Dynamical structure factor of the triangular-lattice antiferromagnet
M. Mourigal,1,* W. T. Fuhrman,1 A. L. Chernyshev,2 and M. E. Zhitomirsky3
1Institute for Quantum Matter and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA2Department of Physics and Astronomy, University of California, Irvine, California 92697, USA
3Service de Physique Statistique, Magnetisme et Supraconductivite, UMR-E9001 CEA-INAC/UJF, 17 rue des Martyrs,38054 Grenoble Cedex 9, France
(Received 3 June 2013; revised manuscript received 14 August 2013; published 5 September 2013)
We elucidate the role of magnon interaction and spontaneous decays in the spin dynamics of the triangular-lattice Heisenberg antiferromagnet by calculating its dynamical structure factor within the spin-wave theory.Explicit theoretical results for neutron-scattering intensity are provided for spins S = 1/2 and S = 3/2. Thedynamical structure factor exhibits unconventional features such as quasiparticle peaks broadened by decays,non-Lorentzian lineshapes, and significant spectral weight redistribution to the two-magnon continuum. This richexcitation spectrum illustrates the complexity of the triangular-lattice antiferromagnet and provides distinctivequalitative and quantitative fingerprints for experimental observation of decay-induced magnon dynamics.
The Heisenberg triangular-lattice antiferromagnet (HTAF)is a prominent model in low-dimensional and frustratedmagnetism and is the subject of significant experimental andtheoretical interest. In zero field, the ground state of the modelis the well-known coplanar 120◦ magnetic structure for allvalues of spin S, including S = 1/2, as evidenced by variousanalytical1–4 and numerical5–7 works. This noncollinear mag-netic order has profound consequences for the spin dynamicsof the HTAF: its elementary excitations (magnons) becomeunstable with respect to spontaneous decay into pairs of othermagnons.8 To describe the excitation spectrum within thespin-wave theory (SWT) the inclusion of magnon interactionis crucial.9–11
Currently, a large number of materials are proposed to befair realizations of the HTAF, although they often deviate fromthe ideal model due to distorted geometry of exchange bondsor additional spin-anisotropy terms. These include the spin-1/2 materials Cs2CuCl4,12 Cs2CuBr4,13 and Ba3CoSb2O9
14,15
and a number of compounds with larger spin values suchas VCl2,16 LuMnO3,17 Rb4Mn(MoO4)3,18 CuCrO2,19 α-SrCr2O4,20 and α-CaCr2O4.21,22 The most comprehensiveexperimental characterization of these materials is done by in-elastic neutron scattering on single crystals,19,22 which directlymeasures the energy and momentum dependence of spin-spincorrelations as described by the dynamical structure factorS(q,ω).
The spin-wave calculation of S(q,ω) in the HTAF is compli-cated by the noncollinear spin arrangement and strong magnoninteraction. Previously, the dynamical structure factor for aquasi-one-dimensional spiral antiferromagnet was calculatedby Ohyama and Shiba.23 Their method was subsequentlyadapted to describe neutron-scattering experiments on theorthorhombically distorted triangular-lattice antiferromagnetCs2CuCl4.24,25 The deficiency of that method is that it operatesdirectly with bare Holstein-Primakoff bosons rather thanwith Bogolyubov quasiparticles, making the systematic 1/S
ranking of different terms difficult and providing results thatare unnecessarily complicated compared to collinear antifer-romagnets. In addition, Ref. 23 does not draw a distinction
between retarded and causal spin Green’s functions, which isimportant for recovering the correct ω → 0 behavior.
One of the goals of the present work is to revisit calculationof the dynamical structure factor for a noncollinear antifer-romagnet, focusing on the 1/S ranking and on the correctω dependence of various contributions to spin correlationfunctions. Our second goal is to provide the first explicit theo-retical results for S(q,ω) of the HTAF for representative valuesof spin to guide experimental inelastic neutron-scatteringmeasurements in realistic materials. Such a reference pointshould allow evaluation of the accuracy and limits of the SWTin various experimental situations and help to identify whenthe latter breaks down in favor of alternative descriptions, forinstance, using spinons.26–28
The 1/S formalism for interacting spin waves in the HTAFwas previously described in detail in Ref. 11. That workfocused on the role of decays in the magnon spectrum andon a classification of singularities appearing in the latter. Thepresent work is concerned with the explicit calculation of thedynamical structure factor for the HTAF within the frameworkof nonlinear SWT.
Section II contains details of the theoretical formalismwhere we use basic notations from Ref. 11. Then, in Sec. III,we use the developed formalism and present high-resolutionpredictions for the dynamical structure factor for S = 1/2 andS = 3/2 along the high-symmetry directions of the Brillouinzone. Our results show a complex excitation spectrum andprovide evidence of the crucial effects of magnon-magnoninteractions on the spin dynamics, demonstrated by broadenedquasiparticle lineshapes, double-peak structures, and contri-butions from the two-particle continuum that dominate a largefraction of the spectrum. We also present the momentum-integrated structure factor and representative constant-ω scansof S(q,ω) and discuss their features. We conclude in Sec. IVand provide various details in the Appendix.
II. DYNAMICAL CORRELATIONS
Neutron scattering experiments provide a direct probe ofthe spin-spin correlation function, otherwise known as the
MOURIGAL, FUHRMAN, CHERNYSHEV, AND ZHITOMIRSKY PHYSICAL REVIEW B 88, 094407 (2013)
FIG. 1. (Color online) (a) Coplanar 120◦ spin structure of theHeisenberg triangular-lattice antiferromagnet. (b) Brillouin zone ofthe triangular lattice with high-symmetry points and paths. Shadedarea is the magnon decay region.
dynamical structure factor:
Sα0β0 (q,ω) =∫ ∞
−∞
dt
2πeiωt
⟨Sα0
q (t)Sβ0−q(0)
⟩, (1)
where α0 and β0 refer to spin components in the laboratoryframe {x0,y0,z0}. The inelastic neutron-scattering cross sectionis proportional to a linear combination of the diagonalcomponents of the correlation function, (1), with momentum-dependent prefactors.29 In the following, we do not assumea particular experimental geometry and consider instead the“total” structure factor in which all three components arecontributing equally:
While the dynamical structure factor is measured in thelaboratory reference frame, the spin-wave calculations areperformed in the rotating frame with z oriented along thelocal magnetization on each site. Using the propagation vectorQ = (4π/3,0) of the 120◦ spin structure [Fig. 1(a)] andchoosing spins to rotate in the x0-z0 plane, the relation betweenspin components in the two frames is S
y0i = S
y
i and
Sx0i = Sz
i sin(Q · ri) + Sxi cos(Q · ri),
(3)S
z0i = Sz
i cos(Q · ri) − Sxi sin(Q · ri).
Then components of the dynamical structure factor in the twocoordinate systems are connected by
k,ω ≡ Sαβ(k,ω) and q± = q ± Q.In Eq. (4) one can readily identify conventional diagonalcontributions of the transverse (xx and yy) and longitudinal(zz) spin fluctuations.30
In addition, the noncollinear nature of the ground state isresponsible for terms with mixed transverse and longitudinal(xz and zx) fluctuations. The frequency dependence and themagnitude of the these off-diagonal correlation functions arediscussed in the Appendix. We find that these off-diagonalcomponents primarily contribute to the singularities withinthe two-magnon continuum that are already present in thediagonal terms, while the dominant features of the structure
factor, arising from the diagonal terms, remain intact. We,thus, conclude that the off-diagonal terms always yield asubleading contribution with respect to the diagonal parts. Asimilar conclusion was reached in the previous work on thedistorted HTAF.25 This allows us to neglect such terms in thefollowing consideration.
Using (4), we rewrite the total structure factor in terms ofthe diagonal and mixed parts and further separate the formerterm into transverse and longitudinal contributions:
S tot(q,ω) = Sdiag(q,ω) + Smix(q,ω),
Sdiag(q,ω) = S⊥(q,ω) + SL(q,ω),
S⊥(q,ω) = Syyq,ω + 1
2
(Sxx
q+,ω + Sxxq−,ω
), (5)
SL(q,ω) = 12
(Szz
q+,ω + Szzq−,ω
),
Smix(q,ω) = i2
(Sxz
q−,ω − Szxq−,ω − Sxz
q+,ω + Szxq+,ω
).
As discussed, we ignore the mixed (off-diagonal) term forthe bulk of this work and use explicitly S tot(q,ω) ≈ Sdiag(q,ω)in Secs. II and III. However, in the Appendix we considerthe exact definition of S tot(q,ω) from Eq. (5) to illustrate thecontribution of the mixed term to the total dynamical structurefactor and justify our decision to neglect it.
The dynamical spin correlator Sαα(q,ω) is related to theretarded Green’s function of spin operators via the fluctuation-dissipation theorem [see, e.g., Ref. 31],
Sαα(q,ω) = − 1
π[1 + nB(ω)] Im
[Gαα
ret (q,ω)], (6)
where nB(ω) = 1/(eω/kBT − 1) is the Bose distribution func-tion. Here we are interested in T = 0 case, for which nB(ω) ≡0 for ω > 0 and nB(ω) ≡ −1 for ω < 0. Hence, S(q,ω) isnonzero only for positive frequencies and
Sαα(q,ω) = − 1
πIm
[Gαα
ret (q,ω)]. (7)
At T = 0, one can use the causal Green’s function, Gαβ(q,t) =−i〈T Sα
q (t)Sβ−q〉, on the right-hand side of Eq. (7) since the
two Green’s functions coincide for ω > 0. This simplifiescalculations, although caution is still needed when dealingwith bosonic Green’s functions at negative frequencies (seeSec. II A). In the next two subsections we consider transverseand longitudinal components of the structure factor.
A. Transverse fluctuations
The spin-wave calculation of the dynamical correlationfunctions proceeds with the Holstein-Primakoff representationof spin operators Sα
i in terms of bosons ai and subsequentexpansion of square roots in boson density a
†i ai ; see Ref. 11
for details on application of the SWT to the HTAF. In order todetermine the leading contributions of order O(1) and O(1/S)to the transverse structure factor in (5), one may use thefollowing expressions:
Sxi =
√S
2(ai + a
†i )�+, S
y
i = −i
√S
2(ai − a
†i )�−, (8)
where
�± = 1 − 2n ± δ
4S(9)
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are the Hartree-Fock factors obtained from the contraction ofboson operators in cubic terms with the expectation valuesn = 〈a†
i ai〉 and δ = 〈aiai〉. Terms beyond the Hartree-Fockapproximation contribute to the transverse structure factor inthe O(1/S2) order and can be neglected.
Substituting (8) into the spin Green’s functions and per-forming the Bogolyubov transformation,11 we obtain
Gxx(q,ω) = S
2�2
+(uq + vq)2[G11(q,ω)+ G11(−q, − ω)
+ 2G12(q,ω)],
Gyy(q,ω) = S
2�2
−(uq − vq)2[G11(q,ω) + G11(−q, − ω)
− 2G12(q,ω)]. (10)
Here G11(q,ω) and G12(q,ω) are the normal and anomalousmagnon Green’s functions and uq and vq are the Bogolyubovcoefficients.
In the harmonic approximation, G12(q,ω) ≡ 0 andG11(q,ω) ≡ G0(q,ω) = (ω − εq + i0)−1, where εq is themagnon energy in the harmonic approximation,
εq = 3JS√
(1 − γq)(1 + 2γq), (11)
with γq = 13 [cos qx + 2 cos( qx
2 ) cos(√
32 qy)]. Hence, in this ap-
proximation, magnon excitations manifest themselves as sharpδ peaks in the dynamical structure factor. However, in spiralantiferromagnets, magnon-magnon interaction substantiallyalters this simplified picture. The complication is mainly dueto three-magnon processes, which are inherent to noncollinearantiferromagnets,8 and produce the ω-dependent self-energyalready in the lowest-order perturbation theory, also leadingto a finite lifetime of magnons in a large part of the Brillouinzone.
For the purpose of the 1/S ranking of various contributionswe note that the magnon energy scales with spin as εq = O(S),and the self-energy as 11,12(q,ω) = O(1). Then, to achievethe O(1/S) accuracy in the structure factor, one can use areduced form of the Belyaev equations for the magnon Green’sfunctions:
G11(q,ω) ≈ 1/[ω − εq − 11(q,ω)],(12)
G12(q,ω) ≈ 12(q,ω)G11(q,ω)G11(−q, − ω).
Clearly, G11(q,ω) = O(1/S) and G12(q,ω) = O(1/S2). In thelowest order, magnon self-energies are
11(q,ω) = HF11 (q) + 1
2
∑k
|V31(k; q)|2ω − εk − εq−k + i0
− 1
2
∑k
|V32(k,q)|2ω + εk + εq+k ∓ i0
,
(13)
12(q,ω) = HF12 (q) + 1
2
∑k
V32(k, − q)V ∗31(k; q)
ω − εk − εq−k + i0
− 1
2
∑k
V32(k,q)V ∗31(k; −q)
ω + εk + εq+k ∓ i0,
where HF(q) are the frequency-independent Hartree-Fockcontributions, V31(k; q) and V32(k,q) are the three-particledecay and source interaction vertices, respectively, and ∓i0correspond to the causal/retarded self-energies.11 We notethat one must use the lower sign in (13) to ensure thecorrect odd-frequency dependence of the imaginary part ofthe magnetic susceptibility. In the following, we use smallδ ≡ 0+ for the numerical evaluation of the self-energiesin (13).
Several important simplifications are in order. The termcontaining anomalous Green’s function on the right-hand sidesof Eqs. (10) is next order in 1/S classification compared to thefirst two. While it does contribute to the structure factor in thesought O(1/S) order, its contribution can be shown to be smallalready for S = 1/2 and also qualitatively redundant to that ofthe other terms (see the Appendix for analysis). We thereforeneglect these terms in the following consideration.
Formally, the first two Green’s functions on the right-handside of Eqs. (10) are of the same order in the 1/S rankingand could contribute equally to Im[Gαα(q,ω)]. However, thesecond term, Im[G11(−q, − ω)], is off-resonance compared toIm[G11(q,ω)] and contains no poles for ω > 0, thus providingno contribution to the structure factor. While this term isimportant to ensure the correct behavior of the spectralfunction for low-energy excitations at ω,εq → 0,32 for allpractical purposes it is negligible.
One should note that the consideration of spectral propertieswithin the SWT always exceeds the nominal 1/S order asthe frequency dependence is automatically “off-shell,” thusincluding contributions of higher 1/S order. A technicalissue arises when evaluating magnon spectral function with11(q,ω) given by Eq. (13). Due to the third term in (13)(“source”), a spurious excitation branch appears in the vicinityof Q with vanishing εq at a q = Q. This unphysical mode isthe manifestation of a pole pushed up from negative valuesof ω. Similar behavior is also present in Im[G11(−q, − ω)],which develops an extension of the same mode in the formallyforbidden ω > 0 region. In principle, these anomalies shouldbe removed by some self-consistent higher-order 1/S expan-sion, an analytically and computationally difficult problem.Here we adopt a more expeditious manner to address theunphysical mode directly by returning the offending sourceself-energy term back on-shell, i.e., taking ω = εq within thisterm. In this way, the effect of magnon-magnon interactionsand decays are maintained while unphysical singularities aresuppressed.
Altogether, for the results in the next section we usethe following expressions for the Syy(q,ω) and Sxx(q,ω)components of the transverse structure factor in (5):
Sxx(q,ω) = S
2�2
+(uq + vq)2A11(q,ω),(14)
Syy(q,ω) = S
2�2
−(uq − vq)2A11(q,ω),
with A11(q,ω) = −(1/π )Im[G11(q,ω)], where G11(q,ω) isdetermined by Eq. (12) with 11(q,ω) given by Eq. (13), inwhich the source term is taken on-shell, ω = εq. We note that inthe harmonic approximation Sxx[Syy](q,ω) ∼ SA11(q,ω) ∼Sδ(ω − εq), which is explicitly of order O(1).
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B. Longitudinal fluctuations
Spin fluctuations in the direction of ordered momentswritten in terms of Holstein-Primakoff bosons are
δSzq = − 1√
N
∑k
a†kak+q. (15)
Then, the inelastic part of the longitudinal neutron crosssection SL(q,ω) in (5) is determined by the correlation func-tion Szz(q,t) = 〈δSz
q(t)δSz−q〉, which probes the two-magnon
continuum.As is clear from the derivation given in the Appendix,
the longitudinal component of the structure factor is oforder O(1/S), a factor of 1/S smaller than the leadingterms in the transverse correlation functions, (14). Therefore,in the spirit of the 1/S expansion it may be calculatedwith bare magnon Green’s functions, neglecting correctionsfrom magnon-magnon interactions. Since this approximationneglects renormalization of the magnon energies, the obtainedwidth of the two-magnon continuum will not be precise butwill still give a fair embodiment of the continuum contributionto the neutron-scattering cross section.
With this we obtain
Szz(q,ω) = 1
2
∑k
(ukvk−q+ vkuk−q)2δ(ω − εk − εk−q);
(16)
see the Appendix for details.Thus, in the following calculations of SL(q,ω) in (5) we
use Szz(q,ω) from Eq. (16).
III. RESULTS AND DISCUSSION
In this section we present high-resolution numerical resultsusing Eqs. (5), (14), and (16) and S tot(q,ω) ≈ Sdiag(q,ω) toprovide specific predictions for the dynamical structure factorof the HTAF for S = 1/2 and S = 3/2, revealing the dramaticeffects of magnon-magnon interactions. We begin with theanalysis of transverse fluctuations related to the normal partof the spectral function and proceed to the comparison ofthe relative weights of transverse and longitudinal fluctuationsin the dynamical structure factor for representative momenta.Finally, we show our results for the total dynamical structurefactors of the HTAF for S = 1/2 and S = 3/2 and concludewith a description of their momentum-integrated forms.
We performed the numerical integration of the self-energiesin Eq. (13) using an artificial broadening of δ = 0.03JS andvarious integration schemes. The intensity plots of the spectralfunction (Fig. 2) and dynamical structure factor (Fig. 4) useda quasi–Monte Carlo integration in MATHEMATICA with anaccuracy goal of four digits. The line plots in Figs. 3 and9 used a Gaussian-quadrature method with 4 × 106 points,while a simple Monte Carlo integration with 5 × 106 pointswas used for Fig. 8. The momentum-integrated S tot(ω) andconstant-energy q scans of S tot(q,ω) in Figs. 5–7 used aGaussian-quadrature method with 1.6 × 105 k and q pointsand a somewhat larger δ = 0.04JS. A higher density mesh of1.44 × 106 points was used in Fig. 5 for the long-wavelengthregion ω/SJ < 0.5, with subsequent extrapolation to ω → 0limit.
A. Transverse fluctuations and spectral function
We begin with an examination of the transverse dynamicalstructure factor S⊥(q,ω) in (5). The dominant contributionto this component originates from the normal part of themagnon Green’s function G11(q,ω), with momenta q andq ± Q. Neglecting the anomalous terms, S⊥(q,ω) is relatedto the spectral function A11(q,ω) = −1/π Im[G11(q,ω)] withmomentum-dependent prefactors [see Eq. (14)]. Therefore,in Fig. 2 we restrict ourselves to A11(q,ω) for S =1/2 andS =3/2 along the high-symmetry directions in the Brillouinzone from Fig. 1(b). This presentation gives the benefit ofrelative simplicity and highlights important features of thespectrum related to magnon interactions and decays, whichare subsequently identified in the more complicated landscapeof the structure factor. The transverse dynamical structurefactor is then obtained by a linear combination of A11(q,ω)and A11(q ± Q,ω) according to (5) and (14).
The effect of magnon interaction is taken into account bythe self-energies, (13), which originate from a direct couplingof the single-particle branch to the two-magnon continuum.Because of such a coupling, an incoherent component ispresent in the intensity plots of A11(q,ω) in Fig. 2, which alsoprovides an insight into the quasiparticle-like behavior of thesingle-particle excitations, potentially broadened by decays.8
A feature of the HTAF spectrum, observed for all momenta,is the downward renormalization of the magnon dispersionfrom its bare value εq (dashed line) for both S =1/2 and S =3/2 (see Fig. 2). This is in agreement with a number of previousworks on the HTAF6,9–11 and on related problems involvingmagnon interaction in noncollinear antiferromagnets.8,33,34
This generic effect is due to an effective repulsion betweenthe single-particle branch and the two-particle continuum fa-cilitated by their coupling. This renormalization is about 18%for S =1/2 in Fig. 2(a) and about 8% for S =3/2 in Fig. 2(b),representing a quantum 1/S effect. The renormalization factorfor S = 1/2 is somewhat less than in the numerical6 andon-shell SWT results9–11 but is in closer agreement with theresults of the off-shell Dyson equation SWT approach.11 Otheraspects of the spectrum renormalization, such as developmentof the “roton-like” minimum at the M point, are also inagreement with earlier studies.6,9,10 Note that the discussedeffect of magnon interaction in the HTAF is in contrast withthe well-known upward spectrum renormalization for collinearantiferromagnets.35
An interesting signature of strong magnon interaction inthe S = 1/2 case is the bright-intensity spot at ω/J ≈ 3in the vicinity of the M point [see Fig. 2(a)]. Upon closerexamination we find that this is an antibonding state: thesingle-magnon state pushed out of the two-magnon continuum.While this state is likely an artifact of our approximationand will broaden significantly if treated self-consistently, itsignifies the strength of the magnon-magnon interaction. Notethat this effect disappears in the S = 3/2 case, where magnoninteraction is weaker.
Another prominent feature is the broadening of the quasi-particle peaks observed for the momenta inside the decayregion [shaded (gray) area of the Brillouin zone in Fig. 1(a)],e.g., along the K� line as well as in the �M and MY directions.A particularly salient broadening occurs in the S = 1/2 case,
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ΓXK
ω/J
(a)
0
1
2
3
M YY1
M X
S =
1/2
01
23
ΓXK
ω/J
(b)
0
2
4
6
8
M YY1
M X
S =
3/2
01
23
FIG. 2. (Color online) Intensity plots of the momentum and energy dependence of the spectral function A11(q,ω) for (a) S = 1/2 and (b)S = 3/2 along the high-symmetry paths in the Brillouin zone in Fig. 1(b). The dashed line is the linear SWT spectrum εq.
as shown in Fig. 2(a). The corresponding magnon excitationsacquire a finite lifetime due to three-particle magnon-magnoninteractions.8 The kinematic conditions required for suchprocesses are discussed in detail elsewhere,11 though we notethat the boundary of the decay region is due to emission of theacoustic magnon εQ. This is distinct from the case of magnetic-field-induced decays in the square-lattice antiferromagnet,where the corresponding decay products are inside the decayregion and thus also unstable.33,36 As a consequence, theboundaries of the HTAF decay region are sharply defined,leading to a spectacular and robust quasiparticle “blowout”when the single-magnon branch enters the decay region andmerges with the two-magnon continuum, as visible alongthe MY path in Fig. 2(a). This effect resembles neutronscattering observations of the so-called termination point for
the excitations of superfluid 4He37 and the triplet excitationsof spin-gap materials.38,39 Similar distortion of the excitationcurve in the vicinity of a continuum boundary was alsoobserved in other spin systems.40
One can see in Fig. 2(a) that the crossing between one-particle spectrum and two-magnon continuum is accompaniedby the “edge” singularity, visible as the lowest-energy branchfor the K� line or as a “double-peak” structure for the �Mpath if cutting along the ω axis. Such features are the vanHove singularities11 due to the bottom of the two-magnoncontinuum (see also Figs. 3, 4, and 8). Within the SWT, theyshould be regularized by the higher order diagrams11 and, inrealistic systems, by a small interlayer coupling.34
While the role of interaction between magnons decreasesfor S = 3/2, magnon decays remain highly visible, in
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S( q
, ω)
q = M(a)
q + Q
q
0
1
2
3
S(q
,ω)
q = Y(b) q ± Q
q
0
0.5
1
1.5
S tot(q,ω)
S L(q,ω)
ω/J
S(q
, ω)
q = Y1
(c)
q − Q
q
1 2 30
0.5
1
1.5
FIG. 3. (Color online) (a–c) Energy dependence of the dynamicalstructure factor for S = 1/2 at representative points of the Brillouinzone: M, Y, and Y1 [see Fig. 1(b)]. The solid line corresponds tothe total structure factor S tot(q,ω); the shaded (gray) area is thelongitudinal part SL(q,ω). Vertical marks indicate positions of thequasiparticle peaks from A11(q,ω) and A11(q ± Q,ω). Horizontalbars indicate the width σ = 0.03J of the Gaussian convolution.
particular, along the K� line and in the blowout region ofthe MY line [see Fig. 2(b)]. The spectral weight transferfrom single-particle excitations to the continuum is, however,strongly reduced.
Finally, we note that many of the spectral features discussedhere, such as spectrum broadening throughout large regionsof the Brillouin zone, dramatic redistributions of spectralweight to the two-magnon continuum, non-Lorentzian two-peak structures, and other features clearly unlike conventionalsingle-particle peaks, have been discussed by us recently forthe quasi-two-dimensional square-lattice antiferromagnet in afield.8,34
B. Total dynamical structure factor
We now proceed with the analysis of the total dynamicalstructure factor S tot(q,ω) ≈ Sdiag(q,ω) in (5) and of the role ofthe longitudinal component SL(q,ω) in it, which we take in theform given by Eq. (16). In Fig. 3 we offer such a considerationfor S =1/2 and for representative high-symmetry points, M,Y, and Y1 [see Fig. 1(b)]. Note that Y1 is the image of the Mpoint shifted by the Q vector.
The contribution of the longitudinal component to the totaldynamical structure factor is shown in Fig. 3 by shaded (gray)areas, while the total structure factor is plotted by solid lines. Inorder to mimic a hypothetical experimental energy resolutionas well as to soften various spurious features such as theedge singularities in S⊥(q,ω) discussed above or the vanHove singularities of the two-magnon density of states inSL(q,ω), the results are convoluted with a Gaussian profileof σ = 0.03J . This is indicated by horizontal bars in Fig. 3and done in addition to the artificial broadening δ used in thenumerical integration.
Several aspects of the results presented in Fig. 3 deservemention. As we discussed above, at each q point the transversecomponent of the structure factor is a linear combina-tion of three spectral functions—A11(q,ω), A11(q − Q,ω),and A11(q + Q,ω)—with different q-dependent weights [seeEqs. (14) and (5)]. For the high-symmetry points of our choice,only two of these terms are distinct. Given the correspondencebetween qM ± Q and the points equivalent to Y1, the similarityof their structure factors, the positions and shapes of thepeaks, and other features in Figs. 3(a) and 3(c) are now easilyunderstood. One can also see that the q-dependent weightsyield different relative intensities of different features at theM and Y1 points. Moreover, using our previous analysis ofthe spectral function, one can observe that the lowest peaksin Figs. 3(a) and 3(c) are resolution limited and both comefrom the sharply defined peak in A11(q,ω) at the M pointin Fig. 2, which is outside the decay region. At the sametime, quasiparticle peaks that are broadened by decays andaccompanied by the non-Lorentzian edge-like features belowthem, labeled q + Q and q in Figs. 3(a) and 3(c), respectively,originate from the same spectral shapes in A11(q,ω) at the Y1
point (see Fig. 2).For the structure factor at the Y point, the transverse part is
dominated by the two well-defined quasiparticle peaks. Whilethe lowest one, q ± Q, corresponds to stable magnons, thesecond peak is from the Y point itself, which is inside thedecay region in Fig. 1(b). A closer inspection of Fig. 2 andthe on-shell analysis in Ref. 11 show that the correspondingbroadening due to decays is small for this point. A similar typeof comprehensive analysis of the structure factor is possiblefor any other q point.
A compelling property of the structure factor for S =1/2in Fig. 3 is the very strong contribution of the longitudinalcomponent SL(q,ω) for each momenta. This is directly relatedto the transfer of part of the static spectral weight (reductionof the ordered moment) to the longitudinal dynamical correla-tions under the action of strong quantum fluctuations. Thus, inaddition to the broad, continuum-like features of the transversestructure factor, the longitudinal component dominates thewide range of ω in each of the plots in Fig. 3. In fact, itcontributes the major portion to the total spectral weight at theY point in Fig. 3(b).
To complete the discussion of Fig. 3, we also note that whilethe transverse parts of the structure factors at the M and Y1
points are related, the corresponding longitudinal componentsare different. This is because, according to Eq. (5), SL(q,ω)at a q point takes two contributions, from Szz(q + Q,ω) andSzz(q − Q,ω). Hence, given the relation between M and Y1,SL(qM,ω) = Szz(qY1 ,ω), but not vice versa.
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ΓXK
ω/J
(a)
0
1
2
3
M YY1
M X
S =
1/2
00.
51
1.5
>2
ΓXK
ω/J
(b)
0
2
4
6
8
M YY1
M X
S =
3/2
01
2>
3
FIG. 4. (Color online) Intensity plots of the momentum and energy dependence of the total dynamic structure factor S tot(q,ω) (5) for (a)S = 1/2 and (b) S = 3/2 along the high-symmetry paths in the Brillouin zone shown in Fig. 1(b).
In Fig. 4 we present the intensity plots of the q and ω
dependence of the total structure factor for both S = 1/2 andS = 3/2 along the high-symmetry paths in Fig. 1(b). One of thebenefits of the insight provided by our preceding discussion ofthe spectral function in Fig. 2 and ofS(q,ω) at selected q pointsin Fig. 3 is that now the complicated view of Fig. 4 is seennaturally as a superposition of three q-modulated A11(q,ω)terms and a background of two Szz(q,ω) terms.
As can be anticipated from the richness in the behaviorof the spectral function, the total dynamical structure factorS tot(q,ω) shows a complex interplay of quasiparticle-like andcontinuum contributions, revealing an abundant broadeningof the peaks coexisting with the sharply defined excitationsthat are brought in by the shifted ±Q branches. For instance,the spectacular “blowout” region of the single-magnon branch
entering the two-magnon continuum along the YM directionnear Y1 in Fig. 2(a) now acquires a “mirror” region aroundthe M point. Note that both of these also coexist with thewell-defined magnon branches at lower energy.
In addition, for S = 1/2, the continuum-like componentdominates the spectrum throughout the Brillouin zone athigher energies. For the S = 3/2 case in Fig. 4(b), the fullstructure factor is composed of three well-defined quasi-particle branches, which also demonstrate some substan-tial continuum-like scattering. Altogether, the total struc-ture factor exhibits a complex landscape consisting ofsharp and broadened quasiparticle peaks as well as sub-stantial continuum contributions from both the transverseand the longitudinal parts of the dynamical structurefactor.
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FIG. 5. (Color online) Energy dependence of the momentum-integrated dynamical structure factor S(ω) for (a) S = 1/2 and(b) S = 3/2. The solid line corresponds to the total momentum-integrated structure factor S tot(ω); the shaded (gray) area is thelongitudinal component SL(ω). Vertical arrows indicate the energiesof the constant-energy scans in Figs. 6 and 7.
In order to analyze the contributions of the continuum andof the quasiparticle-like excitations to the structure factor on amore quantitative level, we consider the momentum-integrateddynamical structure factor
S tot(ω) =∑
q
S tot(q,ω), (17)
which coincides with the spectral density of the spinautocorrelation function. Such a quantity is readily ac-cessible in neutron-scattering experiments on powdersamples.22
Figure 5 shows S tot(ω) (solid line) and its longitudinalcomponent SL(ω) [shaded (gray) area] for S = 1/2 andS = 3/2. Strong peaks are observed for both values of spin.They are clearly identifiable as the van Hove singularitiesin the spectra of dispersive quasiparticle-like modes. Thelowest peak is associated with the high density of states atthe magnon dispersion minimum at the M point (ω ≈ 2JS),which also retains a considerable flatness in the MX direction.The upper peak is the standard van Hove singularity due tothe top of the single-magnon spectrum (ω ≈ 3JS), which alsohas only weak dispersion along certain directions and thuscontributes significantly to the density of states. Althoughthese features appear less pronounced in the momentum-resolved dynamical structure factor in Fig. 4 for S = 1/2compared to S = 3/2, strong peaks in the integrated spectrumin Fig. 5(a) are still present. Their energies can serve toestimate exchange constants and excitation bandwidth, e.g.,from powder-averaged neutron scattering experiments. Wealso note that no sign of the “flat band” feature, advo-cated in Ref. 10 for the renormalized on-shell spectrum ofthe HTAF in the S = 1/2 case, is observed in S tot(ω) inFig. 5(a).
What is most remarkable in the integrated structure factorin Fig. 5 is that a significant spectral weight extends farbeyond the upper edge of the single-magnon spectrum, thelatter identifiable by the van Hove singularity. This behavior isnot unlike the one recently observed in an S = 3/2 triangular-lattice antiferromagnet.22 It also highlights, once more, thenecessity of taking magnon-magnon interaction into accountin going beyond the predictions of the linear SWT for thedynamical structure factor of noncollinear antiferromagnets.For the S = 1/2 case [Fig. 5(a)] the massive contribution ofthe longitudinal SL(ω) to the high-energy spectral weight isalso rather spectacular.
Complementary to both Fig. 4 and Fig. 5, in Figs. 6 and 7 wepresent the constant-energy scans of the dynamical structurefactor S tot(q,ω) for three selected energies (indicated by verti-cal arrows in Fig. 5). Modern neutron-scattering instrumenta-tion is naturally suited for studies of the dynamical correlationsin large regions of momentum space at fixed energies, whichalso motivates such a representation. One of the advantagesof such constant-ω scans is that well-defined spin-waveexcitations and corresponding van Hove singularities appear asbright sharp lines that are easy to distinguish from continuumscattering, manifested as a broadly distributed diffuse intensity.The results for S = 1/2 and S = 3/2 are discussed below.
Figures 6(a) and 7(a) show the spectral weight near thetop of the single-particle spectrum, ω/J = 1.25 and 4.3 forS = 1/2 and S = 3/2, respectively. In Fig. 6(a), a well-definedspectrum is observed only in the vicinity of two-thirds of the�M line, corresponding to excitations outside of the decayregion (see also Fig. 4). This should be compared to Fig. 7(a)for the S = 3/2 case, exhibiting strong signal from almost flatbranches of well-defined excitations contributing to the strongvan Hove singularity observed in S tot(ω) in Fig. 5(b). The restof the Brillouin zone in Fig. 6(a) displays a weaker diffuse con-tinuum, originating from broadened quasiparticle peaks and atwo-magnon continuum. Previously discussed features, suchas blowout around the Y1 and M points, are also clearly visible.
The energy ω/J = 0.85 in Fig. 6(b) corresponds to thevicinity of the first peak in S tot(ω) in Fig. 5(a), associatedwith the roton-like minima at the M points with an almost-flatdispersion along the MX line [see Fig. 4(a)], the latterindicating well-defined magnon excitations via a triangular-shaped intensity. The circular distribution of spectral weightaround the K points corresponds to magnons clearly broadenedby decays. Fainter, diffuse-like contributions are also seenaround the � point. This should be compared to the S = 3/2case in Fig. 7(b), where saddle-point features of the magnondispersion are much sharper.
Finally, a representative cut in the acoustic regime of thespectrum is shown in Figs. 6(c) and 7(c). At these energies,the effect of decays is weaker and concentric distributions ofspectral weight around the K point reveal three distinct acousticspin-wave branches from superposition of various q, q + Qand q − Q contributions. The innermost (circular) distributionof spectral weight corresponds to q → � excitations, whilethe outermost (rounded triangular) contributions are associatedwith spin waves from q → K,K′. Compared to S = 3/2, thespectral intensity in the S = 1/2 case clearly retains somediffuse component. The vicinity of the � point hosts a similarpattern, albeit strongly suppressed by the q-dependent factors.
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FIG. 6. (Color online) Intensity plots of the constant-ω scans ofthe dynamical structure factor S tot(q,ω) in the q plane for S = 1/2.(a) ω/J = 1.25, (b) ω/J = 0.85, and (c) ω/J = 0.5. Energies areindicated in S tot(ω) in Fig. 5(a).
IV. CONCLUSION
We have developed an analytical theory for the dynamicalstructure factor of the triangular-lattice Heisenberg antiferro-magnet and presented explicit numerical results for Sαα(q,ω)in the case of S = 1/2 and S = 3/2. Our treatment includescomparison of different contributions to the dynamical struc-ture factor at 1/S order, ensures the correct form of the Green’sfunctions at low energy, and uses a pseudo-on-shell approachto avoid spurious manifestations of an unphysical pole inthe spectral function. In this way, we have determined thedominant contributions to the dynamical structure factor tofacilitate a thorough computation of the excitation spectrum inthe entire momentum-energy space. In particular, our analysis
FIG. 7. (Color online) Same as in Fig. 6 for S = 3/2. (a)ω/J =4.3, (b) 2.8, and (c) 1.5. Energies are indicated in S tot(ω)in Fig. 5(b).
demonstrates that contributions from anomalous Green’sfunctions and mixed transverse-longitudinal terms can beneglected. The obtained energy dependence of the dynamicalstructure factor displays a rich interplay of quasiparticle- andcontinuum-like features. Although our analysis is purely two-dimensional, we anticipate that further softening of unphysicalsingularities in the energy dependence can be achieved byincreasing the dimensionality, such that our conclusions shouldremain valid for a wide range of realistic materials.
The role of magnon-magnon interactions and the presenceof decays is demonstrated through the energy- and momentum-resolved spectrum as well as the momentum-integratedstructure factor. A multitude of complex phenomena areobservable in both. This includes non-Lorentzian lineshapes,
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MOURIGAL, FUHRMAN, CHERNYSHEV, AND ZHITOMIRSKY PHYSICAL REVIEW B 88, 094407 (2013)
quasiparticle blowout, and roton-like minima, as well as anoverall downward renormalization, a rich density of states,and contributions from van Hove singularities. Both spectrumpresentations, integrated and momentum resolved, highlightthat quantum fluctuations transfer significant spectral weight tothe two-magnon continuum, visible in both the transverse andthe longitudinal components of the dynamical structure factor,with the latter contributing strongly to the overall dynamicalresponse for S = 1/2.
Our results provide the first determination of the fulldynamical structure factor for the isotropic HTAF withinthe framework of nonlinear SWT. They are consistentwith and go beyond prior studies on quasi-one-dimensionalspiral and spatially anisotropic triangular antiferromagnetsby maintaining proper treatment of Green’s functions andachieving a systematic ranking of different 1/S contributions.These detailed calculations provide a guide for experimentalobservation of the effects of magnon interaction and decaysas well as a direct analytical scheme to predict the spindynamics for realistic single-crystalline materials. Moreover,the inclusion of the momentum-integrated dynamical structurefactor provides a guide for observations in materials for whichonly powder samples are available. Thus, this work presentsthe full landscape of the nonlinear spin-wave dynamics in thetriangular lattice Heisenberg antiferromagnet.
ACKNOWLEDGMENTS
This work was supported in part by the U.S. Departmentof Energy under Grant Nos. DE-FG02-08ER46544 (M.M. andW.T.F.) and DE-FG02-04ER46174 (A. L. C.).
APPENDIX: SUBLEADING CORRECTIONS TO THESTRUCTURE FACTOR AND DERIVATION OF S zz
1. Anomalous terms in the transverse structure factor
Here we evaluate the contribution from the anomalousGreen’s function G12(k,ω) to the transverse structure factor.The Green’s functions can be expressed explicitly from (12) as
G11(q,ω) = 1
[ω − εq − 11(q,ω)],
G12(q,ω) = −12(q,ω)
[ω − εq − 11(q,ω)][ω + εq + 11(−q, − ω)].
(A1)
As discussed in Sec. III A, the unphysical mode needs tobe controlled by keeping the source term in (13) on-shell.This form of the self-energies is used in all our numericalevaluations. The relative role of the normal and anomalousGreen’s functions is estimated through their respective spectralfunctions,
A11,12(q,ω) = − 1
πIm[G11,12(q,ω)], (A2)
plotted for representative high-symmetry points in Fig. 8. Thedominant contribution comes from A11(q,ω), while A12(q,ω)yields a much smaller contribution already for S = 1/2.In addition, its contributions are also redundant to that of
FIG. 8. (Color online) (a–c) Energy dependence of the contri-butions to the spectral function for S = 1/2 at the M, Y, and Y1 qpoints; see Fig. 1(b). Solid lines with shaded (gray) area correspondto A11(q,ω), and dashed lines to A12(q,ω). The contributions fromA11(−q, − ω) are vanishingly small.
A11(q,ω) in terms of qualitative features. Overall, we concludethat the contribution from A12(q,ω) is small and that fromA11(−q, − ω) is 0 away from the K point, so that they bothcan be neglected in (10).
2. Longitudinal fluctuations
We begin the derivation of Szz(q,ω) with the expression forthe longitudinal spin fluctuations in (15). At zero temperature,Szz(q,ω) is related to the time-ordered Green’s function by
Gzz(q,t) = −i⟨T δSz
q(t)δSz−q
⟩
= − i
N
∑k1,k2
⟨T a
†k1
(t)ak1+q(t)a†k2
ak2−q⟩. (A3)
This correlator probes the two-particle density of states andthus provides information about the two-magnon continuum.
The longitudinal component of the structure factor is of theorder of O(1/S), a factor of 1/S smaller than the leading termsin the transverse correlation function (10). Then, in the spiritof the 1/S expansion, Gzz in (A3) can be calculated withouttaking into account interaction corrections.
Performing Bogolyubov transformation in (A3) and keep-ing terms only with two creation and two annihilation
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operators, we obtain
Gzz(q,t)
= i∑
k
(ukuk+q + vkvk+q)2G(k, − t)G(k + q,t)
+ i
2
∑k
(ukvk+q + vkuk+q)2G(k, − t)G(−k − q, − t)
+ i
2
∑k
(ukvk−q + vkuk−q)2G(k,t)G(−k + q,t),
(A4)
where G = G11 for brevity. Subsequent transformation to ω
representation shows that the first two terms in (A4) have noimaginary part in the noninteracting approximation. Hence,the leading contribution to the longitudinal structure factor isgiven by the last term:
Szz(q,ω) = 1
2π
∑k
(ukvk−q + vkuk−q)2
× Im∫
dω′
2πiG(k,ω′)G(−k + q,ω − ω′). (A5)
The remaining integral can be taken for G(k,ω) = G011(k,ω)
to yield Eq. (16).To improve on the linear SWT approximation, one can use
(A5) with the interacting Green’s functions. In such an approx-imation the single-magnon energies will be renormalized byinteractions, while still neglecting other effects of interactionsin the correlation function.
3. Mixed transverse-longitudinal fluctuations
Here we provide a few remarks concerning the mixedtransverse-longitudinal correlators. We use the identity
i[Sxz(q,ω) − Szx(q,ω)] = − 1
πIm {Gxz(q,ω) − Gzx(q,ω)},
(A6)
where the corresponding Green’s functions are defined as
Gxz(q,t) = ⟨T Sx
q (t)Sz−q
⟩, Gzx(q,t) = ⟨
T Szq(t)Sx
−q
⟩. (A7)
The above identity is derived by applying the fluctuationdissipation theorem, (7), to time-dependent fluctuations ofthe operator A = Sx
q − iSzq (with A† = Sx
−q + iSz−q) and ex-
cluding parts that are diagonal in spin indices, which havebeen considered before. Using bosonic representation for spinoperators we obtain
Gxz(q,t) = −(1 − �+)
√S
2N
∑k
〈T [aq(t) + a†−q(t)]a†
kak−q〉
(A8)
and a similar expression for Gxz(q,t).The first nonzero contribution to the mixed Green’s func-
tions comes from the first-order perturbation term in the three-particle interaction V3. This means that the mixed correlatorgives an O(1/S) contribution compared to the transversestructure factor, (10), and is, formally, of the same 1/S order asSzz (16). Keeping terms that are nonzero in the noninteracting
S(q
,ω)
q = M(a)
0
0.5
1
1.5
2 S diag(q,ω)
S diag(q,ω) + S mix(q,ω)
S(q
,ω)
q = Y(b)
0
0.25
0.5
0.75
1
ω/J
S(q
,ω)
q = Y1
(c)
1 2 3
0
0.25
0.5
0.75
1
FIG. 9. (Color online) (a–c) Energy dependence of the total dy-namical structure factorS tot(q,ω) = Sdiag(q,ω) + Smix(q,ω) [Eq. (5);dashed lines] compared to the diagonal part [solid lines with shaded(gray) areas] for S = 1/2 at representative q points M, Y, and Y1.Results are convoluted with a Gaussian profile of width σ = 0.03J
as in Fig. 3.
limit and performing standard calculations we obtain
Gxz(q,ω)
= −3S√
3
4(1 − �+)(uq + vq)
∑k
(ukvq−k + vkuq−k)
×{G(q,ω)
[V31(k; q)
ω − εk − εq−k + i0+ V32(−k,q)
ω + εk + εq−k
]
+G(−q, − ω)
[V32(k, − q)
ω − εk − εq−k + i0+ V31(−k; −q)
ω + εk + εq−k
]},
(A9)
while Gzx(q,ω) = −Gxz(q,ω). In Eq. (A9), V31(k; q) andV32(k,q) are the dimensionless cubic vertices from Ref. 11.The final expression for the mixed dynamical structure factorof Eq. (5) reads
Smix(q,ω) = i[Sxz(q − Q,ω) − Sxz(q + Q,ω)], (A10)
where iSxz(q,ω) = −(1/π )Im[Gxz(q,ω)], with Gxz from(A9). For the calculations presented in Fig. 9, the Green’sfunctions are taken as G(q,ω) = G11(q,ω) from Eq. (12),with 11(q,ω) given by Eq. (13). In this approximation,the single- and the two-magnon contributions to the mixedstructure factor are evaluated with the same accuracy as in the
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transverse and the longitudinal components Sxx , Syy and Szz
in (14) and (16), respectively. Because of the nature of ourcalculations, higher order 1/S terms are not self-consistentlyaccounted for, leading to an unphysical overcompensation inFig. 9(c), where a small portion of S tot(q,ω) is slightly below0, presumably an O(1/S2) effect.
Because of the contributions having opposite signs in (A9),it can be anticipated that Smix yields a subleading correctionto the transverse S⊥ in (10) and to the two-magnon continuumSL in (16) despite Szz and Sxz being formally of the same1/S order. This is confirmed in Fig. 9, where Eq. (5) is used toevaluate Sdiag and Smix in the total dynamical structure factorS tot = Sdiag + Smix.
First, the impact of Smix on the dominant peaks in Sdiag
is vanishingly small, showing that the mixed terms do not
affect the leading quasiparticle-like part of the spectrum. Thenwe observe that some of the edge singularities in S tot areenhanced, while some are suppressed, by the inclusion of themixed term, suggesting that the latter would not yield an overallregularization of such singularities. The primary effect of Smix
is a modulation of a relatively small part of the broad two-magnon continuum in Sdiag.
Most importantly, the contribution from Smix(q,ω) tothe momentum-integrated structure factor S(ω) is ex-actly 0 due to the antisymmetric properties of Sxz(q,ω)discussed above. Therefore, we conclude that the off-diagonal term Smix can be neglected compared to theleading transverse S⊥ and longitudinal SL terms in Sdiag.This strongly justifies the choice S tot ≈ Sdiag used inSecs. II and III.
*Correspondence author: [email protected]. Oguchi, J. Phys. Soc. Jpn. Suppl. 52, 183 (1983).2Th. Jolicoeur and J. C. Le Guillou, Phys. Rev. B 40, 2727(1989).
3S. J. Miyake, J. Phys. Soc. Jpn. 61, 983 (1992).4A. V. Chubukov, S. Sachdev, and T. Senthil, J. Phys.: Condens.Matter 6, 8891 (1994).
5L. Capriotti, A. E. Trumper, and S. Sorella, Phys. Rev. Lett. 82,3899 (1999).
6W. H. Zheng, J. O. Fjærestad, R. R. P. Singh, R. H. McKenzie, andR. Coldea, Phys. Rev. B 74, 224420 (2006).
7S. R. White and A. L. Chernyshev, Phys. Rev. Lett. 99, 127004(2007).
8M. E. Zhitomirsky and A. L. Chernyshev, Rev. Mod. Phys. 85, 219(2013).
9A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. Lett. 97,207202 (2006).
10O. A. Starykh, A. V. Chubukov, and A. G. Abanov, Phys. Rev. B74, 180403 (2006).
11A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. B 79, 144416(2009).
12R. Coldea, D. A. Tennant, and Z. Tylczynski, Phys. Rev. B 68,134424 (2003).
13T. Ono, H. Tanaka, H. Aruga Katori, F. Ishikawa, H. Mitamura, andT. Goto, Phys. Rev. B 67, 104431 (2003).
14Y. Shirata, H. Tanaka, A. Matsuo, and K. Kindo, Phys. Rev. Lett.108, 057205 (2012).
15H. D. Zhou, C. Xu, A. M. Hallas, H. J. Silverstein, C. R. Wiebe,I. Umegaki, J. Q. Yan, T. P. Murphy, J.-H. Park, Y. Qiu, J. R. D.Copley, J. S. Gardner, and Y. Takano, Phys. Rev. Lett. 109, 267206(2012).
16H. Kadowaki, K. Ubukoshi, K. Hirakawa, J. L. Martinez, andG. Shirane, J. Phys. Soc. Jpn. 56, 4027 (1987).
17H. J. Lewtas, A. T. Boothroyd, M. Rotter, D. Prabhakaran,H. Muller, M. D. Le, B. Roessli, J. Gavilano, and P. Bourges, Phys.Rev. B 82, 184420 (2010).
18R. Ishii, S. Tanaka, K. Onuma, Y. Nambu, M. Tokunaga,T. Sakakibara, N. Kawashima, Y. Maeno, C. Broholm, D. P.Gautreaux, J. Y. Chan, and S. Nakatsuji, Europhys. Lett. 94, 17001(2011).
19M. Poienar, F. Damay, C. Martin, J. Robert, and S. Petit, Phys. Rev.B 81, 104411 (2010).
20S. E. Dutton, E. Climent-Pascual, P. W. Stephens, J. P. Hodges,A. Huq, C. L. Broholm, and R. J. Cava, J. Phys.: Condens. Matter23, 246005 (2011).
21S. Toth, B. Lake, S. A. J. Kimber, O. Pieper, M. Reehuis, A. T. M.N. Islam, O. Zaharko, C. Ritter, A. H. Hill, H. Ryll, K. Kiefer,D. N. Argyriou, and A. J. Williams, Phys. Rev. B 84, 054452(2011).
22S. Toth, B. Lake, K. Hradil, T. Guidi, K. C. Rule, M. B. Stone, andA. T. M. N. Islam, Phys. Rev. Lett. 109, 127203 (2012).
23T. Ohyama and H. Shiba, J. Phys. Soc. Jpn. 62, 3277 (1993).24M. Y. Veillette, A. J. A. James, and F. H. L. Essler, Phys. Rev. B
72, 134429 (2005).25D. Dalidovich, R. Sknepnek, A. J. Berlinsky, J. Zhang, and
C. Kallin, Phys. Rev. B 73, 184403 (2006).26C.-H. Chung, K. Voelker, and Y. B. Kim, Phys. Rev. B 68, 094412
(2003).27M. Kohno, O. A. Starykh, and L. Balents, Nature Phys. 3, 790
(2007).28A. Mezio, C. N. Sposetti, L. O. Manuel, and A. E. Trumper,
Europhys. Lett. 94, 47001 (2011).29G. L. Squires, Introduction to the Theory of Thermal Neutron
Scattering (Dover, New York, 1996).30C. M. Canali and M. Wallin, Phys. Rev. B 48, 3264 (1993).31G. D. Mahan, Many-Particle Physics (Plenum Press, New York,
1990).32E. F. Talbot, H. R. Glyde, W. G. Stirling, and E. C. Svensson, Phys.
Rev. B 38, 11229 (1988).33M. Mourigal, M. E. Zhitomirsky, and A. L. Chernyshev, Phys. Rev.
B 82, 144402 (2010).34W. T. Fuhrman, M. Mourigal, M. E. Zhitomirsky, and A. L.
Chernyshev, Phys. Rev. B 85, 184405 (2012).35J.-I. Igarashi and T. Nagao, Phys. Rev. B 72, 014403 (2005).36M. E. Zhitomirsky and A. L. Chernyshev, Phys. Rev. Lett. 82, 4536
(1999).37B. Fak and J. Bossy, J. Low Temp. Phys. 112, 1 (1998).38M. B. Stone, I. A. Zaliznyak, T. Hong, C. L. Broholm, and D. H.
Reich, Nature 440, 187 (2006).39T. Masuda, A. Zheludev, H. Manaka, L.-P. Regnault, J.-H. Chung,
and Y. Qiu, Phys. Rev. Lett. 96, 047210 (2006).40R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska,
D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer,Science 327, 177 (2010).
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