UC Irvine UC Irvine Previously Published Works

Home , Roton

Title Field-induced decays in XXZ triangular-lattice antiferromagnets

Permalink https://escholarship.org/uc/item/2b96f2bw

Journal Physical Review B, 94(14)

ISSN 2469-9950

Authors Maksimov, PA Zhitomirsky, ME Chernyshev, AL

Publication Date 2016-10-10

DOI 10.1103/PhysRevB.94.140407

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California Field-induced decays in XXZ triangular-lattice antiferromagnets

P. A. Maksimov,1 M. E. Zhitomirsky,2 and A. L. Chernyshev1 1Department of Physics and Astronomy, University of California, Irvine, California 92697, USA 2CEA, INAC-PHELIQS, F-38000, Grenoble, France (Dated: July 29, 2016) We investigate field-induced transformations in the dynamical response of the XXZ model on the triangular lattice that are associated with the anharmonic magnon coupling and decay phenomena. 1 A set of concrete theoretical predictions is made for a close physical realization of the spin- 2 XXZ model, Ba3CoSb2O9. We demonstrate that dramatic modifications in magnon spectrum must occur in low out-of-plane fields that are easily achievable for this material. The hallmark of the effect is a coexistence of the clearly distinct well-defined magnon excitations with significantly broadened ones in different regions of the k−ω space. The field-induced decays are generic for this class of models and become more prominent at larger anisotropies and in higher fields.

PACS numbers: 75.10.Jm, 75.30.Ds, 75.50.Ee, 78.70.Nx

Triangular-lattice antiferromagnets (TLAFs) are cen- phenomena discussed in this work are substantially more tral to the field of frustrated magnetism as representa- dramatic and should be free from such uncertainties. tives of one of the basic models epitomizing the effect of Model and spectrum.—Owing to frustration and degen- spin frustration [1–4]. They have attracted significant ex- eracies of the model, triangular-lattice antiferromagnets perimental and theoretical interest [5–21] as a potential in external field have a very rich phase diagram [40–43], source of spin-liquid and of a wide variety of intriguing featuring the hallmark plateau, coplanar, and umbrella ordered ground states, see Ref. [22]. Their spectral prop- states, see Ref. [22] for a recent review. We will focus erties have recently emerged as a subject of intense re- on the XXZ Hamiltonian with an easy-plane anisotropy search that has consistently uncovered broad, continuum- whose zero-field ground state is a 120◦ structure like spectral features [6, 23, 24], which are interpreted as ˆ X x x y y z z X z an evidence of fractionalized excitations [6, 20, 25] or of H = J Si Sj + Si Sj + ∆Si Sj − H Si , (1) the phenomenon of magnon decay [26–29]. ij i h i In this work, we outline a theoretical proposal for a where hiji are nearest neighbor cites of the triangular dramatic transformation of the spin-excitation spectrum lattice, J > 0, and 0 ≤ ∆ < 1. In an out-of-plane mag- of the XXZ triangular-lattice antiferromagnet in exter- netic field, the so-called umbrella structure is formed, nal out-of-plane field. This consideration pertains in par- see Fig. 1. In the isotropic limit, ∆ = 1, the coplanar ticular to Ba CoSb O , one of the close physical real- 3 2 9 states are favored instead, but ∆ < 1 always stabilizes izations of the model that has recently been studied by the semiclassical umbrella state for a range of fields, with a variety of experimental techniques [30–34]. The key the H − ∆ region of its stability for S = 1/2 sketched in finding of our work is that a modest out-of-plane field results in a strong damping of the high-energy magnons, affecting a significant part of the k-space. This is dif- H 2.5 ferent from a similar prediction of the field-induced de-

cays in the square- and honeycomb-lattice AFs where z0 K K’ J / 2 ✓ y0 strong spectrum transformations require large fields [35– k x0 38]. In the present case, because the staggered chirality ε of the field-induced umbrella spin structure breaks inver- 1.5 sion symmetry, the resultant k ↔ −k asymmetry of the magnon spectrum opens up a channel for decays of the 1 arXiv:1607.08238v1 [cond-mat.str-el] 27 Jul 2016 high-energy magnons in a broad vicinity of the K0 corners of the Brillouin zone into the two-magnon continuum of the roton-like magnons at the K-points, see Fig. 1. 0.5 H/Hs = 0 ⇤ We note that the recent neutron-scattering work [23] H/Hs = 0.2 0 asserts the existence of an intrinsic broadening in parts of Μ K ´ Γ K ΜΓ the Ba3CoSb2O9 spectrum even in zero field. While scat- terings due to finite-temperature magnon population or FIG. 1: Linear spin wave energy εk of model (1) for ∆ = 0.9, strong effects of disorder in the non-collinear spin struc- S = 1/2 and the fields H = 0 and H = 0.2Hs. Arrows show tures [39] cannot be ruled out as sources of damping ob- schematics of the decay. Insets: umbrella structure in a field, served in Ref. [23], we would like to point out that the 3D plot of εk for H = 0.2Hs, and decay self-energy diagram. 2

Fig. 2(c) [from Ref. [43]]. In Ba3CoSb2O9, estimates of 2 the anisotropy yield ∆ ≈ 0.9 [23, 33] with an additional (a) stabilization of the umbrella-like state provided by the in- M0 K0 terplane coupling [34, 44]. The linear spin-wave (LSW) treatment of the model (1) within the 1/S-expansion is 1.5 X1 L standard, see [45]. The harmonic magnon energies, ε , M k J /

are depicted in Fig. 1 for S =1/2, ∆=0.9, and H =0 and H = 0.2H , where H = 6JS(∆ + 1/2) is the saturation s s K field. The chosen representative field of 0.2Hs is within 1 the umbrella region of Fig. 2(c) and for Ba3CoSb2O9 it (b) 1 corresponds a modest field of about√ 6 T [33]. non-umbrella 0.8 In Fig. 1, one can see the gaps ∝ 1 − ∆ at K and K0 0.5 0.6 points in zero field. In a finite field, the staggered scalar s chirality of the umbrella structure, Si ·(Sj ×Sk), induces H/H 0.4 inversion symmetry breaking. Because of that, magnon 0.2 energy acquires an asymmetric contribution [46], εk =6 (c) 0 0 0 0.2 0.4 0.6 0.8 1 ε k, with the energies at K (K0) points lowered (raised) LXK ´ 1 − proportionally to the field. Note that the K and K0 points FIG. 2: (a) The intensity plot of the spectral function along trade their places in the domain with a shifted pattern 0 MK Γ path with Γk from the self-consistent iDE for ∆ = of the 120◦ order that also corresponds to the flipped 0.9 and H = 0.2Hs. Dashed and solid lines are Γk in the staggered chiralities. It is clear, that the distorted band Born approximation (2) and the iDE solution. (b) The 2D structure brings down the energy of a minimum of the intensity plot of Γk from (2). (c) The H −∆ diagram of the two-magnon continuum associated with the low-energy, decay thresholds in the umbrella state. Shaded are the regions roton-like magnons at K-points. Given the remaining where various forms of decay are allowed, see text. The non- commensurability of the umbrella state, which retains umbrella region for S = 1/2 is sketched from Ref. [43]. The dot marks the values of ∆ and H used in (a) and (b). the 3K=0 property of the 120◦ structure, magnon decays may occur in a proximity of the K0 points via a process 0 εK ⇒ εK +εK( Gi), where Gi’s are the reciprocal lattice ± vectors. While the exact kinematics of such decays is decay vertex Φq,k q;k is derived from the anharmonic − somewhat more complicated, one can simply check where coupling terms of the 1/S-expansion of the model (1), and at what field the on-shell decay conditions, εk = see [45]. It combines the effects of noncollinearity due εq + εk q, are first met for a given ∆. to in-plane 120◦ structure and of the field-induced tilting − This direct verification yields the lower border of the of spins [26, 28, 35]. We show Γk for a representative shaded regions in Fig. 2(c), which is a union of three H = 0.2Hs and for the same ∆ = 0.9 and S = 1/2 as curves. At large anisotropies, ∆ → 0, the decay con- above: in Fig. 2(a) along the MK0Γ path (dashed line) ditions that are fulfilled at the lowest field are the ones and in Fig. 2(b) as a 2D intensity plot. associated with the change of the curvature of the Gold- In addition, we also present the results of the self- stone mode near the Γ point, the kinematics familiar consistent solution of the off-shell Dyson’s equation (DE) from the field-induced decays in the square-lattice [35] for Γk, in which corrections to the magnon energy are and honeycomb-lattice AFs [38], as well as 4He [47]. At ignored but the imaginary part of the the magnon self- larger ∆, the threshold field for decays is precisely deter- energy Σk(ω) due to three-magnon coupling is retained, mined by the “asymmetry-induced” condition εK0 = 2εK referred to as the iDE approach: Γk =−Im Σk (εk + iΓk). discussed above, which is given analytically by H = p ∗ This method accounts for a damping of the decaying (1 − ∆)/(13 − ∆) and is shown by the dashed line in initial-state magnon and regularizes the van Hove singu- Fig. 2(c). Closer to the isotropic limit, ∆ & 0.7, the de- larities associated with the two-magnon continuum that cay conditions are first met away from the high-symmetry can be seen in the Born results of (2) in Fig. 2(a). The points, see some discussion of them for the zero-field case same Figure shows the iDE results for Γk (solid line) and ∆ > 0.92 in Ref. [28]. and the corresponding magnon spectral function in a On(off)-shell decay rate.—To get a sense of the quanti- lorentzian form (intensity plot). We note that the self- tative measure of the field-induced broadening effect and consistency schemes that rely on the broadening of the of the extent of the affected k-space, we first present the decay products, such as iSCBA discussed in Refs. [36, 48], results for the decay rate in the Born approximation are not applicable here because the final-state magnons X 2 remain well-defined in our case. Altogether, our consid- Γk = Γ0 |Φq,k q;k| δ (ωk − ωq − ωk q) , (2) − − eration suggests that a significant field-induced broaden- q ing of quasiparticle peaks due to magnon decays should where Γ0 = 3πJ/4 and εk = 3JSωk. The three-magnon appear in a wide vicinity of the K0 points in low fields. 3

2 (a) 30 (b)

) 20 q =K0 ,

q 0 0 ( M K

S 10

0 1.5 0.96 0.98 1 1.02 1.04 X1 L /J !/J J

/ M 1 K ! =1.05J 30

) 20 q + Q q =X1 0.5 , q ( S 10 q

300 =0.9 w=1.05J q + Q /J ) 20 q =L , q ( H =0.2Hs S 10 (c) q 0 0 LXK ´ 1 K 0.9 1 1.1/!J /J 1.2 1.3

FIG. 3: (a) Intensity plot of S(q, ω) along the MK0ΓKMΓ path. Dotted and dashed lines are constant-energy cut in (b) and 0 ω-cuts in (c). Inset: S(q, ω) vs ω at K . Bars are artiﬁcial width 2δ =0.01J of the calculation [52]. S =1/2, ∆=0.9, H =0.2Hs.

Dynamical structure factor.—Next, we evaluate the spectral functions, A(q, ω) and A(q ± K, ω), with differ- dynamical spin-spin structure factor S(q, ω), the quan- ent weights according to (4) and [45], see also [49]. tity directly observed in the inelastic neutron scattering In our consideration, we include all contributions to experiments. Following Ref. [29], we approximate S(q, ω) the one-loop magnon self-energy Σq(ω) of the 1/S-order as a sum of the diagonal terms [45] of of the non-linear spin-wave theory [26]. Namely, there are two more terms in addition to decay diagram: the Z ∞ α0β0 i iωt α0 β0 source diagram and the Hartree-Fock correction, the lat- S (q, ω) = Im dte hTSq (t)S q(0)i. (3) π − ter comprised of the contributions from the four-magnon −∞ interactions (quartic terms) and from the quantum cor- Transforming to the local (rotating) reference frame of rections to the out-of-plane canting angle of spins, see the ordered moments and keeping terms that contribute [45] for technical details, to the leading 1/S order [29] yields Σ (ω) = ΣHF + Σd (ω) + Σs (ω). (5) h q q q q x0x0 1 yy yy xy xy S (q, ω) = Sq+ + Sq − 2i sin θ Sq+ − Sq 4 − − Having included all one-loop contributions also allows us i 2 xx xx 2 zz zz to consistently take into account the ω-dependence of the + sin θ Sq+ + Sq + cos θ Sq+ + Sq , (4) − − magnon spectral function. Below we demonstrate that z0z0 2 xx 2 zz S (q, ω) = cos θ Sq + sin θ Sq , anharmonic interactions lead to broadening of magnon quasiparticle peaks, redistribution of spectral weight, and y y x x and S 0 0 (q, ω) = S 0 0 (q, ω). Here we used the anti- other dramatic changes in the spectrum. xy yx symmetric nature of the xy contribution Sq =−Sq and In Fig. 3, we present our results for the dynami- αβ αβ introduced shorthand notations for Sq ≡S (q, ω) and cal structure factor S(q, ω) in (4) of the model (1) for “shifted” momenta q± ≡ q ± K, with θ being the out- S = 1/2, ∆ = 0.9, and H = 0.2Hs. First, there is a of-plane canting angle of spins. In the local reference strong downward bandwidth renormalization by about zz frame, Sq components of the dynamical structure fac- 30% compared to the LSW results in Fig. 1, which is tor are “longitudinal”, i.e., are due to the two-magnon characteristic to the TLAFs [18, 27, 28]. The most impor- continuum, having no sharp quasiparticle features [29]. tant result is a significant broadening of magnon spectra The rest of Eq. (4) is “transverse”, i.e., is related to for an extensive range of momenta, accompanied by well- x(y)x(y) the single-magnon spectral function, Sq ∝ A(q, ω), pronounced termination points with distinctive bending with different kinematic q-dependent formfactors, where of spectral lines [51] and other non-Lorentzian features A(q, ω) = −(1/π)ImG(q, ω) and the diagonal magnon that are associated with crossings of the two-magnon con- 1 Green’s function is G(q, ω) = [ω − εq − Σq(ω) + iδ]− . tinuum. The broadening can be seen in a wide proxim- Thus, the dynamical structure factor of the XXZ TLAF ity of the K0 points of the Brillouin zone as well as in in a field should feature three overlapping single-magnon the equivalent regions of the “±K-shifted” components 4

∆ = 0, the non-linear anharmonic coupling of magnons 3.5 is known to result in a very strong spectrum renormal- (a) ization (about 50%), but with no decays kinematically 3 =0 allowed [28]. For S = 1/2 and small enough ∆, Born H =0.4Hs 2.5 approximation and the 1/S, one-loop, ω-dependent self- energy approach are somewhat inconsistent in that the 2 ﬁrst produces unphysically large Γk for H & 0.3Hs and J / the second shows strong spectrum renormalization that 1.5 avoids decays for H . 0.5Hs. Since the reason for this discrepancy is the lack of self-consistency, we resort to the 1 (partially) self-consistent iDE approach described above. In Fig. 4, we show its results for the magnon spectral 0.5 function with the Lorentzian broadening Γk for ∆ = 0 0 and ∆ = 0.5 and for H = 0.4Hs. What is remark- K ´ K able is not only a persistent pattern of a wide k-region 3.5 (b) of the strongly overdamped high-energy magnons [cf., 3 =0.5 Fig. 2(a)], but also the magnitudes of their broadening, which reach the values of almost a half of the magnon H =0.4Hs 2.5 bandwidth even after a self-consistent regularization. Conclusions.—We have provided a detailed analysis of 2 the ﬁeld-induced dynamical response of the XXZ model J /

on the triangular lattice within the umbrella phase. We 1.5 have demonstrated a ubiquitous presence of significant 1 damping of the high-energy magnons already in moder- ate fields, H & 0.2Hs. Other characteristic features, such 0.5 as significant spectral weight redistribution and termination points that separate well-defined excitations from 0 K ´ K the ones that are overdamped, are also expected to occur. The key physical ingredients of this dramatic spectral FIG. 4: Intensity plots of the spectral function with the iDE transformation are a strong spin noncollinearity, which Γk (dashed lines) for S =1/2, H =0.4Hs, and ∆=0 in (a) and is retained by the umbrella state and is essential for the ∆=0.5 in (b). Dotted lines are the LSW spectra for H =0. anharmonic magnon coupling and decays, and the tilted, k ↔ −k asymmetric magnon band structure, owing its origin to the staggered chirality of the umbrella state of the structure factor. Despite the strong renormaliza- that breaks the inversion symmetry. Our consideration tion of the spectrum, the extent of the affected q-region is pertains in particular to Ba3CoSb2O9, which is currently about the same as in the on-shell consideration in Fig. 2. a prime candidate for observing aforementioned proper- The inset of Fig. 3(a) shows S(q, ω) vs ω at a rep- ties in reasonably small fields reachable in experimental resentative K0 point that exhibits a modest broadening setup. Our work should be of a qualitative and quantita- compared with the artificial width (2δ) of the calculation. tive guidance for observations of the dynamical structure The ω-cuts at the X1 and L points near the boundaries of factor in the inelastic neutron-scattering experiments in the decay region in Fig. 3(c) show much heavier damping this and other related systems. in one of the component of S(q, ω), which coexists with Acknowledgments.—We acknowledge useful conversa- the well-defined spectral peak from the “shifted” com- tions with Martin Mourigal and Cristian Batista. We ponent. The enhancement of magnon decays near the are particularly indebted to Martin for his unbiased ex- edge of decay region also correlates well with the on-shell perimental intuition that led to [49]. This work was sup- results in Fig. 2 and points to the van Hove singulari- ported by the U.S. Department of Energy, Office of Sci- ties of the two-magnon continuum as a culprit. The 2D ence, Basic Energy Sciences under Award # DE-FG02- intensity map of the constant-energy cut of S(q, ω) at 04ER46174. A. L. C. would like to thank the Kavli Insti- ω = 1.05J is shown in Fig. 3(b), where one can see mul- tute for Theoretical Physics where part of this work was tiple signatures of the broadening, spectral weight redis- done. The work at KITP was supported in part by NSF tribution around K0, and termination points. Grant No. NSF PHY11-25915. Larger anisotropy.—We complement our consideration of the model (1) by demonstrating the effects of magnetic field on the magnon spectrum for the TLAFs with large easy-plane anisotropy. In the strongly-anisotropic limit, [1] G. H. Wannier, Phys. Rev. 79, 357 (1950). 5

[2] P. W. Anderson, Mater. Res. Bull. 8, 153 (1973); P. Wiebe, I. Umegaki, J. Q. Yan, T. P. Murphy, J.-H. Park, Fazekas and P. W. Anderson, Philos. Mag. 30, 423 (1974). Y. Qiu, J. R. D. Copley, J. S. Gardner, and Y. Takano, [3] D. A. Huse and V. Elser, Phys. Rev. Lett. 60, 2531 (1988). Phys. Rev. Lett. 109, 267206 (2012). [4] D. H. Lee, J. D. Joannopoulos, J. W. Negele, and D. P. [33] T. Susuki, N. Kurita, T. Tanaka, H. Nojiri, A. Matsuo, Landau, Phys. Rev. Lett. 52, 433 (1984). K. Kindo, and H. Tanaka, Phys. Rev. Lett. 110, 267201 [5] M. F. Collins and O. A. Petrenko, Canad. J. Phys. 75, (2013). 605 (1997). [34] G. Koutroulakis, T. Zhou, Y. Kamiya, J. D. Thompson, [6] R. Coldea, D. A. Tennant, A. M. Tsvelik, and Z. Tyl- H. D. Zhou, C. D. Batista, and S. E. Brown, Phys. Rev. czynski, Phys. Rev. Lett. 86, 1335 (2001); R. Coldea, D. B 91, 024410 (2015). A. Tennant, K. Habicht, P. Smeibidl, C. Wolters, and Z. [35] M. E. Zhitomirsky and A. L. Chernyshev, Phys. Rev. Tylczynski, Phys. Rev. Lett. 88, 137203 (2002). Lett. 82, 4536 (1999). [7] L. E. Svistov, A. I. Smirnov, L. A. Prozorova, O. A. Pe- [36] M. Mourigal, M. E. Zhitomirsky, and A. L. Chernyshev, trenko, L. N. Demianets, and A. Ya. Shapiro, Phys. Rev. Phys. Rev. B 82, 144402 (2010). B 67, 094434 (2003). [37] W. T. Fuhrman, M. Mourigal, M. E. Zhitomirsky, and [8] S. Nakatsuji, Y. Nambu, H. Tonomura, O. Sakai, S. Jonas, A. L. Chernyshev, Phys. Rev. B 85, 184405 (2012). C. Broholm, H. Tsunetsugu, Y. Qiu, and Y. Maeno, Sci- [38] P. A. Maksimov and A. L. Chernyshev, Phys. Rev. B 93, ence 309, 1697 (2005). 014418 (2016). [9] A. Olariu, P. Mendels, F. Bert, B. G. Ueland, P. Schiffer, [39] W. Brenig and A. L. Chernyshev, Phys. Rev. Lett. 110, R. F. Berger, and R. J. Cava, Phys. Rev. Lett. 97, 167203 157203 (2013). (2006). [40] H. Kawamura and S. Miyashita, J. Phys. Soc. Jpn. 54, [10] T. Oguchi, J. Phys. Soc. Jpn. Suppl. 52, 183 (1983). 4530 (1985). [11] Th. Jolicoeur and J. C. Le Guillou, Phys. Rev. B 40, [41] S. E. Korshunov, J. Phys. C: Solid State Phys. 19, 5927 2727 (1989). (1986). [12] S. J. Miyake, J. Phys. Soc. Jpn. 61, 983 (1992). [42] A. V. Chubukov and D. I. Golosov, J. Phys.: Condens. [13] A. V. Chubukov, S. Sachdev, and T. Senthil, J. Phys. Matter 3, 69 (1991). Condens. Matter 6, 8891 (1994). [43] D. Yamamoto, G. Marmorini, and I. Danshita, Phys. [14] P. W. Leung and K. J. Runge, Phys. Rev. B 47, 5861 Rev. Lett. 112, 127203 (2014); G. Marmorini, D. Ya- (1993). mamoto, and I. Danshita, Phys. Rev. B 93, 224402 (2016). [15] B. Bernu, P. Lecheminant, C. Lhuillier, and L. Pierre, [44] R. S. Gekht and I. N. Bondarenko, J. Exp. Theor. Phys. Phys. Rev. B 50, 10048 (1994). 84, 345 (1997). [16] L. Capriotti, A. E. Trumper, and S. Sorella, Phys. Rev. [45] See Supplemental Material at Lett. 82, 3899 (1999). http://link.aps.org/supplemental/, for details on the [17] Z. Weihong, J. Oitmaa, and C. J. Hamer, Phys. Rev. B non-linear spin-wave theory. 44, 11869 (1991). [46] M. E. Zhitomirsky and I. A. Zaliznyak, Phys. Rev. B 53, [18] W. Zheng, J. O. Fjærestad, R. R. P. Singh, R. H. McKen- 3428 (1996). zie, and R. Coldea, Phys. Rev. B 74, 224420 (2006). [47] L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 36, 1168 (1959) [Sov. [19] S. R. White and A. L. Chernyshev, Phys. Rev. Lett. 99, Phys. JETP 9, 830 (1959)]. 127004 (2007). [48] T. J. Sluckin and R. M. Bowley, J. Phys. C: Solid State [20] O. A. Starykh, H. Katsura, and L. Balents, Phys. Rev. Phys. 7, 1779 (1974). B 82, 014421 (2010). [49] In experimental systems, one may expect that domains [21] Z. Zhu and S. R. White, Phys. Rev. B 92, 041105 (2015). with opposite patterns of staggered chiralities are present. [22] O. A. Starykh, Rep. Prog. Phys. 78, 052502 (2015). In that case, the dynamical structure factor S(q, ω) is a [23] J. Ma, Y. Kamiya, T. Hong, H. B. Cao, G. Ehlers, W. superposition of Fig. 3 and its mirror image. The single- Tian, C. D. Batista, Z. L. Dun, H. D. Zhou, and M. Mat- domain chiral order can be produced, e.g., by the magne- suda, Phys. Rev. Lett. 116, 087201 (2016). toelectric effect [50]. [24] J. Oh, M. D. Le, J. Jeong, J. Lee, H. Woo, W.-Y. Song, [50] M. Mourigal, M. Enderle, R. K. Kremer, J. M. Law, and T. G. Perring, W. J. L. Buyers, S.-W. Cheong, and J.-G. B. F˚ak,Phys. Rev. B 83, 100409 (2011). Park, Phys. Rev. Lett. 111, 257202 (2013). [51] K. W. Plumb, K. Hwang, Y. Qiu, L. W. Harriger, G. [25] E. A. Ghioldi, A. Mezio, L. O. Manuel, R. R. P. Singh, E. Granroth, A. I. Kolesnikov, G. J. Shu, F. C. Chou, C. J. Oitmaa, and A. E. Trumper, Phys. Rev. B 91, 134423 Rüegg,Y. B. Kim, and Y.-J. Kim, Nat. Phys. 12, 224 (2015). (2016). [26] M. E. Zhitomirsky and A. L. Chernyshev, Rev. Mod. [52] Calculations of the self-energy Σk(ω) were performed Phys. 85, 219 (2013). with a smaller artificial broadening δ0 = 0.0005J. [27] O. A. Starykh, A. V. Chubukov, and A. G. Abanov, Phys. Rev. B 74, 180403 (2006). [28] A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. Lett. 97, 207202 (2006); Phys. Rev. B 79, 144416 (2009). [29] M. Mourigal, W. T. Fuhrman, A. L. Chernyshev, and M. E. Zhitomirsky, Phys. Rev. B 88, 094407 (2013). [30] Y. Doi, Y. Hinatsu, and K. Ohoyama, J. Phys.: Condens. Matter 16, 8923 (2004). [31] Y. Shirata, H. Tanaka, A. Matsuo, and K. Kindo, Phys. Rev. Lett. 108, 057205 (2012). [32] H. D. Zhou, C. Xu, A. M. Hallas, H. J. Silverstein, C. R. 6

Field-induced decays in XXZ triangular-lattice antiferromagnets: Supplemental Material

P. A. Maksimov1, M. E. Zhitomirsky2, and A. L. Chernyshev1 1Department of Physics and Astronomy, University of California, Irvine, California 92697, USA 2CEA, INAC-PHELIQS, F-38000, Grenoble, France (Dated: July 27, 2016)

Here we present the details of the nonlinear spin-wave where the matrix RQ does rotations in the {x0y0}-plane formalism for the XXZ model on a triangular lattice in  cos ϕ − sin ϕ 0  an out-of-plane magnetic field and in the semi-classical i i sin ϕ cos ϕ 0 umbrella state. The formalism bears significant similari- RQ =  i i  , (3) 0 0 1 ties to the Heisenberg triangular antiferromagnet case in zero field [1, 2] and to the square-lattice antiferromagnet where ϕi = Q · ri is an in-plane angle. The out-of-plane in a field [3], with several details that differ from both. rotation is done by Rθ within the {x0, z0}-plane  sin θ 0 cos θ  Rθ =  0 1 0  , (4) Model and spin transformation − cos θ 0 sin θ where θ is the out-of-plane canting angle. The full trans- The nearest-neighbor XXZ Hamiltonian on a triangu- formation is explicitly given by lar lattice in external out-of-plane field is Sx0 = Sx sin θ cos Q · r − Sy sin Q · r + Sz cos θ cos Q · r,

y0 x y z X X S = S sin θ sin Q · r + S cos Q · r + S cos θ sin Q · r, Hˆ = J S · S − (1 − ∆) Sz0 Sz0 − H Sz0 , (1) i j i j i Sz0 = −Sx cos θ + Sz sin θ. (5) ij i h i where the sum is over the nearest-neighbor bonds hiji, Transformed Hamiltonian J > 0 is an exchange coupling constant, 0 ≤ ∆ ≤ 1 is the easy-plane anisotropy parameter, and H is external mag- After performing the axis-rotation transformation (2), netic field in units of gµB. The ground state in zero field we split Hamiltonian into “even” and “odd” parts is a 120 structure with the ordering vector Q= 4π , 0. ◦ 3 ˆ ˆ ˆ In applied field, spins cant towards the field direction to H = Heven + Hodd , (6) form the umbrella structure shown in Fig. 1. Thus, we which become even and odd in bosonic field operators need to align the local spin-quantization axis on each site ˆ X x x 2 2 in the direction given by the spin configuration, with the Heven = J Si Sj sin θ cos δϕij + ∆ cos θ canting angle defined from the energy minimization. ij h i y y The corresponding general transformation of the +Si Sj cos δϕij (7) spin components from the laboratory reference frame z z 2 2 +S S cos θ cos δϕij + ∆ sin θ {x , y , z } to the local reference frame {x, y, z} can be i j 0 0 0 x y y x performed using two consequent rotations + Si Sj − Si Sj sin δϕij sin θ X z −H Si sin θ. 0 Si = RQ · Rθ · Si , (2) i Within the 1/S expansion, this term will yield classical energy, harmonic spectrum, and four-magnon inter- H actions. The three-magnon interactions will be produced by the odd Hamiltonian and originate from the non- collinear spin structure z 0 ˆ X x z z x Hodd = J Si Sj + Si Sj sin θ cos θ (−∆ + cos δϕij) y0 ij x0 ✓ h i z y y z + Si Sj − Si Sj cos θ sin δϕij FIG. 1: Umbrella spin structure on the triangular lattice, θ X x is the out-of-plane canting angle. + HSi cos θ, (8) i 7

◦ 3 where δϕij = ϕi −ϕj = ±120 , i.e., the in-plane spin con-

ﬁguration is retained in the umbrella state. The classical H/Hs = 0 2.5 H/Hs = 0.1 energy can be obtained from (7) as K K’ H/Hs = 0.2

E 1 H sin θ cl = 3 ∆ sin2 θ − cos2 θ − , (9) 2 NJS2 2 JS J /

k 1.5 with the minimization yielding a relation between the ε field H and the canting angle θ as: H = Hs sin θ where 1 1 Hs =6JS ∆ + 2 is the saturation field. The subsequent treatment of the spin Hamiltonian in (7) and (8) involves a standard Holstein-Primakoff transformation 0.5 q + z − † − † 0 Si = ai 2S ai ai ,Si = S ai ai . (10) Μ K ´ Γ K ΜΓ

FIG. 2: Magnon linear spin-wave spectrum for ∆ = 0.9 and Linear spin-wave theory several ﬁelds. Inset: 3D plot of εk for H = 0.2Hs.

The next non-vanishing term in the 1/S expansion of or, explicitly, (7) beyond Ecl is the quadratic Hamiltonian √ 1 kx 3ky (2) X 2λ − 1 γk = cos kx + 2 cos cos , (19) Hˆ = JS a†a + a†a + a†a 3 2 2 i i 4 i j i j √ ij 1 kx 3ky h i γ¯k = sin k − 2 sin cos . (20) 3 x 2 2 − i ai†aj − aj†ai sin δϕij sin θ The Bogolyubov transformation of (11) 2λ + 1 c = u a + v a† , (21) + a a + a†a† , (11) k k k k k 4 i j i j − is standard with the parameters given by

1 2 1 Bk 2 2 Ak where λ = ∆ + 2 cos θ− 2 . Next, we introduce Fourier 2ukvk = , u + v = . (22) p 2 2 k k p 2 2 transformation Ak − Bk Ak − Bk Finally, the excitation spectrum is 1 X ikri ai = √ e ak . (12) N k εk =3JSωk, (23) p 2 2 where ωk = Ak − Bk + Ck. Note that because C k = Reciprocal vectors of triangular lattice are − −Ck, it is not affected by Bogolyubov transformation 4π 2π and uk and vk remain even under k → −k. Fig. 2 b1 = 0, √ , b2 = 2π, −√ . (13) 3 3 shows the linear spin-wave theory spectra in the range of parameters applicable to Ba3CoSb2O9. Since the field- This gives the harmonic Hamiltonian induced staggered scalar chirality of the umbrella structure, Si · (Sj × Sk), breaks the inversion symmetry, it ˆ(2) X Bk H = 3JS (Ak + Ck) ak† ak − ak† a† k + H.c. , leads to an asymmetry of the spectrum [4], εk =6 ε k. 2 − − k Most importantly, it shifts magnon energy at K and K0 (14) corners of the Brillouin zone in the opposite directions. with the parameters 1 2 Cubic vertices Ak = 1 + γk ∆ + cos θ − 1 , (15) 2 1 2 Holstein-Primakoff transformation in (8) yields the Bk = −γk ∆ + cos θ, (16) 2 three-magnon interaction √ Ck = 3¯γk sin θ . (17) r (3) S X 1 Hˆ = J sin 2θ ∆ + (a† + a )a†a 2 2 i i j j Here γk andγ ¯k are the nearest-neighbor amplitudes ij h i

1 X ik δi 1 X ik δi γk = e · , γ¯k = sign(sin δϕ ) e · , (18) − 2i cos θ sin δϕij a†a (a† − a ), (24) 6 6 ij i i j j δi δi 8

⇤ ⌅ ⌅⇤ Magnon decays

(a) (b) Using standard diagrammatic rules, the one-loop decay and source diagrams shown in Fig. 3 are

FIG. 3: Decay (a) and source (b) diagrams. 2 2 (a) 9J S X |Φq,k q;k| Σ11 (k, ω) = − , (32) 4 ω − εq − εk q + iδ q − 2 2 where the first term is due to the out-of-plane spin non- (b) 9J S X |Ξq, k q,k| collinearity and the second is due to the 120 in-plane Σ11 (k, ω) = − − − . (33) ◦ 4 ω + εq + ε k q − iδ order. The Fourier transformation of (24) yields q − − In our numerical calculations we used δ = 0.0005J. r √ (a) (3) S X 1 The on-shell magnon decay rate from ImΣ11 in (32) is Hˆ = 3J γk ∆ + sin 2θ − 3¯γk cos θ 2 2 k,q X 2 Γk = Γ0 |Φq,k q;k| δ (ωk − ωq − ωk q) , (34) − − q × aq† ak† ak+q + H.c. . (25) with the auxiliary constant Γ0 = 3πJ/4. To get a qualitative insight into magnon decays, one Finally, the Bogolyubov transformation (21) of (25) can neglect for a moment the other 1/S-contributions to yields the cubic Hamiltonian for the magnon eigenmodes the spectrum and focus on the effect of the decay part. Given the field-induced asymmetry of the spectrum, it r ˆ(3) 3J S X is clear that the K0 point will be prone to decays above H = Ξqkpcq† ck† cp† + H.c. , (26) 3! 2 a threshold field. Our Fig. 4 shows the on-shell Γk from p=k+q − (34) vs H for this representative k-point at several values r 3J S X of ∆. One can see that at large anisotropies, the on-shell + Φqk;pcq† ck† c p + H.c. , (27) 2! 2 − damping reaches unphysically large values. p=k+q − Kinematics.—A more systematic approach to the on- where the combinatorial factors are due to symmetriza- shell decays involves a consideration of the general decay tion in the source (26) and decay (27) vertices given by conditions [7], εk = εq + εk q. The results are presented in the H−∆ diagram in Fig.− 5. There are two main decay channels here. First, the threshold field value for decays Ξqkp = Fq(uq + vq)(ukvp + vkup) in the proximity of K0 was found numerically (blue line). + F (u + v )(u v + v u ) k k k q p q p However, for ∆ . 0.7 it becomes precisely the threshold + Fp(up + vp)(uqvk + vquk) (28) condition for the decays directly from the K0 point into ¯ − two magnons at the equivalent K points: εK0 = 2εK, + Fq(uq vq)(ukvp + vkup) p which is given analytically by H = (1 − ∆)/(13 − ∆) + F¯ (u − v )(u v + v u ) ∗ k k k q p q p (dashed line). Note that this channel is permitted by the ¯ + Fp(up − vp)(uqvk + vquk) , commensurability of the umbrella state, which retains ηνµ Φqk;p = Fq(uq + vq)(ukup + vkvp) + Fk(uk + vk)(uqup + vqvp) ∆ = 0 6 ∆ = 0.2 + Fp(up + vp)(uqvk + vquk) (29) ∆ = 0.4 ¯ ∆ = 0.6 + Fq(uq − vq)(ukup + vkvp) 5 ∆ = 0.9 + F¯k(uk − vk)(uqup + vqvp) 4 − F¯p(up − vp)(uqvk + vquk), ’ K

Γ 3 with k ↔ −k symmetric and antisymmetric amplitudes

1 2 Fk = γk ∆ + sin 2θ, (30) 2 √ 1 F¯k = − 3¯γk cos θ. (31) 0 0 0.2 0.4 0.6 0.8 1 The terms with F¯k in the vertices have a structure famil- H/Hs iar from the zero-ﬁeld consideration [2], while the sym- 0 metric terms are from the out-of-plane canting. FIG. 4: The on-shell Γk at the K point vs ﬁeld for several ∆. 9

1 3.5

0.8 3 =0.5 H =0.2Hs 0.6 2.5 s 2 J H/H /

0.4 1.5

0.2 1

0.5 0 0 0.2 0.4 0.6 0.8 1 0 K ´ K

FIG. 5: The H−∆ diagram of the decay thresholds in the um- FIG. 6: Intensity plot of the magnon spectral function along brella state. Color shaded regions represent values of param- 0 the MK ΓKMΓ path with the iDE Γk, shown by the dashed eters where decays are allowed by kinematic conditions. The line. S = 1/2, H = 0.2Hs, and ∆ = 0.5. Dotted line is the decay condition εK0 = 2εK is shown by the dashed line. De- LSW spectrum for H =0. cays from a vicinity of the K0 point exist above the blue line. Decays near the Γ point, due to the change of curvature of the Goldstone mode, exist above the black line. For ∆>0.92 Altogether, both the on-shell and the iDE consider- decays exist in zero field [2]. While semiclassically the um- ations suggest a significant field-induced broadening of brella state is stable for any ∆<1, quantum fluctuations lead quasiparticle peaks due to magnon decays in modest to proliferation of coplanar states. The non-umbrella region fields in a wide vicinity of the K0 points. for S =1/2 is sketched from Ref. [6] by a striped pattern.

Hartree-Fock corrections the 3K = 0 property of the 120◦ structure. The second channel is due to the change of curvature of the spectrum to a negative one in the vicinity of the Γ point [3, 8]. For a more consistent consideration of the ω- Expansion of the spectrum to k3 order yields dependence of the magnon spectral function we also need to include all contributions to the one-loop magnon self- r " r √ # λ 3 1 λ 5 1 h 3 energy of the same 1/S-order [7]. For that we need the ε k 0 ' |k|−|k| − + cos 3φ , | |→ 2 16 2 2 λ 24 source diagram (33) and the Hartree-Fock corrections. (35) There are two contributions to the latter. First is from 1 the four-magnon interactions (quartic terms) and the sec- where φ = tan− (ky/kx). The solution of that implicit equation is shown in Fig. 5 as a black line. ond is from the cubic terms due to quantum corrections Dyson’s equation.—In addition to the on-shell damp- to the out-of-plane canting angle of spins. ing, one can employ a straightforward self-consistent ap- Four magnon interaction terms originate from (7) and proach, which we refer to as iDE, which consists of solving can be decoupled using Hartree-Fock averages the off-shell Dyson’s equation (DE) for Γk of the magnon X 2 n = ha†a i = v , (37) pole of the Green’s function where only the imaginary i i k k part of the the magnon self-energy is retained [2] X 2 m = hai†aji = vkγk, (38) Γk = −Im Σk (εk + iΓk) . (36) k X δ = ha a i = u v , (39) This method accounts for a damping of the decaying i i k k k initial-state magnon and regularizes singularities due to ¯ X two-magnon continuum. We present some of the results ∆ = hai aji = ukvkγk, (40) of this approach in Fig. 6, which shows the magnon spec- k tral function (intensity map) with the broadening Γk to obtain correction to the magnon energy spectrum after (dashed line) for representative field and ∆. The self- decoupling and Fourier transformation consistency clearly mitigates the unphysically large val- (4) X (4) (4) ues of the on-shell Γk, but its values remain quite sig- δH2 = 3J δAk + δCk ak† ak (41) nificant. Note that in the iDE self-consistent solution, k decays occur at a lower field than suggested by the on- (4) δBk shell consideration. − ak† a† k + H.c. , 2 − 10

ˆ(3) ˆ(1) where the coeﬃcients are given by should be (re)balanced to zero: δH1 +δH = 0, giving the 1/S correction to the canting angle [3] (4) δA = −d1 − γkd2 k ¯ 1 1 sin θ0 ∆ + m + n d = ∆¯ ∆ + + m ∆ − + n δθ = , (49) 1 2 2 S cos θ0 1 or, using 1/S 1, −h2 ∆ + 2n + ∆¯ + m 2 ∆¯ + m + n 1 δ 1 sin θ = sin θ 1 + . (50) d = m + n ∆ − + ∆ + 0 S 2 2 2 2 1 δ This yields a correction to the harmonic term (11) −h2 ∆ + 2m + + n , (42) 2 2 ∂Hˆ(2) (4) δHˆ(3) = δθ, (51) δBk = d3 + γkd4 2 ∂θ 1 1 1 d = m ∆ + + ∆¯ ∆ − which, after Fourier transform, can be written as 3 2 2 2 2 (3) X (3) (3) h 1 δHˆ = 3JS δA + δC a† a (52) − ∆ + ∆¯ + m 2 k k k k 2 2 k (3) ¯ 1 δ 1 δBk d4 = ∆ + n ∆ + + ∆ − − ak† a† k + H.c. , 2 2 2 2 − 1 δ −h2 ∆ + n + 2∆¯ + (43) with the parameters given by 2 2 (4) √ (3) 1 δC = − 3nhγ¯ (44) δA = − (1 + γk) ∆ + sin 2θ · δθ, (53) k k k 2 and we have introduced a notation h = sin θ. Using Bo- (3) 1 δB = γk ∆ + sin 2θ · δθ, (54) golyubov transformation (21) in this Hamiltonian gives k 2 (3) √ od,(4) δCk = 3¯γk cos θ · δθ. (55) (4) X (4) Vk δH2 = εk ck† ck − ck† c† k + H.c. , (45) 2 − k Similarly to the quartic terms (45), this gives

(3) (3) ! where AkδA − BkδB ε(3) = 3JS k k + δC(3) , (56) k p 2 2 k (4) (4) ! Ak − Bk AkδA − BkδB ε(4) = 3J k k + δC(4) , (46) ! k p 2 2 k (3) (3) A − B od,(3) BkδA − AkδB k k V = −3JS k k . (57) k p 2 2 (4) (4) Ak − Bk BkδA − AkδB V od,(4) = −3J k k , (47) k p 2 2 Ak − Bk Altogether, the Hartree-Fock correction to the magnon energy spectrum is given by (4) and δCk is not aﬀected by diagonalization as it is odd HF (3) (4) in k. Thus, the Hartree-Fock corrections from the four- Σ (k) = εk + εk . (58) (4) magnon interaction to the energy spectrum is εk . Magnon spectral function

Quantum correction from the three-boson terms 1 The spectral function, A(k, ω) = − π ImG(k, ω), with 1 the Green’s function G(k, ω) = [ω − εk − Σ(k, ω) + iδ]− , The Hartree-Fock decoupling of the three-magnon can now be calculated using the full expression for the term (24) gives a correction to the canting angle θ via 1/S, one-loop, ω-dependent self-energy (32), (33), (58) ˆ(1) ˆ(3) X HF (a) (b) H + H1 = (V1 + δV1) ai + ai† , (48) Σ(k, ω) = Σ (k) + Σ11 (k, ω) + Σ11 (k, ω) . (59) i In our calculations, we also kept the source self-energy p ¯ (b) where δV1 =3J S/2(∆ + 1/2) sin 2θ(∆ + m + n), adding term on-shell, i.e., Σ11 (k, εk), in order to avoid interac- to the contribution of the Hamiltonian in (8), Hˆ(1), which tion with negative frequency excitations, see [5]. was vanishing due to energy minimization, with V = The results of the calculations of the spectral function p 1 S/2 cos θ (H − 6JS sin θ (∆ + 1/2)). Now, both terms A (k, ω) are shown as intensity plots in Fig. 7(a)-(c) for 11

2 (a) M0 K0

L 1.5 M

K J / 1 (b) ! =1.05J

M0 K0 0.5 w=1.05J L M

0 K LXK ´ 1 K (c) ! =1.25J

FIG. 7: (a) The k − ω intensity plot of the spectral function A(k, ω) along the MK0ΓKMΓ path. Dotted lines indicate constant energy cuts in (b) and (c), where intensity plots at ω = 1.05J and ω = 1.25J are shown. S = 1/2, ∆ = 0.9, H = 0.2Hs. w=1.25J

y0y0 x0x0 S = 1/2, ∆ = 0.9, and H = 0.2Hs. A cut along the S (q, ω) = S (q, ω), (62) high-symmetry path in Fig. 7(a) exhibits a strong spec- z0z0 2 xx 2 zz S (q, ω) = cos θSq + sin θSq , (63) trum renormalization, ∼ 30%, and a significant broadening of the quasiparticle peaks for an extensive range of xy yx where we used Sq = −Sq and the shorthand notations momenta in the vicinity of K0 corners of the BZ. The lat- from Ref. [5], Sq ≡ S (q ± Q, ω). ± ter is due to an overlap of the one-magnon spectrum with We note that the off-diagonal component, Sx0y0 (q, ω), the two-magnon continuum associated with the roton- is also non-zero, but its contribution to the total structure like excitations at the K points. One can also see well- factor for an unpolarized beam is canceled exactly by pronounced termination points with distinctive bending y0x0 xy its partner, S (q, ω), leaving the Sq contributions in of spectral lines and other non-Lorentzian features in (61) and (62) to be the sole unorthodox± terms due to the Fig. 7(a). The 2D intensity maps of the constant-energy field-induced chiral structure. cuts of A (k, ω) at ω = 1.05J and ω = 1.25J are shown Transverse components of S(q, ω) are given by in Fig. 7(b) and (c), where one can see multiple signatures of the broadening, spectral weight redistribution, xx(yy) S 2 2 and termination points. Sq = Λ (uq ± vq) A(q, ω), (64) 2 ± xy S Sq = i Λ+Λ A(q, ω), (65) Dynamical structure factor 2 − where Λ = 1 − (2n ± δ)/4S, and the longitudinal one is We approximate the dynamical structure factor [5] as ± a sum of the diagonal terms of zz 1 X 2 Sq = (ukvq k + vkuq k) δ (ω − εk − εq k) . Z ∞ 2 − − − α0β0 i iωt α0 β0 k S (q, ω) = Im dte hTSq (t)S q(0)i. (60) π − (66) −∞ The spin rotation to the local reference frame yields Clearly, S(q, ω) should feature three overlapping single- (keeping the leading 1/S terms) magnon spectral functions, A(q, ω) and A(q±Q, ω), with different kinematic formfactors. 1 x0x0 yy yy xy xy In Fig. 8 we present constant energy cuts of S(q, ω) S (q, ω) = Sq+ + Sq − 2i sin θ Sq+ − Sq 4 − − at the same energies and parameters as in Fig. 7(b) and 2 xx xx 2 zz zz (c) to demonstrate proliferation of the unusual features + sin θ Sq+ + Sq + cos θ Sq+ + Sq , (61) − − discussed for the spectral function. 12

M0 K0 [1] O. A. Starykh, A. V. Chubukov, and A. G. Abanov, Phys. Rev. B 74, 180403 (2006). [2] A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. Lett. X1 L 97, 207202 (2006); Phys. Rev. B 79, 144416 (2009). [3] M. E. Zhitomirsky and A. L. Chernyshev, Phys. Rev. Lett. M 82, 4536 (1999); M. Mourigal, M. E. Zhitomirsky, and A. L. Chernyshev, Phys. Rev. B 82, 144402 (2010); W. T. Fuhrman, M. Mourigal, M. E. Zhitomirsky, and A. L. Chernyshev, Phys. Rev. B 85, 184405 (2012). K [4] M. E. Zhitomirsky and I. A. Zaliznyak, Phys. Rev. B 53, 3428 (1996). ! =1.05J [5] M. Mourigal, W. T. Fuhrman, A. L. Chernyshev, and M. E. Zhitomirsky, Phys. Rev. B 88, 094407 (2013). [6] D. Yamamoto, G. Marmorini, and I. Danshita, Phys. Rev. Lett. 112, 127203 (2014); G. Marmorini, D. Yamamoto, and I. Danshita, Phys. Rev. B 93, 224402 (2016). M0 K0 [7] M. E. Zhitomirsky and A. L. Chernyshev, Rev. Mod. Phys. w=1.05J 85, 219 (2013). [8] L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 36, 1168 (1959) [Sov. X1 L Phys. JETP 9, 830 (1959)]. M

! =1.25J

FIG. 8: Intensity plots of the constant energy cuts of S(q, ω) w=1.25J at the same ω and parameters as in Fig. 7(b) and (c).