Dimensional scaling theory

S. S. Schiffman, M.L. Reynolds and F.W. Young, Introduction to Multi-dimensional Scaling Theory, Methods and Applications. Academic Press, New York, 1981. 

Dimensional scaling theory [109] provides a natural means to examine electron-electron correlation, quantum phase transitions [110], and entanglement. The primary effect of electron correlation in the D 00 limit is to open up the dihedral angles from their Hartree-Fock values [109] of exactly 90°. Angles in the correlated solution are determined by the balance between centrifugal effects, which always favor 90°, and interelectron repulsions, which always favor 180°. Since the electrons are localized at the D 00 limit, one might need to add the first harmonic correction in the 1/D expansion to obtain... 

Torgerson, W. S., Multi-dimensional scaling /. Theory and method, Rsychometrika, 17 (1952) 401-419. 

In this book we confine ourselves to three-dimensional systems. According to scaling theory, in two-dimensional systems all states are localized. Although it has been suggested that a transition between power-law and exponential localization can occur (Kaveh 1985c, Mott and Kaveh 1985a, b), this matter is too uncertain at present for discussion in a book. 

The latter assumption distinguishes scaling theory from straightforward dimensional analysis in the continous chain model. 

AC susceptibility measurements can yield valuable information on the dynamical properties of a system. The ac susceptibility of H0B22C2N shows frequency dependence, but could not be analyzed satisfactorily by the dynamical scaling theory of a three dimensional spin glass (Mori and Mamiya, 2003). Therefore, a detailed investigation of the behavior of relaxation times by the Cole-Cole analysis (Cole and Cole, 1941) was performed. The complex susceptibility can be phenomenologically expressed as ... 

S. Greenspoon Finite-size effects in one-dimensional percolation a verification of scaling theory. Canadian J. Phys. 57, 550-552 (1979)... 

Using a scaling theory of localization, adapted for anisotropic materials, Apel and Rice [76] find that the condition Le, c does correspond to the Anderson metal-insulator transition of a quasi-one-dimensional system. 

The finite-size scaling theory combined with transfer matrix calculations had been, since the development of the phenomenological renormalization in 1976 by Nightingale [70], one of the most powerful tools to study critical phenomena in two-dimensional lattice models. For these models the partition function and all the physical quantities of the system (free energy, correlation length, response functions, etc) can be written as a function of the eigenvalues of the transfer matrix [71]. In particular, the free energy takes the form... 

In general, the temperatures at which the heat capacity and the compressibility reach their respective maxima in finite systems are different. From the finite size scaling theory it follows that in the case of a second-order phase transition (e.g. at the critical point) Qm = a/t and 7 = 7/j/, while for any first-order phase transition ocm — Im — d, where d is the dimensionality of the system. The above predictions are often used to determine the nature of phase transitions studied by computer simulation methods [77]. The finite size scaling theory implies also that near the critical point the system free energy is given by... 

SCALING THEORY OF A ONE-DIMENSIONAL FERMI GAS MODEL WITH TWO CHARACTERISTIC ENERGIES... 

In the past few years the method of dimensional scaling [12,22,23] has become increasingly more importemt in quantum theory. Using this technique one can solve the many-particle Schrodinger equation in a space of arbitrary dimension D [17]. By taking the limit of infinite... 

The phenomenon of localisation is readily illustrated by examining the the figure below, which is taken from Dudley Herschbach s introductory article [21] on the theory of dimensional scaling of atomic properties. 

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