Probing Anomalous Longitudinal Fluctuations of the Interacting Bose Gas via Bose-Einstein Condensation of Magnons

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by  Andreas Kreisel

Probing Anomalous Longitudinal Fluctuations of the Interacting Bose Gas via
Bose-Einstein Condensation of Magnons
Andreas Kreisel, Nils Hasselmann, and Peter Kopietz
Institut fu¨r Theoretische Physik, Universit¨at Frankfurt,
Max-von-Laue Strasse 1, 60438 Frankfurt, Germany
7
(Dated: October26, 2006)
0
0 Theemergence of afinitestaggered magnetization in quantumHeisenberg antiferromagnets sub-
2 ject to a uniform magnetic field can be viewed as Bose-Einstein condensation of magnons. Using
n non-perturbativeresults for the infrared behavior of the interacting Bose gas, we present exact re-
a sults for the staggered spin-spin correlation functions of quantum antiferromagnets in a magnetic
J field at zero temperature. In particular, we show that in dimensions 1 < D ≤ 3 the longitudinal
9 dynamicstructurefactorSk(q,ω)describingstaggeredspinfluctuationsinthedirectionofthestag-
geredmagnetizationexhibitsacriticalcontinuumwhoseweightcanbecontrolledexperimentallyby

] varyingthemagnetic field.
h
c PACSnumbers: 75.10.Jm,05.30.Jp,03.75.Kk,75.40.Gb
e
m
- Because magnons in ordered Heisenberg magnets are vergences originating from the gapless dispersion of the
t
a bosonic quasi-particles, it is natural to expect that un- Goldstone mode, giving rise to non-analytic behavior of
t dercertainconditionsthesesystemscanbeusedtostudy some boson correlation functions [7, 8, 9]. Yet, in all
s
. general properties of interacting Bose gases, such as the physical observables of the Bose gas these divergences
t
a phenomenon of Bose-Einstein condensation (BEC). In cancel, so that they seem to be little more than a math-
m
fact, the renewed interest in BEC in recent years has ematical curiosity. Here, we point out that, in contrast
- also led to considerable activity in the field of magnon to the Bose gas, in MBEC the anomalous behavior aris-
d
BEC [1, 2], which we abbreviate MBEC below. ing from the divergencescan be directly observedexper-
n
o Experimentally, MBEC has been observed in two dif- imentally. We shall exploitrecent renormalizationgroup
c results [7] to obtain the infrared behavior of the stag-
ferent classes of systems. The first are spin-singlet sys-
[
tems [1] such as TlCuCl or Haldane gap spin chains, gered spin-spin correlation functions of QAFs in a uni-
3
3 where MBEC corresponds to the collapse of the singlet- form magnetic field at zero temperature. In particular,
v we shall show that in dimensions 1 < D 3 the lon-
tripletgapatacriticalmagneticfieldandthe emergence
5 gitudinal part S (q,ω) of the dynamic stru≤cture factor
7 oflong-rangemagneticorderforlargerfields. Thesecond k
(corresponding to spin-fluctuations parallel to the stag-
5 class of systems, which we shall consider in this work,
gered magnetization) exhibits a critical continuum for
0 are easy-plane quantum antiferromagnets (QAFs) such
1 as Cs CuCl subject to a magnetic field h perpendicular small wave-vectors q and frequencies ω, which can be
2 4 | |
6 measured via neutron scattering.
to the easy plane [2]. If h exceeds some critical field h ,
0 c WestartfromtheHamiltonianofthespin-S QAFona
the ground state is a saturated ferromagnet. For h<h
/ c
at the U(1)-symmetry of the Hamiltonian is spontaneously D-dimensionalhypercubiclatticewithN sitesandlattice
spacing a in a uniform magnetic field in z-direction,
m broken and antiferromagnetic long-range order emerges.
- At zero temperature, the boson density vanishes at the 1
d phase transition so that many-body interactions are not H = 2 JijSi·Sj −h Siz , (1)
n relevant at the zero temperature quantum critical point. Xij Xi
o
c However,thefinitetemperaturetransitionbelongstothe where the sums are over all sites of the lattice, and Si
: same universality class as BEC in the interacting Bose are spin operators with S2 = S(S + 1). We assume
v i
i gas. One advantage of studying BEC in magnetic sys- nearest neighbor coupling, i.e., Jij = J > 0 if i and j
X tems is that the magnetic analogon hc−h of the chemi- are nearest neighbors, and Jij = 0 otherwise. For suf-
r calpotentialcaneasilybecontrolledexperimentally. An- ficiently large h, the ground state of Eq. (1) is a sat-
a
other advantage, which has attracted little attention to urated ferromagnet with magnetization parallel to the
date, is that in MBEC the phase of the condensate has field. Toobtainthelow-lyingmagnonexcitations,weex-
a direct physical interpretation as the orientation of the press the spin operators in terms of canonical boson op-
staggered magnetization within the easy plane. erators b using the Holstein-Primakoff transformation,
i
The concept of MBEC has been formulated theoret- Si− = √2Sb†i[1 − ni/(2S)]1/2 = (Si+)†, Siz = S − ni,
ically many years ago [3, 4, 5, 6], where the focus was where n = b†b . Expanding the square roots and ne-
i i i
mainly on the nature of the quantum phase transition. glecting terms involving six and more boson operators
However, even away from the critical point, perturba- we obtain H E + H + H . The constant part is
0 2 4
tiontheory for the interacting Bosegasis plaguedby di- E = N[J˜S2≈/2 hS], where J˜ = J˜ with J˜ =
0 0 0 k=0 k
−
2
jeik·rjJij. The quadratic and the quartic contribu- y h
tions to the Hamiltonian are
P
x
S
H = J [n +n b†b b†b ]+h n ,(2)
H24 = −41X2ijXiJjij[i2jninij −bj†i−nibij −j −b†jnjibii]. Xi i (3) 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Anticipating antiferromagnetic symmetry breaking, it
of a QAF in a uniform magnetic field for h < h . The local
c
is convenient to define the Fourier transform in the
moments hS i are represented by dashed arrows. We choose
i
reduced (antiferromagnetic) Brillouin zone with two the coordinate system such that the magnetic field h points
branches of magnon operators bkσ labelled by momenta inthez-directionandthestaggeredmagnetizationMs points
k in the reduced zone and σ = , in the y-direction. The sublattices are labelled A and B.
±
b =N−1/2 e−ik·rib +σ e−ik·rib . (4)
kσ i i
valid also for larger canting angles. Of course, if θ is
i∈A i∈B
(cid:2)X X (cid:3) not small it is better to quantize the spin operators in
Herei A/B denotesasummationoverthesitesofonly a suitably defined tilted basis [10, 11]. For simplicity,
∈
one of the sublattices A or B. The quadratic magnon we shall focus here on the regime θ 1 where our ap-
Hamiltonianisthendiagonal,H2 = kσ(ǫkσ−µ)b†kσbkσ, proach based on the Holstein-Primak≪off transformation
with magnon dispersion ǫ = (J˜ + σJ˜ )S and µ = withquantizationaxisalongthedirectionofthemagnetic
kσ 0P k
h h, where h =2J˜S. For small wave-vectorsǫ fieldis thus quantitativelyaccuratefor largeS. Alterna-
c c 0 k,−
− ≈
k2/(2m), with effective mass m = (2JSa2)−1, while the tively, we could represent the spin operators in terms of
symmetric mode ǫ is gapped, ǫ = h , so that projectedbosons[4, 5,12]whichaccountforall1/S cor-
k,+ k=0,+ c
ǫ µ = h. Note that the gapless mode σ = 1 rections at small θ. However, in the spirit of Ref. [13],
k=0,+
− −
describes staggered spin fluctuations. wetakethe finite correctionsdue tohigherordersin1/S
Since inthis workwe areinterestedinthe infraredbe- implicitly into account by expressing our final result in
havior of correlation functions at energy scales smaller terms of the true spin-wave velocity c and the spin sus-
than h, we shall ignore the gapped mode. To derive ceptibility χ, which can be obtained from experiments.
the interaction part of the Hamiltonian of the gapless Toleadingorderin1/S,theclassicaltiltangleθiseas-
magnons, we substitute Eq. (4) into Eq. (3) and neglect ily obtained by substituting ψˆ = (2π)Dδ(k)ψ +∆ψˆ
k 0 k
all terms involving the operators bk,+. Defining the field in Eq. (5) and demanding that the coefficient of the
operatorsψˆk =V1/2bk,−,whereV =NaD isthe volume term linear in ∆ψˆk vanishes. This yields the well known
of the system, we obtain for N , [14] relation ψ 2U = µ. To translate this into a
→∞ | 0| 0
condition for θ, we note from Fig. 1 that for large S
k2
H = E + ( µ)ψˆ†ψˆ the tilt angle is related to the staggered magnetization
0 Zk 2m − k k Ms =V−1 iζihSiyiviaθ ≈Ms/s. Heres=S/aDisthe
+ 12ZqZkZk′Uqψˆk†+qψˆk†′−qψˆk′ψˆk , (5) sfoprini-d∈enAsiatyPndofζith=e−sa1tuforartied∈mBa.gSnientcicesMtas2te=, a2nsd|ψζ0i|2=to1
leading order in spin-wave theory, we obtain for large S
where = dDk/(2π)D and the effective interaction is
k
Ucuqto=ffΘaRn(Λd0χR−=|q|()2χJ−0˜1a.DH)−e1reisΛt0he≈leaa−d1inigs alanrguel-tSrarveisouleltt θ2 ≈Ms2/s2 ≈2ρ0/s=2(1−h/hc), (6)
0 0
for the uniform transverse susceptibility in x-direction, where ρ0 = ψ0 2 is the condensate density. Eq. (6)
| |
which is perpendicular to the staggered magnetization agrees with the small-θ expansion of the classical re-
and to the magnetic field. This completes our mapping sult cosθ = h/hc [15]. To obtain the magnon spectrum
of the spin system onto an interacting Bose gas. for large S, we neglect all terms higher than quadratic
For µ > 0 the field operator ψˆk=0 acquires a vac- in the ∆ψˆk. After a Bogoliubov transformation, one
uum expectation value, corresponding to the emergence finds that for h < hc the magnon spectrum is deter-
of long-range antiferromagnetic order. The spin config- mined by ǫ2 = k2 2 +c2k2 which for small k yields
k 2m 0
uration in the ground state is then canted, as shown in ǫk = c0 k + (k3), with the magnetic field dependent
| | O (cid:0) (cid:1)
Fig.1. Oneshouldkeepinmind,however,thatEq.(5)is spin-wave velocity c0 = µ/m=2√DJSaθ.
accurateonly for smallθ, since otherwisethe Bosegas is Wenowfocusonthe spin-waveinteractions,whichare
no longer dilute such that a three-body interactionmust relevantinD 3. Thespubstitutionψˆk =(2π)Dδ(k)ψ0+
≤
be added. However, the qualitative behavior of corre- ∆ψˆ generates various terms cubic and quartic in ∆ψˆ
k k
lation functions is largely determined by symmetry [7], which are rather tedious to work with. A perturbative
so that our results derivedbelow are expected to remain treatment of these terms is plagued by infrared diver-
3
gences. However, Ward-identities associated with the are within Gaussian approximation given by
U(1)-symmetryoftheHamiltonian(5)canbeusedtoob-
χ−1
tain the infrared behavior of some correlation functions hΠKΠK′i0 = δK,−K′ω2+0c2k2 , (10a)
withoutresortingtoperturbationtheory. Aspointedout 0
ω
in Ref. [7], the constraints imposed on the correlation hΠKΦK′i0 = δK,−K′ω2+c2k2 , (10b)
functions by the Ward identities are more transparent if 0
oinngeteoxptrraensssevserψˆskeainndtelromngsitoufdtiwnoalrfleualctfiuealdtiso,ncso.rRreescpeonntdly- hΦKΦK′i0 = δK,−K′ωχ20+c20ck22k2 , (10c)
0
we haveproposeda similarparameterizationofthe spin-
wave expansion in QAFs [11, 16], which amounts to ex- where δK,−K′ = (2π)D+1δ(ω + ω′)δ(k + k′). It turns
outthattheGaussianapproximationisqualitativelycor-
pressingthefieldoperatorψˆ intermsoftwocanonically
conjugatehermitianoperatokrsΠˆk andΦˆk asfollows[17], rfoerctthfoermthixeetdracnosrvreerlasetioconrfruenlacttiioonnfuΠnKctΦioKn′hΠ, wKhΠeKre′i.a.n.d
h i h i
denotes the full thermal average. Using the results of
ψˆ =ρ1/2Πˆ +i(4ρ )−1/2Φˆ . (7) Ref.[7],we obtainfor the true infraredbehaviorofthese
k 0 k 0 k
correlationfunctions
χ−1
Bothfieldsrepresentstaggeredspinfluctuationsinthedi- hΠKΠK′i = δK,−K′ω2+c2k2 , (11a)
rectionsperpendiculartothe magneticfield,whereΠˆ is
k Z ω
transverse and Φˆk longitudinal with respect to the stag- hΠKΦK′i = δK,−K′ω2+kc2k2 , (11b)
gered magnetization. More precisely, in the coordinate
where c is the renormalized spin-wave velocity, χ =
system shown in Fig. 1 we have for S ,
→∞ M2ρ/(2smc2ρ ) = χ Z /Z2 is the renormalized trans-
s 0 0 ρ c
versesusceptibility,andρ andρarethetruecondensate
0
1 M
Πˆk ≈ Ms i ζie−ik·riSix , Φˆk ≈ ss i ζie−ik·riSiy . ddeuncseidtythaneddtimoteanlsdioennlseitsys.fFacotrocrosnZveρn=ienρc/eρw0,eZhacv=einc/trco0-,
X X (8) and Z = (mc2/ρ)dρ /dµ, all of which approach unity
k 0
This equation becomes quantitatively accurate even for in the classical limit S . We emphasize that in di-
→ ∞
finite S in the dilute limit h h which can be shown mensions D 3 some Feynman diagrams contributing
c
→ ≤
using projected boson operators [12]. The condensed to the abovecorrelationfunctions areinfrareddivergent.
phase corresponds to antiferromagnetic order, Φˆk = However, the Ward identities [7] guarantee that all di-
h i
(2π)Dδ(k)φ , where φ = M2/s for large S. Note that vergencescancel,so that the only difference between the
0 0 s
the total density of the Bose gas corresponds in the un- exact results (11a, 11b) and the corresponding approx-
derlying spin system to ρ=s V−1 Sz , which van- imate expressions (10a, 10b) are finite renormalization
− ih ii
ishes for h h and satisfies ρ s if h is only slightly factors. The exact results (11a, 11b) depend only on
c
≥ ≪ P
smaller than h . Corrections to Eq. (8) would be asso- parameters which can be expressed in terms of thermo-
c
ciated with terms which are of linear or higher order in dynamic derivatives.
the boson density and become negligible for h h . The important point is now that the Ward identities
c
→
donotimposesimilarconstraintsonthelongitudinalcor-
The infrared behavior of correlation functions is most
conveniently derived within a functional integral ap- relationfunction ΦKΦK′ ,whichinD 3isdominated
h i ≤
by a non-analytictermarisingfromthe re-summationof
proach [7, 14]. The system is then described in terms
the leading infrared divergences. Using the results de-
of an action S[Π,Φ], which is a functional of c-number
rived in Ref. [7] we obtain
fields depending on imaginary time τ. Formally, these ω2
fields can be obtained from the field operators by sub- hΦKΦK′i = δK,−K′χ −Zk2ω2+c2k2
stituting Πˆ Π (τ) and Φˆ (2π)Dδ(k)φ +Φ (τ). (cid:20)
Introducingkt→he Fkourier tranksfo→rms in freque0ncy spkace, +K (mc)3 ln[ω2(/mc2c+)2k2] , D =3 , (12)
ΠK = dτeiωτΠk(τ) and ΦK = dτeiωτΦk(τ), the D+1 Zρ3ρ0 ( 3−2D[ωc22 +k2]D2−3 , D <3 #
Gaussian part of the action can be written as
R R whereK =21−Dπ−D/2/Γ(D/2). Thenon-analyticcon-
D
tribution to the longitudinal correlation function of the
1
S [Π,Φ] = χ c2k2Π Π +χ−1Φ Φ interacting Bose gas has first been discussed by Weich-
0 2 0 0 −K K 0 −K K
ZKh man [8], and later in Ref. [9]. The field φK has thus a
+ω(Π Φ Φ Π ) . (9) finite anomalous dimension and its effective action can-
−K K −K K
− not be approximated by a Gaussian. This reflects the
i
generalbehaviorofsystemswithbrokencontinuoussym-
Here K = (k,iω) is a collective label for momenta and metries where Goldstone modes lead to anomalous lon-
frequencies,and = dω . Thecorrelationfunctions gitudinal fluctuations [18, 19, 20, 21, 22]. Note that the
K 2π k
R R R
4
magnetic field dependent microscopic momentum scale czynski, Phys. Rev. Lett. 88, 137203 (2002); M. Mat-
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1 Gu¨del, K. Kr¨amer, H. Mutka, A. Wildes, K. Habicht,
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| | ≈
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∝ | | | |
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| |
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