Anisotropic Interactions in 2 + ϵ Dimensions

Sergio A Albeverio, Friedrich Gesztesy, Raphael J. Høegh-Krohn +1 more

Vladimir V. Prudnikov

The critical dynamics is studied for a spin system with quenched impurities correlated along an epsilon d -dimensional 'line' and randomly distributed in dl- epsilon d dimensions (d=4- epsilon ). The presence of these line defects makes the system anisotropic and the interactions non-local. The renormalisation group (RG) is used to approach the critical region and the dynamical exponents are calculated to the lowest order in a double epsilon , epsilon d expansion. The existence of two correlation lengths leads to the existence of two dynamical exponents Z 1 and Z t , which are connected by the relation (2- eta t )/Z t =(2- eta l )/Z l .

Chongming Yang, Chuan‐Zhang Yang

Ian D. Lawrie, YONKO T. MILLEV, Dimo I. Uzunov

Renormalisation group recursion relations are obtained to two-loop order for an n-component Landau-Ginzburg-Wilson model containing both hypercubic anisotropy and quenched random impurities. The fixed points are enumerated and their stability properties analysed, using the epsilon expansion about four spatial dimensions to second order. The existence of a new fixed point corresponding to an anisotropic impure system is reported. It is found that both isotropic and anisotropic pure systems are unstable to the addition of random impurities for n<n R identical to 4-4 epsilon +O( epsilon 2 ), while both pure and random isotropic systems are unstable to anisotropy for n>n C identical to 4-2 epsilon +O( epsilon 2 ). Singularities in the epsilon expansions for the exponents of the random fixed points are shown to be associated with the onset of focal behaviour (the acquisition of complex eigenvalues). These, together with the slow convergence of the expansions leads one to doubt the reliability of extrapolations to three dimensions.

Ian D. Lawrie, YONKO T. MILLEV, Dimo I. Uzunov

Renormalisation group recursion relations are obtained to two-loop order for an n-component Landau-Ginzburg-Wilson model containing both hypercubic anisotropy and quenched random impurities. The fixed points are enumerated and their stability properties analysed, using the epsilon expansion about four spatial dimensions to second order. The existence of a new fixed point corresponding to an anisotropic impure system is reported. It is found that both isotropic and anisotropic pure systems are unstable to the addition of random impurities for n nC identical to 4-2 epsilon +O( epsilon 2). Singularities in the epsilon expansions for the exponents of the random fixed points are shown to be associated with the onset of focal behaviour (the acquisition of complex eigenvalues). These, together with the slow convergence of the expansions leads one to doubt the reliability of extrapolations to three dimensions.

Sergio A Albeverio, Friedrich Gesztesy, Raphael J. Høegh-Krohn +1 more