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arXiv:1209.0688 · Physical Review B

Magnetic and nematic orderings in spin-1 antiferromagnets with single-ion anisotropy

Keola Wierschem, Yasuyuki Kato, Yusuke Nishida, Cristian D. Batista, Pinaki Sengupta

Published Nov 30, 2012• 22 citations
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Abstract

We study a spin-1 Heisenberg model with exchange interaction $J$, uniaxial single-ion exchange anisotropy $D$, and Zeeman coupling to a magnetic field $B$ parallel to the symmetry axis. We compute the $(D/J,B/J)$ quantum phase diagram for square and simple cubic lattices by combining analytical and quantum Monte Carlo approaches, and find a transition between $XY$ antiferromagnetic and ferronematic phases that spontaneously break the $U(1)$ symmetry of the model. In the language of bosonic gases, this is a transition between a Bose-Einstein condensate (BEC) of single bosons and a BEC of pairs. Our work opens up new avenues for measuring this transition in real magnets.

DOI
10.1103/physrevb.86.201108

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We study a spin-1 Heisenberg model with exchange interaction JJJ, uniaxial single-ion exchange anisotropy DDD, and Zeeman coupling to a magnetic field BBB parallel to the symmetry axis. We compute the (D/J,B/J)(D/J,B/J)(D/J,B/J) quantum phase diagram for square and simple cubic lattices by combining analytical and quantum Monte Carlo approaches, and find a transition between XYXYXY antiferromagnetic and ferronematic phases that spontaneously break the U(1)U(1)U(1) symmetry of the model. In the language of bosonic gases, this is a transition between a Bose-Einstein condensate (BEC) of single bosons and a BEC of pairs. Our work opens up new avenues for measuring this transition in real magnets.
The phase diagram of the XXZXXZXXZ spin-1 quantum magnet on the kagome lattice is studied for all cases where the zzz-component nearest-neighbor spin interaction Jz{J}_{z}Jz​ is antiferromagnetic. Besides Jz{J}_{z}Jz​ and the nearest-neighbor in-plane spin interaction Jxy{J}_{xy}Jxy​, the system is also parametrized by an on-site anisotropic term D(Sz)2D{({S}^{z})}^{2}D(Sz)2. In the zero magnetic field case, the six previously introduced phases, found using various methods, are the nondegenerate gapped photon phase, which breaks no space symmetry or spin symmetry; the sixfold degenerate phase with plaquette order, which breaks both time-reversal symmetry and translational symmetry; the ``superfluid'' (ferromagnetic) phase with an in-plane global U(1) symmetry broken when Jxy<0{J}_{xy}<0Jxy​<0; the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order when Jxy>0{J}_{xy}>0Jxy​>0; the nematic phase when D<0D<0D<0 and large; and a phase with resonating dimers on each hexagon. We obtain all of these phases and partial information about their quantum phase transitions in a single framework by studying condensation of defects in the sixfold plaquette phases. The transition between nematic phase and the sixfold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, another ordered phase with translation symmetry broken is opened up in the nematic phase. Due to the breaking of time-reversal symmetry by the field, a supersolid phase emerges between the sixfold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, organic compound $m\text{\ensuremath{-}}\mathrm{MPYNN}∙\mathrm{B}{\mathrm{F}}_{4}$, or optical lattices of atoms with three degenerate states on every site.
We theoretically establish the mean-field phase diagram of a homogeneous spin-1, spin-orbit-coupled Bose gas as a function of the spin-dependent interaction parameter, the Raman coupling strength, and the quadratic Zeeman shift. We find that the interplay between spin-orbit coupling and spin-dependent interactions leads to the occurrence of ferromagnetic or ferronematic phases which also break translational symmetry. For weak Raman coupling, increasing attractive spin-dependent interactions (as in 87Rb^{87}\mathrm{Rb}87Rb or 7Li^{7}\mathrm{Li}7Li) induces a transition from a uniform to a stripe XYXYXY ferromagnet (with no nematic order). For repulsive spin-dependent interactions (as in 23Na^{23}\mathrm{Na}23Na), however, we find a transition from an XYXYXY spin spiral phase ($\ensuremath{\langle}{S}_{z}\ensuremath{\rangle}=0$ and uniform total density) with uniaxial nematic order to a biaxial ferronematic, where the total density, spin vector, and nematic director oscillate in real space. We investigate the stability of these phases against the quadratic Zeeman effect, which generally tends to favor uniform phases with either ferromagnetic or nematic order but not both. We discuss the relevance of our results to ongoing experiments on spin-orbit-coupled, spinor Bose gases.
We predict the instability of the quantum spin-1 chain material with respect to the spin nematic ordering. For the spin subsystem the spin nematic order parameter is related to the onset of single-ion spin anisotropy. The ordering is caused by the coupling to the elastic subsystem of the crystal or, for spin-1 ultracold bosons in a one-dimensional optical lattice, by the interaction with a Bose-Einstein condensate. Our conclusions are based on the exact integrability of the spin model and on numerical simulations for the nonintegrable cases. Our exact results also describe the behavior of interacting gluons within the lattice toy Yang-Mills-like model, in which spontaneous and field-induced symmetry breaking can be realized.
We study the quadratic Zeeman effect (QZE) in a system of antiferromagnetic spin-1 bosons on a square lattice and derive the ground-state phase diagrams by means of quantum Monte Carlo simulations and mean-field treatment. The QZE imbalances the populations of the magnetic sublevels $\ensuremath{\sigma}=\ifmmode\pm\else\textpm\fi{}1$ and $\ensuremath{\sigma}=0$, and therefore affects the magnetic and mobility properties of the phases. Both methods show that the tip of the even Mott lobes, stabilized by singlet state, is destroyed when turning on the QZE, thus leaving the space to the superfluid phase. Contrariwise, the tips of odd Mott lobes remain unaffected. Therefore, the Mott-superfluid transition with even filling strongly depends on the strength of the QZE, and we show that the QZE can act as a control parameter for this transition at fixed hopping. Using quantum Monte Carlo simulations, we elucidate the nature of the phase transitions and examine in detail the nematic order: the first-order Mott-superfluid transition with even filling observed in the absence of QZE becomes second order for weak QZE, in contradistinction to our mean-field results which predict a first-order transition in a larger range of QZE. Furthermore, a spin nematic order with director along the zzz axis is found in the odd Mott lobes and in the superfluid phase for energetically favored $\ensuremath{\sigma}=\ifmmode\pm\else\textpm\fi{}1$ states. In the superfluid phase with even filling, the xyxyxy components of the nematic director remain finite only for moderate QZE.
The S=1S=1S=1 bilinear-biquadratic Heisenberg exchange model on the triangular lattice with a single-ion anisotropy has previously been shown to host a number of exotic magnetic and nematic orders [Moreno-Cardoner et al., Phys. Rev. B 90, 144409 (2014)], including an extensive region of ``supersolid'' order. In this work, we amend the model by an XXZXXZXXZ anisotropy in the exchange interactions. Tuning to the limit of an exactly solvable S=1S=1S=1 generalized Ising-/Blume-Capel-type model provides a controlled limit to access phases at finite transverse exchange. Notably, we find an additional macroscopically degenerate region in the phase diagram and study its fate under perturbation theory. We further map out phase diagrams as a function of the XXZXXZXXZ anisotropy parameter, the ratio of bilinear and biquadratic interactions, and single-ion anisotropy, and we compute corrections to the total ordered moment in various phases using systematically constructed linear flavor-wave theory. We also present linear flavor-wave spectra of various states, finding that the lowest-energy band in three-sublattice generalized (i.e., with ${S}^{z}=\ifmmode\pm\else\textpm\fi{}1,0)$ Ising/Blume-Capel states, stabilized by strong exchange anisotropies, is remarkably flat, opening up the way to flat-band engineering of magnetic excitation via stabilizing nontrivial Ising-ordered ground states.