Free Fermions

B. Jobs, T. L. Einstein, N. C. Bartelt. Distribution of terrace width on a vicinal surface within the one-dimensional free-fermion model. Phys Rev B 45 8153, 1991. 

The experimental evidence for this scenario, mentioned at the outset of this section, is less compelling since it is harder to control this sort of dosage-sensitive STM experiment than a Monte Carlo simulation. (Specifically, it problematic to convert from CO dosage to the evolution time from initial instability the analysis would be better if the surface could be instantaneously de-oxidized.) From an earlier examination of the terrace-width distribution for Ag(110)2Y—>[001], Ozcomert et al. (1993) concluded that to a good approximation the step-step interactions were purely entropic repulsions (by finding a good fit to a free-fermion form (Jobs et al, 1991)). (But see also Pai et al (1994) for remarkable behavior under different conditions.) From the relationship (Bartelt el a/., 1992)... 

Fluctuations of an isolated step are also suppressed by the microscopic energy cost to form kinks. On coarse-graining, this translates into an effective stiffness or line tension that tends to keep the step straight. Standard microscopic 2D models of step arrays incorporating both of these physical effects include the free-fermion model and the Terrace-Step-Kink (TSK) model. Both models have proved very useful, though their microscopic nature makes detailed calculations difficult. 

Substituting these expressions into (201) we obtain again (212). Note also that if we take to / 0, then Heisenberg operators for free fermions are... 

Now let us consider again free fermions. Heisenberg operators for free fermions... 

We now consider the case of indistinguishable particles, treating explicitly a system of Fermions. It is convenient to write the wave function for a system of free Fermions in the form,... 

It is necessary to first understand the spin of a free fermion. Considered as an isolated dimensionless point particle, no conceivable mechanism can explain the physical origin of its magnetic moment. Even the rotation of a spherically symmetrical indivisible unit charge, associated with a wave structure in the aether, cannot have an intrinsic magnetic moment. 

Figure 2. DDF vs. temperature for bosonic and fermionic Li atoms in an optical lattice. Thin solid line fluctuations due to evaporation (12) (scaling factor 7.8), thin dashed line statistical fluctuations (13). Thick solid (dashed) line total fluctuations (W2) for bosonic (fermionic) Li atoms. Parameters Vo = 5 neV, (ns) = 0.1, d = 0.1 pm, cv Li) 3.6 x 106 J.kg-1.K-1, AF Li) = 6.10 10 m and u)v (Li) 2.106 s-1 [Kastberg 1995]. Inset solid (dashed) line static structure factor vs. n forphonons (nearly-free fermions) in a lattice at finite T.

This clearly reflects the gap in the spin excitation spectrum, which, according to Ref. [4j, is a free fermion spectrum for this particular value of f. The first term describes the response... 

The first term in the second line is the Lagrangian density of the total free fermion field i> x), which represents here a doublet of two fermion fields. The third term corresponds to the Lagrangian density of the free boson... 

In dimensions d > 1, an exact solution is not available because the strong disorder renormalization group can be implemented only numerically. Moreover, mapping the spin system onto free fermions is restricted to one dimension. Therefore, simulation studies have mostly used Monte Carlo methods. The quantum-to-classical mapping for the Hamiltonian in Eq. [30] can be performed analogously to the clean case. The result is a disordered classical Ising model in d + 1 dimensions with the disorder perfectly correlated in one dimension (in 1-1-1 dimensions, this is the famous McCoy-Wu modeF ). The classical Hamiltonian reads... 

Renormalization Group for a Gapless System of Free Fermions. 

Free fermions in a magnetic field Pauli paramagnetism... 

---