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Quantum Triangular Ice: Complete quantum phase diagram

S. A. Owerre

Published Nov 5, 2015• 0 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., [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 $h$, the Hamiltonian possesses competing interactions between a $Z_2$-invariant easy-axis ferromagnetic interaction $J_{\pm\pm}$ 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 $J_{\pm\pm}$, a {\it ferrosolid} phase with broken translational and $Z_2$ symmetries for small $h$ and dominant $J_{z}$, and two lobes with $m=\langle S_z\rangle=\pm 1/6$ 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 {\it ferrosolid} state is the ground state in the dominant Ising limit at zero magnetic field. It remains robust for $J_z\to\infty$. 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.

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Quantum Triangular Ice: Complete phase diagram

S. A. Owerre

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 $h$, the Hamiltonian possesses competing interactions between a $Z_2$-invariant easy-axis ferromagnetic interaction $J_{\pm\pm}$ 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 $J_{\pm\pm}$, a {\it ferrosolid} phase with broken translational and $Z_2$ symmetries for small $h$ and dominant $J_{z}$, and two lobes with $m=\langle S_z\rangle=\pm 1/6$ 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 {\it ferrosolid} state is the ground state in the dominant Ising limit at zero magnetic field. It remains robust for $J_z\to\infty$. 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.

•Nov 5, 2015•0 citations

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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.

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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.
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.
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.
We present the zero-temperature phase diagram of a Heisenberg antiferromagnet on a frustrated triangular lattice with nearest-neighbor (J1)({J}_{1})(J1​) and next-nearest-neighbor (J2)({J}_{2})(J2​) interactions, in a magnetic field. We show that the classical model has an accidental degeneracy for all J2/J1{J}_{2}/{J}_{1}J2​/J1​ and all fields, but the degeneracy is lifted by quantum fluctuations. We show that at large spin SSS, for J2/J1<1/8{J}_{2}/{J}_{1}<1/8J2​/J1​<1/8, quantum fluctuations select the same sequence of three sublattice co-planar states in a field as for J2=0{J}_{2}=0J2​=0, and for 1/8<J2/J1<11/8<{J}_{2}/{J}_{1}<11/8<J2​/J1​<1, they select the canted stripe state for all nonzero fields. The transition between the two states is first order in all fields, with the hysteresis width set by quantum fluctuations. We study the model with arbitrary SSS, including S=1/2S=1/2S=1/2, near the saturation field by exploring the fact that near saturation the density of bosons is small for all SSS. We show that for S>1S>1S>1, the transition remains first order, with a finite hysteresis width, but for S=1/2S=1/2S=1/2 and, possibly, S=1S=1S=1, there appears a new intermediate phase without a quasiclassical long-range order.
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.