3 R ( Ω )  = 0.21 K = 1.2 K = 2.0 K 1/T (K ) -1 Sample C B = 16.9 T,  ν  = 5/3 3 (K) ε  (10 ) -4 0 2 4 6 8 10 10 (a) 2 ε  = 0 ε  = -1.65 ε  = - 0.53 5/3 -2 -1 0 1 2 0 1 2  A, B = 12.0 T  B, B = 14.2 T  C, B = 16.9 T (b) ν  = 5/3 Δ Δ Δ Δ (b) (a) 10 20 30 0 30 60 90 R (k Ω ) B(T) 10 20 30 0 20 40 60 B(T) 1/3 1/3 = 2.0 K ε = 0 10 10 4 3 1/T(K ) -1 R ( Ω ) 1/3 1 2 3 = 5.2 K 1/T(K ) -1 10 10 4 3 ε = -1.35 x 10 -4 R ( Ω ) 1/3 0.0 0.8 1.6 ν  = 2/3 ν  = 2/3 0.3 K T = 4K T = 1.5 K 0.3 K Δ Δ FIG. 4. (a) Arrhenius plots of ν = 5 / 3 resistance minimum for three values of ϵ (shown in units of 10 − $^{4}$) for sample C at n = 6 . 8 × 10 11 cm − $^{2}$. (b) Summary of ν = 5 / 3 gap data for three samples at different B . FIG. 3. Temperature dependence of magnetoresistance around ν = 1 / 2 in sample A for n = 2 . 4 × 10 11 cm − $^{2}$. (a) and (b) correspond to extreme values of ϵ = 0 and − 1 . 35 × 10 − $^{4}$. Corresponding Arrhenius plots are shown as insets. increase linearly with a slope of E$_{2}$ until a LL crossing occurs. After this point, the gap is expected to be a con- stant equal to E c = ¯ heB/m b , where m b = 0 . 46 m$_{0}$ is the band mass for AlAs. A similar fan diagram can be used in the case of single- valley, two-spin systems with the role of E v taken over by E Z . Since this simple picture is inadequate to explain ex- perimental results, a comprehensive theory including the role of interaction was developed for single-valley, two- spin systems [1, 17]. Here we assume that the valley degree of freedom is an isospin, and compare our exper- imental results to the theory developed for the two-spin case. We point out, however, that in our system the Fermi contours are anisotropic at B = 0. This introduces an anisotropic Coulomb interaction, the consequences of which are unknown. (measured in units of V C ) vs. η should be infinite, then gradually decrease with increasing η (signaling finite-size skyrmions) and, beyond a critical η ( ≃ 0 . 054), should be equal to one [17]. In Fig. 2(a) we find an initial slope of 13, implying that if the excitations are skyrmions [11], they are large but finite-sized. Finite skyrmion size has also been reported for spin skyrmions near zero E Z [2], and is presumably because of disorder. We would like to emphasize that the saturation value of the energy gap in Fig. 2(a) ( ≃ 0 . 19 V C ), is close to the cyclotron energy (0 . 17 V C ) in our 2DES. This is not surprising since in our system we expect the gap to be limited by the cyclotron gap (see fan diagram in Fig. 1(b)). We now discuss our gap data at ν = 1 / 3 (Fig. 2(b)). Theoretically, for an ideal single-valley, two-spin 2DES, a ferromagnetic ground state [6] with an energy gap of about ≃ 0 . 024 V C [1] and skyrmionic excitations has been predicted at η = 0 [1, 20, 21]. As in the case of ν = 1, the gap is expected to quickly increase with η , signaling the presence of skyrmions. Our measured ν = 1 / 3 gaps, plotted in Fig. 2(b), are qualitatively consistent with the above expectations. The gap at η = 0 ( ≃ 0 . 006 V C ), however, is only about one-forth of the gap predicted for an ideal single-valley, two-spin 2DES. Also, the slope of Δ /V C vs. η is only slightly greater than one ( ∼ 1 . 5) suggesting that, if the excitations are skyrmions, their size is quite small. Finally, for very large strains the ν = 1 / 3 gap should saturate at a constant, expected to be ≃ 0 . 1 V C for an ideal, single-valley, single-spin 2DES [22]. Our data of Fig. 2(b) are again qualitatively consistent with this expectation except that the saturation value we measure ( ≃ 0 . 015 V C ) is much smaller than 0.1 V C . It is likely that the much reduced values of the gaps and the slope originate from finite-layer thickness, LL mix- ing, disorder [15] and Fermi contour anisotropy. We also remark that in our system an overall strain inhomogene- ity cannot be ruled out. At η = 0, this might lead to The theoretically relevant parameter in the two-spin case is the ratio of the Zeeman energy to Coulomb en- ergy, η = E Z /V C . In our system, we redefine η = E v /V C . In the limit of η = 0, the ground state at ν = 1 is ex- pected to be a QH ferromagnet with an energy gap (for an ideal 2DES) equal to 0 . 62 V C [1, 17]. Experimentally determined gaps, including the value measured in our experiments (0 . 034 V C in Fig. 2(a)), however, are much smaller [2, 9]. Finite layer thickness, LL mixing, and dis- order are possible causes for this reduction [5, 18, 19]. We note that LL mixing is particularly severe for electrons in AlAs because of their large band mass. The proximity of the second LL can be appreciated by noting that E c in our AlAs 2DES (0 . 17 V C ) is a very small fraction of V C . For small enough η , for a single-valley, two-spin system, the excitations above the ferromagnetic ground state are macroscopic spin textures (skyrmions) [1]. The rate at which the ν = 1 QH gap rises as a function of E Z gives direct information about the size of the skyrmions [2, 4, 5, 9]. In an ideal 2DES, the initial slope of Δ