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arXiv:1512.03930 · Physical review. B./Physical review. B

Linear spin-wave study of a quantum kagome ice

S. A. Owerre, A. A. Burkov, Roger G. Melko

Published Apr 1, 2016• 9 citations
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Abstract

We present a large-$S$ 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 $h$, 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-$S$ expansion. We show that the ground state properties generally have the same trends with those observed in QMC simulations. In particular, the large-$S$ ground state phase diagram captures the existence of the disordered lobes.

DOI
10.1103/physrevb.93.144402

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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 SixSjx\ensuremath−SiySjy{S}_{i}^{x}{S}_{j}^{x}\ensuremath{-}{S}_{i}^{y}{S}_{j}^{y}Six​Sjx​\ensuremath−Siy​Sjy​. 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≠0h\ensuremath{\ne}0h\ensuremath=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.
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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.
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.
The S=1/2 kagome-lattice antiferromagnet is one of interesting frustrated quantum spin systems. The system is supposed to exhibit the quantum spin liquid in the ground state, which was proposed as a possible origin of the high-temprature superconductivity. The spin gap is an important physical quantity to characterize the spin liquid behavior. Whether the S=1/2 kagome-lattice antiferromagnet is gapless or has a finite spin gap, is still an unsolved issue. Because any recently developped numerical calculation methods are not enough to determine it in the thermodynamic limit. Our large-scale numerical diazonalization up to 42-spin clusters and a finite-size scaling analysis indicated that the S=1/2 kagome-lattice antiferromagnet is gapless in the thremodynamic limit[1]. It is consistent with the U(1) Dirac spin liquid theory of the kagome-lattice antiferromagnet[2,3]. On the other hand, some density matrix renormalization group (DMRG) calculations supported the gapped Z2 topological spin liquid theory[4,5]. Our recent numerical diagonalization analysis on the magnetization process of a distorted kagome-lattice antiferromagnet indicated that the perfect kagome-lattice system is just on a quantum critical point[6]. It would be a possible reason why it is difficult to determine whether the perfect kagome-lattice antiferromagnet is gapless or gapped. In the present study, a field derivative analysis is applied for the spin gap issue of the kagome-lattice antiferromanget, using the numerical diagonalization data up to 42-spin clusters. It is consistent with our previous conclusion that it is gapless[7]. [1]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 80 (2011) 053704. [2]Y. Ran, M. Hermele, P. A. Lee and X. -G. Wen: Phys. Rev. Lett. 98 (2007) 117205. [3]Y. Iqbal, F. Becca, S. Sorella and D. Poilblanc: Phys. Rev. B 87 (2013) 060405(R). [4]S. Yan, D. A. Huse and S. R. White: Science 332 (2011) 1173. [5]S. Nishimoto, N. Shibata and C. Hotta: Nat. Commun. 4 (2013) 2287. [6]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 83 (2014) 104710. [7]H. Nakano and T. Sakai: in preparation.