Real-space renormalization group method

The lattice gas model of Bell et al. [33] neither gave any detailed mechanism of the orientational ordering nor separated the contributions of the headgroup and the acyl chain. Lavis et al. [34] discussed Ref. 33 critically and concluded that the sharp kink point in the isotherm at transition was an artifact of the mean field approximation used. An improved correspondence to experimental data was claimed by the use of the real-space renormalization group method [35]. The same authors returned to the problem [35] and concluded that in addition to the orientation of the molecules, chain melting had to be included in a model which could interpret the phase transitions. 

In summary, by using a multistage real-space renormalization group method, we show that the finite-size scaling can be applied in Mott MIT. And the dynamic and correlation length critical exponents are found to be z = 0.91 and v = 1, respectively. At the transition point, the charge gap scales with size as Ag 1/L0-91. 

J.-P. MalrieuandN. Guihdry, Phys. Rev. B 63,5110,2001. These authors formulate a renormalization-group procedure where the renormalized Hamiltonian is defined as a Bloch effective Hamiltonian. This procedure is based on the real-space renormalization-group (RSRG) method (a) K. G. Wilson, Rev. Mod. Phys. 47, 773, 1975. (b) S.R. White and R.M. Noack, Phys. Rev. Lett. 68, 3487, 1992. 

Figure 32. Herringbone order parameter for the anisotropic-planar-rotor model (2.5) as a function of the reduced temperature T = TIK. Circles Monte Carlo results [244]. Dotted line mean-field approximation [62, 141]. Solid line triangular cluster-variational method [62]. Arrow first-order transition temperature obtained from a real-space renormalization group treatment of a planar quadrupolar six-state model [345]. (Adapted from Fig. 2 of Ref. 345.)...

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