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arXiv:1511.01843 · Physical review. B./Physical review. B

Ground-state properties of quantum triangular ice

S. A. Owerre

Published Mar 29, 2016• 2 citations
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Abstract

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 $h$, the Hamiltonian possesses competing interactions between a ${Z}_{2}$-invariant easy-axis ferromagnetic interaction ${J}_{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}}$ and a frustrated Ising term ${J}_{z}$. As in the U(1)-invariant model, we obtain four classical distinctive phases, however, the classical phases in the ${Z}_{2}$-invariant model are different. They are as follows: a fully polarized (FP) ferromagnet for large $h$, an easy-axis canted ferromagnet (CFM) with broken ${Z}_{2}$ symmetry for small $h$ and dominant ${J}_{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}}$, a ferrosolid phase with broken translational and ${Z}_{2}$ symmetries for small $h$ and dominant ${J}_{z}$, and two lobes with $m=\ensuremath{\langle}{S}_{z}\ensuremath{\rangle}=\ifmmode\pm\else\textpm\fi{}1/6$ for small $h$ and dominant ${J}_{z}$. We show that quantum fluctuations are suppressed in this model, hence the large-$S$ 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 ${S}_{z}$ order. We present the trends of the ground-state energy and the magnetizations. We also present a detail analysis of the CFM.

DOI
10.1103/physrevb.93.094436

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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.
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 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.
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.
Recent work [Kundu, Damle, Phys. Rev. X 15, 011018 (2025)] has demonstrated that a class of S=1S=1S=1 kagome lattice antiferromagnets with competing anisotropies can exhibit unusual low-temperature behavior that is best described in terms of the statistical mechanics of an interesting interpolation between the fully-packed dimer model and the fully-packed loop model on the bipartite honeycomb lattice. On this and other bipartite lattices in two and three dimension, such fully-packed dimer-loop models generically have two distinct power-law correlated Coulomb phases, both of which admit a low-energy description in terms of a fluctuating divergence-free polarization vector field [Phys. Rev. X 15, 011018 (2025); arXiv:2510.11607]. The transition between these phases corresponds to a confinement transition for the Z2Z_2Z2​ part of the corresponding U(1)U(1)U(1) flux [Phys. Rev. X 15, 011018 (2025); arXiv:2510.11607]. Here we demonstrate that a variant of such a topological confinement transition can also exist in dimer-loop models on non-bipartite lattices such as the kagome and triangular lattices although the low-energy description does not feature any conserved U(1)U(1)U(1) flux. In these non-bipartite dimer-loop models, both phases are short-range correlated liquids, and the topological distinction between them is in terms of a Z2Z_2Z2​-valued winding number instead of the flux of a divergence-free polarization field. In spite of this key difference, this transition appears to be in the same Ising universality class as the analogous transition of the bipartite dimer-loop models. These kagome and triangular lattice dimer-loop models correspond respectively to the low-energy limit of a class of S=1S=1S=1 antiferromagnets with competing anisotropies on the ruby lattice comprising a corner-sharing arrangement of fully-frustrated squares and a lattice of corner-sharing fully-frustrated hexagons.
The spin-12\frac{1}{2}21​ Ising-Heisenberg model on two geometrically related triangles-in-triangles lattices is exactly solved through the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the effective spin-12\frac{1}{2}21​ Ising model on a triangular lattice. Basic thermodynamic quantities were exactly calculated within this rigorous mapping method along with the ground-state and finite-temperature phase diagrams. Apart from the classical ferromagnetic phase, both investigated models exhibit several unconventional quantum ordered and disordered ground states. A mutual competition between two ferromagnetic interactions of basically different character generically leads to the emergence of a quantum ferromagnetic phase, in which a symmetric quantum superposition of three up-up-down states of the Heisenberg trimers accompanies a perfect alignment of all Ising spins. In the highly frustrated regime, we have either found the disordered quantum paramagnetic phase with a substantial residual entropy or a similar but spontaneously ordered phase with a nontrivial criticality at finite temperatures. The latter quantum ground state exhibits a striking coexistence of imperfect spontaneous order with partial disorder, which is evidenced by a quantum reduction of the spontaneous magnetization of Heisenberg spins that indirectly causes a small reduction of the spontaneous magnetization of otherwise classical Ising spins.