In this work we study symmetry fractionalization of vison excitations in topological $\mathbb{Z}_2$ spin liquids. We show that in the presence of the full $\mathrm{SO}(3)$ spin rotational symmetry and if there is an odd number of spin- $\frac12$ per unit cell, the symmetry fractionalization of visons is completely fixed. On the other hand, visons can have different classes of symmetry fractionalization if the spin rotational symmetry is reduced. As a concrete example, we show that visons in the Balents-Fisher-Girvin $\mathbb{Z}_2$ spin liquid have crystal symmetry fractionalization classes which are not allowed in $\mathrm{SO}(3)$ symmetric spin liquids, due to the reduced spin rotational symmetry.In this work we study symmetry fractionalization of vison excitations in topological $\mathbb{Z}_2$ spin liquids. We show that in the presence of the full $\mathrm{SO}(3)$ spin rotational symmetry and if there is an odd number of spin-$\frac12$ per unit cell, the symmetry fractionalization of visons is completely fixed. On the other hand, visons can have different classes of symmetry fractionalization if the spin rotational symmetry is reduced. As a concrete example, we show that visons in the Balents-Fisher-Girvin $\mathbb{Z}_2$ spin liquid have crystal symmetry fractionalization classes which are not allowed in $\mathrm{SO}(3)$ symmetric spin liquids, due to the reduced spin rotational symmetry.

arXiv:1509.02927Sep 9, 201515 citations

Theory of quantum kagome ice and vison zero modes

Yi-Ping Huang, Michael Hermele

We derive an effective ${Z}_{2}$ gauge theory to describe the quantum kagome ice (QKI) state that has been observed by Carrasquilla et al. [Nat. Commun. 6, 7421 (2015)] in Monte Carlo studies of the $S=\frac{1}{2}$ kagome XYZ model in a Zeeman field. The numerical results on QKI are consistent with, but do not confirm or rule out, the hypothesis that it is a ${Z}_{2}$ spin liquid. Our effective theory allows us to explore this hypothesis and make a striking prediction for future numerical studies, namely, that symmetry-protected vison zero modes arise at lattice disclination defects, leading to a Curie defect term in the spin susceptibility, and a characteristic $({N}_{\mathrm{dis}}\ensuremath{-}1)ln2$ contribution to the entropy, where ${N}_{\mathrm{dis}}$ is the number of disclinations. Only the ${Z}_{2}$ Ising symmetry is required to protect the vison zero modes. This is remarkable because a unitary ${Z}_{2}$ symmetry cannot be responsible for symmetry-protected degeneracies of local degrees of freedom. We also discuss other signatures of symmetry fractionalization in the ${Z}_{2}$ spin liquid, and phase transitions out of the ${Z}_{2}$ spin liquid to nearby ordered phases.We derive an effective ${Z}_{2}$ gauge theory to describe the quantum kagome ice (QKI) state that has been observed by Carrasquilla et al. [Nat. Commun. 6, 7421 (2015)] in Monte Carlo studies of the $S=\frac{1}{2}$ kagome XYZ model in a Zeeman field. The numerical results on QKI are consistent with, but do not confirm or rule out, the hypothesis that it is a ${Z}_{2}$ spin liquid. Our effective theory allows us to explore this hypothesis and make a striking prediction for future numerical studies, namely, that symmetry-protected vison zero modes arise at lattice disclination defects, leading to a Curie defect term in the spin susceptibility, and a characteristic $({N}_{\mathrm{dis}}\ensuremath{-}1)ln2$ contribution to the entropy, where ${N}_{\mathrm{dis}}$ is the number of disclinations. Only the ${Z}_{2}$ Ising symmetry is required to protect the vison zero modes. This is remarkable because a unitary ${Z}_{2}$ symmetry cannot be responsible for symmetry-protected degeneracies of local degrees of freedom. We also discuss other signatures of symmetry fractionalization in the ${Z}_{2}$ spin liquid, and phase transitions out of the ${Z}_{2}$ spin liquid to nearby ordered phases.

arXiv:1609.08617Physical review. B./Physical review. BFeb 16, 20177 citations

Quantum Spin Liquids: Signatures of Fractionalization

Nandini Trivedi

Nov 1, 20190 citations

Vison states and confinement transitions of Z 2 spin liquids on the kagome lattice

Yejin Huh, Matthias Punk, Subir Sachdev A

We present a projective symmetry group (PSG) analysis of the spinless excitations of ${\mathbb{Z}}_{2}$ spin liquids on the kagome lattice. In the simplest case, vortices carrying ${\mathbb{Z}}_{2}$ magnetic flux (``visons'') are shown to transform under the 48 element group $\mathit{GL}(2,{\mathbb{Z}}_{3})$ . Alternative exchange couplings can also lead to a second case with visons transforming under 288-element group $\mathit{GL}(2,{\mathbb{Z}}_{3})\ifmmode\times\else\texttimes\fi{}{\mathrm{D}}_{3}$. We study the quantum phase transition in which visons condense into confining states with valence bond solid order. The critical field theories and confining states are classified using the vison PSGs.We present a projective symmetry group (PSG) analysis of the spinless excitations of ${\mathbb{Z}}_{2}$ spin liquids on the kagome lattice. In the simplest case, vortices carrying ${\mathbb{Z}}_{2}$ magnetic flux (``visons'') are shown to transform under the 48 element group $\mathit{GL}(2,{\mathbb{Z}}_{3})$. Alternative exchange couplings can also lead to a second case with visons transforming under 288-element group $\mathit{GL}(2,{\mathbb{Z}}_{3})\ifmmode\times\else\texttimes\fi{}{\mathrm{D}}_{3}$. We study the quantum phase transition in which visons condense into confining states with valence bond solid order. The critical field theories and confining states are classified using the vison PSGs.

arXiv:1106.3330Physical Review BSep 16, 201176 citations

Symmetry fractionalization of visons in

\mathbb Z_2

spin liquids

In this work we study symmetry fractionalization of vison excitations in topological

\mathbb{Z}_2

spin liquids. We show that in the presence of the full

\mathrm{SO}(3)

spin rotational symmetry and if there is an odd number of spin-

\frac12

per unit cell, the symmetry fractionalization of visons is completely fixed. On the other hand, visons can have different classes of symmetry fractionalization if the spin rotational symmetry is reduced. As a concrete example, we show that visons in the Balents-Fisher-Girvin

\mathbb{Z}_2

spin liquid have crystal symmetry fractionalization classes which are not allowed in

\mathrm{SO}(3)

symmetric spin liquids, due to the reduced spin rotational symmetry.

We derive an effective

{Z}_{2}

gauge theory to describe the quantum kagome ice (QKI) state that has been observed by Carrasquilla et al. [Nat. Commun. 6, 7421 (2015)] in Monte Carlo studies of the

S=\frac{1}{2}

kagome XYZ model in a Zeeman field. The numerical results on QKI are consistent with, but do not confirm or rule out, the hypothesis that it is a

{Z}_{2}

spin liquid. Our effective theory allows us to explore this hypothesis and make a striking prediction for future numerical studies, namely, that symmetry-protected vison zero modes arise at lattice disclination defects, leading to a Curie defect term in the spin susceptibility, and a characteristic $({N}_{\mathrm{dis}}\ensuremath{-}1)ln2$ contribution to the entropy, where

{N}_{\mathrm{dis}}

is the number of disclinations. Only the

{Z}_{2}

Ising symmetry is required to protect the vison zero modes. This is remarkable because a unitary

{Z}_{2}

symmetry cannot be responsible for symmetry-protected degeneracies of local degrees of freedom. We also discuss other signatures of symmetry fractionalization in the

{Z}_{2}

spin liquid, and phase transitions out of the

{Z}_{2}

spin liquid to nearby ordered phases.

We present a projective symmetry group (PSG) analysis of the spinless excitations of

{\mathbb{Z}}_{2}

spin liquids on the kagome lattice. In the simplest case, vortices carrying

{\mathbb{Z}}_{2}

magnetic flux (``visons'') are shown to transform under the 48 element group

\mathit{GL}(2,{\mathbb{Z}}_{3})

. Alternative exchange couplings can also lead to a second case with visons transforming under 288-element group $\mathit{GL}(2,{\mathbb{Z}}_{3})\ifmmode\times\else\texttimes\fi{}{\mathrm{D}}_{3}$. We study the quantum phase transition in which visons condense into confining states with valence bond solid order. The critical field theories and confining states are classified using the vison PSGs.