1,006 research outputs found
Dynamical Symmetry Breaking in Fractal Space
We formulate field theories in fractal space and show the phase diagrams of
the coupling versus the fractal dimension for the dynamical symmetry breaking.
We first consider the 4-dimensional Gross-Neveu (GN) model in the
(4-d)-dimensional randomized Cantor space where the fermions are restricted to
a fractal space by the high potential barrier of Cantor fractal shape. By the
statistical treatment of this potential, we obtain an effective action
depending on the fractal dimension. Solving the 1/N leading Schwinger-Dyson
(SD) equation, we get the phase diagram of dynamical symmetry breaking with a
critical line similar to that of the d-dimensional (2<d<4) GN model except for
the system-size dependence. We also consider QED4 with only the fermions
formally compactified to d dimensions. Solving the ladder SD equation, we
obtain the phase diagram of dynamical chiral symmetry breaking with a linear
critical line, which is consistent with the known results for d=4 (the
Maskawa-Nakajima case) and d=2 (the case with the external magnetic field).Comment: 28 pages, 5 figures, LaTeX with epsf macr
Scale-free networks embedded in fractal space
The impact of inhomogeneous arrangement of nodes in space on network
organization cannot be neglected in most of real-world scale-free networks.
Here, we wish to suggest a model for a geographical network with nodes embedded
in a fractal space in which we can tune the network heterogeneity by varying
the strength of the spatial embedding. When the nodes in such networks have
power-law distributed intrinsic weights, the networks are scale-free with the
degree distribution exponent decreasing with increasing fractal dimension if
the spatial embedding is strong enough, while the weakly embedded networks are
still scale-free but the degree exponent is equal to regardless of
the fractal dimension. We show that this phenomenon is related to the
transition from a non-compact to compact phase of the network and that this
transition is related to the divergence of the edge length fluctuations. We
test our analytically derived predictions on the real-world example of networks
describing the soil porous architecture.Comment: 11 pages, 10 figure
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