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arXiv:1609.08617 · Physical review. B./Physical review. B

Theory of quantum kagome ice and vison zero modes

Yi-Ping Huang, Michael Hermele

Published Feb 16, 2017• 7 citations
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Abstract

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.

DOI
10.1103/physrevb.95.075130

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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 (Ndis\ensuremath−1)ln2({N}_{\mathrm{dis}}\ensuremath{-}1)ln2(Ndis​\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.
We present a large-SSS study of a quantum spin ice Hamiltonian, introduced by Huang et al. [Phys. Rev. Lett. 112, 167203 (2014)], on the kagome lattice. This model involves a competition between the frustrating Ising term of classical kagome ice, a Zeeman magnetic field hhh, and a nearest-neighbor transverse spin-flip term ${S}_{i}^{x}{S}_{j}^{x}\ensuremath{-}{S}_{i}^{y}{S}_{j}^{y}$. Recent quantum Monte Carlo (QMC) simulations by Carrasquilla et al. [Nat. Commun. 6, 7421 (2015)], uncovered lobes of a disordered phase for large Ising interaction and $h\ensuremath{\ne}0$---a putative quantum spin liquid phase. Here, we examine the nature of this model using large-SSS expansion. We show that the ground state properties generally have the same trends with those observed in QMC simulations. In particular, the large-SSS ground state phase diagram captures the existence of the disordered lobes.
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 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 $\mathit{GL}(2,{\mathbb{Z}}_{3})\ifmmode\times\else\texttimes\fi{}{\mathrm{D}}_{3}$. 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 recent quantum Monte Carlo (QMC) simulations of the quantum Kagome ice model by Juan Carrasquilla et al., [Nat. Commun., 6, 7421 (2015)], we study the ground-state properties of this model on the triangular lattice. In the presence of a magnetic field hhh, the Hamiltonian possesses competing interactions between a Z2{Z}_{2}Z2​-invariant easy-axis ferromagnetic interaction ${J}_{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}}$ and a frustrated Ising term Jz{J}_{z}Jz​. As in the U(1)-invariant model, we obtain four classical distinctive phases, however, the classical phases in the Z2{Z}_{2}Z2​-invariant model are different. They are as follows: a fully polarized (FP) ferromagnet for large hhh, an easy-axis canted ferromagnet (CFM) with broken Z2{Z}_{2}Z2​ symmetry for small hhh and dominant ${J}_{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}}$, a ferrosolid phase with broken translational and Z2{Z}_{2}Z2​ symmetries for small hhh and dominant Jz{J}_{z}Jz​, and two lobes with $m=\ensuremath{\langle}{S}_{z}\ensuremath{\rangle}=\ifmmode\pm\else\textpm\fi{}1/6$ for small hhh and dominant Jz{J}_{z}Jz​. We show that quantum fluctuations are suppressed in this model, hence the large-SSS expansion gives an accurate picture of the ground-state properties. When quantum fluctuations are introduced, we show that the ferrosolid state is the ground state in the dominant Ising limit at zero magnetic field. It remains robust for ${J}_{z}\ensuremath{\rightarrow}\ensuremath{\infty}$. With nonzero magnetic field the classical lobes acquire a finite magnetic susceptibility with no Sz{S}_{z}Sz​ order. We present the trends of the ground-state energy and the magnetizations. We also present a detail analysis of the CFM.
Motivated by recent quantum Monte Carlo (QMC) simulations of the quantum Kagome ice model by Juan Carrasquilla, et al., [Nature Communications 6, 7421 (2015)], we study the ground state properties of this model on the triangular lattice. In the presence of a magnetic field hhh, the Hamiltonian possesses competing interactions between a Z2Z_2Z2​-invariant easy-axis ferromagnetic interaction J±±J_{\pm\pm}J±±​ and a frustrated Ising term JzJ_zJz​. As in the U(1)-invariant model, we obtain four classical distinctive phases, however, the classical phases in the Z2Z_2Z2​-invariant model are different. They are as follows: a fully polarized (FP) ferromagnet for large hhh, an easy-axis canted ferromagnet (CFM) with broken Z2Z_2Z2​ symmetry for small hhh and dominant J±±J_{\pm\pm}J±±​, a {\it ferrosolid} phase with broken translational and Z2Z_2Z2​ symmetries for small hhh and dominant JzJ_{z}Jz​, and two lobes with m=⟨Sz⟩=±1/6m=\langle S_z\rangle=\pm 1/6m=⟨Sz​⟩=±1/6 for small hhh and dominant JzJ_{z}Jz​. We show that quantum fluctuations are suppressed in this model, hence the large-SSS expansion gives an accurate picture of the ground state properties. When quantum fluctuations are introduced, we show that the {\it ferrosolid} state is the ground state in the dominant Ising limit at zero magnetic field. It remains robust for Jz→∞J_z\to\inftyJz​→∞. With nonzero magnetic field the classical lobes acquire a finite magnetic susceptibility with no SzS_zSz​-order. We present the trends of the ground state energy and the magnetizations. We also present a detail analysis of the CFM.
Motivated by recent quantum Monte Carlo (QMC) simulations of the quantum Kagome ice model by Juan Carrasquilla, et al., [Nature Communications 6, 7421 (2015)], we study the ground state properties of this model on the triangular lattice. In the presence of a magnetic field hhh, the Hamiltonian possesses competing interactions between a Z2Z_2Z2​-invariant easy-axis ferromagnetic interaction J±±J_{\pm\pm}J±±​ and a frustrated Ising term JzJ_zJz​. As in the U(1)-invariant model, we obtain four classical distinctive phases, however, the classical phases in the Z2Z_2Z2​-invariant model are different. They are as follows: a fully polarized (FP) ferromagnet for large hhh, an easy-axis canted ferromagnet (CFM) with broken Z2Z_2Z2​ symmetry for small hhh and dominant J±±J_{\pm\pm}J±±​, a {\it ferrosolid} phase with broken translational and Z2Z_2Z2​ symmetries for small hhh and dominant JzJ_{z}Jz​, and two lobes with m=⟨Sz⟩=±1/6m=\langle S_z\rangle=\pm 1/6m=⟨Sz​⟩=±1/6 for small hhh and dominant JzJ_{z}Jz​. We show that quantum fluctuations are suppressed in this model, hence the large-SSS expansion gives an accurate picture of the ground state properties. When quantum fluctuations are introduced, we show that the {\it ferrosolid} state is the ground state in the dominant Ising limit at zero magnetic field. It remains robust for Jz→∞J_z\to\inftyJz​→∞. With nonzero magnetic field the classical lobes acquire a finite magnetic susceptibility with no SzS_zSz​-order. We present the trends of the ground state energy and the magnetizations. We also present a detail analysis of the CFM.