Donmuş düzensizliğe sahip klasik ve kuantum sistemlerin renormalizasyon grubu kuramı

In the first part of this thesis, the spin-1/2 quantum Heisenberg spin-glass system isstudied in all spatial dimensions d by renormalization-group theory. Strongly asymmetricphase diagrams in temperature and antiferromagnetic bond probability p are obtained indimensions d ? 3. The asymmetry at high temperatures approaching the pure ferromagneticand antiferromagnetic systems disappears as d is increased. However, the asymmetry atlow but finite temperatures remains in all dimensions, with the antiferromagnetic phasereceding from the ferromagnetic phase. A finite-temperature second-order phase boundarydirectly between the ferromagnetic and antiferromagnetic phases occurs in d ? 6, resultingin a new multicritical point at its meeting with the boundaries to the paramagnetic phase.In d = 3, 4, 5, a paramagnetic phase reaching zero temperature intervenes asymmetricallybetween the ferromagnetic and reentrant antiferromagnetic phases. There is no spin-glassphase in any dimension [1].In the second part, we focus on the thermodynamic properties of Apollonian networks(AN) using renormalization-group theory. Our calculations, for (1) ferromagnetic (F) per-colation models and both (2) antiferromagnetic (AF) and (3) weakened-AF percolationmodels, reveal no finite-temperature phase transition for the entire range of bond proba-bility p, exhibiting an infinite F phase for (1) and an infinite SG-like phase for (2) and(3). However, the Ising spin glass studied on an AN small-world network shows a phasetransition as p is increased, between the F and SG-like phases, as a function of disorder, forall temperatures T . We develop an exact recursion matrix method to calculate nonuniformlocal order parameters. The calculation of the average magnetization and the SG orderparameter shows that the phase boundary between F and SG-like phases is first order, gov-erned by a nonuniform fixed distribution. The exact values of the local order parameters,as functions of the interactions, show complicated distributions. Analysis of the parametersreveals short-range correlations of the F phase within the SG-like phase [2].