Dynamical Correlations of the Spin-1/2 Heisenberg XXZ Chain in a Staggered Field

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by  Igor Kuzmenko

Dynamical Correlations of the Spin-1 Heisenberg XXZ Chain in a Staggered Field
2
Igor Kuzmenko1,2 and Fabian H.L. Essler1
1The Rudolf Peierls Centre for Theoretical Physics,
University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK
2Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
(Dated: January 6, 2009)
Weconsidertheeasy-planeanisotropicspin-1 Heisenbergchainincombineduniformlongitudinal
2
andtransversestaggeredmagneticfields. Thelow-energylimitofhismodelisdescribedbythesine-
Gordon quantum field theory. Using methods of integrable quantumfield theory we determine the
variouscomponentsofthedynamicalstructurefactor. Todoso,wederiveexplicitexpressionsforall
9
matrixelementsofthelow-energy projections ofthespinoperatorsinvolvingatmost twoparticles.
0
Wediscuss applications of ourresults toexperiments on one-dimensional quantummagnets.
0
2
PACSnumbers: 75.10.Jm

n
a
J I. INTRODUCTION Heisenberg chain,

6
= J SxSx +SySy +δSzSz
] The field-induced gap problem in anisotropic quasi H j j+1 j j+1 j j+1 −
el one dimensional spin-21 Heisenberg antiferromagnets has Xj h i
- attracted much experimental1–15 and theoretical9,16–28 H Sz+h ( 1)jSx. (2)
r − j − j
t attention in recent years. Two scenarios have been j j
s X X
studied in particular. For isotropic exchange in-
.
at teraction a gap can be induced by the applica- In what follows we will consider the region −1 < δ ≤ 1
which corresponds to an “XY”-like exchange anisotropy.
m tion of a uniform magnetic field in presence of
Itisimportantforouranalysisthatthe staggeredfieldis
a staggered g-tensor and/or a Dzyaloshinskii-Moriya
-
d interaction16. This is the caseformaterialssuchasCop- transverse to the anisotropy whereas the magnetic field
n per Benzoate1–6, CDC [CuCl 2((CD ) SO)]7, Copper- is along the anisotropy axis. Only in this case does the
2 3 2
o Pyrimidine8–12 and Yb As 13,·14. Theoretical studies low-energylimit mapontoanintegrablemodel, the sine-
c have analyzed the excita4tion3spectrum16–18, the dynam- Gordon quantum field theory.
[
ical structure factor17,19, the specific heat20, the mag- The outline of this paper is as follows: in section II

2 netic susceptibility9,21 and the electron-spin resonance we construct the continuum limit of the model (2). In
v lineshape22. In the materials mentioned above appli- section III we derive a spectral representation of the dy-
8 cation of a uniform magnetic field H induces a stag- namical structure factor at low energies. In section IV
9
gered field perpendicular to H. It is the induced stag- we present the calculations for retardedtwo-point corre-
3
lation functions. In section V we present our results for
0 gered field that leads to a spectral gap. The staggered
. field is generated both by a staggered g-tensor29,30 and the components of the dynamical strucure factors. Sec-
9 tion VI summarizes our results. The technical aspects
a Dzyaloshinskii-Moriya(DM) interaction. The simplest
0
of our analysis are summarized in severalappendices: in
8 Hamiltoniandescribingsuchfield-inducedgapsystemsis
0 given by16 Appendix A we discuss how the parameters of the low-
: energy field theory can be determined from the Bethe
v
ansatz solution of the Heisenberg chain in a magnetic
Xi H=J Sj ·Sj+1−H Sjz+h (−1)jSjx, (1) field. Appendices B and C present results for the form
r Xj Xj Xj factors of the operators entering the calculation of the
a dynamical structure factor.
where h = γH. The constant γ is given in terms of the
staggered g-tensor29,30 and the DM interaction. In crit-
II. CONTINUUM LIMIT
ical systems with exchange anisotropy such as the spin-
1/2 Heisenberg XXZ chain a second mechanism for in-
Inthelimit h H,J thestaggeredfieldcanbetaken
ducing a gap by application of a uniform magnetic field | |≪
into accountas a perturbationto the low-energylimit of
exists. While application of a field perpendicular to the
the XXZ chain in a magnetic field. It is well-knownthat
easy plane leaves the system critical, applying a field
the low-energylimitofthe spin-1 HeisenbergXXZ chain
in the easy plane leads to the formation of a spectral 2
gap23–26,31,32. withXY-likeanisotropy|δ|<1isgivenbyafreebosonic
theory33–36
The purpose of the present work is to extend the the-
oretical analysis of the staggered field mechanism for v
= dx (∂ Φ)2+(∂ Θ)2 , (3)
generating a spectral gap to the case of the anisotropic Hh=0 16π x x
Z
(cid:2) (cid:3)
2
wherethe fieldΦ(x)anditsdualfieldΘ(x)arecompact- where
ified
iΦ
1 =exp +iaΘ . (19)
2π Oa 4β
Θ(x) Θ(x)+ , Φ Φ+8πβ. (4) n o
≡ β ≡
We are using normalizations such that
The commutation relation between Φ and Θ reads vτ +ix βa a2 2a2+8β12
1(τ,x) −1(0,0) = 0 .
Θ(x),Φ(x′) =8πiϑ (x x′), (5) hOa O−a i vτ ix v2τ2+x2
H − (cid:20) − (cid:21) (cid:20) (cid:21) (20)
h i
The coefficients a(H), c(H) and (H) have been deter-
where ϑH(x) is the Heaviside step function, equal to 0 mined numerically in Ref. [38]. TAhe staggered magnetic
forx<0,1forx>0and1/2forx=0. Theparameters
field perturbation can be bosonized using (11) – (18),
v, β and k (see below) in the low-energy theory can
F which leads to a sine-Gordon model
be calculated directly from the Bethe ansatz solution on
the XXZ chain37. How this is done is briefly reviewedin v
= dx (∂ Φ(x))2+(∂ Θ(x))2 +
Appendix A. Theresultsaswellasthe otherparameters H 16π x x
used are listed in Table I for the anisotropic parameter Z (cid:26) h i
δ =0.3. Inthecontinuumlimitthelatticespinoperators +µ(h,H)cos(βΘ(x)) , (21)
have the following expansions (cid:27)
whereµ(h,H)=hc(H). Wenotethataswehavechosen
3
Sα = eiQαax α(x)+... , (6) to bosonize in a finite magnetic field, the cutoff of the
j Sa theoryisH ratherthanJ. However,itisstraightforward
a=1
X to recover the zero field limit (where one bosonizes at
where x = ja and a is the lattice spacing. The H = 0 and the cutoff is J) in the expressions for the
0 0
wavenumbers Qα are structure factor we give below.
a
π π
Qx = Qx =Q= 2k , Qx = , (7)
1 − 3 a − F 2 a A. Elementary Excitations
0 0
Qy = Qx , (8)
a a
Qz = 0 , Qz = Qz =2k , (9) The sine-Gordonmodel is integrable and its spectrum
1 2 − 3 F and scattering matrix are known exactly39–44. In the
where the Fermi momentum is given by relevant range of the parameter β (0< β <1) the spec-
trumofelementaryexcitationsconsistsofasoliton–anti-
π
k = (1 2 Sz ). (10) soliton doublet and several soliton–anti-soliton bound
F 2a0 − h ji states called “breathers”. There are altogether [1/ξ]
breathers, where [x] denotes the integer part of x and
Here hSjzi is the magnetization per site. The continuum ξ = β2 . In order to distinguish the various single-
fields α aregivenintermsofthecanonicalbosonΦand 1−β2
Sa particlestatesweintroducelabelssands¯forsolitonsand
its dual field Θ as
anti-solitons respectively and b ,...,b for breathers.
1 [1/ξ]
1 Energy and momentum carried by the elementary exci-
x(x) = (H) 1(x)+ 1 (x) , (11)
S1 2A Oβ O−β tations are expressed in terms of the rapidity θ as
h i
x(x) = c(H)cos βΘ(x) , (12)
S2 vPǫ =∆ǫsinh(θ), Eǫ =∆ǫcosh(θ), (22)
†(cid:16) (cid:17)
x(x) = x(x) , (13) where ∆ = ∆ = ∆, ∆ ∆ = 2∆sin(πξk). The
S3 S1 s s¯ bk ≡ k 2
soliton gap as a function of parameters H and h is20,45
(cid:16) (cid:17)
1+ξ
S1yy((xx)) == c21(iHA)(Hsin)hOββ1Θ((xx))−,O−1β(x)i, ((1154)) ∆J = Ja20v√πΓΓ(1(+ξ22ξ))"Ja0c2(vH)πΓΓ((11++1ξξξ))Jh# 2 . (23)
S2 When δ 1 and the magnetization is small the leading
(cid:16)† (cid:17) ≈
y(x) = y(x) , (16) irrelevant perturbation to the Gaussian model needs to
S3 S1 be taken into account, leading to21
(cid:16) (cid:17)
S1z(x) = 8aπ0β ∂xΦ†(x), 1 (17) ∆J =(cid:18)Jh(cid:19)1+2ξ"B(cid:18)HJ (cid:19)21−2β2(cid:16)2−8β2(cid:17)14#−1+2ξ, (24)
S2z(x) = S3z(x) =−2ia(H)O01(x), (18) where B =0.422169.
(cid:16) (cid:17)
3
B. Scattering States
TABLE I: Amplitudes A, a and c, the dimensionless spin
velocity v/Ja0, the coupling β, and the field H as functions
It is useful to introduce creation and annihilation op- ofthemagnetizationmfortheanisotropicparameterδ=0.3.
erators A†ǫ(θ) and Aǫ(θ) for the elementary excitations. The amplitudes are determined in Ref. [38].
Hereǫ=s,s¯,b ,...,b . Thecreation/annihilationop-
1 [1/ξ]
erators fulfil the so-called Faddeev-Zamolodchikov (FZ)
m A a c v/Ja0 β H/J
algebra
0.02 0.3044 0.3953 0.5275 1.1804 0.386192 0.09093
A (θ )A (θ ) = Sa′b′(θ θ )A (θ )A (θ ); 0.04 0.3065 0.3913 0.5268 1.17114 0.385821 0.18186
a 1 b 2 ab 1− 2 b′ 2 a′ 1
0.06 0.3096 0.3867 0.5256 1.15828 0.385332 0.2598
A†a(θ1)A†b(θ2) = Saab′b′(θ1−θ2)A†b′(θ2)A†a′(θ1); 0.08 0.3130 0.3817 0.5240 1.13738 0.384573 0.35073
0.10 0.3173 0.3769 0.5219 1.11423 0.383768 0.42867
A†(θ )A (θ ) = Sb′a(θ θ )A (θ )A† (θ )
a 1 b 2 ba′ 1− 2 b′ 1 a′ 1 0.12 0.3226 0.3713 0.5194 1.08072 0.38265 0.5196
+2πδ δ(θ θ ). (25)
ab 1− 2 0.14 0.3284 0.3661 0.5164 1.04600 0.381535 0.59754
Here S(θ) is the scattering matrix of the sine- 0.16 0.3354 0.3610 0.5129 1.01244 0.380489 0.66249
Gordon model41–43. Multi-particle scattering states of 0.18 0.3433 0.3559 0.5088 0.966005 0.379084 0.74043
(anti)solitons and breathersare givenin terms ofthe FZ 0.20 0.3527 0.3508 0.5041 0.921509 0.377775 0.80538
creation operators as
0.22 0.3642 0.3460 0.4988 0.870927 0.376322 0.87033
ǫ ,θ =A† (θ )...A† (θ )0 . (26) 0.24 0.3773 0.3415 0.4929 0.813165 0.374702 0.93528
|{ n n}i ǫn n ǫ1 1 | i 0.26 0.3923 0.3371 0.4861 0.760734 0.37326 0.98724
Energy and momentum of these states are 0.28 0.4102 0.3329 0.4785 0.701482 0.371658 1.0392
n n 0.30 0.4321 0.3286 0.4699 0.651491 0.370326 1.07817
E{n} = Eǫi, P{n} = Pǫi. (27) 0.32 0.4596 0.3253 0.4602 0.575147 0.368318 1.13013
Xi=1 Xi=1 0.34 0.493 0.3222 0.4492 0.507976 0.366572 1.16910
The resolution of the identity in the normalization im- 0.36 0.5342 0.3193 0.4367 0.456492 0.365244 1.19508
plied by (25) reads 0.38 0.588 0.3166 0.4222 0.389204 0.363518 1.22431
0.40 0.664 0.3141 0.4053 0.326360 0.361913 1.24704
∞
dθ ...dθ
I = 1 n ǫ ,θ ǫ ,θ . (28) 0.42 0.769 0.3131 0.3851 0.259866 0.360218 1.26652
n!(2π)n |{ n n}ih{ n n}|
nX=0{Xǫj}Z 0.44 0.934 0.3125 0.3602 0.186561 0.358349 1.28276
0.46 1.214 0.3127 0.3279 0.122936 0.356722 1.29251
0.48 1.89 0.3142 0.2796 0.0448071 0.354713 1.299
C. Discrete Symmetries
The Hamiltonian is invariant with respect to charge III. DYNAMICAL STRUCTURE FACTOR
conjugation
Thecentralobjectofourstudyistheinelasticneutron
CΘC−1 = Θ , CΦC−1 = Φ . (29) scattering intensity, which is proportional to46
− −
TheactionofthechargeconjugationoperatorC onphys- kαkα′
ical states follows from I(ω,k) ∝ δαα′ − k2 !Sαα′(ω,k). (31)
α,α′
X
C 0 = 0 ,
| i | i Here α,α′ =x,y,z, k denotes the component of k along
CA†s(θ)C−1 =A†s¯(θ), (30) the chain direction, and the dynamical structure factor
CB†(θ)C−1 =( 1)kB†(θ). on a chain with L sites is defined as
k − k
∞
We see that even breathers are invariant under charge Sαα′(ω,k)= 1 dteiωt−ik(l−l′) 0Sα(t)Sα′ 0 .
conjugation,while oddbreatherschangesign. The topo- L 2π h | l l′ | i
logical charge Xl,l′−Z∞
(32)
∞ Substituting the low-energy expressions (6) into (32) we
β
= dx ∂ Θ(x), obtain
x
Q 2π
−Z∞ Sαα′(ω,k) = 3 1 ∞ dteiωt−i(k−Qαa)l+i(k+Qαb′)l′
is a conserved quantity. We will use the conventions in L 2π
which soliton/antisoliton and breathers have topological aX,b=1 Xl,l′−Z∞
charge 1 and zero respectively. 0 α(t,x) α′(0,y)0 , (33)
∓ × h |Sa Sb | i
4
where x=la , y =l′a and α(x) are the leading terms terms of scattering states of solitons, anti-solitons and
0 0 Sa
in the low energy limits (6) of the lattice spin operators. breathers. Insertingacompletesetofstates(28)between
Usingthattheexpectationvalueisaslowlyvaryingfunc- the operators in (33) and using
tion of x y we see that only terms with
−
0 α(t,x) ǫ ,θ =
k Q Q (34) h |Sa |{ n n}i
≈ a ≈− b =e−iE{n}t+iP{n}xh0|Saα(0,0)|{ǫn,θn}i,
contributeto(33)47. Thedynamicalstructurefactorcan
be expressed by means of a Lehmann representation in we arrive at
∞ 3
2π dθ ...dθ
Sαα′(ω,Qα+q) δ 1 n 0 α ǫ ,θ ǫ ,θ α′ 0 δ(q P )δ(ω E ). (35)
a ≃ a0 nX=1{Xǫn}Xb=1 Qαa,−Qαb′ Z n!(2π)n h |Sa|{ n n}ih{ n n}|Sb | i − {n} − {n}
Sx Sx Sx Sy Sy Sy Sz Sz Sz 2. Sxx, Syy near the point k = π ;
1 2 3 1 2 3 1 2 3 a0
Q −1 0 1 −1 0 1 0 −1 1
3. Szz in the vicinity of the point k =0;
C + − −
TABLE II: Topological charge Q and eigenvalue (where ap- 4. Szz near k = 2k .
F
±
plicable) under charge conjugation C of the continuum spin
operators. In the following we determine these in the “two-particle
approximation”, i.e., keeping only terms with n 2 in
≤
the spectral representation (35). In order to do so we
Here q is assumed to be sufficiently small (q ∆
k , π ). Due to energy-momentum conservation∼onvly≪a makeuseofthe exactformofthe matrixelements enter-
F a0 ing the Lehmann representation, which follow from the
finite number of intermediate states contributes to the form-factor bootstrap approach50,51.
correlator (35). Moreover, at low energies contributions
We note that as a consequence of charge conjugation
of intermediate states with large numbers of particles to
symmetry the components of the structure factor in the
the correlator(35)aregenerallysmall48,49. We therefore
vicinities of k =Q and k = Q are the same.
restrict our following analysis to one and two particle a − a
IV. CALCULATION OF CORRELATION
contributions. Many matrix elements in (35) are in fact
FUNCTIONS: KINEMATICS
zero as can be established by using charge conjugation
symmetry and topological charge conservation. The rel-
evant properties of the continuum spin operators α are The formalism we employ to calculate the dynamical
Ss structurefactorcanbeusedquitegenerallytodetermine
summarized in Table II. Using these properties we fur-
(realandimaginarypartsof) two-pointcorrelationfunc-
thermoreconcludethatatlowenergiesthenon-vanishing
tions. The retarded two-point function of two bosonic
components of the dynamical structure factor are
operators A and B has a spectral representations of the
1. Sxx, Syy in the vicinity of the points k = Q; form
±
2πv ∞ dθ ...dθ δ(vq P )
GAB(ω,q) = 1 n 0A ǫ ,θ ǫ ,θ B 0 − {n}
a n!(2π)n h | |{ n n}ih{ n n}| | iω E +iη
0 nX=1{Xǫn}Z (cid:26) − {n}
δ(vq+P )
{n}
0B ǫ ,θ ǫ ,θ A0 . (36)
n n n n
− h | |{ }ih{ }| | iω+E +iη
{n} (cid:27)
Hereη isapositiveinfinitesimal, ǫ ,θ aren-particle (22) respectively. The leading contribution to the spec-
n n
|{ }i
scattering states of solitons, antisolitons and breathers tral sum in (36) is due to intermediate states with one
(26) with energies and momenta are given by (27) and and two particles. Using momentum conservation it is
5
possible to simplify the expressions for these contribu- butions to GAB
tions as we discuss next.
A. One-particle kinematics
Resolving the momentum conservation delta function
leads to the following result for the one-particle contri-
v δ(vq ∆ sinhθ) δ(vq+∆ sinhθ)
GAB(ω,q) = dθ 0Aθ θ B 0 − a 0B θ θ A0 a
1p a h | | ia ah | | iω ∆ coshθ+iη −h | | ia ah | | iω+∆ coshθ+iη
0 a Z (cid:20) − a a (cid:21)
X
v 0Aθa θa B 0 0B θa θa A0
= h | | 0ia ah 0| | i h | | 0ia ah 0| | i , (37)
a ε (q) ω ε (q)+iη − ω+ε (q)+iη
a 0 a (cid:20) − a a (cid:21)
X
where a runs over all single-particle labels (i.e. soliton, B. Two-particle kinematics
antisoliton and breathers) and
Astwo-particleformfactorsofscalaroperatorsdepend
ε (q) = ∆2 +v2q2 , (38)
a a onlyontherapiditydifference,itisusefultochangevari-
vq
θa = parcsinh . (39) ables to θ± = (θ1 θ2)/2. Resolving the momentum
0 ∆a conservation delta fu±nction then gives
(cid:16) (cid:17)
GAB(ω,q) = v dθ1dθ2 0Aθ ,θ θ ,θ B 0 δ(vq− 2j=1∆ajsinhθj)
2p a0 aX1,a2Z 2(2π)2(cid:20)h | | 2 1ia2a1 a1a2h 1 2| | iω− 2j=P1∆ajcoshθj +iη
0B θ ,θ θ ,θ A0 δ(vPq+ 2j=1∆ajsinhθj)
−h | | 2 1ia2a1 a1a2h 1 2| | iω+ 2j=P1∆ajcoshθj +iη(cid:21)
= v dθ− h0|A|θ0ab−θ−,θ0ab+θ−iba abhθ0ab+θ−,θ0aPb−θ−|B|0i
a 2π ε (q,θ ) ω ε (q,θ )+iη
0 a,b Z (cid:20) ab − − ab −
X
0B θab θ(cid:0) ,θab+θ θa(cid:1)b+θ ,θab θ A0
h | | 0 − − 0 −iba abh 0 − 0 − −| | i , (40)
− ε (q,θ ) ω+ε (q,θ )+iη
ab − ab − (cid:21)
(cid:0) (cid:1)
where where
1
ε (q,θ)= v2q2+∆2+∆2+2∆ ∆ cosh(2θ) 2,
ab a b a b
h i
vq+ε (q,θ ) s2 = ω2 v2q2 , (42)
θab =ln ab − . −
0 ∆ exp(θ )+∆ exp( θ ) s2 ∆2 ∆2
(cid:20) a − b − − (cid:21) θab(s) = arccosh − a− b . (43)
2∆ ∆
The imaginarypartofGAB(ω,q)canbe simplified using (cid:20) a b (cid:21)
2p
1 1
Im
−π ε (q,θ ) [ω ε (q,θ )+iη]
ab − ab −
−
δ(θ θab(s))+δ(θ + θab(s)) Carryingouttheθ− integralusingthedeltafunctionswe
= −− 2 − 2 , (41) obtain
s2 (∆ +∆ )2 s2 (∆ ∆ )2
a b a b
− − −
p p
6
1 v 0Aθσ (ω,q),θ−σ(ω,q θ−σ(ω,q),θσ (ω,q)B 0
ImGAB(ω >0,q) = h | | ba ab iba abh ab ba | | iϑ s ∆ ∆ ,
−π 2p 2πa0 Xa,b σX=± s2−(∆a+∆b)2 s2−(∆a−∆b)2 H(cid:0) − a− b(cid:1)
p p (44)
where ϑ (x) is the Heaviside function and and five breathers with gaps
H
1 ∆ = 0.54098∆, ∆ =1.04162∆, ∆ =1.46461∆,
θ±(ω,q)=arcsinh vq(s2+∆2 ∆2)+ 1 2 3
ab (cid:20)2∆as2 a− b ∆4 = 1.77841∆, ∆5 =1.95962∆. (47)
(cid:16)
ω (s2 (∆ ∆ )2)(s2 (∆ +∆ )2) ,
a b a b Inordertobroadendeltafunctionsappearinginonepar-
± − − −
p (cid:17)(cid:21)(45) ticle contributions, we introduce a small imaginary part
in ω, equal to η =0.01∆.
θσ (ω,q) θ−σ(ω,q)=σθ (s).
ab − ba ab A. Sxx(ω,k)
The two terms in (44)arise from the two delta functions
in (41). Using the results summarized in this section In the continuum limit Sxx(ω,k) is non-vanishing in
we can determine the one and two particle contributions the vicinity of the points k = Q and π . We will con-
to both real and imaginary parts of two point functions. ± a0
sider both cases in turn. As we have noted before, the
Thetwoparticlecontributionstotherealpartinvolveone
response at k = Q is indentical as a result of charge
(principal part) integration, which is readily performed ±
conjugation symmetry, so that it is sufficient to consider
numerically. In order to determine the dynamical struc-
k Q.
ture factor we only require the imaginary part of several ≈−
two point functions.
1. Momenta k≈−Q=−2πhSzi
a0 j
V. RESULTS FOR THE DYNAMICAL
STRUCTURE FACTOR In the continuum limit Sxx(ω, Q+q) with q Q
is given by the two-point function−of x with x≪(11).
S3 S1
Below we present results for the dynamical structure ThisisbecausevQisalargeenergyscaleproportionalto
factorSαβ(ω,Qα+q)(35)in the regime 1<δ 1 and the cutoff in the theory. Using Table II we find that the
a − ≤
for magnetic fields H < H = J(1+δ). We note that if following intermediate states with at most two particles
c
H H or δ 1 the cut-off in the field theory is very contribute
c
≈ ≈−
small, which limits the utility of our approach. For the
sakeofclarityweuseaparticularsetofparametersinall 1. Single-soliton states.
plots
2. Two particle states containing one soliton and one
h breather.
γ = =0.01191, δ =0.3, H =0.2598J. (46)
H
ThecorrespondingmatrixelementsarecalculatedinAp-
These correspond to a magnetization per site of Sz = pendixB. UsingtheresultsofsectionIVtocarryoutthe
h i
0.06 (see Table I) and ξ =0.174371. The spectrum con- rapidityintegralswearriveatthefollowingexpressionfor
sists of soliton and antisoliton with gap ∆ 0.04897J Sxx(ω, Q+q) within the two-particle approximation
≈ −
Sxx(ω >0, Q+q) 2vA˜2v2q2δ(s2 ∆2)+ vA˜2 [1/ξ] Nβ 2 Fsmbkin(θsbk(s)) 2ϑH(s−∆−∆k)
− ≈ a0 ∆2 − 4πa0 Xk=1(cid:16) sbk(cid:17) (cid:12)(cid:12)(s2−(∆−∆k(cid:12)(cid:12))2)(s2−(∆+∆k)2) ×
× Ksβbk(σθsbk(s))eθsσbk(ω,q)+Ks−pbβk(σθsbk(s))e−θsσbk(ω,q) 2. (48)
σX=±h i
Here s is the Mandelstam variable (42), the overall nor- where Z (β) is given by equation (B2), the minimal
1
malization is form factors Fmin(θ) by equation (B6), the pole func-
sbk
1/2
˜= (H) Z (β) , (49)
1
A A
(cid:18) (cid:19)
7
tionsK± (θ)byequation(B22)fork evenand(B23)for where ν = 2(β + 1 )2 > 1. For large ω this in-
sbk 4β
k odd, the normalization factor Nβ by equation (B20) creases as ω2ν, while is goes to zero in a power-law
sbk
and the functions θ (s) and θσ (ω,q) are presented in fashion for ω vq. In presence of a staggered field,
equations (43) ands(b4k5), res pectsibvkely. the dynamical→structure factor (48) has divergence for
ω (∆+∆ )2+v2q2 (k = 1,2,...,[1/ξ]), while the
k
a ∆2S(cid:0)(cid:0) v lar→gefrequencybehavioristhesameaswithoutthestag-
0 0.007 geredpfield.
0.006
7
0.005
6 0.004
0.003
5 0.002
0.001
4
1 2 3 4 5
3
2 2. Vicinity of antiferromagnetic wave number: k≈π/a0
1
ω ∆
1 2 3 4 5 In the continuum limit Sxx(ω,aπ0 +q) with qa0 ≪π is
givenbythetwo-pointfunctionofthe chargeneutralop-
FIG. 1: One and two-particle contributions to Sxx(ω,−Q+ erator x (13). Using Table II and (30) we find that the
∆/v) as a function of ω for δ = 0.3 and H = 0.2598J. The followinSg2 intermediate states with at most two particles
delta-function peak (pink) has been broadened to make it contribute to the two-point function of x
visible. S2
We note thatSxx(ω, Q+q)vanisheswhenq 0. In
− → 1. Single breather states even under charge conjuga-
Figure1wethereforeplotSxx(ω, Q+∆/v)asafunction
− tion, i.e. B† (θ)0 .
ofω. Inordertobroadendelta-functioncontributionswe 2n | i
introduce a small imaginary part in ω. Two features are
clearly visible: there is a coherent peak corresponding
tothecontributionofsingle-solitonexcitationsatenergy
2. Two particle states containing one soliton and one
at∆√2. Athigherenergiesbreather-solitoncontinuaap-
antisoliton.
pear. Theircontributionsgrowwithincreasingωbecause
3. Twoparticlestatescontainingtwoevenortwoodd
x is an irrelevant operator. It is instructive to compare
S3 breathers.
ourresulttothegaplessspin-1/2HeisenbergXXZchain,
see e.g. Ref.[52]. There one has
Using the results of section IV, we obtain the following
ω2+v2q2
Sxx(ω >0, Q+q) , (50) expression in the two-particle approximation
− ∝ (ω2 v2q2)1−ν
−
π vc2(H) [1/2ξ] 2 Fcos(βΘ)(θ (s)) 2ϑ (s 2∆)
Sxx ω >0, +q 2π Fβ δ(s2 ∆2 )+ ss¯ ss¯ H − +
(cid:18) a0 (cid:19) ≈ πa0 ( kX=1 (cid:12) b2k(cid:12) − 2k (cid:12)(cid:12) s√s2−(cid:12)(cid:12)4∆2
(cid:12) (cid:12)
+ [1/ξ] δeven Fβ(cid:12) (θ(cid:12) (s)) 2 ϑH(s−∆k−∆k′) . (51)
k,Xk′=1 k+k′(cid:12) bkbk′ bkbk′ (cid:12) (s2−(∆k−∆k′)2)(s2−(∆k+∆k′)2))
(cid:12) (cid:12) p
Here the single-breather form factors Fβ are given is a scalar operator Sxx(ω, π +q) depends only on the
bk a0
by equation (C14), the soliton-antisoliton form factor Mandelstam variable s (42) rather than on ω and q sep-
Fscs¯os(βΘ)(θ)by(C10)andthebreather-breatherformfac- arately. The first peak in Sxx(ω,aπ0) is due to the b2
torsFbβkbk′(θ) by(C24)respectively. The functionθǫǫ′(s) asinsgecleo-nbdresaitnhgelre-ebxrceiatathtieornc(obnltureibluinteio)n.,Adtuωe=to∆b4.thAebroevies
is given by (43) and 4
ω = 2∆ a strong b b two-breather continuum occurs
1 1 1
1, if k is even, (pink line). Around ω = 2∆ contributions from soliton-
δeven = (52)
k ( 0, overwise. antisoliton and b1b3 and b2b2 two-breather continua are
visible. We note that the thresholds of b b , ss¯and b b
1 3 2 2
In Fig.2 we plot the dynamical structure factor (51) continua all occur around 2∆ is a peculiarity of the pa-
as a function of frequency. We note that because x
S2
8

a ∆2S(cid:1)(cid:1) v vanishing in the vicinity of the points k = Q and π .
03 0.2 We will consider both cases in turn. ± a0
0.15
2.5
0.1
2
0.05
1.5 1. Momenta k≈−Q=−2aπ0hSjzi
1 2 3 4 5
1
0.5 In the continuum limit Syy(ω, Q+q) with q Q
is given by the two-point function−of y with y ≪(14).
ω ∆ S3 S1
1 2 3 4 5 Using Table II we find that the following intermediate
states with at most two particles contribute to the two-
FIG. 2: One and two-particle contributions to Sxx(ω, π ) as
a0 point function
a function of ω for δ=0.3 and H =0.2598J. Delta-function
peaks (blue) havebeen broadened to makethem visible.
1. Single-soliton states.
rameters we have chosen in the plots. 2. Two particle states containing one soliton and one
breather.
B. Syy(ω,k) The corresponding matrix elements are calculated in
Appendix B. Carrying out the rapidity integrals, see
Next we turn to the yy-component of the dynamical section IV, we arrive at the following expression for
structurefactor. InthecontinuumlimitSyy(ω,k)isnon- Syy(ω, Q+q) within the two-particle approximation
−
Syy(ω >0, Q+q) 2vA˜2v2q2+∆2δ(s2 ∆2)+ vA2 [1/ξ] Nβ 2 Fsmbkin(θsbk(s)) 2ϑH(s−∆−∆k)
− ≈ a0 ∆2 − 4πea0 Xk=1(cid:16) sbk(cid:17) (cid:12)(cid:12)(s2−(∆−∆k(cid:12)(cid:12))2)(s2−(∆+∆k)2) ×
× Ksβbk(σθsbk(s))eθsσbk(ω,q)−Ksβbk(−σθsbk(s)p)e−θsσbk(ω,q) 2. (53)
σX=±h i
Here the overall normalization is given by equation make them visible. We see that there is a coherent peak
A
(49), the minimal form factor Fmin(θ) by (B6), the pole corresponding to the contribution of single-soliton exci-
function Kβ (θ) by (B22) for skbekeven and (B23) for k tations at energy at ∆√2. At higher energies breather-
odd,thefunscbktionsθ (s)andθσ (ω,q)by(43)and(45), soliton continua appear. Their contributions grow with
sbk sbk increasing ω because y is an irrelevant operator.
respectively. S3

a ∆2S(cid:2)(cid:2) v
0 0.005
0.004 2. Vicinity of antiferromagnetic wave number: k≈π/a0
12
0.003
10 0.002 In the continuum limit Syy(ω, π +q) with qa π is
8 0.001 givenbythetwo-pointfunctionoaf0the chargeneu0t≪ralop-
6 1 2 3 4 5 erator y (15). Using Table II and (30) we find that the
S2
4 following intermediate states with at most two particles
2 contribute to the two-point function of y
S2
ω ∆
1 2 3 4 5
1. Single breather states odd under charge conjuga-
FIG. 3: One and two-particle contributions to Syy(ω,−Q+ tion, i.e. B2†n+1(θ)|0i.
∆/v) as a function of ω for δ = 0.3 and H = 0.2598J. The
delta-function peak (pink) has been broadened to make it 2. Two particle states containing one soliton and one
visible. Inset: thesoliton-breathertwo-particlecontributions. antisoliton.
WeplotSyy(ω, Q+∆/v)asafunctionofω inFigure 3. Two particle states containing one even and one
−
3. Delta-function contributions have been broadened to odd breather.
9
Using the results of section IV, we obtain the following
expression in the two-particle approximation
π vc2(H) [1/ξ] 2 Fsin(βΘ)(θ (s)) 2ϑ (s 2∆)
Syy ω >0, +q 2π δodd Fβ δ(s2 ∆2)+ ss¯ ss¯ H − +
(cid:18) a0 (cid:19) ≈ πa0 ( Xk=1 k (cid:12) bk(cid:12) − k (cid:12)(cid:12) s√s2−(cid:12)(cid:12)4∆2
(cid:12) (cid:12)
[1/ξ] (cid:12) (cid:12) ϑ (s ∆ ∆ )
+ δodd Fβ (θ (s)) 2 H − k− k′ . (54)
k,Xk′=1 k+k′(cid:12) bkbk′ bkbk′ (cid:12) (s2−(∆k−∆k′)2)(s2−(∆k+∆k′)2))
(cid:12) (cid:12) p
Here the single-breather form factors Fβ are given by breather states are small in comparison. Similarly, the
bk
(C14), the soliton antisoliton form factor Fsin(βΘ)(θ) by two-particleb1b2,ss¯andb1b4 continuashownintheinset
ss¯
of Fig.4 are negligible.
(C11), the two-breatherform factors Fβ (θ) by (C24),
bkbk′
the function θ (s) by (43) and
ab
C. Longitudinal structure factor Szz(ω,k)
1, if k is odd,
δodd = (55)
k ( 0, overwise. Wenowconsiderthezz-componentofdynamicalstruc-
ture factor. In the continual limit Szz(ω,k) is non-
vanishing in the vicinity of the points k = 0 and 2k .
a ∆2S(cid:3)(cid:3) v ± F
0 We will consider both cases in turn.
17.5 0.05 1. Vicinity of ferromagnetic wave number: k≈0
0.04
15
0.03
12.5 0.02 InthecontinuumlimitSzz(ω,q)withqa0 π isgiven
10 ≪
0.01 by the two-point function of the charge neutral opera-
7.55 1 2 3 4 5 tor S1z (17). Using Table II and (30) we find that the
following intermediate states with at most two particles
2.5 contribute to the two-point function of z
ω ∆ S1
1 2 3 4 5
1. Single breather states odd under charge conjuga-
FIG. 4: One and two-particle contributions to Syy(ω, π) tion, i.e. B† (θ)0 .
as a function of ω for δ = 0.3 and H = 0.2598J. Delat0a- 2n+1 | i
function peaks (blue) have been broadened to make them 2. Two particle states containing one soliton and one
visible. Insert: the soliton-antisoliton and breather-breather antisoliton.
two-particle contributions.
3. Two particle states containing one even and one
We plot Syy(ω, π ) as a function of ω in Fig.4. We odd breather.
a0
see that it is dominated by the contribution of the first
breather b (the corresponding delta function has been Using the results of section IV, we obtain the following
1
broadened). The contributions from b and b single- expression in the two-particle approximation
3 5
2a b2ω2 [1/ξ] 2 a b2ω2 FΘ(θ (s)) 2ϑ (s 2∆)
Szz(ω >0,q) 0 δodd FΘ δ(s2 ∆2)+ 0 ss¯ ss¯ H − +
≈ v k bk − k v s√s2 4∆2
(cid:12) (cid:12)
e Xk=1 (cid:12) (cid:12) e (cid:12) (cid:12)−
(cid:12) (cid:12)
+a0b2ω2 [1/ξ] δo(cid:12)dd F(cid:12) Θ (θ (s)) 2 ϑH(s−∆k−∆k′) , (56)
v k+k′ bkbk′ bkbk′ (s2 (∆ ∆ )2)(s2 (∆ +∆ )2)
e k,Xk′=1 (cid:12) (cid:12) − k− k′ − k k′
(cid:12) (cid:12) p
where the single-breather form factor FΘ is given by by equation (C12), the breather-breather form factor
bk
equation(C15),thesoliton-antisolitonformfactorFΘ(θ)
ss¯
10
FΘ (θ) by equation (C26), θ (s) is given by equation pends on the Mandelstam variables (42) rather than on
bkbk′ ab
(43), s is the Mandelstam variable (42), δodd is given in ωandqseparately. Inordertobroadenthedeltafunction
k
(55) and the overallnormalization is contributions we introduce a small imaginary part in ω.
The dominant peak in Szz(ω,q) is due to a b breather
1
1 contribution. Thecontributionsduetob andb breather
b= . (57) 3 5
4πβ states are much smaller. The soliton-antisoliton and
breather-breather contributions to Szz(ω,q) are barely
e visible in the figure.
vS(cid:4)(cid:4) a
0
5 0.0025
0.002
4 0.0015 2. Momenta k≈−2kF
0.001
3
0.0005 InthecontinuumlimitSzz(ω, 2k +q)withqa π
F 0
2 1 2 3 4 5 is given by the two-point functi−on of z with z (≪18).
S3 S2
1 Using Table II and (30) we find that the following inter-
mediate states with at most two particles contribute to
ω ∆
the two-point function of
1 2 3 4 5
FIG. 5: One and two particle contribution to Szz(ω,0) (56) 1. Single soliton state.
as a function of ω for δ = 0.3 and H = 0.2598J. Delta-
function peaks (blue) have been broadened to make them
visible. Insert: the soliton-antisoliton and breather-breather 2. Two particle states containing one soliton and one
two-particle contributions. breather.
Thedynamicalstructurefactor(56)isshowninFigure Using the results of section IV, we obtain the following
5. Note that since z is a scalar operator, Szz(ω,q) de- expression in the two-particle approximation
S1
2
Szz(ω >0, 2k +q) va˜2δ(s2 ∆2)+ va˜2 [1/ξ] N0 2 K0 (θ (s)) 2 Fsmbkin(θsbk(s)) ϑH(s−∆−∆k) .
− F ≈ a0 − 2πa0 kX=1(cid:16) sbk(cid:17) (cid:16) sbk sbk (cid:17) (cid:12)(cid:12)(cid:12)(s2−(∆−∆k(cid:12)(cid:12)(cid:12))2)(s2−(∆+∆k)2)
p (58)
Here the minimal form factor Fmin(θ) is given by (B6), perpendiculartoit. Thequalitativefeaturesofthemodel
sbk
the pole function K0 (θ) by (B22) for k even and (B23) such as a field induced gap and the formation of bound
for k odd, the functsibokn θ (s) by (43), the overall nor- statesaresimilartothecaseofisotropicexchange,which
malization is sbk has been previously studied in detail9,16–18,20–22,27,28.
The main effect of a strong exchange anisotropy is to
Z(0) generate further bound states and increase the bind-
a˜=a(H) . (59)
2 ing energy. We have analyzed these effects on the dy-
r
namic response and determined for the first time all
Thedynamicalstructurefactor(58)isshowninFigure
two-particle contributions, in particular those contain-
6. Here we chose q = 0. The strong low-energy peak
ing one soliton and one breather. The results obtained
in Szz(ω, 2k ) is due to a one-soliton state. Soliton-
− F here can be used to study a quasi one dimensional array
breather continua appear at higher energies.
of anisotropic Heisenberg chains in a uniform magnetic
field by combining a mean-field approach with an RPA-
likeapproximation53–55. Thisisofinterestinviewofneu-
VI. SUMMARY AND CONCLUSIONS
tron scattering experiments on the quasi-1D anisotropic
Heisenberg magnet Cs CoCl 32.
In this work we have determined the low energy dy- 2 4
namicalspinresponseoftheanisotropicspin-1/2Heisen-
berg XXZ chain in the presence of both uniform and
stagered magnetic fields. The uniform field was taken
to be along the anisotropy axis and the staggered field