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Register a new userOrder by Disorder and Energetic Selection of the Ground State in the XY Pyrochlore Antiferromagnet Er2Ti2O7. An Inelastic Neutron Scattering Study
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Examples of materials where an order by disorder mechanism is at play to select a particular ground state are scarce. It has recently been proposed, however, that the antiferromagnetic XY pyrochlore Er2Ti2O7, reveals a most convincing case of this mechanism. Observation of a spin gap at zone centers has recently been interpreted as a corroboration of this physics. In this paper, we argue, however, that the anisotropy generated by the interaction-induced admixing between the crystal-field ground and excited levels provides for an alternative mechanism. It especially predicts the opening of a spin gap of about 15 micro-eV, which is of the same order of magnitude as the one observed experimentally. We report new high resolution inelastic neutron scattering data which can be well understood within this scenario.
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The recent determination of a robust spin Hamiltonian for the anti-ferromagnetic XY pyrochlore Er2Ti2O7 reveals a most convincing case of the order by quantum disorder (ObQD) mechanism for ground state selection. This mechanism relies on quantum fluctuations to remove an accidental symmetry of the magnetic ground state, and selects a particular ordered spin structure below T_N=1.2K. The removal of the continuous degeneracy results in an energy gap in the spectrum of spin wave excitations, long wavelength pseudo-Goldstone modes. We have measured the ObQD spin wave gap at a zone center in Er2Ti2O7, using low incident energy neutrons and the time-of-flight inelastic scattering method. We report a gap of Delta =0.053 +/- 0.006 meV, which is consistent with upper bounds placed on it from heat capacity measurements and roughly consistent with theoretical estimate of ~ 0.02 meV, further validating the spin Hamiltonian that led to that prediction. The gap is observed to vary with square of the order parameter, and goes to zero for T ~ T_N.
We consider the possibility that the discrete long-range ordered states of Er2Ti2O7 are selected energetically at the mean field level as an alternative scenario that suggests selection via thermal fluctuations. We show that nearest neighbour exchange interactions alone are not sufficient for this purpose, but that anisotropies arising from excited single ion crystal field states in Er2Ti2O7, together with appropriate anisotropic exchange interactions, can produce the required long range order. However, the effect of the single ion anisotropies is rather weak so we expect thermal or quantum fluctuations, in some guise, to be ultimately important in this material. We reproduce recent experimental results for the variation of magnetic Bragg peak intensities as a function of magnetic field.
In the pyrochlore lattice Heisenberg antiferromagnet, for large spin length $S$, the massive classical ground state degeneracy is partly lifted by the zero-point energy of quantum fluctuations at harmonic order in spin-waves. However, there remains an infinite manifold of degenerate collinear ground states, related by a gaugelike symmetry. We have extended the spin-wave calculation to quartic order, assuming a Gaussian variational wavefunction (equivalent to Hartree-Fock approximation). Quartic calculations emph{do} break the harmonic-order degeneracy of periodic ground states. The form of the effective Hamiltonian describing this splitting, which depends on loops, was fitted numerically and also rationalized analytically. We find a family of states that are still almost degenerate, being split by the term from loops of length 26. We also calculated the anharmonic terms for the checkerboard lattice, and discuss why it (as well as the kagome lattice) behave differently than the pyrochlore at anharmonic orders.
Several rare earth magnetic pyrochlore materials are well modeled by a spin-1/2 quantum Hamiltonian with anisotropic exchange parameters Js. For the Er2Ti2O7 material, the Js were recently determined from high-field inelastic neutron scattering measurements. Here, we perform high-temperature (T) series expansions to compute the thermodynamic properties of this material using these Js. Comparison with experimental data show that the model describes the material very well including the finite temperature phase transition to an ordered phase at Tc~1.2 K. We show that high temperature expansions give identical results for different q=0 xy order parameter susceptibilities up to 8th order in beta=1/T (presumably to all orders in beta). Conversely, a non-linear susceptibility related to the 6th power of the order parameter reveals a thermal order-by-disorder selection of the same non-colinear psi_2 state as found in Er2Ti2O7.
We investigated the ground state symmetry of the cubic hidden order compound CeB$_6$ by means of core level non-resonant inelastic x-ray scattering (NIXS). The information is obtained from the directional dependence of the scattering function that arises from higher than dipole transitions. Our new method confirms that the ground state is well described using a localized crystal-field model assuming a $Gamma_8$ quartet ground state.
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