Table Of ContentOrder by Disorder Spin Wave Gap in the XY Pyrochlore Magnet Er Ti O
2 2 7
K.A. Ross,1,2 Y.Qiu,2,3 J.R.D. Copley,2 H.A. Dabkowska,4 and B.D. Gaulin5,4,6
1Institute for Quantum Matter and Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, Maryland 21218, USA
2NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
3Department of Materials Science and Engineering,
University of Maryland, College Park, Maryland 20742, USA
4Brockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario, L8S 4M1, Canada
5Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, L8S 4M1, Canada
6Canadian Institute for Advanced Research, 180 Dundas St. W.,Toronto, Ontario, M5G 1Z8, Canada
TherecentdeterminationofarobustspinHamiltonianfortheanti-ferromagneticXYpyrochlore
Er Ti O reveals a most convincing case of the “order by quantum disorder” (ObQD) mechanism
2 2 7
4
for ground state selection. This mechanism relies on quantum fluctuations to remove an acciden-
1
tal symmetry of the magnetic ground state, and selects a particular ordered spin structure below
0
T =1.2K. The removal of the continuous degeneracy results in an energy gap in the spectrum of
2 N
spinwaveexcitations,longwavelengthpseudo-Goldstonemodes.WehavemeasuredtheObQDspin
b wave gap at a zone center in Er Ti O , using low incident energy neutrons and the time-of-flight
2 2 7
e inelastic scattering method. We report a gap of ∆=0.053 ± 0.006 meV, which is consistent with
F upperboundsplacedonitfromheatcapacitymeasurementsandroughlyconsistentwiththeoretical
1 estimate of ∼ 0.02 meV, further validating the spin Hamiltonian that led to that prediction. The
1 gap is observed to vary with square of the order parameter, and goes to zero for T ∼TN.
] PACSnumbers:75.25.-j,75.10.Jm,75.30.Ds,75.50.Ee
l
e
-
r Geometrically frustrated magnetic materials consist thermally or quantum mechanically driven, the latter
t
s of interacting localized magnetic moments arranged on case being called Order-by-Quantum-Disorder (ObQD).
.
t crystalline architectures for which the anisotropies and The same mechanism can quite generally apply to other
a
m interactionsofthemomentsareincompatiblewithasim- types of ensembles, such as ultra-cold atoms and lattice
ple ordered state. Frustration suppresses conventional boson models [6–8]. While there is no lack of theoreti-
-
d phase transitions to long range magnetic order, allowing cal models which definitively display ObD [3–5, 9–13],
n the study of strongly correlated and fluctuating spins the number of real systems which display ObD are few
o at temperature scales much lower than the interaction [14, 15] and even in those cases, their ground state se-
c
energy [1]. Among the exotic ground states which can lection usually cannot be conclusively ascribed to ObD
[
result are quantum entangled states, such as the much (see supplementary material in Ref. 16). The pyrochlore
2
discussed U(1) quantum spin liquid [2]. However, many magnet Er Ti O is a very rare example of a mag-
v 2 2 7
frustrated systems do find their way to long range order netic material that can be compellingly shown to dis-
6
7 via weak energetic terms in the Hamiltonian, ordinar- play ObQD, thanks to a detailed knowledge of its effec-
1 ily insignificant in a non-frustrated system. This pro- tive spin Hamiltonian and the unusually well-protected
1 vides many possibilities for the ground states of such accidental degeneracy it supports [16, 17]. Here, we re-
.
1 magnets. Commonly encountered terms beyond nearest port direct measurements of the defining characteristic
0 neighbor exchange which may be relevant in the effec- oftheexcitationspectrumwhichfurthersupportsObQD
4 tive spin Hamiltonian include long range dipolar inter- in Er Ti O ; a small spin wave gap that is induced by
2 2 7
1
actions,nextnearestneighborexchange,andspin-lattice quantum fluctuations.
:
v coupling terms.
i ObQD introduces an anisotropy into the Hamiltonian
X
“Order-by-Disorder” (ObD) is an appealing, yet elu- throughtheexcitationofspinwavemodessupportedby
r sive, mechanism to drive order in frustrated systems the ordered ground state. This anisotropy manifests it-
a
[1, 3–5]. Generally, ObD breaks an “accidental” contin- self as an energy gap to the long wavelength excitations
uous degeneracy, i.e. one which is not supported by the nearQ(cid:126) =(0,0,0)(andrelatedzonecenters).Theseexci-
symmetry of the Hamiltonian, by accessing low energy tations would otherwise be Goldstone modes within the
fluctuations around these ground states; the favoured continuouslydegeneratemanifold.However,unlikecrys-
states are those with higher densities of low energy talline anisotropy, the ObQD anisotropy is removed as
modes. Thus, ObD selects a ground state by entropic one exits the ordered state. A continuous phase tran-
rather than energetic means. In magnetic systems the sition out of a ground state selected by ObQD is then
selection occurs via low energy spin fluctuations, either expected to be accompanied by a continuous closing of
2
FIG. 1: a) through d): color contour maps of the INS spectrum at the bottom of the spin wave dispersion near the (1,1,1)
Bragg peak, along the [1,1,-1] direction ([H,H,-H] + [K,K,2K], with 0.58 ≤ K ≤ 0.74 r.l.u.), as shown in a representation of
thereciprocalplane(e).Panelsa)andc)showthemeasuredS(Q(cid:126),ω)atT =60mKand700mK,whilepanelsb)andd)show
fitstothesedatausingEq.3,(seetext).f)Energycuts,binningover0.30≤H≤0.36r.l.u.Errorbarsrepresentonestandard
deviation. Note that the factor k /k which appears in the INS cross section is removed in all data presented herein.
f i
the gap. =1.2K[26],althoughremnantdiffusemagneticneutron
scattering near the Bragg peaks [27, 28] indicates some
Being the canonical example of geometric frustration
shortrangedspinstructurepersists.Thisdiffusescatter-
in 3D, the pyrochlore lattice, an array of corner-sharing
ing is not fully understood but it is plausible to ascribe
tetrahedra, often supports large ground state degenera-
ittodomainwallsseparatingthesixpossibledomainsof
cies. Spins interacting via Heisenberg near neighbor an-
the ψ ordered state.
tiferromagnetic (AF) exchange on the pyrochlore lattice 2
are well known to display a macroscopic degeneracy of
While ψ order in Er Ti O is not disputed exper-
2 2 2 7
disordered ground states [18, 19]. A simple modification
imentally, the mechanism leading to it was not fully
to the near-neighbor AF pyrochlore model is the intro-
understood until recently, when an analysis of the full
duction of XY anisotropy. In real pyrochlore materials,
spin wave spectra in the magnetic field-polarized quan-
theanisotropyislocal,resultingfromtheD pointsym-
3d tum paramagnetic state of Er Ti O revealed that an
2 2 7
metryattheR-siteofthecrystallatticeinmaterialssuch
anisotropic exchange Hamiltonian with pseudospin-1/2
as R Ti O (with R being a trivalent rare earth ion).
2 2 7 operators is needed to describe the measured magnetic
For local XY anisotropy and AF exchange interactions,
excitations in Er Ti O [16, 17]. This anisotropic ex-
2 2 7
it was discovered that a large [20] continuous [11, 21],
changemodelandrelatedvariantsisenjoyingmuchcur-
manifold of ordered states arises, but this degeneracy is
rent attention [29–32]. The model with parameters as
lifted by ObD [11, 21, 22], such that a small discrete set
determinedinRef.16wasrecentlystudiedincludingthe
of ordered spin configurations is selected.
effects of finite temperature, which showed that both
Er Ti O is the only known manifestation of an AF thermal and quantum ObD lead to the same ground
2 2 7
XYpyrochloremagnet.LargesinglecrystalsofEr Ti O state selection, i.e. the ψ state [33]. It has also been
2 2 7 2
can be produced via the optical floating zone (OFZ) argued that ψ could be selected energetically through
2
method, allowing a comprehensive study of its magnetic a small admixing of higher crystal field levels [34], al-
properties including anisotropic effects. An estimate of though the magnitude of such an effect has not been
the overall exchange energy comes from Curie-Weiss fits rigorously calculated, and indeed by estimates based on
tothesusceptibility,givingθ =-13Kathightemper- the crystal field splitting in Er Ti O , it is expected to
CW 2 2 7
atures[23,24].Meanwhile,thecrystalfieldHamiltonian benegligible[16].Importantly,Savaryet al showedthat
forEr3+ inEr Ti O isknown todisplaya well-isolated thecontinuousdegeneracyofthelowestenergyspincon-
2 2 7
Kramers ground state doublet with local XY anisotropy figurations at the mean field level cannot be broken by
[24, 25]. Er Ti O is known to order magnetically into longer range interactions or spin-lattice coupling, thus
2 2 7
the ψ basis of the Γ irreducible representation at T cementingObQDasthe mechanismleadingtotheselec-
2 5 N
3
tion of ψ order in Er Ti O [16].
2 2 2 7
AremainingtaskistodirectlyconfirmtheObQDsce-
nario through the measurement of the small spin wave
gapthatisanecessaryconsequenceofit.Herewereport
adirectspectroscopicmeasurementofthespinwavegap
using high resolution inelastic neutron scattering (INS)
fornomaSsainvgalreyectryaslt,awlhoofpErre2dTicit2eOd7a. gWape oufsaedpptrhoxeimthaetoerlyy
0.02meV[16],aswellasupperlimitsof<∼0.05meVfrom
analysisoflowtemperatureheatcapacitymeasurements
[28]andelectronspinresonance[35]inordertocarefully
plan these direct measurements of the ObQD spin wave
gap.
FIG. 3: Temperature dependence of the spin wave gap, ∆,
overlaid on that of the Bragg peak intensity (∝ M(Q(cid:126))2),
mCheaamsupreiodninettahle[1p1r]e.seEnrtrowrboarkrs, oans wtheellgaasptdhaattarhepaolfrtoefdthine
95% confidence interval while those on the Bragg intensity
(one standard deviation) are smaller than the markers.
DmafliicegsonhkntttCppinhrauootodhpuu.pscWeerssoeSumeprmcoeecnptoulrocsoyhimnerdgoemtsteeharv.eteiTcnlohpcwihuseolsstpteipmsreeeors-sfoolinfun-efltutiiohgtnrheoticnnhiscnoisfpdrtpeornuemrt-
soeft1ti0ng˚As,(aEnid=se0le.8c2temdenVe)u.tTrohnisswpriothviidnecdidaenntewnearvgeylernegstoh-
lIluuNttSiioonnexogpfae0vr.ie0m1ae3nmrteedoVun,ceEsderv2ieTnnci2itdOime7nets[2nb7ee].utttTerrohnethiflamunpxtr,ho4ev.e6odritgrieminseoas-l
lower than the previous study, providing a challenge for
the detection of even the strongest inelastic scattering
FIG. 2: a) Intensity vs. energy cuts through the measured (acoustic spin wave modes).
S(Q(cid:126),ω), binning over the region 0.30 ≤ H ≤ 0.36, and 0.58
The experiment was carried out in a 10T supercon-
≤ K ≤ 0.74 r.l.u. b) the binning region overlaid on a map of
ducting magnet with dilution insert. This allowed us to
theelasticscattering((111)Braggpeak).Thefittingfunction
reach well into the ordered state (60mK ∼ T /20). At
allows the gap (∆), intrinsic width (Γ), and scale factors for N
both elastic and inelastic intensities (A, see Eq. 3) to vary. 60 mK the order parameter is approximately saturated
The fitted values are shown in c), where error bars indicate and the ObQD gap should be well-defined and at its
half of the 95% confidence interval. maximum value. The application of a 5T magnetic field
along the [110] direction was used to lift the magnetic
We studied single-crystalline Er Ti O , grown by the excitations out of the energy window of interest [27], al-
2 2 7
OFZ method [36, 37]. The 7 gram crystal studied here lowingaprecisedeterminationoftheinstrumentalback-
is the same as that previously studied [16, 27]. Several ground. This background was subtracted in all figures
other single crystals have also been studied in the lit- shown here.
erature. Unlike some other rare-earth titanate materials Figure 1 shows INS data near the (1,1,1) magnetic
(notably Yb2Ti2O7 [38–40] and Tb2Ti2O7 [41, 42]), dif- zonecenter.Theintensityina)andc)isshownasafunc-
ferent powders and OFZ growths of Er2Ti2O7 appear tion of energy and along the [1,1,-1] direction in recip-
to have consistent magnetic properties down to 50mK, rocal space at T=60 mK and 700 mK, respectively. The
as evidenced by the ordering temperature, ordered spin trajectory in reciprocal space for these scans is shown
structure, and magnetic specific heat [28]. in Fig. 1 e). It is clear that the raw magnetic scatter-
The neutron scattering experiment was carried out ing at T=60 mK in Fig. 1 a) shows a gapped spectrum
at the NIST Center for Neutron Research, using the at (1,1,1), with an approximate gap of ∼ 0.05 meV. On
4
warming up to 700 mK (Fig. 1 c)) the gap has largely The best fit description of this model to the inelastic
closed. Cuts of this data along the energy direction at scattering at T=60 mK is shown in Fig. 1 b), where it
T=60mKand700mKandintegratingover0.30≤H≤ can be directly compared to the measurement in Fig. 1
0.36 r.l.u. are shown in Fig. 1 f). a). The related comparison between the calculated and
To allow a more quantitative estimate of the ObQD measuredS(Q(cid:126),ω)atT=700mKismadeinFig.1d)and
spin wave gap and its temperature dependence, we have c), respectively. Clearly this model provides an excellent
fitthesedataasshowninFigs.1c)andd),andinFig.2 description of the inelastic scattering near the (1,1,1)
a).Tofitthemeasuredspinwavedispersion,werequirea magnetic zone center, and it yields a base temperature
modelforS(Q(cid:126),ω)around(1,1,1),whichwenowdescribe. ObQD spin wave gap of ∆=0.053 ± 0.006 meV.
The magnetic structure of Er Ti O has AF nature, The temperature dependence of the ObQD gap can
2 2 7
allowing us to approximate the dispersion of the low en- be determined by fits to intensity vs energy scans, ap-
ergy acoustic spin waves with the following functional proximating constant-Q(cid:126) scans at the (1,1,1) magnetic
form: zone center, as shown in Fig. 2 a). The inelastic scatter-
ing here has been integrated over the region 0.30 ≤ H ≤
(cid:112)
ωc((cid:126)q)= v2|(cid:126)q|2+∆2, (1) 0.36r.l.u.and0.58≤K≤0.74r.l.u.,asindicatedwithin
the red rectangle around (1,1,1) shown in Fig. 2 b). The
where v = 0.545 meV/˚A−1 is the spin wave veloc-
resulting fits to the data, using a numerical evaluation
ity [16], (cid:126)q is the reciprocal space wavevector measured
of Eq. 3, are shown as the red solid lines overlaid on the
from the zone center (i.e. (cid:126)q = Q(cid:126) −(1,1,1)), and ∆ is
inelastic data in Fig. 2 a). The description of the data
the spin wave gap. This dispersion will be broadened
is clearly excellent, and the temperature dependence of
by the intrinsic (finite) lifetime of the spin excitations,
the three parameters within Eq. 3 extracted from the
which is often modelled by a damped harmonic oscil-
fits, the ObQD gap, ∆, the intrinsic FWHM, Γ, and the
lator. Such damped spin waves produce an imaginary
scale factor, A, are shown in Fig. 2 c). The ObQD gap,
susceptibility, χ(cid:48)(cid:48), which follows a Lorentzian profile in
∆ in the top panel of Fig. 2 c), is observed to decrease
energy with variable full width at half max (FWHM)
fromitsbasetemperaturevalueof∆=0.053±0.006meV
1Γ
Γ: L(x,Γ)= 2 . The imaginary (i.e. absorptive) in what appears to be a continuous fashion. This same
π(x2+(Γ)2)
part of the susceptib2ility is modeled as, temperature dependence is seen in the elastic magnetic
scattering at (1,1,1), while the scale factor A remains
constant (bottom of Fig. 2 c). The energy width of the
(cid:20) (cid:21)
χ(cid:48)(cid:48)((cid:126)q,ω)∝ 1 L(cid:0)ω−ω ((cid:126)q),Γ(cid:1)−L(cid:0)ω+ω ((cid:126)q),Γ(cid:1) . inelastic peak, Γ, is shown in the middle panel of Fig. 2
ω ((cid:126)q) c c c), and it stays roughly constant at 0.04 meV (nearly
c
(2) three times the instrumental resolution, shown as a red
line). The reason for the finite intrinsic energy width of
INS measures the dynamic structure factor, S((cid:126)q,ω), the zone center spin wave is not clear, but it may be re-
which is related to this by S((cid:126)q,ω)∝χ(cid:48)(cid:48)/(1−e−h¯ω/kBT). latedtomagnonsinteractingwithdomainwallsbetween
The resolution of the measurement is broadened in en- the six degenerate ψ domains [16, 27, 28].
2
ergy from instrumental effects, which can be treated as Wecanmakeamoredetailedcomparisonbetweenthe
a Gaussian, R(ω) = exp(−√(h¯ω)2/2c2). For our instrumen- temperature dependence of the ObQD gap, ∆, and the
2πc2
(cid:112) magneticorderparameterinEr Ti O ,andthisiswhat
tal configuration, the resolution FWHM is 2 (2ln(2))c 2 2 7
is shown in Fig. 3. Here we overlay the temperature de-
= 0.013 meV. The intensity is also convolved with the
pendence of ∆, with the measured magnetic Bragg in-
mosaic structure of the crystal, M((cid:126)q), approximated
tensity at the (1,1,1) elastic position in this experiment,
as a three-dimensional Gaussian using fixed widths,
and the previously measured magnetic intensity at the
c , c and c , measured at the (111) Bragg peak;
H K L
M((cid:126)q) = exp(−(qc2H2 + qc2K2 + qc2L2)). Here, cH = 0.068˚A−1, (s2ca,2t,t0e)rienlgasatircepnoosrimtioanliz[1ed1].toTheeacthwootsheetrsoafte7la0s0timcBKr.aAggs
H K L
cK =0.031˚A−1.Theout-of-planemosaicwidth,cL,can- the magnetic Bragg intensity varies as M(Q(cid:126))2, Fig. 3
not be measured using DCS and is taken to be equal to shows the ObQD gap, ∆, to also vary as M(Q(cid:126))2.
c since they both represent widths along transverse (cid:126)q-
H To conclude, inelastic magnetic neutron spectroscopy
directions.
reveals a gap in the spin wave spectrum of Er Ti O of
2 2 7
The expected INS intensity, with scale factor A, is
∆=0.053±0.006meVatthemagneticzonecenter.This
determination is consistent with upper limits placed on
(cid:90)(cid:90)
I((cid:126)q,ω)=A S(q(cid:126)(cid:48),ω(cid:48))R(ω−ω(cid:48))M((cid:126)q−q(cid:126)(cid:48))dω(cid:48)d3q(cid:48). itbyanalysisoflowtemperatureheatcapacity,andwith
theoretical expectations based on a robust microscopic
(3)
spin Hamiltonian. Previous theoretical work shows that
5
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2 2 7 Rev. B 89, (2014).
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