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

Finite-size effects in the Z2 spin liquid on the kagome lattice

Hyejin Ju, Leon Balents

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 occurring 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 nonvanishing dimerization that decays exponentially with circumference.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 occurring 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 nonvanishing dimerization that decays exponentially with circumference.

arXiv:1302.2636Physical Review BMay 8, 20135 citations

Spin Correlations in Quantum Spin Liquids on the Kagome Lattice

Tyler Dodds, Yong Baek Kim

Bulletin of the American Physical SocietyMar 21, 20130 citations

Z 2 spin liquids in the S = 1 2 Heisenberg model on the kagome lattice: A projective symmetry-group study of Schwinger fermion mean-field states

Yuan-Ming Lu, Ying Ran, Patrick A. Lee

Due to strong geometric frustration and quantum fluctuation, the $S=1/2$ quantum Heisenberg antiferromagnet on the $\text{kagome}$ lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg $S=1/2,\text{kagome}$ lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of ${\mathbb{Z}}_{2}$ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of ${\mathbb{Z}}_{2}$ SLs on the $\text{kagome}$ lattice. Among them we found only one gapped ${\mathbb{Z}}_{2}$ SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.Due to strong geometric frustration and quantum fluctuation, the $S=1/2$ quantum Heisenberg antiferromagnet on the $\text{kagome}$ lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg $S=1/2,\text{kagome}$ lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of ${\mathbb{Z}}_{2}$ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of ${\mathbb{Z}}_{2}$ SLs on the $\text{kagome}$ lattice. Among them we found only one gapped ${\mathbb{Z}}_{2}$ SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.

arXiv:1104.1432Physical Review BJun 22, 2011147 citations

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.

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 occurring 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 nonvanishing dimerization that decays exponentially with circumference.

Due to strong geometric frustration and quantum fluctuation, the

S=1/2

quantum Heisenberg antiferromagnet on the

\text{kagome}

lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg

S=1/2,\text{kagome}

lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of

{\mathbb{Z}}_{2}

topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of

{\mathbb{Z}}_{2}

SLs on the

\text{kagome}

lattice. Among them we found only one gapped

{\mathbb{Z}}_{2}

SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.