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Decay of quasiparticles in quantum spin liquids M. E. Zhitomirsky Commissariat à l’Energie Atomique, DSM/DRFMC/SPSMS, 38054 Grenoble, France (Dated: February 4, 2008) Magnetic excitations are studied in gapped quantum spin systems, for which spontaneous two- magnondecaysareallowed bysymmetry. Interactionbetweenone-andtwo-particlestatesacquires 6 nonanalytic frequency and momentum dependence near the boundary of two-magnon continuum. 0 Thisleadstoaterminationpointofsingle-particlebranchforone-dimensionalsystemsandtostrong 0 suppressionofquasiparticleweightintwodimensions. Themomentumdependenceofthedecayrate 2 is calculated in arbitrary dimensions and effect of external magnetic field is discussed. n PACSnumbers: 75.10.Jm,75.10.Pq,78.70.Nx a J 8 A number of magnetic materials with quantum dis- situation. Symmetry of a majority of low-dimensional 1 ordered, or spin-liquid, ground states have been discov- dimersystems,like,forexample,adimerizedchain,13 an ] ered in the past two decades. The well-known exam- asymmetricladderandaplanararrayofdimers,14 allows l ples include the spin-Peierls compound CuGeO ,1 dimer presence of cubic vertices. These should be contrasted e 3 - systems Cs3Cr2Br92 and TlCuCl3,3 integer-spin antifer- with a symmetric ladder,15 where one- and two-magnon tr romagnetic chains,4 and many others. Common prop- states belong to sectors with different parity under per- s erty of all these systems is presence of a spin gap in the mutation of two legs and, consequently, do not interact. . t excitation spectrum, which separates a singlet, S = 0 There is also a suggestion that cubic vertices exists in a m ground state from S = 1 quasiparticles. The low- a Heisenberg antiferromagnetic spin-1 chain beyond the energy triplet of magnons can be split due to intrinsic nonlinearsigma-modeldescription.16Inthefollowing,we - d anisotropiesorunderappliedmagneticfield. Recentneu- shall assume presence of interaction between one- and n tron experiments5,6 on two organic spin-gap materials two-particlesstatesandconsideritsconsequencesforthe o PHCCandIPA-CuCl haverevealeddrastictransforma- dynamic properties of a spin liquid. 3 c tionoftripletquasiparticlesundergoingathighenergies. In Heisenberg magnets, rotational symmetry in the [ Upon entering two-magnoncontinuum, see Fig. 1, spon- spin space fixes uniquely the tensor structure of cubic 1 taneous (T =0) decay of a magnon into a pair of quasi- vertices. Specifically, the decay vertex, Fig. 2a, has the v particles becomes possible, which leads to a rapid de- following form: 5 crease in the quasiparticle life-time in the former case5 0 1 14 ainndthceomlatptleertesydsistaempp.6earanceofthesingle-particlebranch Vˆ3 = 2!Xk,qΓ(k,q)ǫαβγ t†kαt†qβtk+qγ . (1) 0 Quasiparticle instability is well documented for an- 6 other type of quantum liquid—superfluid 4He. Pre- Conservation of the total spin during decay process re- 0 dicted by Pitaevskii nearly fifty years ago,7,8,9 this quires that two created spin-1 quasiparticles must form / t instability was later confirmed by neutron scattering a measurements.10,11 Inliquidhelium,interactionbetween m E one-andtwo-particlestatesisenhancedinthevicinityof - d a decay threshold by a large density of roton states and n leads to avoidedcrossing: the single-particlebranchflat- o tens at energies below twice the roton energy and ceases c to exist completely above that energy scale. ε c : v The aim of present article is to investigate similar ef- i fects in quantum spin liquids. We consider an isotropic X spin system with a quantum disordered ground state, 2∆ r a whichisseparatedbyafinitegap∆fromlow-energyspin- 1 excitations. A bare dispersion of propagating triplets ∆ is given by ε(p). Bosonic operators t and t† destroy p pα pα c and create a quasiparticle with momentum p in one of p the three polarizations α = x,y,z. Interaction between one- and two-particle states is, generally, described by two types of cubic vertices shown in Fig. 2a,b. In super- FIG.1: Schematicstructureoftheenergyspectrumofaquan- fluid 4He presence of such particle non-conserving pro- tum magnet with spin-liquid ground state. Thin solid line is cesses is determined by a Bose condensate, which ab- a bare spectrum ε(p), full solid line is a renormalized single- sorbsoremitsanextraparticle.8,12 Inquantumspinsys- particlebranch,shadedregionistwo-particlecontinuum. The tems with singlet ground states one finds a more diverse decay instability threshold is denoted by pc. 2 ∆p = p p and ∆ω = ω ε and expand accord- c c − − ingly all functions under integral in ∆q= q Q, where Q = 1(p+G). The main difference with −Pitaevskii’s 2 analysis8 is that the antisymmetric vertex vanishes at (a) (b) (c) the decay threshold such that Γ(Q+∆q,Q ∆q) 2∆q Γ(k,q) . (4) k k,q=Q − ≈ ·∇ | Including all dimensional constants in the vertex g 3 ∼ Γ(k,q) and performing angular integration for D > k |∇ | 1 we obtain (d) (e) Λ qD+1dq Σ (ω,p) g2 , (5) FIG. 2: Two cubic vertices of the decay (a) and the source 11 ≃− 3Z0 q2+v2∆p−∆ω−i0 (b) types and their contributions to the normal (c) and (d) and anomalous (e) self-energies. where v = ε(k) and Λ is a lattice cut-off. The 2 k k=Q ∇ | self-energy remains finite as ∆ω, ∆p 0, though it | | → contains an important nonanalytic contribution Σ˜(ω,p). an S = 1 state, which is imposed by the antisymmetric WeshallabsorbaregularpartofΣ (ω,p)intoε(p)and 11 tensor ǫαβγ. The vertex is, consequently, antisymmetric determine a new corrected spectrum from the Dyson’s under permutation of two momenta Γ(k,q) = Γ(q,k). equation: G−1(ω,p)=ω ε(p) Σ˜(ω,p)=0. Similar consideration applies to the source-typ−e vertex, − − Twoothersecond-orderdiagramsconstructedfromcu- Fig. 2b, which is antisymmetric under permutation of bic vertices are presented in Fig. 2d,e. Correction to the any two of three outgoing lines. normal self-energy shown in Fig. 2d is an analytic func- Important kinematic property of the energy spectrum tion of frequency and momentum and can be absorbed ε(p)is a type ofinstability ata decaythresholdmomen- into ε(p). The anomalous self-energy Σ (ω,p), Fig. 2e, 12 tum pc, beyond which the energy conservation hasthesamedependenceonω andpastheone-loopdia- gram (5). Using the Belyaev’s expression for the normal ε(p)=ε(q)+ε(p q) (2) − Green’s function of a Bose gas:8,12 issatisfiedandtwo-magnondecaysbecomepossible. Ex- G−1(ω,p) = ω ε(p) Σ (ω,p) tremum condition imposed on the right hand side of − − 11 Eq. (2) yields that two quasiparticles at the bottom of Σ12(ω,p)Σ21(ω,p) + , (6) continuum always have equal velocities vQ = vp−Q, ω+ε( p)+Σ11( ω, p) − − − where v = ε(k). Then, the simplest possibility is k k that both mo∇menta are also equal with Q = 1(p+G), one can conclude that inclusion of the anomalous self- 2 where G is a reciprocal lattice vector. We have ver- energy simply modifies a coefficient in front of the non- ified the above assertion for several model dispersion analyticcontribution(5) withoutchangingits functional curves17,18 aswellasforafewexperimentalfits.5,6,19 The dependence. Similar result applies to the vertex cor- common feature of all analyzed cases is presence of two rection from multiple two-particle scattering processes. regimes: forasmalltotalmomentumpofamagnonpair Since a one-loop diagramis finite for ω =εc p=pc, the the minimum energy corresponds to equal momenta of renormalizedvertexg˜3 alsoremainsfinite andthe vertex two quasiparticles,whereasfor p nearthe Brillouinzone correction amounts to replacement g32 → g3g˜3 in front boundary two particles of the lowest-energy pair have of the integral in Eq. (5). Thus, the following analysis different momenta. Such a bifurcation is closely related based on analytic properties of the one-particle Green’s to appearance of bound pairs of magnons at large mo- functionisnotrestrictedtothelowest-orderperturbation menta for one-dimensional spin systems.13,20 Kinematic theory used in derivation of Eq. (3) and is quite general. instability at the decay threshold in all considered cases One-dimensional gapped spin systems: Let us begin corresponds, however, to a decay into a pair of quasi- with a stable region, where spontaneous decays are for- particles with equal momenta. In the following we shall bidden: ∆ω < 0, ∆p > 0, such that (v2∆p ∆ω) > 0. − focus on such type of a model-independent instability. Separating the nonanalytic part in the integral (5) and Let us consider the second-order contribution to the expanding ε(p) εc +v1∆p we obtain for the inverse ≈ normal self-energy from the decay processes shown in Green’s function Fig. 2c. Standard calculation yields G−1(ω,p)=∆ω v ∆p λ v ∆p ∆ω , (7) 1 2 − − − dDq Γ(q,p q)2 p Σ11(ω,p)=2Z (2π)D ω ε(q) ε(−p q)+i0 , (3) with λ=πg32/2. The one-particle Green’s function with − − − sucha dependence onω and p has been briefly discussed where D is the number of dimensions. We shall be in- in Ref. 8. In the following we give more details relevant terested in the behavior of energy spectrum at small for the case of Eq. (7). 3 Condition for a pole G−1(ω,p) = 0 is transformed to Two-dimensional gapped spin systems: Below decay quadratic equation, which yields threshold in the stable region, where (v ∆p ∆ω)> 0, 2 − the nonanalytic part of the self-energy becomes λ2 λ4 ε¯(p)=εc+v1∆p− 2 +r 4 +λ2(v2−v1)∆p . (8) Σ˜(ω,p)=λ(v ∆p ∆ω)ln R , (12) 2 − v ∆p ∆ω 2 − Away from the crossing point for ∆p λ2/(v v ), ≫ 2 − 1 where λ = g2/2 and R is an energy cut-off. Though, slopeofthesingle-particlebranchcoincideswiththebare 3 a pole of the Green’s function cannot be found analyti- magnonvelocity dε¯(p)/dp v <0, whereas for ∆p 0 ≈ 1 → cally, it is easy to verify that solution of G−1(ω,p) = 0 slopechangessigntodε¯(p)/dp=v >0. Inthecrossover 2 region ∆p λ2/(v v ) the quasiparticle weight Z, existsallthe waydowntothe crossingpoint. The quasi- 2 1 ∼ − particle branchtouches tangentiallythe boundary of the Z−1= ∂G−1 =1+ λ2+ λ4+4λ2(v2−v1)∆p, cpoanrttiicnlueuwmei∇ghptε¯(ips)gr=advu2alfloyrdpim=inpisch,ewdhaernedasvathneishqeusasait- ∂ω (cid:12) p4(v v )∆p (cid:12)ω=ε¯ 2− 1 the boundary. (cid:12) (9) (cid:12) Above the decay threshold [(v2∆p ∆ω) < 0] the is continuously suppressedand vanishes,once the single- − self-energy acquires imaginary part: particle branch touches the two-magnoncontinuum. On the opposite side of the crossing point, ∆p < 0, R ∆paωrt >of t0h,e asneldf-e(n∆erωgy−bvec2o∆mpe)s >pur0e,lytihmeagnionnaarnyaΣl˜yt=ic Σ˜(ω,p)=−λ(∆ω−v2∆p)(cid:18)ln∆ω−v2∆p +iπ(cid:19)(1.3) y−iieλld√s∆aωfte−rvt2r∆anps.forFmoramtiaolnly,tocoqnudaitdiroanticG−eq1(uωa,tpio)n=th0e dInectahyerpaetretuorfbmataivgenorengsimgreo,wwshleinreeaε¯r(lpy)i≈nsiεdce+tvh1e∆cpon,ttihne- same solution as in the stable region. For small nega- uum: tive ∆p the root (8) is real and satisfies ∆ω v ∆p<0 1 − and∆ω−v2∆p<0. Thelatteroftheabovetworelations γp ≃2πλ(v1−v2)∆p . (14) contradictsthe assumptionmade in the beginning ofthe paragraph for the decay region, whereas the former is Intheclosevicinityofthecrossingpointwewriteinstead inconsistent with zero of G−1(ω,p) in the region, where ε¯(p) v2∆p+a with Rea> 0 and transform equation ≈ v ∆p ∆ω >0,seeEq.(7). Thus,insidethecontinuum, on the pole of the Green’s function to 2 − aphysicalpole ofthe one-particleGreen’sfunction reap- pears only at a finite distance from the crossing point, R (v1 v2)∆p a ln +iπ = − =b>0 . (15) where Re ∆ω >v ∆p or (cid:20) a (cid:21) λ 2 { } λ2 With logarithmic accuracy the solution of the above ∆p< . (10) equation is −2(v v ) 2 1 − b πb Disappearance of single-magnon excitations inside the Rea= , Ima= − . (16) continuum is quite similar to the termination point in ln2R/b+π2 ln2R/b+π2 the energy spectrum of superfluid 4He.7,8,9,10,11 In both q cases the single-particle branch approaches tangentially Single-particle branch of a two-dimensional gapped spin the boundary of the two-particle continuum. system can be, therefore, continued inside the decay re- Further awayfrom the loweredge of the continuum in gion. However, the quasiparticle weight is suppressed in the region, where ε¯(p) εc+v1∆p, one can find for the a range of momenta near the crossing point: ≈ inverse quasiparticle life-time: (v v )∆p Re−1/λ . (17) 1 2 | − |≃ γ 2λ (v v )∆p . (11) p 2 1 ≃ p − | | Suppression of the quasiparticle peak has been exper- Such a square-root dependence of the decay rate follows imentally observed in two-dimensional quantum disor- also from a simple golden rule consideration.16 We have dered antiferromagnet PHCC.5 seen,however,thatthefulldependenceoftheself-energy Three-dimensional gapped spin systems: The nonan- onfrequencyandmomentummustbekeptinthevicinity alytic part of the self-energy is given in this case by ofthedecaythreshold. Besides,ifthecubicverticeshave no special smallness, i.e., λ is of the order of the band- Σ˜(ω,p)= λ(v2∆p ∆ω)3/2, ∆ω <v2∆p , − − width of ε(p), the perturbative regime, where Eq. (11) Σ˜(ω,p)= iλ(∆ω v ∆p)3/2, ∆ω >v ∆p , 2 2 − − applies, is never realized. Such a situation apparently occurs in the quasi one-dimensional gapped spin system whereλ=πg2/2. Suchaweaklynonanalytictermcanbe 3 IPA-CuCl , where no trace of single-magnon excitations treatedperturbativelynearthecrossingpoint. Inpartic- 3 has been observed inside the continuum.6 ular,thequasiparticleweightisnotsuppressedasp p c → 4 and ε¯(p) does not change its slope at the crossing. The components. Such a canted antiferromagnetic structure magnon decay rate in three-dimensional case is opens an additional channel for spontaneous decays of low-energy excitations in the gapless branch, similar to γp 2λ[(v1 v2)∆p]3/2 . (18) the prediction made for ordered antiferromagnets in the ≃ − vicinity of the saturation field.22 Anexperimentallyrelevantquestionishowanexternal Intrinsic magnetic anisotropies, which become impor- magneticfieldaffects the decayprocesses. Ifthe Zeeman tantinmaterialswithspinsS 1,canalsoaffectthede- energyissmallerthanthespingapgµBH <∆,theroleof cay processes. Anisotropy cha≥nges differently dispersion anappliedfieldreducestosplittingatripletoflow-energy of triplet excitations and modifies the resonance condi- excitations. This can, in principle, lead to breaking the tion (2). The decay rate of spin-1 excitations depends, resonance condition (2). To study such a possibility one then, on a polarization, as was experimentally observed shouldtransformfromthevectorbasist tostateswith pα in bond-alternating quasi-one-dimensionalantiferromag- a definite spin projection on the field direction: net NTENP.23 The tensor structure of the decay ver- 1 tex (1) can be also modified due to the absence of spin- tp0 =tpz , tp± = (tpx itpy). (19) rotationalsymmetry. Dynamic properties of spin liquids ∓√2 ∓ with different types of anisotropies deserve further theo- The decay vertex (1) taken in the new basis has the fol- retical investigations. lowing form In conclusion, by studying analytic properties of the one-particleGreen’sfunctionnearthedecaythresholdwe iǫαβγt†pαt†qβtkγ = t†p+t†q−−t†p+t†q− tkz (20) have found that crossing of a single-particle branch into + t† t† t† t(cid:0)† t + t† t† (cid:1) t† t† t . two-magnon continuum is described solely by a growing p+ qz− pz q+ k+ pz q−− p− qz k− line-width of magnons only for three-dimensional quan- (cid:0) (cid:1) (cid:0) (cid:1) tum spin liquids. In two dimensions there is, in addi- In all decay channels a destroyed one-particle and a tion, strong suppression of a one-magnon peak near the created two-particle states experience the same energy shifts. 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