Bethe ansatz

Among the technical methods proper to the one dimensional geometry, one may cite the Bethe ansatz [19], the bosonization techniques [18], and, more recently, the Density-Matrix Renormalization Group (DMRG) method (20, 21] and a closely related scheme which is directy considered in this note, the Recurrent Variational Approach (RVA) [22, 21], The two first methods are analytical and the third one is numerical the RVA method is in between. 

In Fig. 2, we plot the spin gap which we define as the energy gap between the lowest triplet state and the ground state singlet as a function of 1 /N. The Bethe ansatz solution yields a vanishing spin gap in the thermodynamic limit of the uniform Hubbard chain. Polynomial fits to our DMRG data are consistent with this. 

U limit these authors have found that the critical value of Vc is equal to 2t and decreases very slowly with increasing 6. The results of Yu, Saxena and Bishop are in disagreement with the Bethe ansatz results for P = 0, (5 = 0, where the band edge of the excited states should be at 7 — At and not 7 hence, the Vc value from their study is taken with much caution. The results of Gallagher and Mazumdar are exact for J = 0 in the large U limit and are consistent with earlier results. 

For CeCu2 2Si, (Bredl et al. 1985) and CeAlj (de Boer et al. 1985) strong electronic correlations produce a pronounced anomaly at the low-temperature tail of the CF peak in C against T. The whole feature can still be explained via two separated energy scales k T and The shape of this curve can be fitted surprisingly well by a superposition of the Bethe-ansatz result for spin j and Kondo temperature T = = 9 K and 4 K for CeCu2 2 i2 and CeAlj, respective-... 

Even for a concentrated system Uke UBejg this picture seems qualitatively appropriate as apparent from the Bethe-ansatz fit in fig. 29a. For a compound, Ap is also defined as above all resistivities appearing, however, are corrected by the ordinary residual value p = p(B = 0, T = 0), i.e., p->p —pg (Steglich et al. 1987c, Rauchschwalbe 1987). For T> T 8K (Felten et al. 1986), the results can well be described by the universal ciuve... 

Fig. 29. Magnetoresistivity of UBeij. (a) Fit of Bethe-ansatz results by Schlottmann (1983) to Ap against B curves for differing temperatures (top-bottom) 4.0 K, 3.5 K, 3.0 K, 2.5 K, 2.0 K, 1.5 K (Steglich et al. 1987c, Rauchschwalbe 1987). (b) Effective magnetic field = (A b/p,)T (1 + T/T ) as a function of temperature (Steglich et al. 1987c, Rauchschwalbe 1987). Dashed line gives result by Batlogg et al. (1987) as obtained from Ap(T, B), see text.

Fig. 43. Specific heat as CIT against T for CeAlj (Biedl et al. 1978b). Solid line through data points is guide to the eye. Thin horizontal line indicates low-temperature value, yo = 0.135 JK mol" . Dashed line is Bethe-ansatz result for S = j Kondo impurity with = 0.68 T = 3.5 K (Andrei et al. 1983).

Derived from T = 5 K [as obtained from the residual quasielastic neutron line width (HWHM) as well as by using a resonance level model via T = r /0.68, corresponding to the Bethe-ansatz results, see Andrei et al. (1983)], in line with the definition used throughout this article. 

For Bethe ansatz, see e.g. J. Ziman, Models of Disorder, Cambridge Univ. Press, Cambridge, 1979. 

Both the Anderson and the Kondo (or Coqblin-Schrieffer) model have been solved exactly for thermodynamic properties such as the 4f-electron valence, specific heat, static magnetic and charge susceptibilities, and the magnetization as a function of temperature and magnetic field B by means of the Bethe ansatz (see Schlottmann 1989, and references therein). This method also allows one to calculate the zero-temperature resistivity as a function of B. Non-equilibrium properties, such as the finite temperature resistivity, thermopower, heat conductivity or dynamic susceptibility, could be calculated in a self-consistent approximation (the non-crossing approximation), which works well and is based on an /N expansion where N is the degeneracy of the 4f level. 

In the Kondo limit 17oo, 8f > d one has from Bethe ansatz (Andrei and Loewenstein 1981, Andrei et al. 1983, Tsvelick and Wiegmann 1982, 1983) the exact... 

Fig. 6. Temperature dependence of the impurity (or magnetic) specific heat for N = 6 and various values of % in NCA. The result for n,= 1 and the characteristic temperature are calculated by means of the Bethe ansatz (Bickers et al. 1987).

Fig. 7. Temperature dependence of the magnetic susceptibility for N = 6 in NCA The curve for = I is a Bethe-ansatz result. The susceptibility shows a weak maximum at T/T ttiQ.5 which depends on rif (Bickers ct al. 1987).

The theoretical results based on the NCA (or l/N expansion) and on the Bethe ansatz presented so far are generally in rather good agreement with experimental data for dilute R systems. We present a few examples and refer for a more extended (but still rather incomplete) collection of experimental data for HF systems to Stewart (1984) and Grewe and Steglich (1991). 

Fig. 14. Temperature dependence of (a) the susceptibility of YbCuAl and (b) the specific-heat difference between YbCuAl and the non-magnetic analog LuCuAl compared with theoretical results from the Bethe ansatz for a Coqblin-Schrieffer impurity with iv = 8, d = 527 K and gj = f (Schlottmann 1989).

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