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 ${\mathbb{Z}}_{2}$ gauge fields. Requiring $\text{U}(1)$ 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- $\frac{1}{2}$ systems on square, triangular, and kagome lattices, we show that all gapped symmetric ${\mathbb{Z}}_{2}$ spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped ${\mathbb{Z}}_{2}$ spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.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 ${\mathbb{Z}}_{2}$ gauge fields. Requiring $\text{U}(1)$ 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-$\frac{1}{2}$ systems on square, triangular, and kagome lattices, we show that all gapped symmetric ${\mathbb{Z}}_{2}$ spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped ${\mathbb{Z}}_{2}$ spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.

arXiv:1606.05652Physical review. B./Physical review. BMar 19, 201812 citations

Effective field theory, edge states and classification of symmetric Z2 spin liquids

Yuan-Ming Lu, Ashvin Vishwanath

Bulletin of the American Physical SocietyMar 5, 20140 citations

Classifying fractionalization: Symmetry classification of gapped Z 2 spin liquids in two dimensions

Andrew M. Essin, Michael Hermele

We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with ${\mathbb{Z}}_{2}$ topological order, that is, on gapped ${\mathbb{Z}}_{2}$ spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group ${H}^{2}(G,{\mathbb{Z}}_{2})$ . This result leads us to a symmetry classification of gapped ${\mathbb{Z}}_{2}$ spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and $\mathrm{SO}(3)$ spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric ${\mathbb{Z}}_{2}$ spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with ${\mathbb{Z}}_{2}$ topological order, that is, on gapped ${\mathbb{Z}}_{2}$ spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group ${H}^{2}(G,{\mathbb{Z}}_{2})$. This result leads us to a symmetry classification of gapped ${\mathbb{Z}}_{2}$ spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and $\mathrm{SO}(3)$ spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric ${\mathbb{Z}}_{2}$ spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.

arXiv:1212.0593Physical Review BMar 14, 2013240 citations

Detecting symmetry fractionalization in gapped quantum spin liquids by magnetic impurities

Shuangyuan Lu, Yuan-Ming Lu

Abstract We study the Kondo effect of spin-1/2 magnetic impurities in gapped Z2 spin liquids on two-dimensional lattices. We find that if the impurity is placed at a high-symmetry location, a nontrivial spinon fractionalization class of the impurity site symmetry group will necessarily lead to a non-Kramers doublet in the Kondo screening regime, protected by associated crystalline symmetries. This is in sharp contrast to a featureless screening phase in the usual Kondo effect. We demonstrate this symmetry-protected topological degeneracy by an exactly solvable model and by the large-N theory. Based on this effect, we discuss how thermodynamic measurements in the limit of dilute magnetic impurities can be used to detect symmetry fractionalization in gapped Z2 spin liquids.Abstract We study the Kondo effect of spin-1/2 magnetic impurities in gapped Z2 spin liquids on two-dimensional lattices. We find that if the impurity is placed at a high-symmetry location, a nontrivial spinon fractionalization class of the impurity site symmetry group will necessarily lead to a non-Kramers doublet in the Kondo screening regime, protected by associated crystalline symmetries. This is in sharp contrast to a featureless screening phase in the usual Kondo effect. We demonstrate this symmetry-protected topological degeneracy by an exactly solvable model and by the large-N theory. Based on this effect, we discuss how thermodynamic measurements in the limit of dilute magnetic impurities can be used to detect symmetry fractionalization in gapped Z2 spin liquids.

arXiv:2209.09853Physical review. B./Physical review. BSep 3, 20242 citations

Theory of GappedZ2 Spin Liquids

Subir Sachdev A

Cambridge University Press eBooksMar 24, 20230 citations

Realization of symmetry classes for gapped

Z_{2}

spin liquids in simple models

{\mathbb{Z}}_{2}

gauge fields. Requiring

\text{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-

\frac{1}{2}

systems on square, triangular, and kagome lattices, we show that all gapped symmetric

{\mathbb{Z}}_{2}

spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped

{\mathbb{Z}}_{2}

spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.

We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with

{\mathbb{Z}}_{2}

topological order, that is, on gapped

{\mathbb{Z}}_{2}

spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group

{H}^{2}(G,{\mathbb{Z}}_{2})

. This result leads us to a symmetry classification of gapped

{\mathbb{Z}}_{2}

spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and

\mathrm{SO}(3)

spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric

{\mathbb{Z}}_{2}

spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.