Power law singularities

The correlation length is also found to scale in a power-law fashion, and it becomes very large at the transition temperature. One of the most significant results of renormalization group theory is to show that the behavior of the correlation length in the critical region is the basis of the power-law singularities observed in the other thermodynamic properties. 

Spin and charge excitations are thus decoupled by coulombic interactions in the one-dimensional electron gas. However, the one-dimensional Fermion system is not a Fermi liquid, as indicated by the behavior of the momentum distribution function, which does not exhibit a Fermi step at kF and presents a single-particle density of states vanishing according to a power law singularity at EF. This is a Luttinger liquid [29] with... 

The power law singularity is therefore the strongest in the presence of a charge-degrees-of-freedom gap where Tp = 0, which actually corresponds to the Heisenberg universality class. 

The power law singularities in S(k) reflect the power law (algebraic) decay of the Debye-Waller correlation function Cg(R) for large R [47,48],... 

Infrared singular behaviour is characterized by power law singularities in thermodynamic quantities with singular (or critical) exponents which usually vary continuously with system parameters. Depending on the physical interpretation of these parameters in the different situations, this implies 1) and 2) singular ground-state properties, a special particle spectrum and instabilities for correlation functions 3) the phase transition of continuous order for 2-d magnetic systems and other systems with continuous... 

The thermal fluctuations destroy the ideal long-range periodic order of the smectic phase and there are power-law singularities rather than the... 

The sharp Bragg peak characteristic of the long-range ordered crystal lattice is now replaced by strong thermal diffuse scattering with a power-law singularity.The excellent high resolution X-ray measurements of Als-Neilsen et have confirmed this prediction (fig. 5.3.3). 

The spectrum was shown to have a power law singularity at E (Mahan 1967, Nozieres and de Dominicis 1969) ... 

The power-law singularity causes major difficulties when one attempts an ab initio calculation of the spectrum. The spectrum is the Fourier transform of the dynamical response of the conduction electrons to the creation of an f hole. In real time the response is entirely in the transient regime, and thus cannot be analyzed by the steady state methods developed so far in connection with LDA band calculations. 

The choice of a failure criterion is obvious when interface-comer stresses are described by one real-valued, power-law singularity. In this case, a single interface-comer stress-intensity factor /sTa characterizes the magnitude of the stress state in the region of the interface comer. 

One commonly used approach for determining is to match detailed finite-element results with the known form of the asymptotic solution [14,17,18]. For example, displacements normal to the stress-free edge, in the region dominated by the stress singularity, were used to determine K. values for an adhesively bonded butt joint with rigid adherends [18]. The asymptotic problem for this idealized joint is simply bonded rigid and elastic quarter planes, and the asymptotic solution is described by one real-valued, power-law singularity. The asymptotic solution... 

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