Hierarchical lattices that constitute spatially anisotropic systems are introduced. These lattices provide exact solutions for hierarchical models and, simultaneously, approximate solutions for uniaxially or fully anisotropic d = 3 physical models. The global phase diagrams, with d = 2 and d = 1 to d = 3 crossovers, are obtained for Ising and XY magnetic models and percolation systems, including crossovers from algebraic order to true long-range order.
The lower-critical dimension for the existence of the Ising spin-glass phase is calculated, numerically exactly, as d L = 2.520 for a family of hierarchical lattices, from an essentially exact (correlation coefficent R 2 = 0.999 999) near-linear fit to 23 different diminishing fractional dimensions. To obtain this result, the phase transition temperature between the disordered and spin-glass phases, the corresponding critical exponent y T , and the runaway exponent y R of the spin-glass phase are calculated for consecutive hierarchical lattices as dimension is lowered.
The advantages of quantum effects in several technologies, such as computation and communication, have already been well appreciated. Some devices, such as quantum computers and communication links, exhibiting superiority to their classical counterparts, have been demonstrated. The close relationship between information and energy motivates us to explore if similar quantum benefits can be found in energy technologies. Investigation of performance limits for a broader class of information-energy machines is the subject of the rapidly emerging field of quantum thermodynamics. Extension of classical thermodynamical laws to the quantum realm is far from trivial. This short review presents some of the recent efforts in this fundamental direction. It focuses on quantum heat engines and their efficiency bounds when harnessing energy from nonthermal resources, specifically those containing quantum coherence and correlations.
We implement the spectral renormalization group on different deterministic nonspatial networks without translational invariance. We calculate the thermodynamic critical exponents for the Gaussian model on the Cayley tree and the diamond lattice and find that they are functions of the spectral dimension,d. The results are shown to be consistent with those from exact summation and finite-size scaling approaches. Atd = 2, the lower critical dimension for the Ising universality class, the Gaussian fixed point is stable with respect to a ψ 4 perturbation up to second order. However, on generalized diamond lattices, non-Gaussian fixed points arise for 2
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