Finite-dimensional phase-coherent states

If the impurity potential is smooth, the process of scattering on them proceeds quasi-classically. In this case no real scattering takes place and the impurity effect may be reduced to the appearance of a random phase of the electron wave function. As has been shown by Zawadowski (1), such impurities do not affect the thermodynamics of the one-dimensional system, in which, however, no phase transitions exist. The finite temperature of the transition arises due to three-dimensional effects which establish the coherent state in the whole volume. The impurities cause the phase shift on each thread, and, as a result, the coherence drops and the transition temperature diminishes. 

In passing we note that the functions in the set g are completely delocalized over the region of sites defined by the localized particle-antiparticle basis h, while the f-basis contains all possible phase-shifted contributions from each site in accordance with Eqs. (56) and (57) above. Some interrelationships can be recognized here. The first connection concerns Coleman s so-called extreme state [18], cf. the theories of superconductivity and superfluidity based on ODLRO. The second observation relates to the identification of the present finite dimensional representation as a precursor for possible condensations, developing correlations and coherences that may extend over macroscopic dimensions. If h is a set of two-particle determinants and the iV-particle fermionic wave function is constructed from an AGP, antisymmetrized geminal power, based on i, see Eq. (57), then the reduced density matrix can be represented as... 

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