Lifetime of gapped excitations in collinear quantum antiferromagnet: Supplemental Information A. L. Chernyshev, 1 M. E. Zhitomirsky, 2 N. Martin, 2 and L.-P. Regnault 2 1 Department of Physics, University of California, Irvine, California 92697, USA 2 Service de Physique Statistique, Magn´ etisme et Supraconductivit´ e, UMR-E9001 CEA-INAC/UJF, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France (Dated: June 15, 2012) Spin Hamiltonian Here we briefly outline basic steps and main results of the spin-wave calculations for the energy spectrum and the magnon relaxation rates of the J 1 –J 3 antiferromagnet on a honeycomb lattice. The harmonic spin-wave analy- sis of the nearest-neighbor Heisenberg honeycomb-lattice antiferromagnet can be found, for example, in [1]. Geometry of exchange bonds of the considered model is schematically shown in Fig. 1. The unit cell of the antifer- romagnetic structure coincides with the crystal unit cell and contains two oppositely aligned spins S 1,i and S 2,i in positions (0, 0) and ρ =(a/ √ 3, 0). The elementary translation vectors are defined as a 1 = a( √ 3/2, −1/2) and a 2 = a(0, 1). The lattice constant in BaNi 2 (PO 4 ) 2 is equal to a =4.81 ˚ A. The reciprocal lattice basis is b 1 =4π/( √ 3a)(1, 0) and b 2 =2π/( √ 3a)(1, √ 3). The volume of the Brilouin zone is V BZ =8π 2 / √ 3a 2 . The spin Hamiltonian includes Heisenberg exchange interactions between first- and third-neighbor spins to- gether with the single-ion anisotropy: ˆ H = J 1  i S 1,i · (S 2,i + S 2,i−1 + S 2,i+3 ) + J 3  i S 1,i · (S 2,i+2 + S 2,i−2 + S 2,i−1+3 ) (1) + D  i  (S z 1,i ) 2 +(S z 2,i ) 2  . Here S 2,i−1 denotes spin in the unit cell R i − a 1 and so on. The microscopic parameters for BaNi 2 (PO 4 ) 2 3 J 1 1 J 3 a 2 a a FIG. 1: J1–J3 model in a honeycomb lattice. (S = 1) were determined from the magnon dispersion as J 1 =0.38 meV, J 3 =1.52 meV, and D =0.34 meV [2]. The second-neighbor exchange was estimated to be much smaller J 2 =0.05 meV and is neglected in the fol- lowing. Applying the Holstein-Primakoff transformation for two antiferromagnetic sublattices and performing the Fourier transformation a i = 1 N 1/2  k e ikRi a k , b i = 1 N 1/2  k e ik(Ri+ρ) b k , (2) we obtain the harmonic part of the boson Hamiltonian ˆ H 2 = S  k  (3J 13 + D)(a † k a k + b † k b k ) (3) − a k b −k F ∗ k + D 2 (a k a −k + b k b −k ) + h. c.  , where we use the shorthand notations J 13 = J 1 + J 3 , (4) F k = J 1 ( e ik1 + e ik2 + e ik3 ) + J 3 ( e ik4 + e ik5 + e ik6 ) , with k n = k · r n and r 1 = a √ 3 (1, 0) , r 2,3 = a √ 3  − 1 2 , ± √ 3 2  , r 4,5 = a √ 3 (1, ± √ 3) , r 6 = a √ 3 (−2, 0) . (5) Diagonalization of the quadratic form (3) with the help of the canonical Bogolyubov transformation yields ˆ H 2 =  k  ε α (k)α † k α k + ε β (k)β † k β k  , (6) where excitation energies are ε α (k)= S  (3J 13 −|F k |)(3J 13 + |F k | +2D) , (7) ε β (k)= S  (3J 13 + |F k |)(3J 13 −|F k | +2D) . (8) The first magnon branch is gapless, ε α (0) = 0, and reaches the maximum value of ω α max = S  (2J 3 +4J 1 )(4J 3 +2J 1 +2D) ≈ 5.9 meV (9) at k =[q,q] with q ≈ 0.25 in the reciprocal lattice units. The second branch describes optical magnons with a fi- nite energy gap at k =0 Δ=2S  3DJ 13 ≈ 2.8 meV . (10) The maximum of the optical branch ω β max is close to (9).