Spin fluctuation model

Chubukov A.V., Pines D., and Schmatian J., (2003). A Spin Fluctuation Model for d-wave Superconductivity in The Physics of Conventional and Unconventional Superconductors eds. by Bennemann K.H. and Ketterson J.B., Vol. 1 (Springer-Verlag). 

The spin fluctuation model of section 5.2 may be readily extended to describe the magnetic field dependence of this contribution. We define the magnetoresistance above the field Hi required to form a single domain as... 

In conclusion, the spin fluctuation models have gone a long way beyond their He counterparts. On the other hand, the theories do not give a gap function with which one is totally happy (see table 3), and the theories are still probably too simplistic for real heavy-fermion metals. First, they replace the complicated quasiparticle matrix elements, of the form , by a simplistic V k — k ) as used above (in fact, the ambiguities discussed above about the momentum dependence of the pair potential are all due to this simple replacement). Second, the vertex is treated as a product of Pauli (spin) matrices, as in He, and completely ignores any orbital or spin-orbital contributions (in fact, the only thing included at this stage is the directional anisotropy terms in x). Finally, the frequency dependences of the pair interaction, the quasiparticle dispersions, and the gap function have been treated in... 

We summarize the results of some l/N calculations in a set of figures. In figs. 30 and 31 we compare the susceptibility and specific heat results of the l/N calculation with those of the Kondo model. The two sets of results, for N = 6 and 4 respectively, show that the spin fluctuation model virtually reproduces the predictions of the Kondo model. In fig. 32 the specific heat of a dilute alloy of Ce in LaBg is compared... 

Fig. 31. The magnetic susceptibility (upper curves) and specific heat (lower curves) for the single Ce impurity as predicted by the spin fluctuation model for N, = 4 and n, = 0.97. The = 1 curves are the...

In the spin-fluctuation model the tendency towards magnetism is determined by the strength of the effective exchange interaction between electrons in a narrow band. The presence of this exchange interaction leads to an enhanced susceptibility over the Pauli value, predicted for a free-electron gas. At T = 0 K this enhaneement factor, known as the Stoner factor, is given by... 

In the next review (chapter 111), S.H. Liu examines phenomenological approaches to heavy-fermion and mixed-valence materials. He notes that there are two basic approaches to understanding these phenomena the Fermi-liquid models, which are applicable at low temperatures, and the spin fluctuation models, which work well for describing high-temperature properties and behaviors. At the present time Liu concludes that the field is still wide open before we come to even a reasonable understanding of anomalous f-electron behaviors. 

In UPts, a compound for which spin-fluctuations are known to exist, superconduction has recently been reported There is an apparent contradiction with the classical Bar-deen-Cooper model for superconductivity, in which superconduction is hindered by the onset of ordered magnetic phenomena, and is usually not found in very narrow bands. UBei3 which has a y of 1000 mg/mol K, exhibits similar behaviour. 

If the material is ferromagnetic then the entropy, susceptibility and resistance at temperatures just above the Curie point are to be calculated in much the same way (i.e. in terms of spin fluctuations). A treatment of this problem starting from the Stoner-Wohlfarth model is due to Moriya and Kawabata (1973). 

Our model for the density of states is thus as in Fig. 4.7. The total density of states is mainly due to spin fluctuations, and has a maximum for n=1, where n is the number of electrons per atom. The curve for current carriers needs to be used for calculating thermopower and resistance the experimental evidence discussed in the following chapters suggests, however, that the Hall coefficient RH is given by the classical formula 1 jnec. 

The main equation for the d-electron GF in PAM coincides with the equation for the Hubbard model if the hopping matrix elements t, ) in the Hubbard model are replaced by the effective ones Athat are V2 and depend on frequency. By iteration of this equation with respect to Aij(u>) one can construct a perturbation theory near the atomic limit. A singular term in the expansions, describing the interaction of d-electrons with spin fluctuations, was found. This term leads to a resonance peak near the Fermi-level with a width of the order of the Kondo temperature. The dynamical spin susceptibility in the paramagnetic phase in the hydrodynamic limit was also calculated. 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

The impact of this is tremendous. No long-range order (LRO) can exist at finite temperature in one dimension no crystals, no magnets, no superconductors. Only special transitions are possible in two dimensions. The Ising model (n = 1 component) is an example [7]. The Kosterlitz-Thouless transition [8], without LRO, is another case for d = 2 and n = 2, discussed in Section V.C. The thermal fluctuations are very destructive in lower dimensions. Quantum fluctuations (i.e., those associated with the dynamics of a system) also tend to suppress LRO and can sometimes destroy it even at 0 K when the Mermin-Wagner theorem does not apply. Such is the case of the quantum spin- antiferromagnetic models [9] in one dimension. 

Aq is a constant except for the Cu close to Ni. The relation (26) is direct evidence for the attractive force being due to spin fluctuation, irrespective of any theoretical model. In the conventional superconductor, the isotope effect provides direct evidence for phonon-mediated superconductivity. The relation (26) is considered to correspond to the isotope effect in a phonon-mediated superconductivity model. 

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