Modes of magnetic resonance in the spin liquid phase of Cs{}_{2}CuCl{}_{4}

Modes of magnetic resonance in the spin liquid phase of CsCuCl

K. Yu. Povarov P. L. Kapitza Institute for Physical Problems, RAS, 119334 Moscow, Russia    A. I. Smirnov P. L. Kapitza Institute for Physical Problems, RAS, 119334 Moscow, Russia Moscow Institute for Physics and Technology , 141700, Dolgoprudny, Russia    O. A. Starykh Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA Max-Planck-Institut für Physik Komplexer Systeme, D-01187 Dresden, Germany    S. V. Petrov P. L. Kapitza Institute for Physical Problems, RAS, 119334 Moscow, Russia    A. Ya. Shapiro A.V.Shubnikov Institute of Crystallography, RAS, 119333 Moscow, Russia
July 19, 2019

We report the observation of a frequency shift and splitting of the electron spin resonance (ESR) mode of the low-dimensional frustrated antiferromagnet CsCuCl in the spin-correlated state below the Curie-Weiss temperature 4 K but above the ordering temperature 0.62 K. The shift and splitting exhibit strong anisotropy with respect to the direction of the applied magnetic field and do not vanish in zero field. The low-temperature evolution of spin resonance response is a result of the specific modification of one-dimensional spinon continuum under the action of the uniform Dzyaloshinskii-Moriya interaction (DM) within the spin chains. Parameters of the uniform DM interaction are derived from the experiment.

76.30.-v, 75.40.Gb, 75.10.Jm

Quantum antiferromagnets on triangular lattice exhibit a variety of unusual phases with intriguing physical properties. In this work we report results of electron spin resonance (ESR) study of quasi two-dimensional (2D) triangular antiferromagnet CsCuCl in the paramagnetic phase. CsCuCl realizes distorted triangular lattice with strong exchange along the base of the triangular unit and weaker exchange integral along lateral sides of the triangle. The material orders below K into a long-range 3D spiral state ColdeaPRL (). This Néel point is far below the Curie-Weiss temperature =4 K. In the intermediate temperature range a spin-liquid phase with strong in-chain spin correlations takes place. Magnetic properties of CsCuCl are well described by a quasi-one-dimensional (1D) model of weakly coupled spin chains. In particular, inelastic neutron scattering experiments ColdeaPRL (); ColdeaPRB () are nicely explained by a parameter-free theory of dispersing continuum of one-dimensional spinons kohno2007 (). Moreover, the same description, extended in Ref. Starykh () to include residual symmetry-allowed Dzyaloshinskii-Moriya (DM) interactions between the Cu-spins, is able to consistently explain majority of the phases in the phase diagram, for three different directions of the magnetic field TokiwaPhase (). In addition, several numerical investigations sheng (); Sorella () of the spatially anisotropic Heisenberg model also find that in a wide region of the parameter space the spin chains are almost decoupled and inter-chain spin correlations are exponentially weak. This behavior follows from highly frustrated structure of the inter-chain exchange and provides additional justification to the 1D-based approach Starykh () to CsCuCl.

Electron spin resonance (ESR) represents a sensitive probe of low-energy magnetic excitations with zero momenta, . It is well known as a method providing fine details of low-energy spin spectra in both ordered and paramagnetic esrbook (); Ajiro (); OshikawaAffleck () phases. Importantly, both the position and the linewidth of the ESR signal are crucially sensitive to anisotropies of the spin Hamiltonian. Here we report a dramatic manifestation of the uniform Dzyaloshinskii-Moriya (DM) interaction on the ESR spectrum of CsCuCl in the paramagnetic phase. We observe and explain splitting of the ESR mode into two modes with lowering the temperature, significantly extending previous study semeno (), and also find strong dependence of the ESR absorption on the polarization of the microwave magnetic field with respect to the DM vector in zero external field.

The crystal samples were grown by two methods. First set of samples was prepared by the crystallization from the melt of a stoichiometric mixture of CsCuCl and CsCl in an ampule, which was slowly pulled from the hot (T= 520C) area of the oven. The second method was crystallization from solution Soboleva (). The room temperature crystal lattice parameters for both sets of crystals are a=9.75 Å, b=7.607 Å, c=12.394Å. The samples obtained by last method have natural faces and are lengthened along b-axes. Both sets of crystals demonstrated identical ESR signals. The study in the frequency range 9-90 GHz was performed with a set of homemade spectrometers equipped with cryomagnets. The crystal samples were placed within microwave resonators at the maximum of the microwave magnetic field. The resonance lines were taken at the fixed frequency, as dependences of microwave power, transmitted through the resonator, on the external magnetic field. Temperatures down to 1.3 K (0.4 K), were obtained by pumping vapor of He (He), correspondingly. Cooling down to 0.1 K was achieved with the help of a dilution cryostat Kelvinox-400.

At temperatures above K crystals of CsCuCl exhibit conventional paramagnetic resonance of Cu magnetic ions with a single narrow line corresponding to -factor values , , for the magnetic field applied along , and axes correspondingly. These -factor values agree well with those reported earlier at K and K Sharnoff (), and, for , at K semeno ().

Figure 1: ESR frequency at K for . Dotted line is paramagnetic resonance with , solid line – theory (see text). Empty circles stand for resonances loosing intensity with cooling. Upper insert: gap vs temperature, various symbols correspond to different cryostats. Lower insert: evolution of the resonance line with cooling.
Figure 2: ESR frequencies at K for . Dotted line is paramagnetic resonance with , dashed line – theory (see text). Upper insert: lineshape at K, GHz. Points – experimental data, solid line is the result of a fit by the sum of two Lorentzians (dashed lines). Lower insert: evolution of the resonance line with cooling.

Significant evolution of the ESR spectrum was found on cooling below 6 K. This evolution depends strongly on the microwave frequency and orientation of the external magnetic field . (i) For , we observe a single Lorentzian line shifting with cooling to lower fields (see Fig. 1). At =14 GHz the resonance field is near zero. At lower frequencies, we observe a strong decrease of the intensity with cooling, and no visible shift from the paramagnetic resonance field, in contrast to ESR lines taken at 17 GHz, which exhibit a shift and do not loose intensity at cooling. At =9 GHz the integral intensity at T=1.3 K is a half of that at T=4 K. The resonant frequencies above 14 GHz may be well fitted by the frequency-field relation of an ordered antiferromagnet with “gap” GHz,


(ii) For , the ESR line strongly broadens on cooling, and its shape distorts, as shown in the inserts of Fig. 2. The absorption at =1.3 K may be fitted by a sum of two Lorentzians, indicating the splitting of the resonance mode. The dependence at is presented in Fig. 2. Two modes at 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 doublet, while at lower frequencies the signal becomes too broad. The angular dependence of the resonance field within the plane, taken at =27 GHz and =1.3 K, is shown in Fig.3. For the splitting of about 0.35 T is resolved for frequencies 27 and 31 GHz.

(iii) We have also studied polarization dependence of the ESR absorption at . Fig.  4 shows transmission vs field dependences at K and K for different orientations of the microwave and static fields with respect to the crystal axes, but with . The measurements are done at GHz. At K both the position and the lineshape of the paramagnetic resonance line are the same for all three principal polarizations of the microwave field. However, at K, the absorption in zero field for is at least three times more intensive than for .

The splitting of the ESR line for and the polarization dependence of the signal were also detected previously in Ref. semeno ().

Figure 3: The angular dependence of the resonance field in the plane for GHz at K. Thick line is the theoretical prediction with and GHz. The measured values of the resonance field deep in the paramagnetic phase (at K) are presented by crosses, the dashed line is a theoretical fit of a paramagnetic resonance with of CsCuCl.
Figure 4: The polarization effect at GHz for . Circles with error bars show the uncertainty resulting from normalizing the signals to zero absorption at T.

We now turn to explanation of the data and show that all unusual features of the ESR signal reported above are naturally explained by quasi-1D nature of CsCuCl. The relevant Hamiltonian


contains three terms: the first describes intrachain exchange ( runs along crystal axis), the second – uniform DM interaction between chain spins, and the third is the usual Zeeman term, allowing for anisotropic -factor. The dots stand for the omitted interchain exchange as well as DM interactions on interchain bonds. Detailed symmetry analysis of the allowed DM interactions Starykh () shows that there are four different orientations of the DM vector (see Fig. 6 in Starykh ()) depending on chain’s integer coordinates : . Here indices magnetic layers, while numerates chains within a layer. Crucially, crystal symmetry forbids DM vector to have component along the chain axis.Starykh ()

The essence of the observed ESR line splitting can be explained by considering a single Heisenberg chain with uniform DM interaction with vector along the local axis, when the field is lined up along the same axis (, ). In this geometry the DM interaction can be gauged away by position-dependent rotation of lattice spins,


The rotation angle is chosen so as to eliminate the DM coupling from the Hamiltonian. The transformed describes spin chain with quadratic in easy-plane anisotropy. To linear in accuracy the ESR response of this model coincides with that of an isotropic Heisenberg chain, detailed study of which is described in Ref.OshikawaAffleck (). ESR absorption is determined by the transverse structure factor = of the chain, evaluated at . This is produced by characteristic to the spin-1/2 chain spinon continuum and is described by a sum of delta-function peaks at frequencies , see eq.(3.16) of OshikawaAffleck () and Fig.5. Here is the zero-field spinon velocity and is the lattice spacing support (). The response in the original (un-rotated) basis is obtained by un-doing the momentum boost (3), which corresponds to setting in the expression for the structure factor . This immediately implies the splitting of the ESR line into two lines, at frequencies . Note that for small the splitting is linear in which justifies our neglect of the easy-plane anisotropy.

While the four-sublattice structure of the DM vectors makes it impossible to study the configuration in CsCuCl, the above argument allows one to immediately understand polarization-dependent absorption (finding (iii) above) in zero magnetic field, at the frequency . Since the microwave absorption is proportional to the square of the microwave field component perpendicular to the effective field, we conclude that configuration with (that is, ) should result in maximal possible absorption. For along the () axis, absorption is a factor () smaller. Using numerical estimates of derived below, the absorption for different polarizations should obey the relations: and . This agrees qualitatively with our data presented in Fig. 4 – the zero field absorption at and GHz is a factor of 3 more intensive for than for .

To understand our findings (i) and (ii), one needs to analyze the arbitrary orientation of the and vectors, which is the problem solved in Starykhchains (). The main physical point of Ref.Starykhchains () consists in the observation that a uniform DM interaction, much like a spin-orbit Rashba interaction in 2D conductors kalevich (); jantsch (); folk (), acts on spinon excitations of 1D chains as an internal momentum-dependent magnetic field. Technically, this observation follows from spin-current formulation support () of the problem (2), which is valid below the strong-coupling temperature scale ( for the spin chain here)buragohain (). The spin currents represent spin density fluctuations of spinons near the right/left (R/L) Fermi points of a 1D system. It follows that the right and left-moving currents experience different, in magnitude and direction, total magnetic fields: and . It is then natural that the ESR response of such a chain consists of two peaks at frequencies .Starykhchains ()

For the CsCuCl-specific configuration of chain DM interactions and our analysis predicts


Observe that while for fixed chain indices , these chiral ESR frequencies transform into each other under lattice translations . These equations naturally explain the difference between and situations. For the external and the internal DM field are mutually perpendicular and the vector sum has the the same absolute value for both signs. One then finds single ESR frequency of the form (1) where the gap is in fact determined by DM interaction, .

For the DM field has a component along and the frequencies of the two spin excitations are different. In principle, because of the four values of the vector, four different frequencies may be present when the field is oriented within the plane. We attempted to check this experimentally but failed to resolve separate lines within the broad band of () For within the plane (4) and (5) predict two different frequencies. Solving equations (4,5) at the fixed frequency for the magnetic field we were able to fit the experimental angular dependence in Fig. 3. In this way we obtained and GHz, which corresponds to and T. We have to note here that for the observed splitting of the frequency doublet T is about smaller than the prediction of (4,5) with these values.This deviation from the predicted value is dicussed in support ().

Figure 5: Spinon spectrum of Heisenberg chain for with (solid lines) and without (dashed lines) DM interaction. When , momentum is boosted by .

A more detailed comparison between the experiment and the theory would require extending the latter to finite temperatures since the reported measurements are performed in the intermediate temperature range . Temperature dependence of the gap for in a wider range including is shown on Fig.1. The variation of the gap in the range 1 K2 K is much slower than right near . This supports our main assumption that the reported spectral features of CsCuCl reflect specific spin chain physics which dominates the temperature range .

In conclusion, we demonstrated that the uniform DM interaction, which is a distinctive feature of CsCuCl, results in a new kind of spin resonance in a S=1/2 chain antiferromagnet. The observed spectrum is a consequence of the splitting of the chain’s spinon continuum by the internal magnetic field which is produced by the uniform DM interaction. We believe that a similar phenomenon is possible in a higher-dimensional magnetic systems with fractionalized spinon excitations. Our findings differ strongly from the well-known single-frequency resonance of antiferromagnetic spin chain with staggered DM interaction OshikawaAffleck (). It also differs from a conventional 3D antiferromagnet which acquires an energy gap in the spectrum only below the ordering transition.

We would like to thank K.O. Keshishev for guidance in work with the dilution refrigerator, and L. Balents, S. Gangadharaiah, V.N. Glazkov, M. Oshikawa, and R. Moessner for insightful discussions. We thank RFBR Grant 09-02-736 and NSF Grant No. DMR-0808842 (O.A.S.) for financial support. Part of the work has been performed within the ASG on “Unconventional Magnetism in High Fields” at the MPIPKS (Dresden).


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Supporting material for
“Modes of magnetic resonance in the spin liquid phase of CsCuCl.”

In this work we report results of electron spin resonance (ESR) study of quasi two-dimensional (2D) antiferromagnet CsCuCl in the paramagnetic phase. CsCuCl realizes distorted triangular lattice structure, shown in Fig.1, with the strong exchange along the base of the triangular cell and with the weaker exchange integral for the exchange paths along lateral sides of the triangle, their ratio is . In-chain DM interaction

derived in Ref. Starykh2, , describes four possible orientations of the vector on the chain with integer indices , see Figure 1. Here and axes are aligned along and crystallographic axis of an orthorombic Pnma lattice, respectively.

Figure 1: Parameters of spin interactions in CsCuCl. Four different orientations of the chain DM vector are illustrated by red arrows.

Appendix A Angular dependences of the resonance fields

In addition to the angular dependence of the resonance field on the angle within the -plane, presented in the main text, we present in Fig.2 the angular dependence of the resonance field on the angle within the the -plane. In this case the agreement is only qualitative: the splitting at , obtained by a two-Lorentzian fit constitutes about 50% of the predicted by the DM-theory. Nonetheless, the predicted splitting of the resonance fields is within the broad band of absorption, observed at , as indicated by blue arrows in the upper panel of Fig. 2.

Fig.3 presents the evolution of the absorption band for the field oriented within the -plane. According to our theory (see main text) the ESR line should split into four resonances, see Fig.4. We failed to resolve four lines because the width of the individual lines is too broad. Nevertheless, the expected positions of four lines, marked by arrows in Fig.3, do lie within the measured absorption bands.

Figure 2: Upper panel: GHz resonance lines for the magnetic field in the -plane. The curves are taken at the equidistant angles. Blue arrows for curve indicate the calculated resonance fields for the splitted ESR line. Lower panel: The angular dependence of the resonance field in -plane for GHz at K. Thick line is the theoretical calculation (see text) with and GHz. Dashed line shows calculated angular dependence for a paramagnet with g-factors of CsCuCl, crosses represent experimental values at K (deep in the paramagnetic phase).

Appendix B Frequency-field and temperature dependences for

The frequency-field dependence for at is shown in the Fig.5. At K the splitting is resolved only for two frequencies in the middle of the frequency range. At low frequencies, the large linewidth masks the splitting, while at higher frequencies the splitting predicted by the DM-theory is not observed at this temperature. However, at lower temperatures the splitting of the ESR line for becomes more visible. It is clearly seen even at the high frequency edge of the range, GHz, as is seen in Fig. 6. The splitting observed at K equals T, which corresponds satisfactory to the predicted value of T (horizontal shift between the asymptotic parts of dashed lines in Fig.5). For and at K, we observe similar splitting of T for the frequency GHz.

Appendix C Temperature dependence of ESR lines at frequencies above and below the gap for

It was mentioned in the main text that at the frequencies below the DM gap, i.e. GHz, the ESR line does not exhibit a shift but instead looses the intensity. This observation is illustrated in Fig. 7, where the data for GHz ESR signal are presented. One can clearly see that the resonance field does not change with lowering the temperature, while the intensity of the line does diminish strongly.

To contrast this with quite different evolution of the signal at higher frequencies , we present in Fig.7 detailed temperature evolution of the ESR line at GHz. Here the shift of the line’s center to the lower values of the resonance magnetic field is obvious. The signal also broadens such that the overall intensity is conserved.

For other field orientations, in particular for , the ESR line completely disappears for low frequencies .

Figure 3: The ESR lines for GHz and K at different orientations of the static field within the -plane. Arrows mark the resonances, expected at for decoupled spin chains according to the DM theory (see main text) with and GHz.
Figure 4: Resonance fields for GHz ESR as a function of the angle in the -plane according to the DM-theory with and GHz. Dashed line shows calculated angular dependence for a paramagnet with g-factors of CsCuCl, crosses represent experimental values at K (deep in the paramagnetic phase)

Appendix D Theoretical details

d.1 Lattice Hamiltonian for

When , unitary rotation (3) of the paper transforms chain Hamiltonian (2) into


where for brevity we do not write chain’s integer coordinates . As claimed in the paper the obtained Hamiltonian (1) possesses an easy-plane anisotropy of the strength . Neglecting it takes us to the Heisenberg limit discussed in the paper and illustrated in Fig.5 there.

d.2 Low-energy theory

The obtained Hamiltonian (1) can also be used to derive the low-energy form of the DM coupling. This is done by bosonizing (1) (with ) which results in gogolin_book2 ()


where is the pair of conjugated bosonic fields which describes low-energy excitations in the rotated basis. In particular, the transverse component of the spin density operator (at momentum ) is expressed via as gogolin_book2 ()


where determines scaling dimension of the spin operator in terms of the known compactification radius , see Ref.oa1999-2, , and is non-universal amplitude. Also note that spinon velocity in (2) is function of magnetic field. It monotonically decreases with and reaches zero at the saturation oa1999-2 (). Near the dependence is weak and we can safely approximate by its zero-field value everywhere in this study. Thus, when necessary, we use


for the velocity . Here is the lattice constant in units of which we measure coordinate .

Comparing (3) with equation (3) of the paper we immediately observe that at low energies that unitary rotation simply corresponds to the shift of momentum by (which is exactly the physics discussed in the main paper). Hence bosonic field of the original theory and of the rotated one are simply related as


Hence the low-energy theory of the original Heisenberg chain with uniform DM interaction can be obtained from (2) via (5) so that


Using that at and we find that the last term in (6) is proportional to the difference of the right and left spin currents. This allows us to identify the lattice and the low-energy forms of the DM interaction as


This result is written in terms of fluctuating spin densities (spin currents gogolin_book2 ()) defined in terms of (Dirac) fermions near the right/left (R/L) Fermi points, correspondingly. More precisely,


It turns out that the spin-current formulation is particularly convenient for the problem at hand (see osnano2 () for pedagogical explanation of the technique).

To complete the low-energy description we also need the low-energy form of the Zeeman coupling, which is well-known and reads (for the moment we keep magnetic field along axis as well)


In writing (2) we have omitted a number of marginal terms which account for the easy-plane anisotropy as well as residual backscattering interaction between right- and left- spin-currents Starykhchains2 (); garate2 (). These terms do not affect our main result - the splitting of the ESR signal into two due to the interplay of the external and the internal DM fields. They are, most likely, important for understanding the width of the individual ESR line, which is the problem of much greater theoretical complexity and is left for future studies.

With the help of (7) and (9) we can now write down the low-energy Hamiltonian of the Heisenberg chain subject to both Zeeman and DM interactions acting along different directions,


where Hamiltonian of the ideal Heisenberg chain is written in Sugawara form gogolin_book2 () as


To connect with previous expressions, it is worth noting that (11) is in fact equivalent to the abelian bosonization form (2). The more complicated-looking (11) has one significant advantage over (2): it makes evident a remarkable emerging SU(2) SU(2) symmetry gogolin_book2 (), 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 scale ( for the spin chain here)buragohain2 (), when the spin current formulation becomes appropriate. Above that temperature a more conventional semi-classical description of the spin-1/2 chain in terms of fluctuating Néel vector order parameter field is valid buragohain2 (). In fact, one can think of scale as of ‘Haldane’ temperature below which the difference between integer and half-integer spin chains becomes pronounced. It is the temperature (or, equivalently, energy) scale below which the true spinon nature of the quantum ground state of the critical spin-1/2 chain is manifest.

Note that upper indices and on currents in (10) denote directions of the vectors and . Observe that DM coupling (last term in ) is odd under spatial inversion which interchanges and – this symmetry distinction is already evident in the lattice Hamiltonian (2) of the main paper, see also (7).

d.3 ESR as a chiral probe

The remaining steps closely follows Ref. Starykhchains2, . We now perform independent rotations of the right- and left- currents in (10), so that in the rotated basis


the Hamiltonian describes unusual situation whereby right and left-moving currents experience different, both in magnitude and direction, magnetic fields: and . Observe that such a rotation does not affect which retains its quadratic in spin currents form (11).

It is now rather natural (see Refs. Starykhchains2, ; demartino2, for details) that the ESR response the spin chain consists of two peaks at frequencies (here we make use of (4))


The obtained result includes and situations discusses in the main paper as special cases.

It is interesting to note that in the described case of a single spin chain with constant vector the ESR experiment becomes chiral probe: (13) shows that right- and left- moving excitations can be accessed independently by simply varying (and rotating) external magnetic field with respect to DM axis . In the case of CsCuCl this interesting point is lost as the DM axis takes on four different directions, depending on the coordinates of the spin chain. As a result the right ESR frequency, equation (4) of the main paper, for, say, chain with even coordinates coincides with the left frequency (equation (5)) from chains with odd indices. Thus, the crystal structure of the material masks the chiral nature of the ESR probe.

d.4 Frequency dependence of the ESR response

Figure 5: ESR frequencies at K for . Dotted line is paramagnetic resonance with , dashed line – DM theory (see text), and GHz.
Figure 6: GHz ESR lines at different temperatures for .

We would like to describe here one possible reason for a pronounced frequency dependence of the ESR signal as described in Section C above. Consider first arrangement when our analysis predicts that the signal is centered about the single frequency


This is just equation (1) of the main paper (see also discussion following equations (4,5) there). This equation clearly states that there is a minimal ESR frequency, given by , below which no resonance is possible in this geometry. Experimentally the resonance is observed both above and below , but with markedly different temperature dependencies. The low-frequency () response remains paramagnetic-like and quickly looses intensity on lowering , see Figure 7. On the contrary, the high-frequency response (Figure 7) follows prediction (14) rather nicely, as detailed in the main text.

Figure 7: Upper panel: set of 9.63 GHz ESR lines at different temperatures for . The narrow resonance at T is the signal of g=2.0 DPPH marker; a tiny resonance, appearing only at low temperatures, is due to a paramagnetic impurities. Lower panel: set of 26.92 GHz ESR lines at different temperatures for . The narrow resonance at T is the DPPH signal.

We’d like to argue that the observed difference is a finite-T effect. Eq. (14) describes idealized situation of completely decoupled spin chains, to which CsCuCl is only an approximation in the intermediate temperature window between discussed above and the ordering temperature . In this temperature interval one expects sizable thermal population of spinons at pretty much all momenta, not restricted to the 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 neutral 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 in Fig. 7. According to this argument the intensity of the paramagnetic part of the signal should diminish with lowering T, simply following diminishing of the population of thermal spinons, in agreement with our observation in Fig. 7. The higher-frequency response () is instead the intrinsic property of the spin chain.

It is easy to extend this argument to other field directions. Let, for example, : then according to equation (5) of the paper we can achieve the situation when and , while . Thus in this case and this condition is realized for of chains in the system. Similar argument shows that for the minimal resonance frequency is . To reiterate, below the corresponding minimal resonance 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 of the temperature.

It is interesting to note that there is one ‘magic’ orientation of the external field for which . This happens for and , in which case of the chains experiences zero total field. It would be interesting to study this special orientation in more details in the future.


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