Spontaneous Magnon Decays

5 MB·English

by  M. E. Zhitomirsky

Colloquium: Spontaneous Magnon Decays
M. E. Zhitomirsky
∗
Service de Physique Statistique, Magn´etisme et Supraconductivit´e, UMR-E9001
CEA-INAC/UJF, 17 rue des Martyrs, 38054 Grenoble cedex 9, France
A. L. Chernyshev
†
Department of Physics, University of California, Irvine, California 92697, USA
(Dated: 23 January 2013)
A theoretical overview of the phenomenon of spontaneous magnon decays in quantum antifer-
romagnets is presented. The intrinsic zero-temperature damping of magnons in quantum spin
3 systems is a fascinating many-body effect, which has recently attracted significant attention in
1 view of its possible observation in neutron-scattering experiments. An introduction to the the-
0 oryofmagnoninteractionsandadiscussionofnecessarysymmetryandkinematicconditionsfor
2 spontaneous decays are provided. Various parallels with the decays of anharmonic phonons and
excitationsinsuperfluid4Heareextensivelyused. Threeprincipalcasesofspontaneousmagnon
n
decays are considered: field-induced decays in Heisenberg antiferromagnets, zero-field decays in
a spiralantiferromagnets,andtriplondecaysinquantum-disorderedmagnets. Analyticalresultsare
J
compared with available numerical data and prospective materials for experimental observation

6 ofthedecay-relatedeffectsarebrieflydiscussed.
2

]
PACSnumbers: 75.30.Ds75.10.Jm75.50.Ee
l
e
-
r
t Contents B. Terminationpointofthemagnonbranch 21
s C. Decayoflongitudinalmagnons 23
.
t I. Introduction 1
a
VIII. Summary and outlook 24
m
II. Magnon-magnon interactions 3
- A. Four-magnoninteractions 3 Acknowledgments 25
d B. Three-magnoninteractions 3
n C. Magnonself-energies 4 References 25
o D. Three-bosoninteractionsinspinliquids 5
c
[ III. Kinematics of two-magnon decays 5
I. INTRODUCTION
A. Decaythresholdboundary 6
3
B. Decayofacousticexcitations 6
v
1. Singleacousticbranch 6 According to conventional wisdom a quasiparticle is
8
2. Severalacousticbranches 7 presumed well-defined until proven not to be. The text-
7
C. Decaysingularities 7
bookpictureofmagnonsaslong-livedexcitationsweakly
2
5 IV. Field-induced decays 8 interacting with each other works very well for many
. A. Generaldiscussion 8 magneticmaterials. Nevertheless,inanumberofrecently
5
B. Square-latticeantiferromagnet 9 studiedspinsystemsmagnonsneitherinteractweaklynor
0
2 1. Modelandformalism 9 remainwell-definedevenatzerotemperature. Thisisdue
2. Decayregions 10
1 to the so-called spontaneous quasiparticle decay, a spec-
C. Spectralfunctionandself-consistentBorn
: tacular quantum-mechanical many-body effect, which is
v approximation 10
i D. DecaysingularitiesandSCBA 12 the topic of the current Colloquium.
X E. Decaysinthreedimensions 14 Magnetic excitations have been the subject of inten-
r sive theoretical and experimental studies since the first
a V. Zero-field decays 15
half of the last century. Historically, the concept of spin
A. Generaldiscussion 15
waves was introduced by Bloch (1930) and, indepen-
B. Triangular-latticeantiferromagnet 16
dently, by Slater (1930) for the wavelike propagation of
VI. Damping of low-energy magnons 18 spin flips in ordered ferromagnets; see also Hoddeson et
al. (1987). Already in his earliest work Bloch (1930) de-
VII. Magnon decay in quantum spin liquids 19
scribedspinwavesasweaklyinteractingexcitationsobey-
A. Bond-operatorformalism 20
ingBosestatistics. Subsequently,HolsteinandPrimakoff
(1940) introduced second quantization of spin waves in
terms of bosonic operators, the approach widely used to
∗Electronicaddress: [email protected] thisday. Thefirstuseoftheword“magnon,”whichcon-
†Electronicaddress: [email protected] cisely unifies the notions of an elementary quasiparticle
2
and a magnetic quantum, can be traced back to Pomer- of a quantitative theory for the 1D to 3D crossover in
anchuk (1941), who ascribed this term to Lev Landau, weakly coupled chains is still in progress, we shall not
see Walker and Slack (1970). Although the terms “spin discuss this physics here.
waves”and“magnons”aresometimesusedtodistinguish The second class is, in a sense, simpler and more puz-
between long- and short-wavelength excitations, respec- zling at the same time. It is comprised of quantum mag-
tively, in the following we use both names interchange- nets with well-ordered ground states whose dynamical
ably. properties are yet quite different from the predictions of
In addition, the fundamental contribution to the the- harmonic spin-wave theory (Zheng et al., 2006a). In the
oryofspinwavesinmagneticinsulatorswasmadebyAn- following,wefocusonthissecondclassofunconventional
derson(1952),Kubo(1952),andDyson(1956a,b). These magnetic systems. It includes quantum magnets with
and subsequent studies developed a comprehensive pic- noncollinear spin ordering. The key role in their unusual
tureofspinwavesinconventionalquantumferromagnets dynamicalpropertiesisplayedbycubicanharmonicities,
and antiferromagnets, which can be found in various re- orthree-magnonprocesses,whichoriginatefromthenon-
views and books, see Van Kranendonk and Van Vleck collinearity of spins. Three-magnon processes are ab-
(1958), Akhiezer et al. (1968), Mattis (1981), White sent in conventional collinear antiferromagnets, making
(1983),Borovik-RomanovandSinha(1988),Manousakis their spin-wave excitations intrinsically weakly interact-
(1991), and Majlis (2007). Invented in the 1950s, inelas- ing. The most striking effect of three-magnon interac-
tic neutron-scattering spectroscopy (Brockhouse, 1995) tions is the finite lifetime of spin waves: not only at
has quickly become a major experimental method of in- nonzero temperatures when magnon linewidth is deter-
vestigating spin waves and other magnetic excitations in mined by scattering off thermal excitations, but also at
solids. Withavarietyofspecializedtechniquescurrently T =0whendampingisproducedbyspontaneousmagnon
used at numerous neutron facilities around the world, decay.
it remains the most powerful and versatile research tool In order to put the phenomenon of spontaneous
for studies of magnetic spectra. Other microscopic tech- magnon decays in a broader context, we note that two
niques to investigate excitations in magnetic materials othersubfieldsofcondensedmatterphysicsalsodealwith
areelectronspinresonance(GurevichandMelkov,1996) decays of bosonic excitations. These are anharmonic
and light scattering methods, such as resonant inelastic crystals(Kosevich,2005)andsuperfluidliquidsandgases
X-ray scattering (Beaurepaire, 2006). (LifshitzandPitaevskii,1980;PethickandSmith,2008).
Usually,themagnon-magnoninteractionsplayaminor Cubicanharmonicitiesarenaturallypresentinthesesys-
role at low temperatures. For example, as T 0 devia- tems, but their role differs both quantitatively and qual-
→
tion of the spontaneous magnetization of a ferromagnet itativelyfromthatinquantumspinsystems. Incrystals,
fromitssaturatedvaluefollowsBloch’slaw∆M T3/2, the anharmonic corrections are generally small due to
∝
the result obtained by considering magnons as noninter- small mean fluctuations of ions around equilibrium posi-
acting bosons. Interactions yield a correction to Bloch’s tions,theconditionsatisfiedforthemajorityofsolidsbe-
lawthatscalesasT4,whichisstillsmallerthanthe“kine- low the Debye temperature. In superfluids, the strength
matic” correction proportional to T5/2 due to deviation of cubic interactions is determined by the condensate
of magnon dispersion from its parabolic form (Dyson, fraction and may again become small, as is the case of
1956b). In antiferromagnets, effects of magnon-magnon superfluid 4He. Quantum magnets are unique in that (i)
interactionsurviveatzerotemperatureduetozero-point cubic anharmonicities may be tuned on and off by an
motion in the ground state. An estimate of their role is externalmagneticfieldorbychangingthelatticegeome-
given by the magnitude of the quantum correction to try, and (ii) the existing variety of magnetic compounds
the magnon dispersion law obtained in the harmonic, provides diverse forms of the magnon dispersion law ε ,
k
or semiclassical, approximation. Even in low dimension which plays key role in spontaneous quasiparticle decay.
(2D)andeveninthe“extremequantum”caseofS =1/2, Still, various parallels and analogies between the three
such a correction for the square-lattice Heisenberg anti- branchesofcondensedmatterphysicsprovetobehelpful
ferromagnet amounts to a mere 16% rescaling of the andwillbeextensivelyusedinthesubsequentdiscussion.
∼
harmonic dispersion (Manousakis, 1991). Therestofthearticleisorganizedasfollows. SectionII
Recently, therehasbeenagrowingbodyoftheoretical containsageneraldiscussionoftheoriginofcubicaswell
and experimental studies showing that the conventional asquarticanharmonicitiesinquantummagnets. Theen-
pictureofmagnonsasweaklyinteractingquasiparticlesis ergy and momentum conservation conditions for spon-
not always correct. The first class of systems exhibiting taneous decays and the decay-induced singularities in
significantdeviationsfromtheconventionalsemiclassical two-dimensionalspinsystemsareanalyzedinSec.III.In
dynamicsisweaklycoupledantiferromagneticchains,for Sec. IV we provide a description of spontaneous magnon
which the spin-wave approximation is not a good start- decays in the square-lattice Heisenberg antiferromagnet
ing point. Their high-energy spectra are well-described in strong external field, while Sec. V deals with another
by propagating one-dimensional spinons, the natural ba- prototypical noncollinear magnet: the triangular-lattice
sis for excitations in spin-1 chains (Coldea et al., 2001; Heisenberg antiferromagnet in zero field. A few general
2
Kohnoet al.,2007;Schulz,1996). Sincethedevelopment scenarios of the decay of long-wavelength excitations are
3
considered in Sec. VI. Section VII is devoted to the dis- As pointed out by Anderson (1984), the symmetry-
cussion of the role of cubic anharmonicities in gapped broken antiferromagnetic ground state corresponds to a
quantum spin liquids. Finally, a summary and outlook superposition of states with different values of the total
are presented in Sec. VIII. spin. Hence, elementary excitations cannot be assigned
with a definite value of spin: the spin of the spin wave
ceases to exist. Because of that, the effective Hamilto-
II. MAGNON-MAGNON INTERACTIONS nian of a quantum antiferromagnet contains additional
interaction terms,
In the classical picture, the interaction between
manadgncoannsbisecroemlaeteadrbtiotrtahrielyamsmpaliltluadsetohfespaimnpplirteucdeessdioen- Hˆ4 = 31! V4(1)(k1,k2,k3;k4) a†k1a†k2a†k3ak4+h.c.
creases. Zero-point motion of spins puts a lower bound (cid:88)ki (cid:0) (cid:1)
1
oeffnetchtse ianmqpuliatnutduemanmdalgenaedtss.toUnsouna-lnlye,gltihgeibcleoninsitdeeraracttiioonn +4! V4(2)(k1,k2,k3,k4) a†k1a†k2a†k3a†k4+h.c. , (2)
of magnon-magnon interactions begins with a discussion (cid:88)ki (cid:0) (cid:1)
ofbosonicrepresentationsforspinoperators. Herewein- that do not conserve the number of excitations; see,
tentionallyskipthisstep, postponingtechnicaldetailsto for example, Harris et al. (1971). The first term in
the subsequent sections. Instead, we discuss the general Eq. (2) describes decay and recombination processes of
structureofeffectivebosonicHamiltoniansdefinedsolely one magnon into three and vice versa, whereas the sec-
by the symmetry of the magnetic ground state and cor- ond, so-called source term, corresponds to creation and
responding excitations. annihilation of four particles out of (into) an antiferro-
magnetic vacuum.
A. Four-magnon interactions
B. Three-magnon interactions
To draw an analogy between different bosonic systems
we begin with the Hamiltonian of a normal Bose gas
Another type of anharmonicity that may appear in an
Hˆ0 = εka†kak (1) eaffcteicotnivteerbmos.oInnicaHgaemneilrtiocnqiuananitsutmhesythstreeme-pwairtthicnleoninctoenr--
k
(cid:88) served number of particles anharmonicities of all orders
1
+ 4 V4(0)(k1,k2;k3,k4)a†k1a†k2ak3ak4 +..., arepresent,beginningwiththelowest-ordercubicterms,
which couple one- and two-particle states. For instance,
(cid:88)ki
cubic terms represent the dominant anharmonicity for
where ε is the kinetic energy, V(0)(k ,k ;k ,k ) is the lattice vibrations (Kosevich, 2005). However, additional
k 4 1 2 3 4
two-particle scattering amplitude, the ellipsis stands for symmetryrestrictionsmayforbidsuchcubicinteractions
n-particle interactions, and momentum conservation is in antiferromagnets.
assumedfromnowonforvariousk-summations. Thein- In the case of a collinear antiferromagnet, the remain-
teraction term in Eq. (1) conserves the number of parti- ing SO(2) rotational symmetry about the N´eel vector
cles,anaturalconstraintforliquidheliumorcoldatomic direction (z axis) prohibits cubic terms. Although the
gases. At the same time, an effective bosonic Hamilto- single-magnon state is not an eigenstate of the Sz op-
tot
nian with exactly the same structure as in Eq. (1) de- erator, such a state still preserves an odd parity under
scribes the quantum Heisenberg ferromagnet (Holstein the π rotation about the z axis. In contrast, the two-
and Primakoff, 1940; Oguchi, 1960). Although the num- magnon states are invariant under this symmetry op-
ber of quasiparticles may not be conserved, in a ferro- eration. Consequently, the coupling between one- and
magnet the conservation is enforced by the invariance two-particlesectorsisstrictlyforbiddenincollinearanti-
of the ground state under an arbitrary rotation about ferromagnets and cubic terms do not occur in their ef-
the magnetization direction. Because of this symmetry fective bosonic description. In hindsight, this lack of
the ground state and all excitations are characterized by low-degree anharmonicities explains why the spin-wave
definite values of the z-projection of the total spin Sz, theoryworksquantitativelywellevenforspin-1 systems:
2
from Sz = NS and down. Consequently, the matrix magnets with collinear spin structures have excitations
tot
elements of the Hamiltonian vanish for states with dif- that are intrinsically weakly-interacting.
ferent spin projections, resulting in the particlelike form On the other hand, three-magnon interaction terms
of the interaction term in Eq. (1). In other words, in must be present if the spin-rotational symmetry is com-
the Heisenberg ferromagnet every magnon has an intrin- pletely broken in the ground state, i.e., when the mag-
sic quantum number ∆Sz = 1, which is conserved in netic structure becomes noncollinear. This is realized,
−
magnon scattering processes. for example, due to spin canting in an applied magnetic
The quasiparticle-number conservation does not hold field or because of competing interactions in frustrated
for interaction processes in quantum antiferromagnets. antiferromagnets. The general form of the three-boson
4
interaction is given by (a) (b)
q q
1
Hˆ3 = 2! V3(1)(k1,k2;k3) a†k1a†k2ak3 +h.c. k k k k
(cid:88)ki (cid:0) (cid:1) k − q − k − q
1
(2)
+ 3! V3 (k1,k2,k3) a†k1a†k2a†k3 +h.c. ,(3) (c) (d)
(cid:88)ki (cid:0) (cid:1) q q
where the first term describes the two-particle decay k p k k p k
and recombination processes while the second is another
k − q − p − k − q − p
source term.
There exists a deep analogy between the noncollinear
FIG.1 (coloronline). Zero-temperatureself-energydiagrams
quantum magnets and superfluid boson systems regard-
due to the (a) and (b) three-magnon, and (c) and (d) four-
ing the presence of cubic interactions in their effective
magnon interactions.
Hamiltonians. The Bose gas in the normal state is in-
variant under the group of gauge transformations U(1)
and is described by the Hamiltonian (1). The macro-
based on the Holstein-Primakoff transformation for spin
scopic occupation of the lowest-energy state a = 0
(cid:104) k=0(cid:105) (cid:54) operatorsproducesaninfiniteseriesofinteractionterms.
below the superfluid transition leads to breaking of the
Typically, these higher-orders terms are small due to the
U(1)symmetryand,atthesametime,totheappearance
spin-waveexpansionparameter a a /2S 1associated
†
of the cubic terms by virtue of the Bogolyubov substitu- (cid:104) (cid:105) (cid:28)
with each order of expansion. We disregard them in the
tion
following analysis as they do not alter our results, either
qualitatively or quantitatively.
a†k1a†k2ak3ak4 →a†k1a†k2ak3(cid:104)a0(cid:105)+... . (4)
In other words, the vacuum state with the Bose conden-
sateabsorbsoremitsanextraparticle, enablingnoncon- C. Magnon self-energies
serving processes between excitations.
An analogous consideration of the noncollinear mag- Magnon interactions play a dual role. On the one
netically ordered state is possible, leading to the same hand,theyrenormalizemagnonenergyand,ontheother
qualitativeanswer. Infact,itwasrealizedalongtimeago hand, they can lead to magnon decay and result in a fi-
by Matsubara and Matsuda (1956) and by Batyev and nite lifetime. Both effects are treated on equal footing
Braginskii (1984) that there is a one-to-one correspon- within the standard Green’s function approach (Lifshitz
dence between breaking the SO(2) rotational invariance and Pitaevskii, 1980; Mahan, 2000). The lowest-order
about the field direction for Heisenberg and planar mag- self-energydiagramsproducedbythree-andfour-particle
nets and breaking the U(1) gauge symmetry at the su- vertices at T = 0 are shown in Fig. 1. Analytic expres-
perfluidtransition. Noncollinearmagneticstructuressta- sions for the diagrams in Figs. 1(a) and 1(b) are
bilized by competing exchange interactions in zero field
belong to a different class as they spontaneously break (1) 2
1 V (q;k)
the full SO(3) rotational symmetry without the help of Σ (k,ω) = 3 , (5)
a
2 ω ε ε +i0
an external field. Still, the absence of any remaining (cid:88)q −(cid:12)(cid:12) q− k−q(cid:12)(cid:12)
symmetry constraints for their elementary excitations, (2) 2
1 V (q,k)
apart from energy and momentum conservation, permits Σ (k,ω) = 3 ,
b
−2 ω+ε +ε
all possible anharmonic terms including the cubic ones. q (cid:12) q k(cid:12)+q
(cid:88) (cid:12) (cid:12)
It is necessary to mention that cubic anharmonicities
due to the long-range dipolar interactions have been dis- whereasthethree-magnondecayprocessofFig.1(c)gives
cussed already in the early works on the quantum the-
ory of spin waves in ferromagnets (Akhiezer et al., 1968; (1) 2
1 V (q,p;k)
HolsteinandPrimakoff,1940)wheretheyplayanimpor- Σ (k,ω)= 4 . (6)
c
6 ω ε ε ε +i0
tant role in the low-frequency magnetization dynamics (cid:88)q,p − q(cid:12)(cid:12)− p− k−q(cid:12)(cid:12)−p
(Chernyshev, 2012; Gurevich and Melkov, 1996). How-
ever, the effect of cubic terms is completely negligible at Weincludedonlyindependentmomentainthearguments
higher energies because of the smallness of the dipole- of vertices in Eqs. (5) and (6).
dipole interaction compared to the exchange energy. In Inthelowestorderofperturbationtheory,therealpart
contrast, cubic anharmonicities discussed in this work of the on-shell self-energy Σ(k,ω=ε ) gives a correction
k
are of exchange origin and have a profound effect on the to the dispersion δε , and the imaginary part yields the
k
magnon excitation spectra. decay rate Γ = ImΣ(k,ε ). The source diagrams in
k k
−
Anharmonic interactions are not restricted to cubic Figs. 1(b) and 1(d) have no imaginary parts, while the
and quartic processes. The usual spin-wave expansion magnon damping resulting from the decay diagram in
5
Fig. 1(a) is given by III. KINEMATICS OF TWO-MAGNON DECAYS
(3) π (1) 2 The aim of this section is to consider kinematic con-
Γ = V (q;k) δ(ε ε ε ) . (7)
k 2 3 k− q− k−q straintsinthetwo-particledecayprocessthatfollowfrom
q
(cid:88)(cid:12) (cid:12) energy conservation:
(cid:12) (cid:12)
Thethree-particledecayprocessinFig.1(c)yieldsasim- ε =ε +ε . (10)
k q k q
ilar expression −
For a magnon with the momentum k, the relation (10)
(4) π (1) 2 is an equation for an unknown q with k being an ex-
Γ = V (q,p;k) δ(ε ε ε ε ) .
k 6 4 k− q− p− k−q−p ternal parameter. Solutions of Eq. (10) form a decay
(cid:88)q (cid:12) (cid:12) surface, the locus of momenta of quasiparticles created
(cid:12) (cid:12) (8) in the decay. As a function of the initial momentum k
(1) (1)
Existence of the decay amplitudes V3 and V4 is a the decay surface expands or shrinks and may disappear
necessary, but by no means sufficient condition for spon- completely. In the latter case, the magnon with momen-
taneous magnon decays. The decay rate Γk is nonzero tumkbecomesstable. Theregioninkspacewithstable
onlyiftheenergyconservationissatisfiedforatleastone excitations is separated from the decay region where de-
decay channel. A general analysis of the kinematic con- cays are allowed by the decay threshold boundary.
ditions following from the energy conservation for two- For any given k the two-particle excitations form a
particle decays is given in Sec. III. continuum of states in a certain energy interval
Emin(k) E (k,q) ε +ε Emax(k) . (11)
2 ≤ 2 ≡ q k−q ≤ 2
With the help of this statement it is straightforward to
D. Three-boson interactions in spin liquids
see that nontrivial solutions of Eq. (10) exist only if the
single-particle branch and the two-particle continuum
Apart from magnetically ordered antiferromagnets,
overlap. From this perspective it also becomes evident
thereisawideclassofquantum-disorderedmagnetswith
that the decay threshold boundary must be the surface
spin-liquid-like ground states. At zero temperature such
of intersections of the single-particle branch ε with the
magnetsremaincompletelyisotropicunderspinrotations k
bottom of the continuum Emin(k).
and their ground-state wave function is a singlet state of 2
Suppose for a moment that spontaneous two-particle
the total spin S = 0; see Sec. VII for more detailed
tot
decays are completely forbidden, i.e., the one-magnon
discussion and references. In spin-dimer systems and in
branch lies below the two-magnon continuum in the
some gapped chain materials, the low-energy excitations
whole Brillouin zone
are S = 1 magnons separated by a finite gap from the
singlet ground-state. This means that at any given mo- ε ε +ε for k,q , (12)
k q k q
mentum k there is a triplet of excited states with the ≤ − ∀
where the equality may be realized only for a trivial so-
same energy εk. We use bosonic operators t†kα and tkα lution q=0, provided that ε = 0. Applying the same
with α = x,y, and z to describe creation or annihila- 0
relation (12) to ε on the right-hand side gives
tionofsuchquasiparticles,whicharealsocalledtriplons. k−q
Having an intrinsic quantum number, spin polarization εk εq+εp+εk q p for q,p , (13)
≤ − − ∀
α,doesnotchangeappreciablythestructureofthequar-
whichmeansthattheone-magnonbranchalsoliesbelow
tic terms in Eqs. (1) and (2): the SO(3)-invariant anhar-
the three-magnon continuum as well as any n-magnon
monic interactions are constructed by making all pos-
continua. Thus, if two-particle decays of Eq. (10) are
sible convolutions for two pairs of polarization indices.
prohibitedforanyvalueofk,theenergyandmomentum
An interesting observation is that in this case the spin-
conservation also forbid decays in all n-particle channels
rotational symmetry does not forbid the cubic vertices.
with n 3 (Harris et al., 1971). This is why it is im-
Thetotalspinisconservedsincetwospin-1magnonscan ≥
portant to analyze kinematic conditions for two-particle
form a state with the same total spin S = 1. The in-
tot decays as the first step even if the cubic terms are not
variant form of the cubic interaction for triplons is then
present.
given by
Aside from finding whether spontaneous decays exist
ornot,thereisanotherreasonforconsideringkinematics
1
Hˆ3= 2! V3(k1,k2;k3)(cid:15)αβγ t†k1αt†k2βtk3γ+h.c. , (9) oflfuidde4cHayes,.PIintaveevsstikgiait(i1n9g59t)wfoo-urontdonthadtectahyeseninhatnhceesdudpeenr--
(cid:88)ki (cid:0) (cid:1)
sity of states (DoS) near the bottom of the two-particle
where the conservation of total spin is imposed by the continuummayproducestrongsingularitiesinthesingle-
antisymmetrictensor(cid:15)αβγ. Inaddition,therearecertain particle spectrum at the decay threshold boundary. Ad-
lattice symmetries affecting the structure of the decay ditional singularities may also occur inside the decay re-
vertex in Eq. (9), which may still forbid the two-particle gion due to topological transitions of the decay surface
decaysoftriplons. Specificexamplesofthatarediscussed (Chernyshev and Zhitomirsky, 2006). A general analysis
in Sec. VII. of these two effects is given in Sec. III.C.
6
�⇡ The resultant decay region is given by the union of all
0.90H.95Hs XX⇡0 ssuidberreignigonaslletnhcelodseedcabyychthaennbeolusninda(rii)e–s(iovb)t.aiSnoemdebyofcothne-
0.85Hs calculated surfaces define the decay boundary while the
X⇡ 0.8Hs s rFeisgtucreor2resshpoownsdatnoesxaadmdplel-epoofintthseidnescidaeytthheredshecoaldybreoguinodn-.
M
⇡
ariesforthecubicHeisenbergantiferromagnetinapplied
magnetic field, the problem discussed in more detail in
Sec. IV. Magnetic fields are given in units of the satura-
tion field H and below H 0.76H decays are strictly
� s ∗ ≈ s
/ forbidden. ForH >H thedecayregiongrowsoutofthe
kz ∗
M point,fillingoutthewholeBrillouinzoneatH =H .
π s
X0
kx/� B. Decay of acoustic excitations
X
ky/� Forasufficientlycomplicateddispersionlawεk thede-
M
cay threshold boundary has to be found numerically by
solving Eq. (10) and Eqs. (14)–(17). However, it is often
FIG. 2 (color online). Magnon decay threshold boundaries
for the cubic antiferromagnet in an external field shown in possibletodrawconclusionsaboutthepresenceofdecays
one-eighthoftheBrillouinzoneforseveralvaluesofthefield. by analyzing only the low-energy part of the spectrum.
Decays are possible within the part of the Brillouin zone en- Suchanapproachisfrequentlyusedintheoreticalstudies
closing the M point. of phonon decays (Kosevich, 2005) and we briefly review
π
thecorrespondingarguments, whichwillalsoberelevant
for the subsequent treatment of magnon decays.
A. Decay threshold boundary
The bottom of the two-particle continuum can
be found by imposing the extremum condition
1. Single acoustic branch
E (k,q) = 0, which is satisfied if the velocities of
q 2
∇
two magnons are equal,
In the decay process of a long-wavelength acoustic ex-
v =v . (14) citation the incident and the two emitted quasiparticles
q k q
− have the same velocity c. The linear approximation for
Using this, the decay threshold boundary is determined theexcitationenergy,εk =ck,doesnotprovidesufficient
by the system of equations (10) and (14). One can dis- information on the possibility of decays and one has to
tinguishseveraltypesofgeneralsolutions,ordecaychan- include a nonlinear correction to the linear dispersion:
nels, for such boundaries (Chernyshev and Zhitomirsky,
2006; Lifshitz and Pitaevskii, 1980): εk ck+αk3 . (18)
≈
(i) The emission of two equivalent magnons with equal
For a weak nonlinearity αk3 ck, the momenta of de-
momentaq=k q= 1(k+G),whereGisareciprocal (cid:28)
− 2 cayingandemittedquasiparticlesarenearlyparallelsuch
lattice vector. In this case Eq. (14) is satisfied automat-
that
ically while Eq. (10) is reduced to
kqϕ2
ε =2ε . (15) k q k q+ , (19)
k (k+G)/2 | − |≈ − 2(k q)
−
(ii) The emission of an acoustic magnon with q 0. In
→ where ϕ is a small angle between q and k. The energy
this case Eq. (10) is fulfilled while Eq. (14) is equivalent
conservation (10) within the same approximation is
to
ckqϕ2
vk = v0 . (16) 3αkq(k q)= . (20)
| | | | − 2(k q)
−
(iii) The emission of an acoustic magnon with q Q
i
→ The nontrivial solutions q,ϕ=0 exist only for the posi-
(for magnets with several acoustic branches):
(cid:54)
tive sign of the cubic nonlinearity α. While the asymp-
ε =ε . (17) toticform(18)isrelevanttomanyphysicalexamples,an
k k−Qi evenmoregeneralconditioncanbeformulatedtoinclude
(iv) The emission of two finite-energy magnons with dif- other cases of gapless energy spectra. If the low-energy
ferentenergiesbutthesamevelocities. Nosimplification partofthespectrumisaconcavefunctionofthemomen-
occurs in this case. tum ∂2ε/∂k2 <0, Fig. 3(a), magnons remain stable. On
7
ε (a) ε (b) C. Decay singularities
Wenowdemonstratethefactthatthedecaythreshold
boundary should generally correspond to a nonanalyt-
icity in the particle’s spectrum because of its coupling
to the two-particle continuum. Consider the decay self-
k k
energyΣ (k,ω)fromEq.(5)inthevicinityofthethresh-
a
old boundary for two-particle decays. Generally the de-
cayvertexhasnoadditionalsmallnessintermsofthemo-
ε (c)
menta of participating quasiparticles, i.e., V(1) = (J).
3 O
Focusingonthe2DcaseandexpandingEq.(10)insmall
∆k=k k and ∆q=q q , where k is the crossing
∗ ∗ ∗
− −
pointofthesingle-magnonbranchandthebottomofthe
k
continuum and q is the minimum point of E (k ,q),
Q Q Q ∗ 2 ∗
2 1 3 energy conservation gives
FIG. 3 (color online). Sketches of the quasiparticle spectra ∆q2 ∆q2
εk with (a) negative and (b) positive curvature of the acous- εk−εq−εk−q ≈(v1−v2)·∆k− a2x− b2y =0, (22)
tic mode, respectively. (c) Spectrum with several acoustic
branches. where v and v are the velocities of the initial and final
1 2
magnons and a and b are constants. Next, the singular
part of the self-energy is given by
the other hand, for the upward curvature ∂2ε/∂k2 > 0,
Fig. 3(b), the low-energy excitations are unstable with d2q
respect to spontaneous decays. Σ(k,εk)∝ (v v )∆k q2/a2 q2/b2+i0, (23)
Notethataccordingtotheabovecriteriontwo-magnon (cid:90) 1− 2 − x − y
decaysarekinematicallyallowedinHeisenbergferromag-
and a straightforward integration in Eq. (23) yields a
nets. Indeed, the energy conservation equation (10) can
logarithmicsingularityinthespectrum(Zhitomirskyand
be satisfied for quasiparticles with a ferromagnetic dis-
Chernyshev, 1999)
persion law ε k2. However, as discussed in Sec. II,
k
∝ (1) (1)
the decay vertices V and V vanish exactly in the Λ
3 4 ReΣ(k,ε ) ln , Γ = ImΣ(k,ε ) Θ(∆k),
Heisenberg ferromagnet and magnons remain well de- k (cid:39) ∆k k − k (cid:39)
fined. Nevertheless,decaysdoexistinanisotropicplanar | | (24)
ferromagnets,forwhichthemagnonnumberconservation where Θ(x) is the step function. The imaginary part of
(1)
doesnotholdanymoreandV isnonzero(Stephanovich Σ(k,ε ) in Eq. (7) is directly related to the two-particle
4 k
and Zhitomirsky, 2011; Villain, 1974). densityofstates. Therefore,intwodimensionsitisnatu-
raltohaveajumpinΓ uponenteringthecontinuum,as
k
obtainedinEq.(24). Ineffectthisdemonstratesthatthe
2. Several acoustic branches vanHovesingularityinthecontinuum’sdensityofstates
gets transferred directly onto the quasiparticle spectrum
Lattice vibrations in crystals have three acoustic via the three-particle coupling.
brancheswithdifferentsoundvelocitiescorrespondingto Consider now the other type of 2D van Hove singu-
one longitudinal and two transverse phonons. Heisen- larity, a saddle-point inside the continuum. Expansion
berg antiferromagnets with a helical or spiral magnetic of the energy conservation in the vicinity of the saddle
structure also have three Goldstone modes, Fig. 3(c), at point yields the same form as in Eq. (22) with a change
k=0 and Q, where Q is the ordering wave vector; see of sign in front of either ∆q2 or ∆q2. Therefore, the
± x y
Sec. V.A for more details. Energy conservation for the singular part of the self-energy is
decay of a fast quasiparticle into two slow quasiparticles
can be satisfied already in the linear approximation for d2q
Σ(k,ε ) (25)
the energy k ∝ (v v )∆k q2/a2+q2/b2+i0
(cid:90) 1− 2 − x y
c1k =c2q+c3 k q , for c1 >c2,c3, (21) andthelogarithmicsingularityappearsnowintheimag-
| − |
inary part:
where all momenta are measured relative to the cor-
responding Goldstone Q points. Thus, if this decay Λ
ReΣ(k,ε ) sign(∆k), Γ ln , (26)
channeliscompatiblewithmomentumconservation,i.e., k (cid:39) k (cid:39) ∆k
Q = Q +Q , then the fast excitation can decay re- | |
1 2 3
gardless of the curvature of the corresponding acoustic againinagreementwiththerelationbetweenthe2Dtwo-
branch. magnon DoS and Γ .
k
8
Geometric consideration of the decay surface near sin- H
gularities (Chernyshev and Zhitomirsky, 2006) offers a
usefulalternativeperspective: whilecrossingofthedecay
threshold boundary obviously corresponds to nucleation
of the decay surface, the saddle-point singularity corre- z y
θ θ
sponds to splitting of the decay surface into two disjoint x
pieces. Altogether, both types of singularities (24) and
FIG.4 (coloronline). Field-inducedcantedspinstructurein
(26) result from topological transitions of the decay sur-
the square-lattice Heisenberg antiferromagnet.
face with a change in the number of connected sheets.
In 3D the decay self-energy generally exhibits a less
singular but still nonanalytic behavior. The logarith- ing mostly on the S = 1/2 case, we demonstrate that
mic peak at the decay threshold boundary is replaced magnons become damped and even overdamped in most
by a continuous square-root behavior ∆k for both oftheBrillouinzoneforarangeoffieldsnearthesatura-
(cid:39) | |
ReΣ(k,εk)andΓk. Aremarkableexceptionisthethresh- tion field (Zhitomirsky and Chernyshev, 1999). Confir-
(cid:112)
old to the two-roton decay in the spectrum of superfluid mation of these results from numerical (Sylju˚asen, 2008)
4He. The dispersion of roton excitations has a minimum and experimental studies is highlighted. We also illus-
onasphereinthemomentumspace, leadingtothesame trate the occurrence of the threshold and saddle-point
logarithmicdivergenceoftheself-energyasinEq.(24)at singularities discussed in Sec. III.C in the perturbative
the energy twice the roton gap (Lifshitz and Pitaevskii, treatment of the magnon spectrum and offer two ap-
1980). This strong anomaly produces the “endpoint” in proaches to regularize them self-consistently. Detailed
thedispersioncurveεkofthesuperfluid4Hepredictedby results of one such self-consistent regularization method
Pitaevskii(1959). Suchaterminationpointwaslaterob- (Mourigal et al., 2010) are presented and an example
servedininelasticneutron-scatteringexperiments(Glyde of the higher-dimensional extension of the square-lattice
et al., 1998; Graf et al., 1974). case is touched upon (Fuhrman et al., 2012).
The analysis of this section can be straightforwardly
extended to the three-particle decay self-energy Σ (k,ω)
c
given by Eq. (6). The general outcome is similar to the
A. General discussion
effects of higher dimensions: singularities are replaced
by continuous albeit nonanalytic behavior. Overall, the
The evolution of an ordered magnetic structure of the
two-magnon decays play a more significant role than the
two-sublattice antiferromagnet in an external field is rel-
higher-orderdecayprocesses. Decaysandassociatedsin-
atively trivial. Spins orient themselves transverse to the
gularities are also enhanced by the low dimensionality of
fielddirectionandtiltgraduallytowardsitsdirectionun-
a magnetic system.
til they become fully aligned at H H , where H is re-
s s
≥
ferredtoasthesaturationfield,seeFig.4. Intheabsence
ofanisotropyandfrustrationthereisnootherphasetran-
IV. FIELD-INDUCED DECAYS sitioninthewholerangefromH =0toH =H . Asfirst
s
argued by Zhitomirsky and Chernyshev (1999), it is the
We devote this section to spontaneous decays in the dynamical properties of quantum antiferromagnets that
square-lattice Heisenberg antiferromagnet in strong ex- undergoanunexpectedlydramatictransformationonthe
ternal field. In zero field the ground state of this model way to full saturation. The phenomenon responsible for
is a two-sublattice N´eel structure, which, due to its this transformation is spontaneous magnon decay, which
collinearity, prohibits cubic anharmonicities and two- becomes possible in sufficiently strong magnetic fields.
magnon decays; see Sec. II.B. The three-magnon and all Thefield-inducedmagnondecaysaregenerallypresent
othern-magnondecaysareforbiddenbyenergyconserva- in a broad class of quantum antiferromagnets. Indeed,
tion,asinthecubic-latticemodelstudiedbyHarrisetal. three-magnon interactions that couple one- and two-
(1971) and in agreement with the arguments of Sec. III. magnon states always exist in canted antiferromagnetic
Thus, in zero field and at T = 0 magnons have an infi- structures created by an external field; see Sec. II. Usu-
nite lifetime. In the opposite limit, when spins are fully ally, kinematic constraints prevent spontaneous decays
polarizedbyexternalfield,thegroundstateisequivalent from taking place in zero or weak magnetic fields. It is
to that of a ferromagnet with all particle-nonconserving easy to see that from the acoustic branch of the magnon
termsstrictlyforbiddensothatthemagnonsarewellde- spectrum (18), which retains the concave shape (α < 0)
fined again. It comes as a surprise that there exists a in small fields and thus is stable with respect to spon-
regime between these two limits where magnons become taneous decays according to Sec. III.B. However, in a
heavily damped throughout the Brillouin zone. strong enough magnetic field the convexity of the acous-
Next we discuss general arguments for the existence tic spectrum must change. This is because at the satu-
of the field-induced decays in a broad class of antiferro- ration field, H = H , the magnon velocity vanishes and
s
magnets and provide details of the spin-wave approach the dispersion at low energies is that of a ferromagnet,
to the square-lattice antiferromagnet in a field. Focus- ε k2, which has an upward curvature. By continuity,
k
∝
9
the spectrum has to preserve its positive curvature for et al., 2004; Nikuni et al., 2000; Regnault et al., 2006)
a certain range of magnetic fields H < H < H , where and theoretically (Affleck, 1991; Affleck and Wellman,
∗ s
thethresholdfieldH correspondstoα=0. Thisimplies 1992;GiamarchiandTsvelik,1999;Mila,1998)andcon-
∗
that the two-particle decays are present at H >H . tinue to attract attention (Giamarchi et al., 2008). We
∗
point out that the mechanism of spontaneous magnon
decays discussed here equally applies to the vicinity of
the triplet condensation field H in these magnets be-
B. Square-lattice antiferromagnet c
cause of the duality between H and the saturation field
c
H (Mila, 1998). This is highly advantageous because in
TheHeisenbergmodelonasquarelattice,atoymodel s
many BECmagnetsthe lower condensationfieldis read-
intheearlydaysofthetheoryofaniferromagnets(Ander-
ily accessible, which makes them prime candidates for
son,1952;Kubo,1952),becameaparadigmaticmodelin
observing magnon decays directly in inelastic neutron-
the late 1980s due to its direct relevance to the parental
scattering experiments. It can be argued that the re-
materials of the high-T cuprates (Chakravarty et al.,
c
cently observed suppression of thermal conductivity in
1989; Manousakis, 1991). It is also viewed as one of
the vicinity of critical fields in one such material (Ko-
theprototypicalstronglycorrelatedmodelsincondensed
hama et al., 2011) is due to magnon decays.
matter physics, used for benchmarking various theoret-
Extension of the spin-wave theory to finite magnetic
ical methods against the more precise numerical tech-
fields was developed by Osano et al. (1982), Zhitomirsky
niques (Sandvik, 1997; White and Chernyshev, 2007).
and Nikuni (1998), Zhitomirsky and Chernyshev (1999),
The standard spin-wave theory provides an accurate
Chernyshev and Zhitomirsky (2009b), and Mourigal et
description of the static and dynamical properties of
al. (2010). For alternative approaches applicable in the
the square-lattice antiferromagnet in zero field even for
vicinity of the saturation field, see also Batyev and Bra-
S = 1/2 (Hamer et al., 1992; Igarashi, 1992; Igarashi
ginskii (1984), Gluzman (1993), Kreisel et al. (2008),
and Nagao, 2005; Weihong and Hamer, 1993). In the
and Syromyatnikov (2009). Close agreement of the spin-
words of Sandvik (1997): “The presently most accurate
wave calculations with exact diagonalization and quan-
calculations [for the S = 1/2 case]... indicate that the
tum Monte Carlo (QMC) results for the ground-state
true values of the ground-state parameters deviate from
properties in the whole range of fields 0 < H < H
their 1/S2 spin-wave values by only 1-2% or less.” The s
was established by Lu¨scher and L¨auchli (2009). The
spin-wave calculations also offer a fine fit to the overall
same work has also confirmed the earlier QMC results
magnondispersion(IgarashiandNagao,2005;Syromyat-
by Sylju˚asen (2008) that demonstrated clear signatures
nikov,2010;WeihongandHamer,1993)withtheonlyde-
of magnon decays in the high-field regime. We provide
viation from the numerical results along the (π,0)-(0,π)
more details on that in the following.
path (Sandvik and Singh, 2001; Zheng et al., 2005).
Heisenbergmodelsinstrongfieldshavereceivedsome-
whatlessattentionuntilrecently[see,however,deJongh
1. Model and formalism
andMiedema(1974)anddeGrootanddeJongh(1986)],
primarilybecausethemajorityofavailablematerialshad
WebeginwiththeHeisenbergHamiltonianofnearest-
exchange constants much larger than the strength of
neighbor spins on a square lattice in a magnetic field
available magnetic fields. The constraints are even more
directed along z axis in the laboratory reference frame,
0
stringentonneutron-scatteringstudiesinthefield,where
the practical limit is currently at about 14T. Neverthe- ˆ =J S S H Sz0 . (27)
less, recent developments in the synthesis of molecular H i· j − i
based antiferromagnets with moderate strength of ex- (cid:88)(cid:104)ij(cid:105) (cid:88)i
changecouplingbetweenspins(Coomeretal.,2007;Lan- Here, J is the nearest-neighbor coupling and H is the
caster et al., 2007; Woodward et al., 2002; Xiao et al., field in units of gµ . With the details of the technical
B
2009) have opened the high-magnetic-field regime to ex- approach explicated by Zhitomirsky and Nikuni (1998)
perimentalinvestigationsforanumberoflayeredsquare- and Mourigal et al. (2010), we summarize here the key
lattice materials (Tsyrulin et al., 2009, 2010). Further- steps of the spin-wave theory approach to this problem.
more,newfield-induceddynamicaleffectscanbepresent First we align the local spin quantization axis on each
inantiferromagnetswithotherlatticegeometries(Coldea site in the direction given by the canted spin configu-
et al., 2001). ration shown in Fig. 4. The corresponding transforma-
Another class of antiferromagnets directly relevant to tion of the spin components from the laboratory frame
our discussion incorporates quantum spin systems with (x ,y ,z ) to the local frame is
0 0 0
singlet ground states and gapped triplet excitations.
They are often called BEC magnets because the field- Six0 =Sixsinθ+SizeiQ·ricosθ , Siy0 =Siy ,
itnhdeuBceodse-oErdinesrtineignicnonthdeemnsactainonb(eBdEeCsc)riobfetdripinletteerxmcsitao-f Siz0 =−SixeiQ·ricosθ+Sizsinθ , (28)
tions. In recent years these quantum spin-gap magnets whereθisthecantingangletobedefinedfromtheenergy
havebeenintensivelystudiedbothexperimentally(Jaime minimization and Q=(π,π) is the ordering wave vector
10
X’ M
of the two-sublattice N´eel structure. Next the standard
Holstein-Primakofftransformationbosonizesthespinop-
eSri±at=orsSixS±iz i=Siy.SE−xpaa†inadi,inSgi−squ=area-†ir(o2oSts−,oan†ieaoi)b1t/a2i,nswtihthe 0.76
bosonic Hamiltonian as a sum,
ˆ = ˆ + ˆ + ˆ + ˆ + ˆ +..., (29)
SW 0 1 2 3 4
H H H H H H
each term being of the nth order in bosonic operators 0.80
afo†irmanpdroaviidaensdthcearbraysinisgfoarntheexp1l/icSitexfapcatnosrioSn2.−Inm/2p.orTtahnist 0.86
consequences of the noncollinear spin structure are the
terms SzSx(y), which describe coupling between trans- Γ X
j i
verse Sx(y) and longitudinal Sz oscillations andyield cu-
bic anharmonicities (ˆ3) in the bosonic Hamiltonian. FIG. 5 (color online). Decay regions in one-fourth of the
H
Minimization of the classical energy ˆ0 in Eq. (29) BrillouinzoneforrepresentativevaluesofH/Hs indicatedby
fixes the canting angle of the classical spHin structure to numbers. Solidanddashedlinesarethresholdboundariesfor
two- and three-magnon decays, respectively.
sinθ = H/H , where the saturation field is H = 8JS.
s s
This procedure also cancels the ˆ term that is linear in
1
H
a†i and ai. After subsequent Fourier transformation, the where ϕ is the azimuthal angle of k = (k ,k ). The
harmonic part of the Hamiltonian (ˆ ) is diagonalized x y
H2 curvature of the spectrum α changes its sign for k along
by the Bogolyubov transformation, ak = ukbk+vkb†k, the diagonal (ϕ=π/4) at the decay threshold field
yielding −
2
HˆSW = ε˜kb†kbk (30) H∗ = √7 Hs ≈0.7559Hs . (33)
k
(cid:88)
1 (1) Note that this expression remains valid for the cubic-
+ 2 V3 (k,q) b†k q+Qb†qbk+h.c. +..., and for the layered square-lattice antiferromagnet with
−
k,q
(cid:88) (cid:0) (cid:1) arbitrary antiferromagnetic interlayer coupling.
where ε˜ = ε + δε with ε being the magnon dis- StaggeredcantingofspinsinEq.(28)“shifts”themo-
k k k k
persion given by the linear spin-wave theory and δε mentum in the two-magnon decay condition in Eq. (10),
k
is from the 1/S corrections due to angle renormaliza- which now reads as εk = εq+εk q+Q. Using the Bose-
tion and Hartree-Fock decoupling of cubic and quartic condensateperspectiveofSec.II.B−,suchachangeisnat-
terms in Eq. (29), respectively. The ellipsis stands for uralasthemagnongoingintothecondensatenowcarries
(2) themomentumQ;seeEq.(4). Apartfromthatthekine-
the classical energy, the three-boson source term V as
3 matic analysis of Sec. III remains the same. Numerical
in Eq. (3), and the higher-order terms in the 1/S ex-
results for the decay region at H > H are shown in
pansion. Although some of these terms do contribute to ∗
Fig. 5. Up to H 0.84H the boundary of the decay
thesubsequentresults, thisabbreviatedformconstitutes ≈ s
region is determined entirely by the decays into a pair
the essential part of the Hamiltonian and is sufficient for
of magnons with equal momenta; see Sec. III.A, type
our discussion. For the explicit expressions of δε and
k
vertices V(1) and V(2), see Mourigal et al. (2010). (i) in Eq. (15). At fields higher than 0.85Hs the decay
3 3 channel with emission of an acoustic magnon, type (ii)
in Eq. (16), creates protrusions leading to more compli-
catedshapesofthedecayregion. Above0.9H thedecay
2. Decay regions s
regionspreadsalmostovertheentireBrillouinzone,sim-
ilarlytothe3DdecayregionsinFig.2. InFig.5,wealso
One can already gain detailed insight into the decay
show boundaries for the three-magnon decays, which are
conditions in the manner discussed in Sec. III by analyz-
lessrestrictiveandcoverlargerareasoftheBrillouinzone
ing the magnon spectrum in the harmonic approxima-
than their two-magnon counterparts.
tion, which is given by
ε =4JS (1+γ )(1 cos2θγ ) , (31)
k k k
−
C. Spectral function and self-consistent Born
whereγ = 1(cosk (cid:112)+cosk ). TheacousticmodenearQ approximation
k 2 x y
follows the asymptotic form (18) ε ck+αk3 with
Q+k
≈
Information on the magnon energy renormalization
c 9+cos4ϕ and decay-induced lifetime due to interactions can
c=2√2JScosθ, α= tan2θ ,(32)
16 − 6 be obtained from the single-particle spectral function
(cid:18) (cid:19)