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Journal of Physics C Solid State Physics

Magnetisation relaxation in a one-dimensional semi-infinite chain with an enhanced surface exchange interaction

Fumihiko Tanaka

Published Dec 28, 1979• 4 citations
Publisher

Abstract

An exact solution of the kinetic Ising model in a one-dimensional semi-infinite chain with an enhanced surface exchange interaction J'=J(1+ Delta ) is obtained. For a sufficiently large Delta , two localised modes with even and odd symmetries appear near the surface, the eigenvalues of which are located outside the band of the bulk relaxation spectrum. The critical value of Delta is obtained and the Green function G m,n ( omega ) is also derived.

DOI
10.1088/0022-3719/12/24/025

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An exact solution of the kinetic Ising model in a one-dimensional semi-infinite chain with an enhanced surface exchange interaction J'=J(1+ Delta ) is obtained. For a sufficiently large Delta , two localised modes with even and odd symmetries appear near the surface, the eigenvalues of which are located outside the band of the bulk relaxation spectrum. The critical value of Delta is obtained and the Green function G m,n ( omega ) is also derived.
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The surface critical behaviour of the semi-infinite one-dimensional quantum Ising model in a transverse field is studied in the presence of an aperiodic surface-extended modulation. The perturbed couplings are distributed according to a generalized Fredholm sequence, leading to a marginal perturbation and varying surface exponents. The surface magnetic exponents are calculated exactly, whereas the expression of the surface energy density exponent is conjectured from a finite-size scaling study. The system displays surface order at the bulk critical point, above a critical value of the modulation amplitude. It may be considered as a discrete realization of the Hilhorst-van Leeuwen model.
We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large mmm, mmm being the distance from the edge. The case A<0A<0A<0 of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when A>0A>0A>0. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.
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We study the two-dimensional semi-infinite Ising model with a free surface at or near bulk criticality. Special attention is paid to the influence of a boundary magnetic field h1 on the surface-near regime and the crossover between the fixed points at h1=0 and h1=∞. Near the surface, a smallh1 causes a steeply increasing magnetization m(z)~z3/8 log z as the distance z increases away from the surface. By means of a phenomenological scaling analysis, this phenomenon can be related to the well-known logarithmic dependence of the surface magnetization m1 on h1. Our analysis provides a deeper understanding of the existing exact results on m(z) and relates the short-distance phenomena in d=2 to those in higher dimensions. Both the results of the scaling analysis and the exact analytic profiles are corroborated by Monte Carlo simulations.