Zero- and longitudinal-field measurements

The magnetic behavior of the 2-leg ladder system differs markedly from that of the 3-leg system. In Fig.32a, we show typical $\mu$SR spectra in the 2-leg ladder system. Note that Fig.32 has a horizontal scale about 10 times larger than that for Fig.29a and Fig.30. In the 2-leg system, weak relaxation below $\sim$50 K was observed. However, from the longitudinal field (LF) measurements (Fig.32b), the source of this relaxation was found to be fluctuating rather than static internal fields. If the slow relaxation in the zero-field measurement were due to a static field distribution, the characteristic field distribution width would be $\delta H\sim 10$ G, as the relaxation rate suggests. In this case, the $\mu$SR spectrum would be flattened in a LF$\sim$100 G. As shown in Fig.32b, the relaxation persists up to LF$\sim$2 kG, proving that this slow relaxation is a T₁-like dynamical one (see section 3.4).

  

Figure 32: a: LF=100 G $\mu$SR spectra in the 2-leg ladder system. b: Longitudinal field decoupling measurements at 20 mK. For both panels the solid lines are fits with a square-root exponential function. Note that the horizontal scale is $\sim$10 times larger than that in Fig.29a and Fig.30.

We analyzed the muon spin relaxation in the 2-leg ladder system using a square-root exponential function ($P_{\mu}(t)$=$\exp(-\sqrt{\lambda t})$), which is appropriate for dilute fluctuating moments (see section 3.2). The temperature dependence of the relaxation rate ($\lambda$) is shown in Fig.33. The increase of $\lambda$ down to $\sim$40 K indicates a slowing down of field fluctuations. Still the field fluctuations persists down to the milli-Kelvin regime, as shown by the saturation of $\lambda$. This result indicates that there is no static order in the 2-leg ladder material, supporting the theoretical expectations for the non-magnetic ground state of this system.

  

Figure 33: Temperature dependence of muon spin relaxation rate ($\lambda$) in the 2-leg ladder system. The solid lines are guide to the eye. The inset is the longitudinal field dependence of the relaxation rate at 20 mK, with the fit to the T₁ relaxation theory (eq.33).