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Non-linear spin wave theory in the strong easy-axis limit of the triangular XXZ model

Achille Mauri, Siebe Roose, Frédéric Mila

Abstract

Motivated by recent experimental studies, we investigate the spectrum of the nearest-neighbour triangular XXZ model within the $1/S$ expansion, in the limit in which the exchange couplings present a strong easy-axis anisotropy $J_{xy}/J_{zz} \ll 1$. We show that in the limit in which $1/S \to 0$ and $J_{xy} \to 0$ at fixed $V = J_{zz}/(S J_{xy})$, the triangular spin model can be reduced to an effective boson model with quartic interactions on the honeycomb lattice. This effective model interpolates between a spin-wave ($V \to 0$) and a strong-coupling limit ($V \to \infty$), and encodes in a simple framework the regimes discussed by Kleine et al. [Z. Phys. B Condens. Matter 86, 405 (1992); 87, 103 (1992)]. For zero field, the classical ground state of the model presents an accidental degeneracy, which has a particularly simple form and which can be expressed in terms of a simple symmetry of the classical energy. The model thus offers a particularly transparent realization of a theory with quantum order-by-disorder and a pseudo-Goldstone mode. We analyze the spectrum at zero magnetic field by calculating the self-energy at one-loop order, using a self-consistent renormalization of the gap and the energy scale. Within the self-consistent approximation considered here, the corrections present a complex evolution as a function of $V$. We discuss the one-loop corrections in comparison with the spectrum observed experimentally in K$_{2}$Co(SeO$_{3}$)$_{2}$.

Non-linear spin wave theory in the strong easy-axis limit of the triangular XXZ model

Abstract

Motivated by recent experimental studies, we investigate the spectrum of the nearest-neighbour triangular XXZ model within the expansion, in the limit in which the exchange couplings present a strong easy-axis anisotropy . We show that in the limit in which and at fixed , the triangular spin model can be reduced to an effective boson model with quartic interactions on the honeycomb lattice. This effective model interpolates between a spin-wave () and a strong-coupling limit (), and encodes in a simple framework the regimes discussed by Kleine et al. [Z. Phys. B Condens. Matter 86, 405 (1992); 87, 103 (1992)]. For zero field, the classical ground state of the model presents an accidental degeneracy, which has a particularly simple form and which can be expressed in terms of a simple symmetry of the classical energy. The model thus offers a particularly transparent realization of a theory with quantum order-by-disorder and a pseudo-Goldstone mode. We analyze the spectrum at zero magnetic field by calculating the self-energy at one-loop order, using a self-consistent renormalization of the gap and the energy scale. Within the self-consistent approximation considered here, the corrections present a complex evolution as a function of . We discuss the one-loop corrections in comparison with the spectrum observed experimentally in KCo(SeO).

Paper Structure

This paper contains 20 sections, 102 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model, linear spin-wave approximation, and perturbation theory
  3. Effective model on the honeycomb lattice
  4. Semiclassical limit $V \to 0$
  5. Self-energy in the one-loop approximation
  6. Bare propagator
  7. Self-energy at one-loop order
  8. Pseudo-Goldstone gap at zero magnetic field
  9. Self-consistent theory of the spectrum at zero field
  10. Numerical results and discussion
  11. Comparison with the zero-field spectrum of K$_{2}$Co(SeO$_{3}$)$_{2}$
  12. Summary and conclusions
  13. Non-interacting theory and bare propagators
  14. Correction to the order parameter
  15. Calculation of the self-energy
  16. ...and 5 more sections

Figures (9)

  • Figure 1: (a) Structure of the "Y" configuration. (b) Brillouin zone and special $\mathbf{k}$ points in momentum space. The coordinates of the $\Gamma$, $K$, $M$, and $Y$ points are respectively $\Gamma = (0, 0)$, $K = (4\pi/3, 0)$, $M = \pi(1, \sqrt{3}/3)$, $Y = 2 \pi/3 (1, \sqrt{3}/3)$. The gray area shows the reduced Brillouin zone, folded due to the $\sqrt{3} \times \sqrt{3}$ unit cell.
  • Figure 2: Dispersions of the two lowest LSWT branches, calculated for $\alpha = 0.08$ and for magnetic fields $b = 0$, $b = 0.02$, $b = 1$, and $b = 3$, along the path $\Gamma K M \Gamma$ shown in Fig. \ref{['3subl']}. The blue solid lines show the dispersions obtained by solving the complete LSWT equations for the triangular XXZ model; the red dashed lines show the analytical dispersions \ref{['LSWT_ap']}, which are the $\alpha \to 0$ limits of the LSWT. Due to the small value of $\alpha$, the dispersions are well approximated by the analytical expressions. The dispersions at zero field present saddle points at $M = (\pi, \sqrt{3}\pi/3 )$. In addition, the two dispersions cross at the point $Y = (2 \pi/3, 2\sqrt{3}\pi/9)$, where $f_{\mathbf{k}} = 0$ for all values of $b$. The Y point is akin to a "Dirac" node. Due to this reason, the separation of the spectrum into a Goldstone and a pseudo-Goldstone part is only meaningful at small momentum, and cannot be continuously extended to the full Brillouin zone.
  • Figure 3: The ground state for $\alpha \to 0$ and zero field hypothesized by Ref. kleine_zpb_1992b. For $S\geq 3/2$, the spins at the C sublattice are frozen in the state $S^{z} = -S$. The remaining spins fluctuate between $S^{z} = +S$ and $S^{z} = +(S-1)$ and form a superfluid of bosons hopping on the honeycomb lattice. In the figure, the $-S$ spins are represented by black circles, the $+S$ spins by gray circles, and the $+(S -1)$ spins by open circles. The Ising constraint for $\alpha \to 0$ is equivalent to an infinite nearest-neighbour repulsion between the $S^{z} = +(S-1)$ spins.
  • Figure 4: Self-energy diagrams contributing at one-loop order. The graphs (b) and (c) are one-line irreducible. The tadpole graph (a) instead is one-line reducible. The graph (a) reflects the change of the magnon Hamiltonian $\partial^{2} H/(\partial m_{i}^{\alpha} \partial m_{j}^{\beta})|_{\bar{m}}$ due to the one-loop renormalization of the order parameter.
  • Figure 5: Red solid lines: dispersion relation at zero physical field within the renormalized one-loop approximation, Eq. \ref{['C_k']}, for growing values of the coupling constant $V$, corresponding to the renormalized fields $\tilde{b} = 0.0001, 0.03, 0.2, 0.5$. Blue dashed lines: dispersions $z\epsilon_{\lambda}(\mathbf{k}, \tilde{b})$. Grey lines: bare spectrum at $b = 0$, $\epsilon(\mathbf{k}) = \sqrt{9 - |f_{\mathbf{k}}|^{2}}$.
  • ...and 4 more figures