Modes of magnetic resonance in the spin liquid phase of Cs2CuCl4

0.88 MB·English

by  K. Yu. Povarov

Modes of magnetic resonance in the spin liquid phase of Cs CuCl
2 4
K. Yu. Povarov,1,∗ A. I. Smirnov,1,2 O. A. Starykh,3,4 S. V. Petrov,1 and A. Ya. Shapiro5
1P. L. Kapitza Institute for Physical Problems, RAS, 119334 Moscow, Russia
2Moscow Institute for Physics and Technology , 141700, Dolgoprudny, Russia
3Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA
4Max-Planck-Institut fu¨r Physik Komplexer Systeme, D-01187 Dresden, Germany
5A.V.Shubnikov Institute of Crystallography, RAS, 119333 Moscow, Russia
(Dated: January 28, 2011)
Wereport theobservationofafrequencyshiftandsplittingoftheelectron spinresonance(ESR)
1 mode of the low-dimensional S = 1/2 frustrated antiferromagnet Cs2CuCl4 in the spin-correlated
1 statebelowtheCurie-Weisstemperature4Kbutabovetheorderingtemperature0.62K.Theshift
0 and splitting exhibit strong anisotropy with respect to the direction of the applied magnetic field
2 anddonotvanishinzerofield. Thelow-temperatureevolutionofspinresonanceresponseisaresult
n of the specific modification of one-dimensional spinon continuum under the action of the uniform
a Dzyaloshinskii-Moriya interaction (DM) within the spin chains. Parameters of the uniform DM
J interaction are derived from theexperiment.

7
2
Quantum antiferromagnets on triangular lattice ex- are crucially sensitive to anisotropies of the spin Hamil-

] hibitavarietyofunusualphaseswithintriguingphysical tonian. Here we report a dramatic manifestation of the
l
e properties. In this work we report results of electron uniform Dzyaloshinskii-Moriya (DM) interaction on the
- spin resonance (ESR) study of quasi two-dimensional ESR spectrum of Cs CuCl in the paramagnetic phase.
r 2 4
t (2D) triangular antiferromagnet Cs CuCl in the para- We observe and explain splitting of the ESR mode into
s 2 4
t. magnetic phase. Cs2CuCl4 realizes distorted triangular two modes with lowering the temperature, significantly
a lattice with strong exchange J along the base of the tri- extending previous study [11], and also find strong de-
m angular unit and weaker exchange integral J′ = 0.34J pendence of the ESR absorption on the polarization of
- along lateral sides of the triangle. The material orders the microwave magnetic field h with respect to the DM
d
below T =0.62 K into a long-range 3D spiral state [1]. vector D in zero external field.
n N
o This N´eel point is far below the Curie-Weiss tempera- Thecrystalsamplesweregrownbytwomethods. First
c ture TCW=4 K. In the intermediate temperature range set of samples was prepared by the crystallization from
[ T < T < T a spin-liquid phase with strong in-chain
N CW themeltofastoichiometricmixtureofCsCuCl3andCsCl
1 spin correlations takes place. Magnetic properties of inanampule, whichwas slowlypulledfromthe hot(T=
v Cs2CuCl4 are well described by a quasi-one-dimensional 520◦C) area of the oven. The second method was crys-
5 (1D) model of weakly coupled spin S = 1/2 chains. In
tallization from solution [12]. The room temperature
7
particular,inelasticneutronscatteringexperiments[1,2]
2 crystal lattice parameters for both sets of crystals are
5 are nicely explained by a parameter-free theory of dis- a=9.75˚A,b=7.607˚A,c=12.394˚A.Thesamplesobtained
. persingcontinuumofone-dimensionalspinons[3]. More- by last method have natural faces and are lengthened
1
over, the same description, extended in Ref. [4] to in-
0 along b-axes. Both sets of crystals demonstrated identi-
clude residual symmetry-allowed Dzyaloshinskii-Moriya
1 cal ESR signals. The study in the frequency range 9-90
1 (DM)interactionsbetweentheCu-spins,isabletoconsis- GHz was performed with a set of homemade spectrom-
: tently explainmajorityofthe phasesinthe T H phase
v − eters equipped with cryomagnets. The crystal samples
i diagram, for three different directions of the magnetic were placed within microwave resonators at the maxi-
X
field [5]. In addition, several numerical investigations
mum of the microwave magnetic field. The resonance
r [6,7]ofthespatiallyanisotropicJ J′ Heisenbergmodel
a − linesweretakenatthefixedfrequency,asdependencesof
also find that in a wide region of the parameter space
microwavepower,transmitted throughthe resonator,on
0 J′/J 0.6 the spin chains are almost decoupled
theexternalmagneticfield. Temperaturesdownto1.3K
≤ ≤
andinter-chainspin correlationsareexponentially weak. (0.4 K), were obtained by pumping vapor of 4He (3He),
This behaviorfollows fromhighly frustratedstructureof
correspondingly. Cooling down to 0.1 K was achieved
the inter-chain exchange J′ and provides additional jus-
with the help of a dilution cryostatKelvinox-400.
tification to the 1D-based approach[4] to Cs CuCl .
2 4 AttemperaturesaboveT =10KcrystalsofCs CuCl
2 4
Electron spin resonance (ESR) represents a sensitive exhibit conventional paramagnetic resonance of Cu2+
probe of low-energy magnetic excitations with zero mo- S = 1/2 magnetic ions with a single narrow line cor-
menta, q 0. It is well known as a method pro- responding to g-factor values g = 2.20 0.02, g =
a b
≈ ±
viding fine details of low-energy spin spectra in both 2.08 0.02,g =2.30 0.02forthemagneticfieldapplied
c
± ±
ordered and paramagnetic [8–10] phases. Importantly, along a, b and c axes correspondingly. These g-factor
both the position and the linewidth of the ESR signal values agree well with those reported earlier at T = 300
2
T=1.3KisahalfofthatatT=4K.Theresonantfrequen-
cies above 14 GHz may be well fitted by the frequency-
field relation of an ordered antiferromagnet with “gap”
∆/(2π~)=14 GHz,
2π~ν = (gµ H)2+∆2. (1)
B
p
(ii) For H a,c, the ESR line strongly broadens on
k
cooling, and its shape distorts, as shown in the inserts
of Fig. 2. The absorption at T=1.3 K may be fitted
by a sum of two Lorentzians, indicating the splitting of
the resonance mode. The ν(H) dependence at H a
k
is presented in Fig. 2. Two modes at T = 1.3 K were
resolved for measurements in the frequency range 14-50
GHz. At higher frequencies the splitting is masked by
natural linewidth, which prevents the resolution of the
FIG. 1. ESR frequency at T =1.3 K for H kb. Dotted line doublet, while at lower frequencies the signal becomes
is paramagnetic resonance with g = 2.08, solid line – theory too broad. The angular φ dependence of the resonance
(see text). Empty circles stand for resonances loosing inten-
fieldH =H(sinφ,cosφ,0)withinthea bplane,taken
r
sity with cooling. Upper insert: gap vs temperature, various −
atν=27GHzandT=1.3K,isshowninFig.3. ForH c
symbols correspond todifferent cryostats. Lower insert: evo- k
thesplittingofabout0.35 0.1Tisresolvedforfrequen-
lution of the resonance line with cooling.
±
cies 27 and 31 GHz.
(iii) We have also studied polarization dependence of
the ESR absorption at H 0. Fig. 4 shows transmis-
→
sion vs field dependences at T =1.3 K and T =6 K for
different orientations of the microwave and static fields
with respect to the crystal axes, but with h H. The
⊥
measurements are done at ν = 14 GHz. At T > 6 K
both the position and the lineshape of the paramagnetic
resonance line are the same for all three principal polar-
izations of the microwave field. However, at T = 1.3 K,
the absorption in zero field for h b is at least three
k
times more intensive than for h a,c.
k
The splitting of the ESR line for H c and the po-
k
larization dependence of the signal were also detected
previously in Ref. [11].
Wenowturntoexplanationofthedataandshowthat
all unusual features of the ESR signal reported above
FIG. 2. ESR frequencies at T = 1.3 K for H k a. Dotted arenaturallyexplainedbyquasi-1DnatureofCs CuCl .
2 4
line is paramagnetic resonance with g = 2.20, dashed line –
The relevant Hamiltonian
theory (see text). Upper insert: lineshape at T = 1.3 K,
ν = 27 GHz. Points – experimental data, solid line is the = JS S D S S +
x,y,z x+1,y,z y,z x,y,z x+1,y,z
H · − · ×
result of a fit by the sum of two Lorentzians (dashed lines). X
x,y,z
Lower insert: evolution of the resonance line with cooling.
gµ H S +... (2)
B x,y,z
− ·
contains three terms: the first describes intrachain ex-
K and 77 K [13], and, for g , at T =4.2 K [11]. changeJ (x runs alongcrystalb axis),the second– uni-
c
Significant evolution of the ESR spectrum was found form DM interaction D between chain spins, and the
y,z
oncoolingbelow6K.Thisevolutiondependsstronglyon third is the usual Zeeman term, allowing for anisotropic
the microwave frequency ν and orientation of the exter- g-factor. The dots stand for the omitted interchain ex-
nal magnetic field H. (i) For H b, we observe a single change as well as DM interactions on interchain bonds.
k
Lorentzian line shifting with cooling to lower fields (see Detailed symmetry analysis of the allowed DM interac-
Fig. 1). At ν=14 GHz the resonance field is near zero. tions [4] shows that there are four different orientations
Atlowerfrequencies,we observea strongdecreaseofthe ofthe DMvector(see Fig. 6 in[4])depending onchain’s
intensitywithcooling,andnovisibleshiftfromthepara- integer coordinates y,z: D = D ( 1)zaˆ+D ( 1)ycˆ.
y,z a c
− −
magnetic resonance field, in contrast to ESR lines taken Here z indices magnetic b c layers, while y numerates
−
at ν >17 GHz, which exhibit a shift and do not loosein- chainswithinalayer. Crucially,crystalsymmetryforbids
tensity at cooling. At ν=9 GHz the integral intensity at DM vector to have component along the chain b axis.[4]
3
as to eliminate the DM coupling from the Hamiltonian.
The transformed ˜ describes spin chain with quadratic
H
1.2 in D easy-plane anisotropy. To linear in D/J accuracy
Experiment, T=1.3 K
Theory the ESR response of this model coincides with that of
1.1
, Paramagnetic, T=10 K
an isotropic Heisenberg chain, detailed study of which is
T)1.0 described in Ref.[10]. ESR absorption is determined by
eld (0.9 the transverse structure factor Sxx(ω,q˜) = Syy(ω,q˜) of
Fi the chain, evaluated at q˜=0. This is produced by char-
c 0.8
eti acteristic to the spin-1/2 chain spinon continuum and is
gn0.7 described by a sum of delta-function peaks at frequen-
a
M cies ~ω = ~vq˜ gµ H , seeeq.(3.16)of [10]andFig.5.
0.6 ± B
Here v =πJ|a /(±2~) is th|e zero-field spinon velocity and
0
0.5
a isthelatticespacing[14]. Theresponseintheoriginal
0
0.4 b a b a b (un-rotated) basis is obtained by un-doing the momen-
0 30 60 90 120 150 180 210 240 270 300 330 360
tum boost (3), which corresponds to setting q˜= D/Ja
0
Angle (deg.)
in the expression for the structure factor . This
xx/yy
S
immediately implies the splitting of the ESR line into
FIG. 3. Theangular dependenceof theresonance field Hr in two lines, at frequencies ~ω = gµ H πD/2. Note
± B
the ab plane for ν =27 GHz at T =1.3 K. Thick line is the that for small D/H the splitting π|D =π±(D2+D|2)1/2 is
theoretical prediction with Da/(4~) = 8 and Dc/(4~) = 11 linear in D which justifies our neglect of thae easyc-plane
GHz. Themeasured valuesof theresonance field deep in the
anisotropy.
paramagnetic phase (at T = 10 K) are presented by crosses,
thedashedlineisatheoreticalfitofaparamagneticresonance While the four-sublattice structure of the DM vec-
with ga,b,c of Cs2CuCl4. tors makes it impossible to study the D H configu-
k
ration in Cs CuCl , the above argument allows one to
2 4
immediately understand polarization-dependent absorp-
tion(finding(iii)above)inzeromagneticfield,atthefre-

0.005 quency2πν =πD/2~. Sincethemicrowaveabsorptionis
proportionaltothe squareofthemicrowavefieldcompo-
0.000
nentperpendiculartotheeffectivefield,weconcludethat
s)
el. unit-0.005 =14.44 GHz cmoanxfiigmuarlatpioosnsiwbiltehahbs⊥orpDti(otnh.aFtoisr,hhaklobn)gshthoeulad(rces)ualtxiisn,
ge (r-0.010 H || c ,T h= |1| .a3 K absorption is a factor Dc2/D2 (Da2/D2) smaller. Using
Chan-0.015 H || c ,T h= |6| bK ntiuomneforircadliffeesrteimntapteoslaorfizDataio,cnsdesrhiovuedldboebleoyw,thteherealabtsioornps-:
wer -0.020 TT==16. 3K K Pb/Pa 1.6 and Pb/Pc 2.8. This agrees qualitatively
Po H || a, h || c withou≃r datapresented≃inFig.4 – the zerofield absorp-
T=1.3 K
-0.025
T=6 K tionatν =14and17GHz isafactorof3moreintensive
for h b than for h a,c .
-0.030
k k
0.0 0.5 1.0 1.5 2.0 To understand our findings (i) and (ii), one needs to
Magnetic Field (T) analyzethearbitraryorientationoftheHandDvectors,
which is the problem solved in [15]. The main physi-
FIG.4. Thepolarization effect at ν =14.44 GHzfor H →0. cal point of Ref.[15] consists in the observation that a
Circles with error bars show the uncertainty resulting from uniform DM interaction, much like a spin-orbit Rashba
normalizing the signals to zero absorption at H >1.5 T. interactionin2Dconductors[16–18],actsonspinonexci-
tationsof1Dchainsasaninternalmomentum-dependent
magneticfield. Technically,thisobservationfollowsfrom
The essence of the observed ESR line splitting can be
spin-currentformulation[14]oftheproblem(2),whichis
explained by considering a single Heisenberg chain with
valid below the strong-coupling temperature scale T
uniform DM interaction with vector D along the local 0 ∼
Je−πS (S = 1/2 for the spin chain here)[19]. The
zˆ axis, when the field is lined up along the same axis
spin currents M represent spin density fluctuations
(D Dzˆ, H Hzˆ). In this geometry the DM inter- R/L
y,z → → of spinons near the right/left (R/L) Fermi points of a
action can be gauged away by position-dependent rota-
1D system. It follows that the right and left-moving
tion of lattice spins,
currents experience different, in magnitude and direc-
S+ =S˜+ eiαx, Sz =S˜z . (3) tion, total magnetic fields: BR = H+~vD/gµBJ and
x,y,z x,y,z x,y,z x,y,z B = H ~vD/gµ J. It is then natural that the ESR
L B
−
The rotation angle tan(α) α = D/J is chosen so responseofsuchachainconsistsoftwopeaksatfrequen-
≈ −
4
1 K< T <2 K is much slower than right near T . This
N
supportsourmainassumptionthatthereportedspectral
features of Cs CuCl reflect specific spin chain physics
2 4
which dominates the temperature rangeT <T <T .
N CW
In conclusion, we demonstrated that the uniform DM
interaction,whichisadistinctivefeatureofCs CuCl ,re-
2 4
sultsinanewkindofspinresonanceinaS=1/2chainan-
tiferromagnet. The observed spectrum is a consequence
of the splitting of the chain’s spinon continuum by the
FIG.5. SpinonspectrumofS =1/2Heisenbergchainforq∼ internal magnetic field which is produced by the uni-
0with(solidlines)andwithout(dashedlines)DMinteraction. form DM interaction. We believe that a similar phe-
When DkH, momentum is boosted byq=D/Ja0. nomenon is possible in a higher-dimensional magnetic
systemswithfractionalizedspinonexcitations. Ourfind-
cies 2π~ν =gµ B .[15] ingsdiffer stronglyfromthe well-knownsingle-frequency
R/L B R/L
resonance of S = 1/2 antiferromagnetic spin chain with
For the Cs CuCl -specific configuration of chain DM
2 4
interactions and H=(H ,H ,H ) our analysis predicts staggeredDMinteraction[10]. Italsodiffers froma con-
a b c
ventional 3D antiferromagnet which acquires an energy
(2π~νR)2 =(gbµBHb)2+(gaµBHa+( 1)zπDa/2)2+ gap in the spectrum only below the ordering transition.
−
(g µ H +( 1)yπD /2)2, (4) We would like to thank K.O. Keshishev for guidance
c B c c
−
(2π~ν )2 =(g µ H )2+(g µ H ( 1)zπD /2)2+ in work with the dilution refrigerator, and L. Balents,
L b B b a B a a
− − S. Gangadharaiah, V.N. Glazkov, M. Oshikawa, and R.
(g µ H ( 1)yπD /2)2, (5)
c B c− − c Moessner for insightful discussions. We thank RFBR
Observethatwhileν =ν forfixedchainindices(y,z), Grant 09-02-736 and NSF Grant No. DMR-0808842
R L
6
these chiral ESR frequencies transform into each other (O.A.S.)forfinancialsupport. Partoftheworkhasbeen
under lattice translations (y,z) (y 1,z 1). These performed within the ASG on “Unconventional Mag-
→ ± ±
equations naturally explain the difference between H netism in High Fields” at the MPIPKS (Dresden).
k
a,c and H b situations. For H b the external and
k k
theinternalDMfieldaremutuallyperpendicularandthe
vector sum H D has the the same absolute value for
±
bothsigns. OnethenfindssingleESRfrequencyν =ν
R L ∗
[email protected]
of the form (1) where the gap is in fact determined by
[1] R. Coldea, D. A. Tennant, A. M. Tsvelik, Z. Tylchynski,
DM interaction, ∆=π D2+D2/2.
a c Phys. Rev.Lett. 86, 1335 (2001).
For H a,c the DMpfield has a component along H
[2] R.Coldea,D.A.Tennant,Z.TylczynskiPhys.Rev.B68,
k
andthe frequenciesofthe twospinexcitationsarediffer- 134424 (2003).
ent. In principle, because of the four values of the Dy,z [3] M. Kohno, O. A. Starykh, and L. Balents, Nat. Phys. 3,
vector,fourdifferentfrequenciesmaybepresentwhenthe 790 (2007).
fieldHisorientedwithinthea cplane. Weattempted [4] O. A. Starykh, H. Katsura, L. Balents, Phys. Rev. B82,
− 014421 (2010).
to check this experimentally but failed to resolve sepa-
[5] Y. Tokiwa, T. Radu, R. Coldea, H.Wilhelm, Z. Tylczyn-
rate lines within the broad band of absorption.[14] For
ski, F. Steglich, Phys.Rev.B73, 134414 (2006).
H within the a b plane (4) and (5) predict two dif-
[6] M. Q.Weng, D.N. Sheng,Z.Y. Weng,and R.J. Bursill,
−
ferent frequencies. Solving equations (4,5) at the fixed Phys. Rev.B74, 012407 (2006).
frequency for the magnetic field we were able to fit the [7] D. Heidarian, S. Sorella, F. Becca, Phys. Rev. B 80,
experimental angular dependence in Fig. 3. In this way 012404 (2009).
weobtainedD /(4~)=8 2andD /(4~)=11 2GHz, [8] K. Katsumata, J. Phys.: Condens. Matter 12, R589
a c
± ± (2000).
which corresponds to 0.29 0.07 and 0.39 0.07 T. We
± ± [9] T. Asano, H.Nojiri, Y.Inagaki, J. P. Boucher, T. Sakon,
havetonoteherethatforH ctheobservedsplittingof
k Y. Ajiro, and M. Motokawa, Phys. Rev. Lett.84, 5880
the frequencydoublet∆ν =0.35Tisabout50%smaller
(2000).
than the prediction of (4,5) with these Da/c values.This [10] M.Oshikawa,I.Affleck,Phys.Rev.B65,134410 (2002).
deviation from the predicted value is dicussed in [14]. [11] J. M. Schrama, A. Ardavan, A. V. Semeno, P. J. Gee,
A more detailed comparison between the experiment E. Rzepniewski, J. Suto, R. Coldea, J. Singleton, P. Goy
and the theory would require extending the latter to fi- Physica B 256-258, 637 (1998).
[12] L. V. Soboleva, L. M. Belyaev, V. V. Ogadzhanova, M.
nite temperatures since the reported measurements are
G. Vasil’eva, Kristallografia 24, 817 (1981).
performed in the intermediate temperature range T =
N [13] M. Sharnoff,J. Chem. Phys.44, 3383 (1964).
0.62K < T < T = 4K. Temperature dependence of
CW [14] see Supporting material for more details.
the gap ∆ for H b in a wider range including TN is [15] S. Gangadharaiah, J. Sun, O. A. Starykh, Phys. Rev.
k
shown on Fig.1. The variation of the gap in the range B78, 054436 (2008).
5
[16] V. K. Kalevich and V. L. Korenev, JETP Lett. 52, 230 [18] S. M. Frolov et al., Nature458, 868 (2009).
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[17] Z.Wilamowski,H.Malissa,F.Sch¨affler,andW.Jantsch, (1999).
Phys. Rev.Lett.98, 187203 (2007).
6
Supporting material for
“Modes of magnetic resonance in the spin liquid
phase of Cs CuCl .”
2 4
In this work we report results of electron spin reso-
nance (ESR) study of quasi two-dimensional (2D) an-
tiferromagnet Cs CuCl in the paramagnetic phase.
2 4
Cs CuCl realizes distorted triangular lattice structure,
2 4
shown in Fig.1, with the strong exchange J along the
base of the triangularcell and with the weaker exchange
integral J′ for the exchange paths along lateral sides of
the triangle, their ratio is J′/J = 0.34. In-chain DM
interaction
D =D ( 1)zaˆ+D ( 1)ycˆ,
y,z a c
− −
derived in Ref. 1, describes four possible orientations of
the D vector on the chain with integer indices y,z, see
Figure 1. Here y and z axes are aligned along a and
c crystallographic axis of an orthorombic Pnma lattice,
respectively.
FIG. 1. Parameters of spin interactions in Cs2CuCl4. Four
differentorientationsofthechainDMvectorDareillustrated FIG. 2. Upper panel: ν = 27 GHz resonance lines for the
by red arrows. magnetic field H in the bc-plane. The curves are taken at
the equidistant angles. Blue arrows for H k c curve indi-
catethecalculated resonance fieldsfor thesplitted ESRline.
Lower panel: The ψ angular dependence of the resonance
field in bc-plane for ν = 27 GHz at T = 1.3 K. Thick line
ANGULRAERSODNEPAENNCDEEFNICELEDSSOF THE iasndthDect/h(e4o~r)et=ica1l1cGaHlcuz.laDtioanshe(dseelinteexsth)owwsitchalDcuala/t(e4d~)an=gu8-
lardependenceforaparamagnet withg-factors ofCs2CuCl4,
Inadditiontotheangulardependenceoftheresonance crosses represent experimental values at T = 10 K (deep in
fieldontheangleφwithintheab-plane,presentedinthe theparamagnetic phase).
maintext,wepresentinFig.2theangulardependenceof
the resonance field on the angle ψ within the the bc-
plane. In this case the agreement is only qualitative: because the width of the individual lines is too broad.
the splitting at H c, obtained by a two-Lorentzian Nevertheless,theexpectedpositionsoffourlines,marked
fit constitutes aboutk50% of the predicted by the DM- byarrowsinFig.3,doliewithinthemeasuredabsorption
theory. Nonetheless, the predicted splitting of the reso- bands.
nance fields is within the broad band of absorption, ob-
servedatH c,asindicatedbybluearrowsintheupper
k
panel of Fig. 2. FREQUENCY-FIELD AND TEMPERATURE
Fig.3presentstheevolutionoftheabsorptionbandfor DEPENDENCES FOR Hkc
the field oriented within the ac-plane. According to our
theory (see main text) the ESR line should split into Thefrequency-fielddependenceforH catT =1.3K
k
four resonances,see Fig.4. We failedto resolvefourlines is shown in the Fig.5. At T = 1.3 K the splitting is
7
resolved only for two frequencies in the middle of the
frequency range. At low frequencies, the large linewidth

masksthesplitting, while athigherfrequenciesthesplit-
ting predicted by the DM-theory is not observed at this
temperature. However, at lower temperatures the split-
a
tingoftheESRlineforH cbecomesmorevisible. Itis o
k = 180
clearlyseenevenatthehighfrequencyedgeoftherange,
ν =76 GHz, as is seen in Fig. 6. The splitting observed
o
atT =0.79K equals0.6 0.1T, whichcorrespondssat- = 163
±
isfactorytothepredictedvalueof0.75T(horizontalshift o
= 146
between the asymptotic parts of dashed lines in Fig.5).
For H a andat T =0.8K, we observesimilar splitting o
k s) = 129
of 0.6 0.1 T for the frequency ν =72.9 GHz. nit
± u
b. o
ar = 111
er ( o
TEMPERATURE DEPENDENCE OF ESR LINES w = 90
Po
AT FREQUENCIES ABOVE AND BELOW THE d c
GAP ∆ FOR Hkb mitte = 77o
s
n o
Itwasmentionedinthe maintextthatatthe frequen- Tra = 60
cies below the DM gap, i.e. ν < ∆ = 14 GHz, the ESR o
= 43
line does not exhibit a shift but instead looses the inten-
o
sity. This observation is illustrated in Fig. 7, where the = 26
o
data for ν = 9.63 GHz ESR signal are presented. One = 9
can clearly see that the resonance field does not change
withloweringthe temperature,while the intensityofthe
line does diminish strongly.
To contrast this with quite different evolution of the =26.95 GHz
signalathigherfrequenciesν >∆,wepresentinFig.7de-
tailed temperature evolutionof the ESR line at ν =26.9
GHz. Here the shift of the line’s center to the lower val- 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
uesoftheresonancemagneticfieldisobvious. Thesignal Magnetic Field (T)
alsobroadenssuchthattheoverallintensityisconserved.
For other field orientations, in particular for H a,c, FIG. 3. The ESR lines for ν = 27 GHz and T = 1.3 K at
the ESR line completely disappears for low frequkencies different orientations of the static field within the ac-plane.
Arrowsmarktheresonances,expectedatT =0fordecoupled
ν <∆.
spin chainsaccording totheDMtheory (seemain text)with
Da/4~=8 and Dc/4~=11 GHz.
THEORETICAL DETAILS
Low-energy theory
Lattice Hamiltonian for DkH
The obtained Hamiltonian (1) can also be used to de-
rive the low-energy form of the DM coupling. This is
When H D, unitary rotation (3) of the paper trans- done by bosonizing (1) (with H =0) which results in [3]
k
forms chain Hamiltonian (2) into
v
˜ = dx (∂ ϕ˜)2+(∂ θ˜)2 , (2)
H Z 2{ x x }
˜ = J2+D2[S˜xS˜x +S˜yS˜y ]+JS˜zS˜z
H Xx {p x x+1 x x+1 x x+1 where(ϕ˜,θ˜)isthepairofconjugatedbosonicfieldswhich
gµ HS˜z, (1) describes low-energy excitations in the rotated basis. In
− B x
particular, the transverse component of the spin density
where for brevity we do not write chain’s integer co- operator (at momentum π) is expressed via θ˜as [3]
ordinates y,z. As claimed in the paper the obtained S˜+ =A eiβθ˜ (3)
Hamiltonian(1)possessesaneasy-planeanisotropyofthe x 3
strengthD2/2J. NeglectingittakesustotheHeisenberg where β =2πR determines scaling dimension ofthe spin
limitdiscussedinthepaperandillustratedinFig.5there. operator in terms of the known compactification radius
8
difference of the right andleft spin currents. This allows
us to identify the lattice andthe low-energyforms ofthe

1.4 Theory DM interaction as
1.3 , Paramagnetic, T=10 K
vD
1.2 Dzˆ S S dx (Mz Mz) (7)
· x× x+1 →− J Z R− L
T)1.1 Xx
d (1.0
el This result is written in terms of fluctuating spin densi-
gnetic Fi000...789 tfeiersm(isopnisnΨcuRr/rLenntsea[3r])thMeRari/gLhtd/elfienfted(Rin/Lt)erFmesrmofi(pDoiirnatcs),
Ma correspondingly. More precisely,
0.6
0.5 σa
0.4 MRa/L =Ψ†R/L;s s2s′ΨR/L;s′. (8)
0.3 a c a
0 30 60 90 120 150 180 It turns out that the spin-current formulation is partic-
Angle (deg.) ularly convenient for the problem at hand (see [5] for
pedagogicalexplanation of the technique).
FIG. 4. Resonance fields for ν = 27 GHz ESR as a function To complete the low-energy description we also need
oftheangleintheac-planeaccordingtotheDM-theorywith the low-energy form of the Zeeman coupling, which is
Da/4~ = 8 and Dc/4~ = 11 GHz. Dashed line shows calcu- well-knownandreads(forthemomentwekeepmagnetic
lated angular dependence for a paramagnet with g-factors of field along zˆaxis as well)
Cs2CuCl4,crossesrepresentexperimentalvaluesatT =10K
(deep in theparamagnetic phase) gµ HSz gµBH dx ∂ ϕ=
X B x → √2π Z x
x
R, see Ref.4, and A3 is non-universal amplitude. Also =gµ H dx (Mz +Mz). (9)
B Z R L
note that spinonvelocity v in (2) is function ofmagnetic
field. ItmonotonicallydecreaseswithH andreacheszero
In writing (2) we have omitted a number of marginal
at the saturation [4]. Near H = 0 the v(H) dependence
termswhichaccountfortheeasy-planeanisotropyaswell
is weakandwe cansafely approximatev byits zero-field
asresidualbackscatteringinteractionbetweenright-and
value everywhere in this study. Thus, when necessary,
left- spin-currents [2, 6]. These terms do not affect our
we use
main result - the splitting of the ESR signal into two
πJa due to the interplay of the external H and the internal
0
v = 2 ~ (4) DMDfields. Theyare,mostlikely,importantforunder-
standing the width of the individual ESR line, which is
forthevelocityv. Herea0 isthe latticeconstantinunits the problem of much greater theoretical complexity and
of which we measure coordinate x. is left for future studies.
Comparing (3) with equation (3) of the paper we im- With the help of (7) and (9) we can now write down
mediately observe that at low energies that unitary ro- thelow-energyHamiltonianoftheHeisenbergchainsub-
tation simply corresponds to the shift of momentum by ject to both Zeeman and DM interactions acting along
α= D/J (whichis exactlythe physicsdiscussedinthe different directions,
−
mainpaper). Hencebosonicfieldθ oftheoriginaltheory
and θ˜of the rotated one are simply related as = dx gµ H(Mh +Mh)+
Hchain H0−Z B R L
D
θ =θ˜ x. (5) dx (vD/J)(Md Md) (10)
− βJ −Z R− L
Hence the low-energy theory of the original Heisenberg where Hamiltonian of the ideal Heisenberg chain is
0
H
chainwithuniformDMinteractioncanbeobtainedfrom written in Sugawara form [3] as
(2) via (5) so that
2πv
= dx v (∂ ϕ)2+(∂ θ+ D )2 = H0 = 3 Z dx {MR·MR+ML·ML} (11)
H Z 2{ x x βJ }
v vD To connect with previous expressions, it is worth noting
=Z dx 2{(∂xϕ)2+(∂xθ)2}+ βJ∂xθ+const.(6) that (11) is in fact equivalent to the abelian bosoniza-
tion form (2). The more complicated-looking (11) has
Using that at H =0 β =√2π and Mz Mz = ∂ θ/β one significant advantage over (2): it makes evident
R− L − x
we find that the last term in (6) is proportional to the a remarkable emerging SU(2) SU(2) symmetry [3],
R L
×
9
which allows for independent rotations of the right- and
left- spin currents. This key for the following symmetry

emerges below the strong-coupling energy/temperature 90
scale T0 Je−πS (S = 1/2 for the spin chain here)[7], 80 H || c
∼
when the spin current formulation becomes appropri-
70
ate. Above that temperature a more conventional semi-
classical description of the spin-1/2 chain in terms of Hz)60
G
fluctuating N´eel vector order parameter field is valid [7]. y (50
c
In fact, one can think of T0 scale as of ‘Haldane’ tem- en40
u
perature below which the difference between integer and q
half-integer spin chains becomes pronounced. It is the Fre30
20
temperature (or, equivalently, energy) scale below which
the true spinon nature of the quantum ground state of 10
the critical spin-1/2 chain is manifest. 0
Note that upper indices d and h on currents M 0.0 0.5 1.0 1.5 2.0 2.5 3.0
R/L
in (10) denote directions of the vectors D and H. Ob- Magnetic Field (T)
servethatDMcoupling(lasttermin )isoddunder
chain
H
spatialinversionwhichinterchangesRandL–this sym- FIG.5. ESRfrequenciesat T =1.3 Kfor H kc. Dotted line
metry distinction is alreadyevidentin the lattice Hamil- is paramagnetic resonance with g = 2.30, dashed line – DM
tonian (2) of the main paper, see also (7). theory (see text),Da/4~=8 and Dc/4~=11 GHz.
ESR as a chiral probe
The remaining steps closely follows Ref. 2. We now nits) T=2.6 K
u
performindependentrotationsoftheright-andleft-cur- b. =75.88 GHz
rents in (10), so that in the rotated basis ar
er ( H || c
w T=1.27 K
o
Hchain =H0−gµBZ dx(BRMRz +BLMLz) (12) mitted P T=0.79 K
the Hamiltonian describes unusual situation whereby ns
right and left-moving currents experience different, both Tra T=0.68 K
in magnitude and direction, magnetic fields: B =
R
H +~vD/gµ J and B = H ~vD/gµ J. Observe
B L B
− 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
that such a rotation does not affect which retains its
0
H Magnetic Field (T)
quadratic in spin currents form (11).
It is now rather natural (see Refs. 2 and 8 for de-
tails) that the ESR response the spin chain consists of FIG. 6. ν = 75.88 GHz ESR lines at different temperatures
for H kc.
two peaks at frequencies (here we make use of (4))
2π~ ν =gµ B = gµ H πD/2. (13)
R/L B R/L B
| | | ± | fromchainswithoddy,zindices. Thus,thecrystalstruc-
TheobtainedresultincludesH DandH=0situations ture of the material masks the chiral nature of the ESR
k probe.
discusses in the main paper as special cases.
It is interesting to note that in the described case of a
single spin chainwith constantD vectorthe ESRexper-
iment becomes chiral probe: (13) shows that right- and Frequency dependence of the ESR response
left-movingexcitationscanbeaccessedindependentlyby
simply varying (and rotating) external magnetic field H We would like to describe here one possible reason for
withrespecttoDMaxisD. Inthe caseofCs2CuCl4 this a pronounced frequency dependence of the ESR signal
interestingpoint islostasthe DMaxis takesonfour dif- as described in Section above. Consider first H b
k
ferentdirections,dependingonthey,zcoordinatesofthe arrangement when our analysis predicts that the signal
spinchain. AsaresulttherightESRfrequency,equation is centered about the single frequency
(4) of the main paper, for, say, chain with even coordi-
natesy,z coincideswiththeleftfrequency(equation(5)) 2π~ ν = (gµ H)2+π2(D2+D2)/4. (14)
R/L B a c
p
10
pretty much all momenta, not restricted to the q = 0
limit in which the current field theory is formulated.
These thermal spinons are not particularly correlated
and can be thought of as forming an ideal gas of neu-
tral fermions. It seems physically sensible to think that
their ESR response is paramagnetic and not sensitive to
the low-energy effects due to the internal DM field. It is
this response that is measured at ν < ν in Fig. 7.
b,min
According to this argument the intensity of the param-
agnetic part of the signal should diminish with lowering
T,simplyfollowingdiminishingofthepopulationofther-
malspinons,inagreementwithourobservationinFig.7.
Thehigher-frequencyresponse(ν >ν )isinsteadthe
b,min
intrinsic property of the spin chain.
It is easy to extend this argument to other field direc-
tions. Let, for example, H a: then according to equa-
k
tion (5) of the paper we can achieve the situation when
g µ H = πD /2 and 2π~ν (z = odd) = 2π~ν (z =
a B a a R L
even) = πD /2, while 2π~ν (z = even) = 2π~ν (z =
c R L
odd) = π D2+4D2/2. Thus in this case ν =
c a a,min
D /(4~) anpd this condition is realized for 1/2 of chains
c
in the system. Similar argument shows that for H c
the minimal resonance frequency is ν = D /(4k~).
c,min a
Toreiterate,belowthe correspondingminimalresonance
frequency we expect standard paramagnetic ESR signal
due to thermally excited population of spinons, and the
intensity of such a signal should decrease with lowering
FIG. 7. Upper panel: set of 9.63 GHz ESR lines at different
temperatures for H k b. The narrow resonance at ∼ 0.35 T of the temperature.
is the signal of g=2.0 DPPH marker; a tiny resonance, ap-
It is interesting to note that there is one ‘magic’ ori-
pearing only at low temperatures, is due to a paramagnetic
entation of the external field for which ν = 0. This
impurities. Lowerpanel: setof26.92GHzESRlinesatdiffer- min
ent temperatures for H k b. The narrow resonance at ∼ 0.9 happens for gaµBHa = πDa/2,Hb = 0 and gcµBHc =
T is the DPPH signal. πDc/2, in which case 1/4 of the chains experiences zero
total field. It would be interesting to study this special
orientation in more details in the future.
This is just equation (1) of the main paper (see also dis-
cussion following equations (4,5) there). This equation
clearly states that there is a minimal ESR frequency,
given by ν = ∆/(2π~) = D/(4~), below which
b,min
∗
no resonance is possible in this geometry. Experimen- [email protected]
tally the resonance is observed both above and below [1] O. A. Starykh, H. Katsura, L. Balents, Phys. Rev. B82,
014421 (2010).
ν , but with markedly different temperature depen-
b,min [2] S.Gangadharaiah,J.Sun,O.A.Starykh,Phys.Rev.B78,
dencies. The low-frequency (ν < ν ) response re-
b,min 054436 (2008).
mains paramagnetic-like and quickly looses intensity on
[3] A. O. Gogolin, A. A. Nersesyan, and A. M. Tsve-
lowering T, see Figure 7. On the contrary, the high- lik, Bosonization approach to strongly correlated systems,
frequency response (Figure 7) follows prediction (14) Cambridge Univ.Press (1999).
rather nicely, as detailed in the main text. [4] I.AffleckandM.Oshikawa,Phys.Rev.B60,1038(1999).
[5] O. A. Starykh, Ch. 30 in Handbook of nanophysics: nan-
We’d like to argue that the observed difference is a
otubes and nanowires, Ed. K. D. Sattler, CRC Press
finite-T effect. Eq. (14) describes idealized T = 0 sit-
(2010).
uation of completely decoupled spin chains, to which
[6] I. Garate and I.Affleck, Phys.Rev.B81, 144419 (2010).
Cs2CuCl4 is only an approximation in the intermedi- [7] C. Buragohain and S. Sachdev, Phys. Rev. B59, 9285
atetemperaturewindowbetweenT0 discussedaboveand (1999).
the orderingtemperature T . In this temperature inter- [8] A. De Martino, R. Egger, F. Murphy-Armando, and K.
N
val one expects sizable thermal population of spinons at Hallberg, J. Phys.: Condens. Matter 16, S1437 (2004).