Nature of the low energy excitations in the short range ordered region of Cs$_2$CuCl$_4$ as revealed by $^{133}$Cs NMR

We report nuclear magnetic resonance measurements of the spin-1/2 anisotropic triangular lattice antiferromagnet Cs$_2$CuCl$_4$ as a function of temperature and applied magnetic field. The observed temperature and magnetic field dependence of the NMR relaxation rate suggests that low energy excitations in the short-range ordered region stabilized over a wide range of intermediate fields and temperatures of the phase diagram are gapless or nearly gapless fermionic excitations. An upper bound on the size of the gap of 0.037 meV $\approx J/10$ is established. The magnetization and NMR relaxation rate can be qualitatively described either by a quasi-1D picture of weakly coupled chains, or by mean-field theories of specific 2D spin liquids; however, quantitative differences exist between data and theory in both cases. This comparison indicates that 2D interactions are quantitatively important in describing the low-energy physics.


Introduction
The emergence of particles with fractional quantum numbers is among the most remarkable phenomena that may arise in strongly correlated electron systems. In one spatial dimension the exact Bethe ansatz solution of the spin-1/2 Heisenberg antiferromagnetic chain provides a prototype as the solution exhibits a non-magnetic spin liquid ground state with deconfined spin-1/2 (spinon) excitations [1,2]. These excitations carry half of the local spin degree of freedom ∆S = ±1.
The existence of spinons as fractional elementary excitations has also been well established experimentally in quasi-1D antiferromagnets [3].
The search for examples of spin fractionalization has turned to dimensions greater than one. Inelastic neutron scattering measurements on the two dimensional frustrated quantum antiferromagnet (AF) Cs 2 CuCl 4 with spins on an anisotropic triangular lattice have shown dominant continua of excitations as characteristic of spin-1/2 spinon quasiparticles [4,5] and this has stimulated intense theoretical work to explain these findings [6,7,8,9,10,11,12]. Cs 2 CuCl 4 is a hard insulator with orthorhombic space group where magnetic spin-1/2 Cu atoms form a linear chain, with coupling J = 0.375 meV, in theb-direction. Chains are stacked together along theĉ-axis separated by a distance of b/2 and with coupling J = 0.125 meV [13,14]. Thus, the spins form a frustrated anisotropic triangular lattice, as illustrated in Fig. 1(b). Small interplane coupling J = 0.017 meV (a-axis) stabilizes long range spiral order below 0.62 K in zero applied field [14]. Schematic phase diagram of Cs 2 CuCl 4 is shown in Fig. 1(a). A small Dzyaloshinskii-Moriya (DM) interaction D ∼ 5%J (which we neglect in the calculations presented below) is present along the interchain links. At intermediate temperatures (T ≈ 400 mK to 2.5 K) a short-range ordered (SRO) region is stabilized upon application of a magnetic field (H = Hẑ) of sufficient strength along any of the three crystalline axes [4,5,15]. The magnetic field breaks the full spinrotational symmetry of an isotropic system, but the subgroup of U(1) rotations in the plane perpendicular to the field remains unbroken in the SRO region.
Kohno, Starykh, and Balents [11] have shown that many features of the experimental data in Cs 2 CuCl 4 can be understood by viewing the anisotropic triangular lattice of spins as a system of weakly-coupled one-dimensional Heisenberg chains. The work was recently extended to include the effects of an external magnetic field [12] but not yet non-zero temperatures. As our focus here is on the SRO regime at intermediate temperatures, we also compare data to alternative relevant models of spin liquid states previously proposed for the 2D anisotropic triangular lattice, where specific predictions of non-zero temperature properties are available. We consider two different spin liquid states, with elementary excitations that obey either bosonic or fermionic statistics, proposed earlier to describe the spin-1/2 Heisenberg AF on anisotropic triangular lattices. The bosonic Sp(N) large-N mean-field theory supports bosonic spinons that generically have a gap in the excitation spectrum [6,17,18]. By contrast another family of mean-field theories support fermionic spinons with no gap in the excitation spectrum  [15,16]. The green lines indicate the maximum in the temperature dependence of susceptibility curves and are interpreted as indication of the cross-over from the paramagnetic (PM) to the short range ordered (SRO) region. First and second order phase transitions are presented by red and blue lines, respectively. At temperatures below those delineated by blue lines different long range order phases are stabilized. For field above 8 T at low temperatures fully polarized (FP) state is formed. (b) Triangular magnetic lattice formed by Cu spins (Cu 2+ ions) displayed as small blue spheres with exchange couplings of J = 0.375 meV along solid lines and J = 0.125 meV along dashed lines. Our NMR measurements discussed in this paper were performed on Cs(A) site depicted by red spheres. [6,7,19,20]. In the context of those theories (which assume non-interacting spinons), one can ask which description (bosons or fermions) yields a better mean-field description. Those two models have different temperature-dependence of the NMR relaxation rate, which we probe directly in the experiments.
Here we measure the local magnetization and the nuclear magnetic resonance (NMR) relaxation rate to probe the magnetic behavior across the full phase diagram in applied field including the magnetically-ordered phase at low temperatures, the shortrange ordered (SRO) region above T N , and the high-temperature paramagnetic region. We focus on the properties in the SRO region of the phase diagram where spins are strongly correlated, but not long-range ordered. By considering the temperature and field dependence of the rate we deduce that the low energy excitations in the SRO region are best characterized as gapless fermionic excitations. Furthermore, we compare our results with previous data on 1D chain materials [21,22] and available theoretical models. The quasi-1D picture proposed in Ref. [11] that includes in a consistent way the effects of the strong, but frustrated interchain couplings has not yet been extended to cover the intermediate temperature and field range. As most of our data was collected in this intermediate temperature and field range, we compare the results with alternative models of spin liquid phases for the anisotropic triangular lattice antiferromagnet with fermionic excitations. Comparison of the NMR data to mean-field descriptions based upon variational calculations using Gutzwiller-projected wavefunctions implies that in Cs 2 CuCl 4 in the SRO region (at non-zero temperature and applied field) 2D interactions are important for a quantitative understanding of the low energy properties. The rest of the paper is organized as follows. In Sec. II technical details of the NMR experimental setup are described. In Sec. III we present the results for the temperature and magnetic field dependence of the NMR line and comparison with experiments on 1D materials and with theoretical models. Finally conclusions are summarized in Sec. IV.

Experiment
We used solution-grown single crystals of Cs 2 CuCl 4 . The measurements were conducted at the National High Magnetic Field Laboratory (NHMFL) using a 17 T sweepable magnet. We present data on one of the two magnetically inequivalent Cs sites, labeled Cs(A) and believed to be a better probe of the magnetism of Cu +2 ions due to its stronger hyperfine coupling [23]. At low temperatures (T 20 K) quadrupolar effects on 133 Cs (I = 7/2) NMR are masked by the dominant magnetic broadening [23]. The NMR relaxation rate (T −1 1 ) was measured as described in detail in Ref. [23]. In essence, the magnetization was saturated by applying a train of pulses equally spaced by a time t < T 2 at different frequencies across the magnetically broadened line. Following the saturation pulse train, the signal was detected after a variable delay time using a standard spin echo sequence (π/2 − τ − π). The Knight shift K ≡ (ω N − Hγ)/Hγ is obtained from the frequency of the first spectral moment ω N using the gyromagnetic ratio 133 γ = 5.5844 MHz/T. The shift also provides a direct measure of the local magnetization m loc = K · H/A zz , since they are linearly related via the strength of the transferred hyperfine tensor (A zz ) [24]. For H||ĉ, the relevant component of the hyperfine tensor ((A cc )) for determining m loc equals to 1.23 T/µ B [24]. The only nonzero off-diagonal element of this symmetric hyperfine tensor is A ac = ±0.185 T/µ B [24].

Results and Discussion
Details of the temperature and applied field dependence of the NMR shift and the rate are discussed in order to delineate different regions in the phase diagram of this frustrated magnetic system. Notably, we identify the temperature and applied field region where a state characterized by short-range antiferromagnetic correlations is stabilized and we find good agreement with results from bulk magnetization [15] and neutron scattering experiments [4]. Although this region cannot be distinguished from the paramagnetic one by any change of symmetry, it is appropriate to refer to it as SRO because spins are strongly correlated but not in a long range ordered state. We now discuss some further ways that our experiment distinguishes between the SRO and paramagnetic regions.

Temperature Dependence
The temperature dependence of the shift at different H is illustrated in Fig. 2. It exhibits features typical of magnetization of any low dimensional AF with short-range order. That is, in the SRO region the shift increases with increasing T , as evident in the H = 3 T data. This is in contrast to the shift in the paramagnetic (PM) state which decreases with increasing T , as apparent in the H = 6.6 T data. Thus, the characteristic T dependence delineates the boundary to the SRO region. The maximum in the temperature dependence of the shift signals a cross-over from PM to SRO in the vicinity of T ≈ 2.4 K. This finding is consistent with that from bulk magnetization measurements [15]. That is, the low field susceptibility, χ(T ) = M/H displays a broad maximum at T ≈ 2.8 K characteristic of short-range antiferromagnetic correlations [15]. Moreover, the overall temperature dependence in zero field is well described by high-T series expansion models for the partially frustrated triangular lattice with J = 4.46 K and J /J = 1/3 [15].
The temperature dependence of the NMR relaxation rate also serves to distinguish each region of the phase diagram as shown in Fig. 3. In the PM region the rate increases as T is lowered. The rate reaches its maximum near T ≈ 2.4 K at H = 3 T as evident in the inset to Fig. 3. The maximum in T −1 1 , like the maximum in the T dependence of the shift, also indicates a cross-over from PM to SRO. On further lowering the temperature in the SRO region, the relaxation rate decreases linearly in temperature. Below 500 mK the rate increases due to an enhancement of fluctuations associated with the transition to the long-range ordered (LRO) magnetic state. The exponential decrease of the rate below T ≈ 320 mK may be due to the opening of a spin gap in the LRO state (at H = 3 T).

Field Dependence
The applied field dependence of the full width at half maximum (FWHM) of spectra at T ≈ 495 mK is plotted in Fig. 4. A typical spectrum, from which FWHM data were extracted, is shown in the inset to Fig. 4. At T ≈ 495 mK for fields below ≈ 7 T the system is in the short-range ordered region as discussed above. The FWHM smoothly increases with increasing H up to ≈ 7 T. As the FWHM measures the variation of the electron spin operator projected along the direction of H, the increase of the FWHM implies the increased static short-range correlation along the direction of H||c in the SRO region. We cannot exclude the possibility that some portion of the FWHM increase with H is due to a DM interaction. However, careful considerations of NMR spectra obtained when H is oriented away from theĉ-axis lead us to conclude that the contribution of DM interaction to the FWHM is not a dominant one. For 7 T H 8.4 T, the FWHM decreases with increasing H as expected in the paramagnetic region, where static shortrange correlations are suppressed. The field at which the FWHM reaches its maximum coincides with that where the maximum of the differential susceptibility (dm loc /dH) occurs [15]. Thus, the maximum in the FWHM indicates the onset of the cross-over from SRO to PM. In the fully polarized state the FWHM is constant as all spins are aligned with the field. By examining the T and H dependence of the NMR observables we are able to clearly identify the cross-over from the short-range ordered region. To gain further insight into the microscopic nature of this region, we proceed to the analysis of the H dependence of the relaxation rate. In Fig. 5, we plot the relaxation rate as a function of applied field. At low fields, as H increases, the rate rises steadily and attains its maximum at a field H M . The maximum of the rate is smeared out by increasing temperature. At a given temperature, maxima of both the rate and FWHM occur at nearly equal fields.
In the SRO region, the rate is T independent for H up to ∼ 5 T and, in the limit of H → 0, it extrapolates to a non-zero value of ∼ 0.5 ms −1 . Furthermore, in the limit of T → 0 the rate extrapolates to a non-zero value of ∼ 1 ms −1 at 3 T in the SRO region as evident from the data plotted in Fig. 3. The fact that the rate extrapolates to a nonzero value in the limit of H → 0 and T → 0 indicate that the rate is dominated by gapless excitations. Nonetheless, since the SRO region is stabilized at non-zero temperature we  can only place an upper bound on the value of the gap (∆) at H = 0. Specifically, the gap is smaller than the energy scale set by the smallest temperature (T 430 mK) probed in the SRO region in our experiment, k B T 0.037 meV ≈ J/10. Strictly speaking, the evidence of gapless excitations clearly exists only for fields above 1.7 T (for technical reasons this was the lowest field probed in our experiment). For one Bohr magneton (Cu 2+ , S = 1/2) a field of 1 Tesla corresponds to 0.06 meV ≈ J/6 which is the same order of magnitude as the energy scale of some of the magnetic couplings in Cs 2 CuCl 4 . It is indeed possible that applying fields larger than 1 T might close a low energy gap due to these weak couplings. However, as we will discuss in the next section, our data agree qualitatively with an observed evolution well-described by gapless 1D fermionic excitations. Thus, we do not expect additional low energy gaps to open up below 1 T, and even if that is the case such a gap does not dominate the nature of the quasiparticles over the wide field range of the SRO region. This suggests that the observed behavior of the rate in the SRO is dominated by gapless or nearly-gapless excitations.
Although the spin dynamics above the ordering temperature is commonly governed by gapless critical fluctuations, the issue of the existence of a gap in this compound is non trivial given the fact that the dominant excitations at base temperature occur in a dispersive continuum of excited states manifested at intermediate and high energies, that remains largely unchanged upon crossing the temperature-driven transitions from the ordered to the SRO region [4,5]. Moreover, since neutron scattering experiments were not able to determine a tight bound on the gap, it is important to attempt to determine its value with a low energy probe such as NMR as is done in this work [4,5,6].
Having established that the excitations in the SRO region are gapless or nearly gapless, we proceed to examine the statistics of the excitations. The temperature dependence of the NMR rate is particularly sensitive to the statistics of the excitations [22,25]. Assuming that the excitations are weakly interacting, the question is which statistics (bosonic or fermionic) yields a better mean-field description of the temperature dependence of the rate. The absence of T dependence of the rate for H up to ∼ 5 T and the fact that the rate extrapolates to a non-zero value in the limit of H → 0 and T → 0 imply that gapless excitations are fermionic [21]. By contrast bosonic excitations would lead to a much stronger dependence on temperature [8,21,25], in disagreement with the data. As discussed in the introduction, the gapless excitations are predicted by fermionic treatments. A similarly small value of the gap can be obtained from a bosonic treatment if the system is close to a second-order phase transition [6]; however, fermionic spin liquids offer a more natural explanation of the data.

Comparison with the 1D Spin Chain
The observed field and temperature dependence of the relaxation rate resembles that measured in the 1D spin-1/2 AF chain compound α-CuNSal, as illustrated in Fig. 6 [21,25]. For H < H sat , our data agree qualitatively with an observed evolution well-described by gapless 1D fermionic excitations [21,22] suggesting that the rate  Figure 6. The magnetic field dependence of the NMR relaxation rates measured in Cs 2 CuCl 4 and in the 1D spin-1/2 AF chain, α-CuNSal [21,22].

H/H sat
in Cs 2 CuCl 4 may be dominated by conventional spinons found in 1D. On the other hand, there are significant quantitative differences indicating that the strong interchain interactions produce measurable effects. In particular, the peak in the rate is found at H significantly lower than H sat , and at low fields the data display a stronger dependence on the field than expected from the model. Furthermore, for comparable values of k B T /J both the measured and calculated 1D rates [21,22] exhibit a sharper peak than that evident in the Cs 2 CuCl 4 data. In Fig. 7 we show the comparison of the magnetization data with the zerotemperature quasi-1D model proposed by Starykh and Balents in Ref. [10]. This treats the 1D chain exactly and interchain interaction J at mean-field level. Thus, if external field is H then the effective field felt by spins on the 1D chain is slightly smaller, H 1d = H−4J ·S z , where S z is magnetization in field H 1d , S z = 1/π arcsin(1/(1−π/2+πJ/H 1d )), g = 2.3, J = 0.374 meV, J = 0.128 meV, and S z = 1/2 at H 1d = 2J. The disagreement between data and the quasi-1D model appears at high fields near saturation as the quasi-1D model predicts saturation at a field 2J + 2J and not at 2J + 2J + J 2 /2J, as follows from the spin-wave theory [15,26]. Furthermore effects of the interlayer and DM interaction are not included in the quasi-1D model, nor is rounding due to non-zero temperature (k B T /J = 0.14). For completeness in the section below we also compare the data with alternative models of spin liquids relevant for the anisotropic triangularlattice antiferromagnet, where the effects of temperature can be treated explicitly.

2D Spin Liquid Models
For the triangular lattice antiferromagnet a number of spin liquid states have been theoretically proposed based on the symmetry arguments [7]. Relevant to the anisotropic triangular lattice and possessing gapless fermionic spinons are two U (1) spin liquids, one with commensurate SRO (U1B) and one with incommensurate SRO (U1C). A variational study of related states is described in Ref. [9]. Here, we compare the results of static (magnetization) and dynamic (relaxation rate) measurements to predictions based on a combination of Gutzwiller-projected wavefunctions and mean-field theory. Parameters of the mean-field theory are chosen to minimize the ground state energy of the corresponding Gutzwiller-projected wavefunctions. For simplicity, the parameters are optimized for the case of zero applied magnetic field and then kept constant over the entire phase diagram. For J/J ≈ 3, the optimized parameters exhibit an enhanced one dimensional character as the ratio of intra-chain to inter-chain hopping amplitudes is λ/χ = 7 and λ/χ = 8 for the U1B and U1C states, respectively. The enhancement of 1D correlations is broadly consistent with the quasi-1D picture of Kohno, Starykh, and Balents [11]. At the mean-field level, the spinons are non-interacting and the magnetic field acts via Zeeman coupling as a chemical potential of opposite sign for spin-up and down spinons, changing their relative populations. We set the bandwidth of the spinon dispersion so that the critical field required for full polarization of the spins matches that of the experiment in the limit of T → 0. A comparison between the calculated magnetization in the different spin liquid states as well as for an anisotropic tightbinding model, and the experimentally measured values is shown in Fig. 7. Similar to the quasi-1D model, both 2D models give a good qualitative description of the data, and the U1C model fits best quantitatively for H 7 T.  Dependence of the magnetization on applied field as calculated at T = 600 mK in the two different U(1) spin liquid models (see text) and for the purposes of comparison, a tight-binding model on the anisotropic triangular lattice and a quasi-1D model at zero-temperature described in Ref. [10] for J = 0.374 meV and J = 0.128 meV. The calculated magnetization is compared to experimental measurements. In the PM regime 7 T H 8 T, the mean-field theory does not apply and consequently a comparison between measurements and calculations is not meaningful.
is not meaningful because there is a cross-over from the SRO to PM regime at ≈ 7 T prior to full polarization at ≈ 8 T. The PM state is not captured by mean-field theory nor by the tight-binding model because the bonds have been held constant at their zerotemperature values. In addition, the slight disagreement between calculations and data near saturation may also be due to the small inter-layer coupling and DM interaction, both of the order of ∼ 5%J, that are not included in the theoretical model and assumed to lead only to a small upward rescaling of the saturation field.
The relaxation rate for a field oriented along the z-direction is given by [27]: where the A i,αβ denote components of the hyperfine coupling tensor at site i and δS i = S i − S i . Since the 133 Cs(A) ions are located close to the center of a triangle of Cu +2 spins [24], momentum dependent form factors are important and have been included in the calculation. With the assumption that the Cs(A) ions are located at the center of the triangle, the form factor reduces to |A(q)| 2 = A 2 (3+2 cos(q x )+2 cos(q y )+2 cos(q x −q y )) [28]. However, the most important term in the sum in Eq. 1 is due to the transverse (β = x, y) autocorrelation function (i = j) which gives the following approximate form where f (E) is the Fermi-Dirac distribution, N the density of states (DOS), and µ is the Zeeman-shifted chemical potential. The dominant contribution to the correlation function is proportional to the product of two Fermi functions and is the result of scattering two spin-1/2 spinons. The relaxation rate of the two U (1) SL states is plotted in Fig. 8. As evident from Eq. 2, peaks in the relaxation rate correspond to maxima in the DOS. For the U1C state, the peak occurs prior to saturation at H ≈ 0.9H sat , closer to the observed value of H ≈ 0.8H sat . This is because for U1C the DOS is a maximum before saturation is reached, while for U1B the peak is at the H sat . Unlike the U1B phase, the DOS for the U1C phase reaches a constant value at the top and bottom of the bands at saturation. In contrast, the rate vanishes at low fields in T → 0 limit in both states because the DOS vanishes at the Fermi surface. As calculations are performed at a non-zero temperature of 600 mK, the rate reaches a non-zero value at zero field. Both models agree with data at intermediate fields but not near saturation where data shows no evidence for a sharp peak/divergence. This may be due to oversimplification in the mean-field approximation for the theoretical models, or the fact that the mean-field parameters are optimized for zero field (see above) so that the high-field behavior can be no more than qualitatively correct.  Important quantitative and qualitative differences between the calculated and the experimentally measured rates remain. The derivative of f (E) in Eq. 2 leads to a stronger temperature dependence of the relaxation rate than evidenced experimentally in Fig. 3. In particular, the rate as calculated in the mean-field approximation vanishes at zero temperature. By contrast, an extrapolation of the T −1 1 SL data plotted in Fig. 3 down to zero temperature yields T −1 1 ≈ 1 ms −1 at H = 3 T. The discrepancy may stem from the fact that the calculated rate, unlike the (one-point) magnetization, is affected by correlations neglected in the mean-field approximation [10]. Gutzwillerprojection is known to change the power-law exponent of the algebraically decaying spin-spin correlations in SL phases [29]. In 1D, where the rate can be calculated using a Jordan-Wigner transformation to represent spin excitations as spinless fermions, there is only a weak dependence on temperature [22]. Thus, the observed weak dependence of the relaxation rate on temperature may also point towards the dynamics in the SL state of Cs 2 CuCl 4 being more like that of 1D spinons traveling along individual chains [11].

Conclusions
We have probed the low energy excitations in the spin-1/2 anisotropic triangular antiferromagnet Cs 2 CuCl 4 . We have found that the short-range ordered region stabilized at intermediate temperatures and applied fields is best characterized by gapless or nearly gapless fermionic spinon excitations, with experiments yielding an upper bound on the size of the gap value of 0.037 meV ≈ J/10. We have compared the observed field dependence of the magnetization and NMR rate with mean-field theories proposed for anisotropic triangular lattice antiferromagnet and found good quantitative agreement for the field-dependence of the NMR relaxation rate at intermediate fields. Our results call for an extension of the self-consistent quasi-1D picture (Ref. [11]) and other theoretical models for spin liquid states in the anisotropic triangular lattice quantum antiferromagnet to include quantitative predictions for the field and temperature dependence of the NMR relaxation rate.