Planar Pyrochlore, Quantum Ice and Sliding Ice

Abstract

We study quantum antiferromagnetism on the highly frustrated checkerboard lattice, also known as the square lattice with crossings. The quantum Heisenberg antiferromagnet on this lattice is of interest as a two-dimensional analog of the pyrochlore lattice magnet. By combining several approaches we conclude that this system is most likely ordered for all values of spin, S, with a Néel state for large S giving way to a two-fold degenerate valence-bond solid for smaller S. We show next that the Ising antiferromagnet with a weak four-spin exchange, equivalent to square ice with the leading quantum dynamics, exhibits long range "anti-ferroelectric" order. As a byproduct of this analysis we obtain, in the system of weakly coupled ice planes, a sliding phase with XY symmetry. Introduction: On the heels of recent progress in understanding highly frustrated classical magnets, coupled with a substantial experimental effort, 1 renewed attention is now focused on the behaviour of their quantum counterparts. In particular, the Heisenberg pyrochlore antiferromagnet is being studied with view to the question of whether frustration-enhanced quantum fluctuations might lead to unconventional ordering -or complete absence thereof -especially for small values of the quantum spin, S. 2-8 This model classically is special in that frustration prevents any sort of ordering or dynamical phase transition down to the lowest temperatures, 9-11 for which reason it is termed a cooperative paramagnet or classical spin liquid.The challenge of this problem arises from the small energy scale generated by the frustration: in a semiclassical picture, any linear combination of the classically degenerate ground states -the collection of which is of extensive dimensionality 11 -may be selected as the quantum ground state. For the highly frustrated two-dimensional magnet on the related kagome lattice, exact diagonalisations of small clusters 12 have provided crucial benchmarks. This system has turned out to be particularly well suited to this approach as it appears to have a very short correlation length, although one does find a large number of low-lying singlet excitations.For the pyrochlore magnet, it looks as if such results will elude us for some time to come. The pyrochlore lattice, being three-dimensional, displays a more inclement scaling of the Hilbert space dimension with linear system size. Moreover, its unit cell contains four spins and its structure implies that the smallest system without spurious boundary condition effects contains at least 16 sites.To evade this, attention has shifted to a system which avoids some of these complications, namely the checkerboard lattice (Fig. 1). It is expected to have similar properties to the pyrochlore as it has the same local structure -both can be thought of as networks of corner-sharing tetrahedra. Further, the size and topology of its ground state manifold for Heisenberg magnets are identical to

New physics in frustrated magnets: Spin ices, monopoles, etc. (Review Article)

“…That is why, there exists a possibility of spreading of a local spin flip to other places due to non-Ising components of the particle-particle interaction [222,223,224,225,226,227,228,229,230,231,232]. In general, such a possibility can yield magnetic ordering, hence, the level of magnetic frustration for quantum spin ices is lower than for usual ones (sometimes called classical).…”

Low-Temperature Low-Field Phases of the Pyrochlore Quantum MagnetTb2Ti2O7

“…On the other hand, a muon-spin-rotation (µSR) frequency shift in low magnetic fields, 20 mT and 60 mT, applied along the [110] direction suggests a transition at a lower temperature of about 150 mK, a transition which has not been identified [12]. The static magnetic susceptibility of a zero-field-cooled polycrystalline sample in a 1 mT field shows history dependence suggestive of spin-glass behavior below about 100 mK, with an anomaly at about 70 mK interpreted as the spin-glass transition from the high-temperature phase [9].Numerical diagonalization of a single-tetrahedron fourspin model predicts [24,25] that the zero-field ground state of Tb 2 Ti 2 O 7 is a quantum spin liquid, dubbed a quantum spin ice [24,26], which-with low increasing field along the [111] direction-gradually turns into a partially polarized state akin to the kagome-ice state [27,28] of classical spin-ice magnets. At higher fields, H ≥ 82 mT, it evolves into a "three-in one-out" state, with three spins pointing into and one pointing out of every tetrahedron.…”

“…Numerical diagonalization of a single-tetrahedron fourspin model predicts [24,25] that the zero-field ground state of Tb 2 Ti 2 O 7 is a quantum spin liquid, dubbed a quantum spin ice [24,26], which-with low increasing field along the [111] direction-gradually turns into a partially polarized state akin to the kagome-ice state [27,28] of classical spin-ice magnets. At higher fields, H ≥ 82 mT, it evolves into a "three-in one-out" state, with three spins pointing into and one pointing out of every tetrahedron.…”

Classical correlations of defects in lattices with geometrical frustration in the motion of a particle

“…For the 3D diamond lattice it took about 10 10 steps until convergence. We also applied an alternative Monte Carlo algorithm with loop updates 14 which is known to be ergodic and unbiased but shows considerable slower convergence.…”

“…To test the implementation of the algorithm, we first reproduced the known monomer two-point correlations on the hard-core dimer covering on the square and triangular lattice ͑see Figs. 3 and 5͒ as well as the dipolar correlations in the undoped system on a square lattice 14,15 The square lattice is a bipartite lattice and the two defects are on different sublattices. We extracted the exponent from the numerical data by linear interpolation of log-log plots and verified the results by finite-size scaling C͑x / L͒ = L ␥ ͑x / L͒ ␥ c͑x / L͒ with exponent ␥ and system size L. Figure 3 compares the numerical data and the analytical results.…”