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arXiv:1106.3330 · Physical Review B

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

Yejin Huh, Matthias Punk, Subir Sachdev A

Published Sep 16, 2011• 76 citations
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Abstract

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.

DOI
10.1103/physrevb.84.094419

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We present a projective symmetry group (PSG) analysis of the spinless excitations of Z2{\mathbb{Z}}_{2}Z2​ spin liquids on the kagome lattice. In the simplest case, vortices carrying Z2{\mathbb{Z}}_{2}Z2​ magnetic flux (``visons'') are shown to transform under the 48 element group GL(2,Z3)\mathit{GL}(2,{\mathbb{Z}}_{3})GL(2,Z3​). Alternative exchange couplings can also lead to a second case with visons transforming under 288-element group GL(2,Z3)\ifmmode×\else\texttimes\fiD3\mathit{GL}(2,{\mathbb{Z}}_{3})\ifmmode\times\else\texttimes\fi{}{\mathrm{D}}_{3}GL(2,Z3​)\ifmmode×\else\texttimes\fiD3​. 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.
Quantum spin liquids are exotic many-body states featured with long-range entanglement and fractional anyon quasiparticles. Quantum phase transitions of spin liquids are particularly interesting problems related with novel phenomena of anyon condensation and anyon confinement. Here we study a quantum dimer model, which implements a transition between a Z 2 spin liquid ( Z 2 SL ) and a valence bond solid (VBS) on the kagome lattice. The transition is driven by the condensation of vison excitation of the Z 2 spin liquid, which impacts on other anyon excitations especially leading to the confinement of spinon excitations. By numerical exact diagonalization of the dimer model, we directly measure the vison condensation using vison string operators, and explicitly check a confining potential acting on spinon excitations in the VBS state. It is observed that topological degeneracy of the spin-liquid state is lifted concomitantly with the vison condensation. The dimer ordering pattern of the VBS state is identified by investigating dimer structure factor. Furthermore, we find an interesting state that exhibits features of spin liquid and VBS simultaneously. We discuss the origin of the mixed behaviors and possible scenarios expected in thermodynamic limit. This work complements the previous analytical studies on the dimer model [ and ] by providing numerical evidences on the vison condensation and the spinon confinement in the Z 2 SL -to-VBS transition. Published by the American Physical Society 2024
We consider Z2{Z}_{2}Z2​ spin liquids on the kagome lattice on the verge of a valence bond solid (VBS) transition, where vortex excitations carrying Z2{Z}_{2}Z2​ magnetic flux---so-called visons---condense. We show that these vison excitations can couple directly to the external electromagnetic field, even though they carry neither spin nor charge. This is possible via a magnetoelastic coupling mechanism recently identified. [Potter, Senthil, and Lee, arXiv:1301.3495; Hao, Phys. Rev. B 85, 174432 (2012)] For the case of transitions to a 36-site unit cell VBS state, the corresponding finite ac conductivity has a specific power-law frequency dependence, which is related to the crossover exponent of the quantum critical point. The visons' contribution to the optical conductivity at transitions to VBS states with a 12-site unit cell vanishes, however.
Due to strong geometric frustration and quantum fluctuation, the S=1/2S=1/2S=1/2 quantum Heisenberg antiferromagnet on the kagome\text{kagome}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,kagomeS=1/2,\text{kagome}S=1/2,kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2{\mathbb{Z}}_{2}Z2​ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2{\mathbb{Z}}_{2}Z2​ SLs on the kagome\text{kagome}kagome lattice. Among them we found only one gapped Z2{\mathbb{Z}}_{2}Z2​ 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.
Gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids have been proposed as candidates for the ground state of the S=1/2S=1/2S=1/2 quantum antiferromagnet on the kagome lattice. We extend the use of projected entangled pair states to construct (on the cylinder) resonating valence bond (RVB) states including both nearest-neighbor and next-nearest-neighbor singlet bonds. Our ansatz---dubbed ``simplex spin liquid''---allows for an asymmetry between the two types of triangles (of order 22%--3%2 in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological Z2{\mathbb{Z}}_{2}Z2​ structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB minimally entangled states. The spinon and vison coherence lengths are extracted from a finite size scaling with regard to the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless, suggesting the occurrence, on a cylinder with real open boundaries, of gapless edge modes formally similar to Luttinger liquid (nonchiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalization studies.
A symmetry-based analysis (projective symmetry group) is used to study spin-liquid phases on the triangular and Kagom\'e lattices in the Schwinger boson framework. A maximum of eight distinct Z2{Z}_{2}Z2​ spin-liquid states are found for each lattice, which preserve all symmetries. Out of these only a few have nonvanishing nearest-neighbor amplitudes, which are studied in greater detail. On the triangular lattice, only two such states are present---the first (zero-flux state) is the well-known state introduced by Sachdev, which on condensation of spinons leads to the 120\ifmmode^\circ\else\textdegree\fi{} ordered state. The other solution, which we call the $\ensuremath{\pi}$-flux state has not previously been discussed. Spinon condensation leads to an ordering wave vector at the Brillouin zone edge centers, in contrast to the 120\ifmmode^\circ\else\textdegree\fi{} state. While the zero-flux state is more stable with just nearest-neighbor exchange, we find that the introduction of either next-neighbor antiferromagnetic exchange or four-spin ring exchange (of the sign obtained from a Hubbard model) tends to favor the $\ensuremath{\pi}$-flux state. On the Kagom\'e lattice four solutions are obtained---two have been previously discussed by Sachdev, which on spinon condensation give rise to the q=0q=0q=0 and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ spin-ordered states. In addition we find two states with significantly larger values of the quantum parameter at which magnetic ordering occurs. For one of them this even exceeds unity ${\ensuremath{\kappa}}_{c}\ensuremath{\approx}2.0$ in a nearest-neighbor model, indicating that if stabilized, could remain spin disordered for physical values of the spin. This state is also stabilized by ring-exchange interactions with signs as derived from the Hubbard model.
We derive an effective Z2{Z}_{2}Z2​ 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=12S=\frac{1}{2}S=21​ 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 Z2{Z}_{2}Z2​ 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 Ndis{N}_{\mathrm{dis}}Ndis​ is the number of disclinations. Only the Z2{Z}_{2}Z2​ Ising symmetry is required to protect the vison zero modes. This is remarkable because a unitary Z2{Z}_{2}Z2​ symmetry cannot be responsible for symmetry-protected degeneracies of local degrees of freedom. We also discuss other signatures of symmetry fractionalization in the Z2{Z}_{2}Z2​ spin liquid, and phase transitions out of the Z2{Z}_{2}Z2​ spin liquid to nearby ordered phases.