A simple phenomenological real-space renormalization group method for quantum Heisenberg spins with nearest and next nearest neighbour interactions on a pyrochlore lattice is presented. Assuming a scaling law for the order parameter of two clusters of different sizes a set of coupled equations that gives the fixed points of the renormalization group transformation and, thus, the critical temperatures and ordered phases of the system is found. The particular case of spins $\frac{1}{2}$ is studied in detail. Furthermore, to simplify the mathematical details, from all the possible phases arising from the renormalization group transformation, only those phases in which the magnetic lattice is commensurate with a subdivision of the crystal lattice into four interlocked fcc sublattices are considered. These correspond to a quantum spin liquid, ferromagnetic order, or non-collinear order in which the total magnetic moment of a tetrahedral unit is zero. The corresponding phase diagram is constructed and the differences with respect to the classical model are analysed. It is found that this method reproduces fairly well the phase diagram of the pyrochlore lattice under the aforementioned constrains.
We numerically compute the temperature dependence of spin structure factor and thermodynamic quantities in the antiferromagnetic quantum Ising model in the pyrochlore lattice. This model exhibits spin disorder ground state with exponentially-decayed spin correlation. We reproduce the temperature dependence of the pinch point structure in the neutron scattering experiment and correct entropy obtained from the measurement of the specific heat.
I consider the class of "depleted pyrochlore" lattices of corner-sharing triangles, made by removing spins from a pyrochlore lattice such that every tetrahedron loses exactly one. Previously known examples are the "hyperkagome" and "kagome staircase". I give criteria in terms of loops for whether a given depleted lattice can order analogous to the kagome \sqrt{3} \times \sqrt{three} state, and also show how the pseudo-dipolar correlations (due to local constraints) generalize to even the random depleted case.
We study corrections to single tetrahedron based approximations for the entropy, specific heat and uniform susceptibility of the pyrochlore lattice Ising antiferromagnet, by a Numerical Linked Cluster (NLC) expansion. In a tetrahedron based NLC, the first order gives the Pauling residual entropy of ${1\over 2}\log{3\over 2}\approx 0.20273$. A 16-th order NLC calculation changes the residual entropy to 0.205507 a correction of 1.37 percent over the Pauling value. At high temperatures, the accuracy of the calculations is verified by a high temperature series expansion. We find the corrections to…
We perform the high-performance computation of the ferromagnetic Ising model on the pyrochlore lattice. We determine the critical temperature accurately based on the finite-size scaling of the Binder ratio. Comparing with the data on the simple cubic lattice, we argue the universal finite-size scaling. We also calculate the classical XY model and the classical Heisenberg model on the pyrochlore lattice.
The cactus approximation in the cluster variation method is applied to the spin ice system with nearest neighbor ferromagnetic coupling. The temperature dependences of the entropy and the specific heat show qualitatively good agreement with those observed by Monte Carlo simulations and experiments, and the Pauling value is reproduced for the residual entropy. The analytic expression of the q-dependent magnetic susceptibility is obtained, from which the absence of magnetic phase transition is confirmed. The neutron scattering pattern is also evaluated and found to be consistent with that obtain…
Quantum Heisenberg antiferromagnets on pyrochlore and checkerboard lattices in a strong external magnetic field are mapped onto hard-core lattice gases with an extended exclusion region. The effective models are studied by the exchange Monte Carlo simulations and by the transfer matrix method. The transition point and the critical exponents are obtained numerically for a square-lattice gas of particles with the second-neighbor exclusion, which describes a checkerboard antiferromagnet. The exact structure of the magnon crystal state is determined for a pyrochlore antiferromagnet.
The ordering of the Ising model on a pyrochlore lattice interacting via the long-range RKKY interaction, which models a metallic pyrochlore magnet such as Pr_2Ir_2O_7, is studied by Monte Carlo simulations. Depending on the parameter k_F representing the Fermi wavevector, the model exhibits rich ordering behaviors.
For Ising spin models which bear the spin-ice type macroscopic (quasi-)degeneracy, conventional classical Monte Carlo (MC) simulation using single spin flips suffers from dynamical freezing at low temperatures ($T$). A similar difficulty is seen also in a family of Heisenberg spin models with easy-axis anisotropy or biquadratic interactions. In the Ising case, the difficulty is avoided by introducing a non-local update based on the loop algorithm. We present an extension of the loop algorithm to the Heisenberg case. As an example of its application, we review our recent study on spin-glass (SG…
Effects of local lattice distortions on the spin ordering are investigated for the antiferromagnetic classical Heisenberg model on the pyrochlore lattice. It is found by Monte Carlo simulations that the spin-lattice coupling (SLC) originating from site phonons induces a first-order transition into two different types of collinear magnetic ordered states. The state realized at stronger SLC is cubic symmetric characterized by the magnetic (1/2,1/2,1/2) Bragg peaks, while that at weaker SLC is tetragonal symmetric characterized by the (1,1,0) ones, each accompanied by the commensurate local latti…
The ground state of the S=1/2 Heisenberg antiferromagnet on the pyrochlore lattice is theoretically investigated. Starting from the limit of isolated tetrahedra, I include interactions between the tetrahedra and obtain an effective model for the spin-singlet ground state multiplet by third-order perturbation. I determine its ground state using the mean-field approximation and found a dimerized state with a four-sublattice structure, which agrees with the proposal by Harris et al. I also discuss chirality correlations and spin correlations for this state.
A class of three dimensional classical lattice systems with macroscopic ground state degeneracies, most famously the spin ice system, are known to exhibit "Coulomb" phases wherein long wavelength correlations within the ground state manifold are described by an emergent Maxwell electrodynamics. We discuss a new example of this phenomenon-the four state Potts model on the pyrochlore lattice-where the long wavelength description now involves three independent gauge fields as we confirm via simulation. The excitations above the ground state manifold are bions, defects that are simultaneously char…
The highly frustrated Heisenberg antiferromagnet on Checkerboard and Pyrochlore lattices is subject to strong quantum fluctuations. This problem is amenable to the Contractor Renormalization (CORE) algorithm, which systematically computes the effective interactions in a basis of local cluster eigenstates. CORE was recently used to derive the Plaquette Boson-Fermion model for the square lattice Hubbard model (cond-mat/0108087). For the spin half Checkerboard and Pyrochlore cases, we find singlet ground states which break lattice symmetry, and a gap to spin one excitations. The low excitations a…
We theoretically investigate effects of lattice distortions on the spin ordering of classical Heisenberg antiferromagnets on the breathing pyrochlore lattice. In the model, local lattice distortions originating from the site vibration are taken into account to yield effective spin interactions via the spin-lattice coupling (SLC). The breathing lattice-alternation is characterized by the ratio of the nearest-neighbor interaction for large tetrahedra to that for small ones, J'/J. It is found by Monte Carlo simulations that the system exhibits a first-order transition into four different types of…
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 Neel 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 q…
The ground state ensemble of the highly frustrated pyrochlore-lattice antiferromagnet can be mapped to a coarse-grained ``polarization'' field satisfying a zero-divergence condition From this it follows that the correlations of this field, as well as the actual spin correlations, decay with separation like a dipole-dipole interaction ($1/|R|^3$). Furthermore, a lattice version of the derivation gives an approximate formula for spin correlations, with several features that agree well with simulations and neutron-diffraction measurements of diffuse scattering, in particular the pinch-point (pseu…
The hedgehog lattice, a three-dimensional periodic array of magnetic monopoles and antimonopoles, is known to be realized in the presence of the Dzyaloshinskii-Moriya (DM) interaction. Here, we demonstrate by means of Monte Carlo simulations that the hedgehog lattice is induced by not the DM interaction but frustration in classical Heisenberg antiferromagnets on the breathing pyrochlore lattice. In the model, the breathing bond-alternation is characterized by the ratio of the nearest-neighbor (NN) antiferromagnetic exchange interaction for large tetrahedra to that for small ones, J1'/J1. A qua…
The geometrically-frustrated spin glass Y2Mo2O7 has been considered widely to be crystallographically ordered with a unique nearest neighbor magnetic exchange interaction, J. To test this assertion, we present x-ray-absorption fine-structure results for the Mo and Y K edges as a function of temperature and compare them to results from a well-ordered pyrochlore, Tl2Mn2O7. We find that the Mo-Mo pair distances are significantly disordered at approximately right angles to the Y-Mo pairs. These results strongly suggest that lattice disorder nucleates the spin-glass phase in this material.
We study a large-N deformation of the S=1/2 pyrochlore Heisenberg antiferromagnet which leads to a soluble quantum dimer model at leading non-trivial order. In this limit, the ground state manifold -- while extensively degenerate -- breaks the inversion symmetry of the lattice, which implies a finite temperature Ising transition without translational symmetry breaking. At lower temperatures and further in the 1/N expansion, we discuss an effective Hamiltonian within the degenerate manifold, which has a transparent physical interpretation as representing dimer potential energies. We find mean-f…
The ordering of the classical Heisenberg model on the pyrochlore lattice with the antiferromagnetic nearest-neighbor interaction J1 and the ferromagnetic next-nearest-neighbour interaction J2 is investigated by means of a mean-field analysis and a Monte Carlo simulation. For a moderate J2/J1-value, the model exhibits a first-order transition into an incommensurate multiple-q ordered state where multiple Bragg peaks coexist in the spin structure factor. We show that there are two types of metastable multiple-q states, a cubic symmetric sextuple-q state and a non-cubic symmetric quadruple-q stat…
The recent identification of the dysprosium titanate compound $Dy_2 Ti_2 O_7$ as a ``Spin-Ice'', i.e. the spin analog of regular entropic ice of Pauling, has created considerable excitement. The ability to manipulate spins using magnetic fields gives a unique advantage over regular ice in these systems, and has been used to study the recovery of entropy. Predicted magnetization plateaus have been observed, testing the underlying model consisting of a competition between short ranged super exchange, and long ranged dipolar interactions between spins. I discuss other compounds that are possibly …
We study the problem of partially ordered phases with periodically arranged disordered (paramagnetic) sites on the pyrochlore lattice, a network of corner-sharing tetrahedra. The periodicity of these phases is characterized by one or more wave vectors k=(1/2 1/2 1/2). Starting from a general microscopic Hamiltonian including anisotropic nearest-neighbor exchange, long-range dipolar interactions and second- and third-nearest neighbor exchange, we identify using standard mean-field theory (s-MFT) an extended range of interaction parameters that support partially ordered phases. We demonstrate th…
We theoretically investigate a $J_1$-$J_3$ classical Heisenberg model on the breathing pyrochlore lattice, where the nearest-neighbor (NN) exchange interactions for small and large tetrahedra, $J_1$ and $J_1'$, take different values due to the breathing bond-alternation and $J_3$ is the third NN antiferromagnetic interaction along the bond direction. It is found by means of Monte Carlo simulations that for large $J_3$, a hedgehog lattice, a three-dimensional periodic array of magnetic monopoles and antimonopoles, emerges in the form of a quadruple-${\bf Q}$ state characterized by the ordering …
We study the low temperature dynamics of the classical Heisenberg antiferromagnet with nearest neighbour interactions on the pyrochlore lattice. We present extensive results for the wavevector and frequency dependence of the dynamical structure factor, obtained from simulations of the precessional dynamics. We also construct a solvable stochastic model for dynamics with conserved magnetisation, which accurately reproduces most features of the precessional results. Spin correlations relax at a rate independent of wavevector and proportional to temperature.
We study the effects of a uniform electric field on the the ground state and excitations of the three-dimensional U(1) spin liquid phase of a breathing pyrochlore lattice, arising due to the coupling between the conventional (Maxwell) electric field and the emergent electrodynamics of the quantum spin ice material. This is an extension of the studies for isotropic pyrochlores in Phys. Rev. B 96, 125145 (2017) to the anisotropic case, as the lattice inversion symmetry is broken in breathing pyrochlores. The emergent photons are found to exhibit birefringence, analogous to the isotropic case. Ho…
