Session H20: Frustration in 3D

Spin-nematic order in the frustrated pyrochlore-lattice quantum rotor model

As an example of ordering due to quantum fluctuations, we examine the nearest-neighbor antiferromagnetic quantum $O(N)$ rotor model on the pyrochlore lattice. Classically, this system remains disordered even at zero temperature; we find that adding quantum fluctuations induces an ordered phase that survives to positive temperature, and we determine how its phase diagram scales with the coupling constant and the number of spin components. We demonstrate, using quantum Monte Carlo simulations, that this phase has long-range spin-nematic order, and that the phase transition into it appears to be first order.   

Engineering order--from--disorder

One of the most intriguing features of frustrated magnetism is the prevalence of {\it order--from--disorder} effects, where fluctuations conspire to select a unique state from a disordered manifold of degenerate classical ground states. Quite generally, order--from--disorder effects favour collinear spin configurations, and therefore lead to magnetization plateaux in many two--dimensional frustrated spin systems. Much less is known about order from disorder in three dimensions. Here we use classical Monte Carlo and low--T expansion techniques to show how thermal fluctuations stabilize a half--magnetization plateau in a Heisenberg model on the highly frustrated, three--dimensional Pyrochlore lattice if --- and only if --- they are given a ``helping hand'' by additional longer--ranged interactions. Our results provide the first example of fluctuation--driven magnetization plateau which can be tuned at will between different ordered states.   

Inelastic Neutron Scattering Studies of GeCo$_{2}$O$_{4}$

GeCo$_{2}$O$_{4}$ is a normal spinel in which the magnetic spin- 3/2 Co$^{2+}$ ions are located on the crystallographic $B$-sites. GeCo$_{2}$O$_{4 }$has a N\'{e}el transition (T$_{N}$ = 20.6 K) that coincides closely with a cubic-to-tetragonal structural phase transition, below which $c$/$a \quad >$ 1. Inelastic neutron scattering measurements show that there is an energy gap in the N\'{e}el state of magnitude 3 meV. We have measured the dispersion of the lowest energy spin wave branch along the (100), (110) and (111) directions for single crystals grown by the floating zone technique. In addition, inelastic neutron powder diffraction data have been obtained to study the higher energy crystal field levels of Co$^{2+}$. These data will be described and compared with results for other $B$-site spinels.   

Suppressing magnetic order in GeNi$_{2}$O$_{4}$ and GeCo$_{2}$O$_{4}$

Spin S = 3/2 GeCo$_{2}$O$_{4}$ and spin S = 1 GeNi$_{2}$O$_{4}$ are both frustrated spinels, but they nevertheless order magnetically a low temperatures. GeCo$_{2}$O$_{4}$ achieves this by undergoing a structural phase transition, but the mechanism adopted by GeNi$_{2}$O$_{4}$ is still an open question. We measured the magnetization, specific heat and magnetocaloric effect in polycrystalline samples of the title materials in high fields up to 45 T, and obtained detailed H,T phase diagrams. In the case of GeNi$_{2}$O$_{4}$ an almost degenerate double phase transition in zero field splits into two transitions separated by a wide magnetic field gap. The possibility of a spin-flop transition driven by single-ion anisotropy that also relieves the frustration in the Ni compound will be discussed.   

Spin singlet state due to heptamer formation in AlV$_2$O$_4$

Spinel AlV$_2$O$_4$ is a mixed-valent system, where V ions construct a pyrochlore lattice with an average valence of 2.5+. This compound shows a phase transition at $ T_{\rm c} \sim $ 700K, and a doubling of the unit cell along the [111] direction is observed below $ T_{\rm c} $. A 3-1 type ``valence-skipping'' charge-ordering(CO) due to the strong correlation has been proposed to describe this transition.[1] Recently, a more careful analysis reported a trimer formation on [111] Kagom\'e layers.[2] It is also pointed out that the temperature dependence of the magnetic susceptibility can be decomposed into the Curie-Weiss contribution from $S=1$ local moments on 1/8 of V sites and a spin-gap contribution from the rest V sites.[2] In the present study, instead of the CO scenario, we focus on the pairing of the Kagom\'e layers which leads to seven V-sites clusters (heptamers) by two trimers and a V site between the trimers. On the basis of the exact diagonalization study, we found that this model explains well the magnetic susceptibility in experiments. Hence, the heptamer formation appears to be the origin of the phase transition in this compound rather than the CO due to the electron correlation. [1]K. Matsuno $et$ $al$., J. Phys. Soc. Jpn. {\bf 70}, 1456(2001) [2]Y. Horibe, $et$ $al$., unpublished   

Entropy and susceptibility of ``stuffed'' spin ice -- Ho$_{2}$(Ho$_{x}$Ti$_{2-x})$O$_{7-x/2}$

The spin ice material Ho$_{2}$Ti$_{2}$O$_{7}$ has been studied extensively due to its apparent residual ground state entropy, which is similar to that seen in water ice. This material has a pyrochlore structure in which the Ho$^{3+}$ and Ti$^{4+}$ cations form two interpenetrating sets of corner sharing tetrahedra. Here we present thermodynamic measurements on Ho$_{2}$(Ho$_{x}$Ti$_{2-x})$O$_{7-x/2}$, with 0 $\le \quad x \quad \le $ 0.67, where we have replaced some Ti with Ho -- effectively stuffing the lattice with more magnetic ions. We find that the zero field magnetic entropy remains essentially unchanged with stuffing. AC susceptibility measurements show the $T$ = 2 K peak associated with the spin ice freezing decreases in magnitude with increasing $x$, indicating that spin freezing has been suppressed. While the residual entropy in Ho$_{2}$Ti$_{2}$O$_{7}$ is reduced with the application of a magnetic field, our measurements show that the entropy becomes less sensitive to applied field as $x$ is increased. This work is supported by the NSF.   

Spin dynamics in the kagom\'{e} ice state

Macroscopic degeneracy of the spin ice state in the geometrically frustrated pyrochlore oxide Dy$_{2}$Ti$_{2}$O$_{7}$ is partly or fully released under magnetic fields. For field along a [111] axis, it was proposed that another macroscopic degeneracy with a finite zero- temperature entropy occurs in the frustrated kagom\'{e} layers, which are built in the pyrochlore lattice. We have investigated this kagom\'{e} ice state using single-crystalline neutron scattering experiments. Spin correlations, i.e. scattering intensity maps, will be discussed based on Monte-Calro simulations of the spin-ice model with the dipolar interatction.   

Spontaneous Strains in the Ground State of the Frustrated Pyrochlore Tb$_2$Ti$_2$O$_7$

The geometrically frustrated pyrochlore antiferromagnet Tb$_2$Ti$_2$O$_7$ has been the focus of much recent attention due to the spin liquid physics it displays at low temperatures. We have carried out high resolution x-ray scattering measurements of the principal Bragg peaks of Tb$_2$Ti$_2$O$_7$ which show substantial longitudinal broadening below approximately 20 K. Measurements on the non-magnetic pyrochlore Y$_2$Ti$_2$O$_7$ show no such broadening, suggesting this effect is driven by strong magnetoelastic coupling, as had been concluded from earlier anomolous low temperature Young's modulus measurements [Mamsurova et.al,JETP Letters,1986]. We compare our x-ray scattering results to similar measurements in TbVO$_4$, which undergoes a cooperative Jahn-Teller phase transition near 34 K, and conclude that the longitundinal broadening is due to cubic-tetragonal fluctuations near a zero temperature phase transition, which acts to relieve the magnetic frustration.   

Crystal Field Fluctuations in a Frustrated Pyrochlore Antiferromagnet Tb$_2$Ti$_2$O$_7$.

The antiferromagnetic pyrochlore Tb$_2$Ti$_2$O$_7$ presents a challenging puzzle to experimentalists and theorists studying frustrated magnets. Results from muon spin resonance and neutron scattering experiments for Tb$_2$Ti$_2$O$_7$ reveal a paramagnetic structure down to $50 $mK despite an antiferromagnetic Curie-Weiss temperature, $\theta_{\rm CW}=-20 $K. Crystal field calculations show that the Tb$^{3+}$ ion in Tb$_2$Ti$_2$O$_7$ is a ground state doublet with local $\langle 111 \rangle$ anisotropy and is separated from the first excited doublet state by a gap of 20K. We apply the Rayleigh-Schrodinger method to map the four states problem with exchange and dipole-dipole interactions onto an effective Hamiltonian with two states per ion. We give some properties of this effective Hamiltonian and discuss the possible classical and quantum phases of Tb$_2$Ti$_2$O$_7$.   

Nematic order in square lattice frustrated ferromagnets

We present a new scenario for the breakdown of ferromagnetic order in a two--dimensional quantum magnet with competing ferromagnetic and antiferromagnetic interactions. In this, dynamical effects lead to the formation of two--magnon bound states, which undergo Bose--Einstein condensation, giving rise to bond--centered nematic order. This scenario is explored in some detail for an extended Heisenberg model on a square lattice. In particular, we present numerical evidence confirming the existence of a state with d-- wave nematic correlations but no long range magnetic order, lying between the saturated ferromagnetic and collinear antiferromagnetic phases of the ferromagnetic $J_1$--$J_2$ model. We argue by continuity of spectra that this phase is also present in a model with 4--spin cyclic exchange. The case of the multiple spin exchange model on a triangular lattice, relevant to magnetism of solid ${}^3$He films, is also discussed.   

A metamagnetic critical point in a three dimensional frustrated antiferromagnet

The competition between different forms of order is central to the problem of strong correlation. This is particularly true of frustrated systems, which frequently exist at or near to a zero--temperature critical point. Here we show that a state with a half--magnetization plateau but no long range order can arise when a three dimensional frustrated antiferromagnet is tuned to a critical point bordering a metamagnetic state. We use classical Monte Carlo simulation and low--temperature expansion techniques to accurately characterize this ``spin pseudogap'' state, and show how its properties relate to those of the critical point. Our results provide an example of three dimensional spin model which can be used to study the relationship between gap and ``pseudogap'' --- i.e. long range and preformed local order --- near a metamagnetic critical point.   

Transverse susceptibility of Ising spins

The ac vector magnetic susceptibility of the Ising magnet LiHo$_x$Y$_{1-x}$F$_4$ is studied as a function of frequency, excitation amplitude, transverse field and temperature. In the spin liquid ($x=4.5$\%), application of a dc magnetic field projects the linear susceptibility transverse to the Ising axis. In addition, pumping the system along the Ising axis produces non-linear excitations in the transverse plane; these excitations can encode information and are potentially useful for quantum computation. In the ferromagnet ($x=65$\%), measurements of the transverse susceptibility are used to probe the dynamics of domain wall tunneling and to estimate the mass of the domain walls.   

$LiHo_xY_{1-x}F_4$ and the quantum Ising spin glass

$LiHo_xY_{1-x}F_4$ in transverse field is considered to be the experimental realization of the transverse field Ising model, and a main tool in the study of quantum magnetism. For different values of $x$, ranging from $1$ to $10^-3$, the system exhibits ferromagnetism, spin glass order, single large spin tunneling and various other interesting phenomena. Here we consider the spin glass regime, and in particular address two long standing puzzles posed by the experiments, i.e. the large disordering transverse field and the diminishing of the cusp in the nonlinear susceptibility as temperature is decreased. The solution of these puzzles emphasizes the significance of the hyperfine and offdiagonal dipolar interactions in this system. We then show that the latter destroys the spin glass order at any finite transverse field in this system, as well as in a general dipolar Ising spin glass. The nature of the disordering of the spin glass phase by quantum fluctuations will be discussed.   

Random Field Effects in the Transverse Field Ising Spin-Glass LiHo$_x$Y$_{1-x}$F$_4$ Magnetic Material.

Magnetic LiHo$_x$Y$_{1-x}$F$_4$ in a transverse field has attracted a lot of attention recently because it is an ideal system to study quantum effects in disordered systems. When a magnetic field is applied transversely to the Ising magnetic moments in LiHo$_x$Y$_{1-x}$F$_4$, the field-induced transition behaves quite differently from theoretical expectations. For $x = 0.167$ a paramagnet to spin glass transition has been studied experimentally in some detail. In particular,experiments find that the cusp in the nonlinear susceptibility signaling the glass state decreases in size as the temperature is lowered and the critical transverse field increases. We will show that the combination of applied transverse field and random off-diagonal dipolar interactions lead to random longitudinal field in LiHo$_x$Y$_{1-x}$F$_4$. Using a simple model, we will show how this random field phenomenology accounts semi-quantitatively for the behavior of the nonlinear susceptibility in LiHo$_x$Y$_{1-x}$F$_4$.