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arXiv:hep-th/9311073 · Modern Physics Letters A

2D ISING MODEL WITH A DEFECT LINE

D. C. Cabra, Carlos Naón

Published Jul 30, 1994• 10 citations
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Abstract

We study the two-dimensional Ising model with a defect line and evaluate multipoint energy correlation functions using nonperturbative field-theoretical methods. We also discuss the evaluation of the two-spin correlator on the defect line.
DOI
10.1142/s0217732394001969

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The 2n2n2n-spin correlation function for the two-dimensional Ising model at T=TcT={T}_{c}T=Tc​ is evaluated for the special case in which all the spins lie along a straight line, separated by many lattice constants. The resulting 2n2n2n-spin function is simply a quotient of products of two-spin correlations. A hypothesis of reducibility of fluctuations in the critical state is introduced. This hypothesis asserts that the product of any two local fluctuating quantities in the same neighborhood of space may be effectively replaced by a finite sum of local fluctuating quantities in this neighborhood. As a result, the previously found form for the 2n2n2n-spin function may be used to evaluate the correlation function of nnn energy densities when all nnn points lie on a line. The nnn-energy correlation function is simply a sum of products of two-energy correlations. The quotient form for the spin correlation plus scaling is shown to immediately imply the logarithmic specific heat.
Critical phenomena in the two-dimensional Ising model with a defect line are studied using boundary conformal field theory on the c=1c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1c=1 orbifold. Novel features of the boundary states arising from the orbifold structure, including continuously varying boundary critical exponents, are elucidated. New features of the Ising defect problem are obtained including a novel universality class of defect lines and the universal boundary to bulk crossover of the spin correlation function.
The energy-density---energy-density and energy-density---spin correlation functions are evaluated for the two-dimensional square Ising model. Results are obtained at zero magnetic field and near the critical point, i.e., for $\ensuremath{\epsilon}=(\frac{4J}{\mathrm{kT}})|(T\ensuremath{-}\frac{{T}_{c}}{{T}_{c}})|\ensuremath{\ll}1.$ The correlation functions are ${〈{E}_{R}{E}_{0}〉\ensuremath{-}{〈E〉}^{2},={(\frac{2J}{\ensuremath{\pi}})}^{2}{\ensuremath{\epsilon}}^{2}[{{K}_{1}}^{2}(\ensuremath{\epsilon}R)\ensuremath{-}{{K}_{0}}^{2}(\ensuremath{\epsilon}R)],}{〈{E}_{R}{\ensuremath{\sigma}}_{0}〉\ensuremath{-}〈E〉〈\ensuremath{\sigma}〉,=(\frac{2J}{\ensuremath{\pi}})\ensuremath{\epsilon}〈\ensuremath{\sigma}〉\ensuremath{\int}{2\ensuremath{\epsilon}R}^{\ensuremath{\infty}}\mathrm{ds} {s}^{\ensuremath{-}2}{e}^{\ensuremath{-}8},}$ where RRR is the distance between the points in question and K0{K}_{0}K0​ and K1{K}_{1}K1​ are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.