PEPS explicitly breaking either (i) particular point-group lattice symmetries (lattice nematics) or (ii) time-reversal symmetry (chiral spin liquids) or (iii) SU(2) spin rotation symmetry down to U(1) (spin nematics or N\'eel antiferromagnets) can also be constructed. We apply this framework to search for new topological chiral spin liquids characterized by well-defined chiral edge modes, as revealed by their entanglement spectrum. In particular, we show how the symmetrization of a double-layer PEPS leads to a chiral topological state with a gapless edge described by a SU(2)2 Wess-Zumino-Witten model.
A projective symmetry group (PSG) has been regarded as a classification theory of spin liquids.However, it does not include a symmetry-protected topological order of fermionic spinon excitations, and thus the classification of gapped spin liquids is incomplete.We demonstrate the classification beyond PSG by utilizing the Kitaev model on the square-octagon lattice, where two gapped spin liquids are distinguished by a topological Z 2 invariant.This Z 2 invariant can be defined solely by the time-reversal and translation symmetries on condition that the time-reversal symmetry is implemented projectively.Thus, it is a hidden class of topological Kitaev spin liquids with helical edge states, which has been ignored for a long time.This suggests that there exists an unknown classification scheme of gapped spin liquids beyond PSG.