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Register a new userQuantum order by disorder and accidental soft mode Er2Ti2O7
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Motivated by recent neutron scattering experiments, we derive and study an effective pseudo-dipolar spin-1/2 model for the XY pyrochlore antiferromagnet Er2Ti2O7. While a bond-dependent in-plane exchange anisotropy removes any continuous symmetry, it does lead to a one-parameter `accidental classical degeneracy. This degeneracy is lifted by quantum fluctuations in favor of the non-coplanar spin structure observed experimentally -- a rare experimental instance of quantum order by disorder. A non-Goldstone low-energy mode is present in the excitation spectrum in accordance with inelastic neutron scattering data. Our theory also resolves the puzzle of the experimentally observed continuous ordering transition, absent from previous models.
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Here we establish the systematic existence of a U(1) degeneracy of all symmetry-allowed Hamiltonians quadratic in the spins on the pyrochlore lattice, at the mean-field level. By extracting the Hamiltonian of Er2Ti2O7 from inelastic neutron scattering measurements, we then show that the U(1)-degenerate states of Er2Ti2O7 are its classical ground states, and unambiguously show that quantum fluctuations break the degeneracy in a way which is confirmed by experiment. This is the first definitive observation of order-by-disorder in any material. We provide further verifiable consequences of this phenomenon, and several additional comparisons between theory and experiment.
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.
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.
If magnetic frustration is most commonly known for undermining long-range order, as famously illustrated by spin liquids, the ability of matter to develop new collective mechanisms in order to fight frustration is no less fascinating, providing an avenue for the exploration and discovery of unconventional properties of matter. Here we study an ideal minimal model of such mechanisms which, incidentally, pertains to the perplexing quantum spin ice candidate Yb2Ti2O7. Specifically, we explain how thermal and quantum fluctuations, optimized by order-by-disorder selection, conspire to expand the stability region of an accidentally degenerate continuous symmetry U(1) manifold against the classical splayed ferromagnetic ground state that is displayed by the sister compound Yb2Sn2O7. The resulting competition gives rise to multiple phase transitions, in striking similitude with recent experiments on Yb2Ti2O7 [Lhotel et al., Phys. Rev. B 89 224419 (2014)]. Considering the effective Hamiltonian determined for Yb2Ti2O7, we provide, by combining a gamut of numerical techniques, compelling evidence that such multiphase competition is the long-sought missing key to understanding the intrinsic properties of this material. As a corollary, our work offers a pertinent illustration of the influence of chemical pressure in rare-earth pyrochlores.
We examine the Si(111) multi-valley quantum Hall system and show that it exhibits an exceptionally rich interplay of broken symmetries and quantum Hall ordering already near integer fillings $ u$ in the range $ u=0-6$. This six-valley system has a large $[SU(2)]^3rtimes D_3$ symmetry in the limit where the magnetic length is much larger than the lattice constant. We find that the discrete ${D}_3$ factor breaks over a broad range of fillings at a finite temperature transition to a discrete nematic phase. As $T rightarrow 0$ the $[SU(2)]^3$ continuous symmetry also breaks: completely near $ u =3$, to a residual $[U(1)]^2times SU(2)$ near $ u=2$ and $4$ and to a residual $U(1)times [SU(2)]^2$ near $ u=1$ and $5$. Interestingly, the symmetry breaking near $ u=2,4$ and $ u=3$ involves a combination of selection by thermal fluctuations known as order by disorder and a selection by the energetics of Skyrme lattices induced by moving away from the commensurate fillings, a mechanism we term order by doping. We also exhibit modestly simpler analogs in the four-valley Si(110) system.
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